HORATIO WARD STEBBINS
1878-1933
ENGINEERING LIBRARY
HORATIO WARD STEBBINS received the A.B. degree at
the University of California in 1899, and the B.S. de-
gree at the Massachusetts Institute of Technology in
1902. After twelve years of professional practice he
entered the Department of Mechanical Engineering at
Leland Stanford Jr. University where he became Asso-
ciate Professor. He was a member of the American
Society of Mechanical Engineers, Sigma Xi, and Phi
Beta Kappa. He was an ardent student and a beloved
teacher. This book is given in memory of him.
THE THEORY OF OPTICS
LONGMANS, GREEN AND CO.
5J FIFTH AVENUE, NEW YORK
221 EAST 20TH STREET, CHICAGO
88 TREMONT STREET, BOSTON
LONGMANS, GREEN AND CO. LTD.
39 PATERNOSTER ROW, LONDON, E C 4
6 OLD COURT HOUSE STREET, CALCUTTA
J3 NICOL ROAD, BOMBAY
3$A MOUNT ROAD, MADRAS
LONGMANS, GREEN AND CO.
128 UNIVERSITY AVENUE, TORONTO
THE
THEORY OF OPTICS
BY
PAUL DRUDE
' Professor of Physics at the University of Giessen
TRANSLATED FROM THE GERMAN
BY
C. RIBORG MANN AND ROBERT A. MILLIKAN
NEW IMPRESSION
LONGMANS, GREEN AND CO.
LONDON NEW YORK TORONTO
1933
DRUDE
THE THEORY OF OPTICS
COPYRIGHT 1901
BY LONGMANS, GREEN AND CO.
ALL RIGHTS RESERVED, INCLUDING THE
RIGHT TO REPRODUCE THIS BOOK, OR
ANY PORTION THEREOF, IN ANY FORM
First Edition April 1902
Reprinted November 1907
April 1913, November 1916, January 19
June 1922, May 1925, October 1929
September 1933
ENGINEERING LIBRARY
MADE IN THE UNITED STATES OF AMERICA
PREFACE TO THE ENGLISH TRANSLATION
THERE does not exist to-day in the English language a
general advanced text upon Optics which embodies the im-
portant advances in both theory and experiment which have
been made within the last decade.
Preston's " Theory of Light " is at present the only gen-
eral text upon Optics in English. Satisfactory as this work
is for the purposes of the general student, it approaches the
subject from the historical standpoint and contains no funda-
mental development of some of the important theories which
are fast becoming the basis of modern optics. Thus it touches
but slightly upon the theory of optical instruments a branch
of optics which has received at the hands of Abbe and his fol-
lowers a most extensive and beautiful development ; it gives
a most meagre presentation of the electromagnetic theory
a theory which has recently been brought into particular
prominence by the work of Lorentz, Zeeman, and others ; and
it contains no discussion whatever of the application of the
laws of thermodynamics to the study of radiation.
The book by Heath, the last edition of which appeared in
1895, well supplies the lack in the field of Geometrical Optics,
and Basset's " Treatise on Physical Optics " (1892) is a valua-
ble and advanced presentation of many aspects of the wave
theory. But no complete development of the electromagnetic
theory in all its bearings, and no comprehensive discussion of
iii
903646
iv PREFACE TO THE ENGLISH TRANSLATION
the relation between the laws of radiation and the principles of
thermodynamics, have yet been attempted in any general text
in English.
It is in precisely these two respects that the " Lehrbuchder
Optik " by Professor Paul Drude (Leipzig, 1900) particularly
excels. Therefore in making this book, written by one who
has contributed so largely to the progress which has been
made in Optics within the last ten years, accessible to the
English-speaking public, the translators have rendered a very
important service to English and American students of
Physics.
No one who desires to gain an insight into the most mod-
ern aspects of optical research can afford to be unfamiliar with
this remarkably original and consecutive presentation of the
subject of Optics.
A. A. MiCHELSON.
UNIVERSITY OF CHICAGO,
February, 1902.
AUTHOR'S PREFACE
THE purpose of the present book is to introduce the reader
who is already familiar with the fundamental concepts of the
differential and integral calculus into the domain of optics
in such a way that he may be able both to understand the
aims and results of the most recent investigation and, in addi-
tion, to follow the original works in detail.
The book was written at the request of the publisher a
request to which I gladly responded, not only because I
shared his view that a modern text embracing the entire
domain was wanting, but also because I hoped to obtain for
myself some new ideas from the' deeper insight into the sub-
ject which writing in book form necessitates. In the second
and third sections of the Physical Optics I have advanced some
new theories. In the rest of the book I have merely endeav-
ored to present in the simplest possible way results already
published.
Since I had a text-book in mind rather than a compen-
dium, I have avoided the citation of such references as bear
only upon the historical development of optics. The few refer-
ences which I have included are merely intended to serve the
reader for more complete information upon those points
which can find only brief presentation in the text, especially
in the case of the more recent investigations which have not
yet found place in the text-books.
vi AUTHOR'S PREFACE
In order to keep in touch with experiment and attain the
simplest possible presentation of the subject I have chosen a
synthetic method. The simplest experiments lead into the
domain of geometrical optics, in which but few assumptions
need to be made as to the nature of light. Hence I have
begun with geometrical optics, following closely the excellent
treatment given by Czapski in " Winkelmann's Handbuch der
Physik " and by Lommer in the ninth edition of the " Miiller-
Pouillet " text.
The first section of the Physical Optics, which follows the
Geometrical, treats of those general properties of light from
which the conclusion is drawn that light consists in a periodic
change of condition which is propagated with finite velocity in
the form of transverse waves. In this section I have included,
as an important advance upon most previous texts, Sommer-
feld's rigorous solution of the simplest case of diffraction,
Cornu's geometric representation of Fresnel's integrals, and,
on the experimental side, Michelson's echelon spectroscope.
In the second section, for the sake of the treatment of the
optical properties of different bodies, an extension of the
hypotheses as to the nature of light became for the first time
necessary. In accordance with the purpose of the book I have
merely mentioned the mechanical theories of light ; but the
electromagnetic theory, which permits the simplest and most
consistent treatment of optical relations, I have presented in
the following form :
Let X, Y, Z, and a, fi, y represent respectively the com-
ponents of the electric and magnetic forces (the first measured
in electrostatic units); also letj x ,j y ,j z , and s x , s y , s z represent
the components of the electric and magnetic current densities,
i.e. times the number of electric or magnetic lines of force
4?f
which pass in unit time through a unit surface at rest with
reference to the ether ; then, if c represent the ratio of the
AUTHOR'S PREFACE vii
electromagnetic to the electrostatic unit, the following funda-
mental equations always hold :
_ ~ _ , ere .
The number of lines of force is defined in the usual way.
The particular optical properties of bodies first make their
appearance in the equations which connect the electric and
magnetic current densities with the electric and magnetic
forces. Let these equations be called the substance equations
in order to distinguish them from the above fundamental
equations. Since these substance equations are developed
for non-homogeneous bodies, i.e. for bodies whose properties
vary from point to point, and since the fundamental equa-
tions hold in all cases, both the differential equations of the
electric and magnetic forces and the equations of condition
which must be fulfilled at the surface of a body are imme-
diately obtained.
In the process of setting up " substance and fundamental
equations " I have again proceeded synthetically in that I
have deduced them from the simplest electric and magnetic
experiments. Since the book is to treat mainly of optics this
process can here be but briefly sketched. For a more com-
plete development the reader is referred to my book " Physik
des Aethers auf elektromagnetische Grundlage " (Enke, 1894).
In this way however, no explanation of the phenomena of
dispersion is obtained because pure electromagnetic experi-
ments lead to conclusions in what may be called the domain
of macrophysical properties only. For the explanation of
optical dispersion a hypothesis as to the microphysical proper-
ties of bodies must be made. As such I have made use of
the ion-hypothesis introduced by Helmholtz because it seemed
to me the simplest, most intelligible, and most consistent way
of presenting not only dispersion, absorption, and rotary
viii AUTHOR'S PREFACE
polarization, but also magneto-optical phenomena and the
optical properties of bodies in motion. These two last-named
subjects I have thought it especially necessary to consider
because the first has acquired new interest from Zeeman's dis-
covery, and the second has received at the hands of H. A.
Lorentz a development as comprehensive as it is elegant.
This theory of Lorentz I have attempted to simplify by the
elimination of all quantities which are not necessary to optics.
With respect to magneto-optical phenomena I have pointed
out that it is, in general, impossible to explain them by the
mere supposition that ions set in motion in a magnetic field
are subject to a deflecting force, but that in the case of the
strongly magnetic metals the ions must be in such a continuous
motion as to produce Ampere's molecular currents. This
supposition also disposes at once of the hitherto unanswered
question as to why the permeability of iron and, in fact, of all
other substances must be assumed equal to that of the free
ether for those vibrations which produce light.
The application of the ion-hypothesis leads also to some
new dispersion formulae for the natural and magnetic rotation
of the plane of polarization, formulae which are experimentally
verified. Furthermore, in the case of the metals, the ion-
hypothesis leads to dispersion formulae which make the con-
tinuity of the optical and electrical properties of the metals
depend essentially upon the inertia of the ions, and which have
also been experimentally verified within the narrow limits thus
far accessible to observation.
The third section of the book is concerned with the rela-
tion of optics to thermodynamics and (in the third chapter) to
the kinetic theory of gases. The pioneer theoretical work in
these subjects was done by KirchhofT, Clausius, Boltzmann,
and W. Wien, and the many fruitful experimental investiga-
tions in radiation which have been more recently undertaken
show clearly that theory and experiment reach most perfect
development through their mutual support.
AUTHOR'S PREFACE ix
Imbued with this conviction, I have written this book in the
endeavor to make the theory accessible to that wider circle of
readers who have not the time to undertake the study of the
original works. I can make no claim to such completeness as
is aimed at in Mascart's excellent treatise, or in Winkelmann's
Handbuch. For the sake of brevity I have passed over many
interesting and important fields of optical investigation. My
purpose is attained if these pages strengthen the reader in
the view that optics is not an old and worn-out branch of
Physics, but that in it also there pulses a new life whose further
nourishing must be inviting to every one.
Mr. F. Kiebitz has given me efficient assistance in the
reading of the proof.
LEIPZIG, January, 1900.
INTRODUCTION
MANY optical phenomena, among them those which have
found the most extensive practical application, take place in
accordance with the following fundamental laws :
1 . The law of the rectilinear propagation of light ;
2 . The law of the independence of the different portions of
a beam of light ;
3. The law of reflection ;
4. The law of refraction.
Since these four fundamental laws relate only to the
geometrical determination of the propagation of light, conclu-
sions concerning certain geometrical relations in optics may
be reached by making them the starting-point of the analysis
without taking account of other properties of light. Hence
these fundamental laws constitute a sufficient foundation for
so-called geometrical optics, and no especial hypothesis which
enters more closely into the nature of light is needed to make
the superstructure complete.
In contrast with geometrical optics stands physical optics,
which deals with other than the purely geometrical properties,
and which enters more closely into the relation of the physical
properties of different bodies to light phenomena. The best
success in making a convenient classification of the great
multitude of these phenomena has been attained by devising
particular hypotheses as to the nature of light.
From the standpoint of physical optics the four above-men-
tioned fundamental laws appear only as very close approxima-
XI
xii INTRODUCTION
tions. However, it is possible to state within what limits the
laws of geometrical optics are accurate, i.e. under what cir-
cumstances their consequences deviate from the actual facts.
This circumstance must be borne in mind if geometrical
optics is to be treated as a field for real discipline in physics
rather than one for the practice of pure mathematics. The
truly complete theory of optical instruments can only be
developed from the standpoint of physical optics; but since,
as has been already remarked, the laws of geometrical optics
furnish in most cases very close approximations to the actual
facts, it seems justifiable to follow out the consequences of
these laws even in such complicated cases as arise in the
theory of optical instruments.
TABLE OF CONTENTS
PART I. -GEOMETRICAL OPTICS
CHAPTER I
THE FUNDAMENTAL LAWS
ART. PAGE
1. Direct Experiment , i
2. Law of the Extreme Path 6
3. Law of Malus n
CHAPTER II
GEOMETRICAL THEORY OF OPTICAL IMAGES
1. The Concept of Optical Images 14
2. General Formulae for Images 15
3. Images Formed by Coaxial Surfaces 17
4. Construction of Conjugate Points 24
5. Classification of the Different Kinds of Optical Systems 25
6. Telescopic Systems 26
7. Combinations of Systems 28
CHAPTER III
PHYSICAL CONDITIONS FOR IMAGE FORMATION
1. Refraction at a Spherical Surface 32
2. Reflection at a Spherical Surface 36
3. Lenses 40
4. Thin Lenses 42
5. Experimental Determination of Focal Length 44
6. Astigmatic Systems 46
7. Means of Widening the Limits of Image Formation 52
8. Spherical Aberration 54
xiii
xiv TABLE OF CONTENTS
ART. PAGE
9. The Law of Sines 58
10. Images of Large Surfaces by Narrow Beams 63
11. Chromatic Aberration of Dioptric Systems 66
CHAPTER IV
APERTURES AND THE EFFECTS DEPENDING UPON THEM
1. Entrance- and Exit-Pupils 73
2. Telecentric Systems 75
3. Field of View 76
4. The Fundamental Laws of Photometry 77
5. The Intensity of Radiation and the Intensity of Illumination of
Optical Surfaces 84
6. Subjective Brightness of Optical Images 86
7. The Brightness of Point Sources 90
8. The Effect of the Aperture upon the Resolving Power of Optical
Instruments 91
CHAPTER V
OPTICAL INSTRUMENTS
1. Photographic Systems 93
2. Simple Magnify ing-glasses 95
3. The Microscope 97
4. The Astronomical Telescope 107
5. The Opera Glass 109
6. The Terrestrial Telescope 112
7. The Zeiss Binocular 112
8. The Reflecting Telescope 113
PART II. PHYSICAL OPTICS
SECTION I
GENERAL PROPERTIES OF LIGHT
CHAPTER I
THE VELOCITY OF LIGHT
1. Romer's Method 114
2. Bradley 's Method 115
TABLE OF CONTENTS xv
ART. PAC;H
3. Fizeau's Method 1 1 6
4. Foucault's Method 1 1 8
5. Dependence of the Velocity of Light upon the Medium and the
Color 120
6. The Velocity of a Group of Waves 121
CHAPTER II
INTERFERENCE OF LIGHT
1. General Considerations 124
2. Hypotheses as to the Nature of Light 124
3. Fresnel's M irrors 1 30
4. Modifications of the Fresnel Mirrors 134
5. Newton's Rings and the Colors of Thin Plates 136
6. Achromatic Interference Bands 144
7. The Interferometer 144
8. Interference with Large Difference of Path 148
9. Stationary Waves 1 54
10. Photography in Natural Colors 1 56
CHAPTER III
HUYGENS' PRINCIPLE
1. Huygens' Principle as first Conceived 1 59
2. Fresnel's Improvement of Huygens' Principle 162
3. The Differential Equation of the Light Disturbance 169
4. A Mathematical Theorem i7 2
5. Two General Equations 174
6. Rigorous Formulation of Huygens' Principle 179
CHAPTER IV
DIFFRACTION OF LIGHT
1. General Treatment of Diffraction Phenomena 185
2. Fresnel's Diffraction Phenomena 188
3. Fresnel's Integrals 1 88
4. Diffraction by a Straight Edge 1 92
5. Diffraction through a Narrow Slit 1 98
6. Diffraction by a Narrow Screen 201
y. Rigorous Treatment of Diffraction by a Straight Edge 203
xvi TABLE OF CONTENTS
ART. PAGE
8. Fraunhofer's Diffraction Phenomena 213
9. Diffraction through a Rectangular Opening 214
10. Diffraction through a Rhomboid 217
1 1. Diffraction through a Slit 217
12. Diffraction Openings of any Form 219
13. Several Diffraction Openings of like Form and Orientation 219
14. Babinet's Theorem 221
1 5. The Diffraction Grating 222
1 6. The Concave Grating 225
17. Focal Properties of a Plane Grating 227
1 8. Resolving Power of a Grating 227
19. Michelson's Echelon 228
20. The Resolving Power of a Prism 233
21. Limit of Resolution of a Telescope 235
22. The Limit of Resolution of the Human Eye 236
23. The Limit of Resolution of the Microscope 236
CHAPTER V
POLARIZATION
1. Polarization by Double Refraction 242
2. The Nicol Prism 244
3. Other Means of Producing Polarized Light 246
4. Interference of Polarized Light 247
5. Mathematical Discussion of Polarized Light 247
6. Stationary Waves Produced by Obliquely Incident Polarized
Light 251
7. Position of the Determinative Vector in Crystals 252
8. Natural and Partially Polarized Light 253
9. Experimental Investigation of Elliptically Polarized Light 255
SECTION II
OPTICAL PROPERTIES OF BODIES
CHAPTER I
THEORY OF LIGHT
1. Mechanical Theory 259
2. Electromagnetic Theory 260
3. The Definition of the Electric and of the Magnetic Force 262
TABLE OF CONTENTS xvii
4. Definition of the Electric Current in the Electrostatic and the
Electromagnetic Systems 263
5. Definition of the Magnetic Current 265
6. The Ether 267
7. Isotropic Dielectrics 268
8. The Boundary Conditions 271
9. The Energy of the Electromagnetic Field 272
10. The Rays of Light as the Lines of Energy Flow 273
CHAPTER II
TRANSPARENT ISOTROPIC MEDIA
1. The Velocity of Light 274
2. The Transverse Nature of Plane Waves 278
3. Reflection and Refraction at the Boundary between two Trans-
parent Isotropic Media 278
4. Perpendicular Incidence ; Stationary Waves 284
5. Polarization of Natural Light by Passage through a Pile of
Plates 285
6. Experimental Verification of the Theory 286
7. Elliptic Polarization of the Reflected Light and the Surface or
Transition Layer 287
8. Total Reflection 295
9. Penetration of the Light into the Second Medium in the Case of
Total Reflection 299
10. Application of Total Reflection to the Determination of Index
of Refraction 301
11. The Intensity of Light in Newton's Rings 302
12. Non-homogeneous Media ; Curved Rays 306
CHAPTER III
OPTICAL PROPERTIES OF TRANSPARENT CRYSTALS
1. Differential Equations and Boundary Conditions 308
2. Light-vectors and Light-rays 311
3. Fresnel's Law for the Velocity of Light 314
4. The Directions of the Vibrations 316
5. The Normal Surface 317
6. Geometrical Construction of the Wave Surface and of the Direc-
tion of Vibration 320
xviii TABLE OF CONTENTS
ART. PAGE
7. Uniaxial Crystals 323
8. Determination of the Direction of the Ray from the Direction of
the Wave Normal . 324
9. The Ray Surface 326
10. Conical Refraction 331
11. Passage of Light through Plates and Prisms of Crystal 335
12. Total Reflection at the Surface of Crystalline Plates 339
13. Partial Reflection at the Surface of a Crystalline Plate 344
14. Interference Phenomena Produced by Crystalline Plates in
Polarized Light when the Incidence is Normal 344
15. Interference Phenomena in Crystalline Plates in Convergent
Polarized Light 349
CHAPTER IV
ABSORBING MEDIA
1. Electromagnetic Theory 358
2. Metallic Reflection 361
3. The Optical Constants of the Metals 366
4. Absorbing Crystals 368
5. Interference Phenomena in Absorbing Biaxial Crystals 374
6. Interference Phenomena in Absorbing Uniaxial Crystals 380
CHAPTER V
DISPERSION
1. Theoretical Considerations 382
2. Normal Dispersion 388
3. Anomalous Dispersion 392
4. Dispersion of the Metals < 396
CHAPTER VI
OPTICALLY ACTIVE SUBSTANCES
1 . General Considerations 4
2. Isotropic Media 40 T
3. Rotation of the Plane of Polarization 404
4. Crystals , 408
5. Rotary Dispersion 41 2
6. Absorbing Active Substances 415
TABLE OF CONTENTS xix
CHAPTER VII
MAGNETICALLY ACTIVE SUBSTANCES
A. Hypothesis of Molecular Currents
ART. PAGE
1. General Considerations 418
2. Deduction of the Differential Equations 420
3. The Magnetic Rotation of the Plane of Polarization 426
4. Dispersion in Magnetic Rotation of the Plane of Polarization. . 429
5. Direction of Magnetization Perpendicular to the Ray 433
B. Hypothesis of the Hall Effect
1 . General Considerations 433
2. Deduction of the Differential Equations 435
3. Rays Parallel to the Direction of Magnetization 437
4. Dispersion in the Magnetic Rotation of the Plane of Polarization. 438
5. The Impressed Period Close to a Natural Period 440
6. Rays Perpendicular to the Direction of Magnetization 443
7. The Impressed Period in the Neighborhood of a Natural Period. 444
8. The Zeeman Effect 446
9. The Magneto-optical Properties of Iron, Nickel, and Cobalt... 449
10. The Effects of the Magnetic Field of the Ray of Light 452
CHAPTER VIII
BODIES IN MOTION
1 . General Considerations 457
2. The Differential Equations of the Electromagnetic Field Re-
ferred to a Fixed System of Coordinates 457
3. The Velocity of Light in Moving Media 465
4. The Differential Equations and the Boundary Conditions Re-
ferred to a Moving System of Coordinates which is Fixed
with Reference to the Moving Medium 467
5. The Determination of the Direction of the Ray by Huygens'
Principle 470
6. The Absolute Time Replaced by a Time which is a Function of
the Coordinates 471
7. The Configuration of the Rays Independent of the Motion 473
8. The Earth as a Moving System 474
9. The Aberration of Light 475
10. Fizeau's Experiment with Polarized Light 477
11. Michelson's Interference Experiment 478
xx TABLE OF CONTENTS
PART III. RADIATION
CHAPTER I
ENERGY OF RADIATION
ART. PACK
1. Emissive Power 483
2. Intensity of Radiation of a Surface 484
3. The Mechanical Equivalent of the Unit of Light 485
4. The Radiation from the Sun 487
5. The Efficiency of a Source of Light 487
6. The Pressure of Radiation 488
7. Prevost's Theory of Exchanges 491
CHAPTER II
APPLICATION OF THE SECOND LAW OF THERMODYNAMICS TO
PURE TEMPERATURE RADIATION
1. The Two Laws of Thermodynamics 493
2. Temperature Radiation and Luminescence 494
3. The Emissive Power of a Perfect Reflector or of a Perfectly
Transparent Body is Zero 495
4. Kirchhoff's Law of Emission and Absorption 496
5. Consequences of Kirchhoff's Law 499
6. The Dependence of the Intensity of Radiation upon the Index
of Refraction of the Surrounding Medium 502
7. The Sine Law in the Formation of Optical Images of Surface
Elements 505
8. Absolute Temperature 506
9. Entropy 510
10. General Equations of Thermodynamics 511
11. The Dependence of the Total Radiation of a Black upon its Ab-
solute Temperature 512
12. The Temperature of the Sun Calculated from its Total Emission 515
13. The Effect of Change in Temperature upon the Spectrum of
a Black Body 516
14. The Temperature of the Sun Determined from the Distribution
of Energy in the Solar Spectrum 523
15. The Distribution of the Energy in the Spectrum of a Black
Body 524
TABLE OF CONTENTS xxi
CHAPTER III
INCANDESCENT VAPORS AND GASES
IRT. PAGE
1. Distinction between Temperature Radiation and Luminescence. 528
2. The Ion-hypothesis 529
3. The Damping of Ionic Vibrations because of Radiation 534
4. The Radiation of the Ions under the Influence of External
Radiation 535
5. Fluorescence 536
6. The Broadening of the Spectral Lines Due to Motion in the Line
of Sight 537
7. Other Causes of the Broadening of the Spectral Lines 541
INDEX e .. 543
PART I
GEOMETRICAL OPTICS
CHAPTER I
THE FUNDAMENTAL LAWS
I. Direct Experiment. The four fundamental laws stated
above are obtained by direct experiment.
The rectilinear propagation of light is shown by the shadow
of an opaque body which a point source of light P casts upon
a screen 5. If the opaque body contains an aperture L, then
the edge of the shadow cast upon the screen is found to be the
intersection of 5 with a cone whose vertex lies in the source P
and whose surface passes through the periphery of the aper-
ture L.
If the aperture is made smaller, the boundary of the shadow
upon the screen 5 contracts. Moreover it becomes indefinite
when L is made very small (e.g. less than i mm.'), for
points upon the screen which lie within the geometrical shadow
now receive light from P. However, it is to be observed
that a true point source can never be realized, and, on account
of the finite extent of the source, the edge of the shadow could
never be perfectly sharp even if light were propagated in
straight lines (umbra and penumbra). Nevertheless, in the
case of a very small opening L (say of about one tenth mm.
diameter) the light is spread out behind L upon the screen so
far that in this case the propagation cannot possibly be recti-
linear.
_ _ THEORY OF OPTICS
The same result is obtained if the shadow which an opaque
?'- ^ csts lUpQR ,the screen S is studied, instead of the
spreading out of the light which has passed through a hole in
an opaque object. If S' is sufficiently small, rectilinear
propagation of light from P does not take place. It is there-
fore necessary to bear in mind that the law of the rectilinear
propagation of light holds only when the free opening through
which the light passes, or the screens which prevent its passage,
are not too small.
In order to conveniently describe the propagation of light
from a source P to a screen S, it is customary to say that P
sends rays to 5. The path of a ray of light is then defined
by the fact that its effect upon 5 can be cut off only by an
obstacle that lies in the path of the ray itself. When the
propagation of light is rectilinear the rays are straight lines,
as when light from P passes through a sufficiently large open-
ing in an opaque body. In this case it is customary to say
that P sends a beam of light through L.
Since by diminishing L the result upon the screen 5 is the
same as though the influence of certain of the rays proceeding
from P were simply removed while that of the other rays
remained unchanged, it follows that the different parts of a
beam of light are independent of one another.
This law too breaks down if the diminution of the open-
ing L is carried too far. But in that case the conception of
light rays propagated in straight lines is altogether untenable.
The concept of light rays is then merely introduced for
convenience. It is altogether impossible to isolate a single
ray and prove its physical existence. For the more one tries
to attain this end by narrowing the beam, the less does light
proceed in straight lines, and the more does the concept of
light rays lose its physical significance.
If the homogeneity of the space in which the light rays exist
is disturbed by the introduction of some substance, the rays
undergo a sudden change of direction at its surface: each ray
splits up into two, a reflected and a refracted ray. If the sur-
THE FUNDAMENTAL LAWS 3
face of the body upon which the light falls is plane, then the
plane of incidence is that plane which is defined by the incident
ray and the normal N to the surface, and the angle of
incidence is the angle included between these two direc-
tions.
The following laws hold : The reflected and refracted rays
both lie in the plane of incidence. The angle of reflection (the
angle included between A^and the reflected ray) is equal to the
angle of incidence. The angle of refraction
i.e. the index of A with respect to B is the reciprocal of the
index of B with respect to A .
The law of refraction stated in (i) permits, then, the con-
clusion that 0' may also be regarded as the angle of incidence
in the body, and as the angle of refraction in the surround-
ing medium; i.e. that the direction of propagation may be
reversed without changing the path of the rays. For the case
of reflection it is at once evident that this principle of reversi-
bility also holds.
Therefore equation (i), which corresponds to the passage
of light from a body A to a. body B or the reverse, may be
put in the symmetrical form
;* a .sin 0* = n b - sin 6 , ..... (3)
in which a and
n b , and if
(6)
then sin 0^ > I ; i.e. there is no real angle of refraction 4 .
In that case no refraction occurs at the surface, but reflection
only. The whole intensity of the incident ray must then be
contained in the reflected ray; i.e. there is total reflection.
In all other cases {partial reflection] the intensity of the
incident light is divided between the reflected and the re-
fracted rays according to a law which will be more fully
considered later (Section 2, Chapter II). Here the observa-
tion must suffice that, in general, for transparent bodies the
refracted ray contains much more light than the reflected.
Only in the case of the metals does the latter contain almost
the entire intensity of the incident light. It is also to be
observed that the law of reflection holds for very opaque bodies,
like the metals, but the law of refraction is no longer correct
in the form given in (i) or (3). This point will be more fully
discussed later (Section 2, Chapter IV).
The different qualities perceptible in light are called colors.
The refractive index depends on the color, and, when referred
to air, increases, for transparent bodies, as the color changes
from red through yellow to blue. The spreading out of white
light into a spectrum by passage through a prism is due to this
change of index with the color, and is called dispersion.
If the surface of the body upon which the light falls is not
plane but curved, it may still be looked upon as made up of
very small elementary planes (the tangent planes), and the
paths of the light rays may be constructed according to the
6 THEORY OF OPTICS
above laws. However, this process is reliable only when the
curvature of the surface does not exceed a certain limit, i.e.
when the surface may be considered smooth.
Rough surfaces exhibit irregular (diffuse) reflection and
refraction and act as though they themselves emitted light.
The surface of a body is visible only because of diffuse reflec-
tion and refraction. The surface of a perfect mirror is invisi-
ble. Only objects which lie outside of the mirror, and whose
rays are reflected by it, are seen.
2. Law of the Extreme Path.* All of these experi-
mental facts as to the direction of light rays are comprehended
in the law of the extreme path. If a ray of light in passing
from a point P to a point P' experiences any number of reflec-
tions and refractions, then the sum of the products of the
index of refraction of each medium by the distance traversed
in it, i.e. 2nl, has a maximum or minimum value; i.e. it
differs from a like sum for all other paths which are infinitely
close to the actual path by terms of the second or higher order.
Thus if 6 denotes the variation of the first order,
$2nl =o (7)
The product, index of refraction times distance traversed,
is known as the optical length of the ray.
In order to prove the proposition for a single refraction let
POP' be the actual path of the light (Fig. i), OE the inter-
section of the plane of incidence PON with the surface (tan-
gent plane) of the refracting body, O' a point on the surface
of the refracting body infinitely near to O, so that OO'
makes any angle with the plane of incidence, i.e. with the
line OE. Then it is to be proved that, to terms of the second
or higher order,
. . (8)
* ' Extreme ' is here used to denote either greatest or least (maximum or
minimum). TR.
THE FUNDAMENTAL LAWS
in which n and ri represent the indices of refraction of the
adjoining media.
If a perpendicular OR be dropped from O upon PO' and a
perpendicular OR' upon P ' ', then, to terms of the second
order,
PO' = PO + RO', OP' = OP' - O'R'. . . (9)
Also, to the same degree of approximation,
RO' = OO'.cos POO', O'R' = OO'.cos POO. (10)
ri
FIG. i.
In order to calculate cos POO' imagine an axis OD perpen-
dicular to ON and OE, and introduce the direction cosines of
the lines PO and OO' referred to a rectangular system of
coordinates whose axes are ON, OE, and OD, If represent
the angle of incidence PON, then, disregarding the sign, the
direction cosines of PO are
those of OO are
cos 0, sin 0, o,
o, cos , sin $.
According to a principle of analytical geometry the cosine
of the angle between any two lines is equal to the sum of the
8 THEORY OF OPTICS
products of the corresponding direction cosines of the lines with
reference to a system of rectangular coordinates, i.e.
cos POO' sin 0-cos fl,
and similarly
cos P'OO 1 = sin 0'-cos $,
in which 0' represents the angle of refraction.
Then, from (9) and (10),
n.PO' + ri-O'P' = n.PO + n-OO'.sm 0-cos
+ n'-OP' n'-OO'-sm 0'-cos fl.
Since now from the law of refraction the relation exists
n-sin = ;z'-sin 0',
it follows that equation (8) holds for any position whatever
of the point 0' which is infinitely close to O.
For the case of a single reflection equation (7) may be
more simply proved. It then takes the form
6(PO + OP') = o, (u)
in which (Fig. 2) PO and OP' denote the actual path of the
ray. If P l be that point which is symmetrical to P with
fP'
FIG. 2.
respect to the tangent plane OE of the refracting body, then
for every point O' in the tangent plane, PO' = P^O' . The
length ^f the path of the light from P to P' for a single reflec-
THE FUNDAMENTAL LAWS 9
tion at the tangent plane OE is, then, for every position of the
point O f , equal to P^O 1 -f- O'P' '. Now this length is a mini-
mum if P l , O' t and P' lie in a straight line. But in that case
the point O' actually coincides with the point O which is
determined by the law of reflection. But since the property
of a minimum (as well as of a maximum) is expressed by the
vanishing of the first derivative, i.e. by equation (n), there-
fore equation (7) is proved for a single reflection.
It is to be observed that the vanishing of the first derivative
is the condition of a maximum as well as of a minimum. In
the case in which the refracting body is actually bounded by a
plane, it follows at once from the construction given that the
path of the light in reflection is a minimum. It may also be
proved, as will be more fully shown later on, that in the case
of refraction the actual path is a minimum if the refracting
body is bounded by a plane. Hence this principle has often
been called the law of least path.
When, however, the surface of the refracting or reflecting
body is curved, then the path of the light is a minimum or a
maximum according to the nature of the curvature. The
vanishing of the first derivative is the only property which is
common to all cases, and this also is entirely sufficient for the
determination of the path of the ray.
A clear comprehension of the subject is facilitated by the
i'ntroduction of the so-called aplanatic surface, which is a sur-
face such that from every point upon it the sum of the optical
paths to two points P and P' is constant. For such a surface
the derivative, not only of the first order, but also of any
other order, of the sum of the optical paths vanishes.
In the case of reflection the aplanatic surface, defined by
PA + P'A = constant C, .... (12)
is an ellipsoid of revolution having the points Pand P' as foci.
If SOS' represents a section of a mirror (Fig. 3) and
a point upon it such that PO and P'O are incident and
reflected rays, then the aplanatic surface AOA', which
io THEORY OF OPTICS
passes through the point O and corresponds to the points P
and P r , must evidently be tangent to the mirror SOS' at 0,
since at this point the first derivative of the optical paths
vanishes for both surfaces. If now, as in the figure, the mirror
SOS' is more concave than the aplanatic surface, then the
optical path PO -f- OP is a maximum, otherwise a minimum.
FIG. 3.
The proof of this appears at once from the figure, since for all
points O' within the ellipsoid AOA' whose equation is given
in (12), the sum PO + OP' is smaller than the constant C,
while for all points outside, this sum is larger than C, and for
the actual point of reflection (9, it is equal to C.
In the case of refraction the aplanatic surface, defined by
H .PA -\-ri -P'A = constant C,
is a so-called Cartesian oval which must be convex towards
the less refractive medium (in Fig. 4 n < '), and indeed more
convex than a sphere described about P' as a centre.
This aplanatic surface also separates the regions for whose
points 0' the sum of the optical paths n-PO' + n'-P'O' > C
from those for which that sum < C. The former regions lie
on the side of the aplanatic surface toward the less refractive
medium (left in the figure), the latter on the side toward the
more refractive medium (right in the figure).
If now SOS' represents a section of the surface between the
THE FUNDAMENTAL LAWS n
two media, and PO, P'O the actual path which the light takes
in accordance with the law of refraction, then the length of the
path through O is a maximum or a minimum according as
SOS' is more or less convex toward the less refracting medium
FIG. 4.
than the aplanatic surface AOA'. The proof appears at once
from the figure.
If, for example, SOS' is a plane, the length of the path is
a minimum. In the case shown in the figure the length of the
path is a maximum.
Since, as will be shown later, the index of refraction is
inversely proportional to the velocity, the optical path nl is
proportional to the time which the light requires to travel the
distance /. The principle of least path is then identical with
Fermat 's principle of least time, but it is evident from the
above that, under certain circumstances, the time may also be
a maximum.
Since d^nl = o holds for each single reflection or refrac-
tion, the equation $2nl = o may at once be applied to the
case of any number of reflections and refractions.
3. The Law of Malus. Geometrically considered there
are two different kinds of ray systems : those which may be
cut at right angles by a properly constructed surface F (ortho-
12
THEORY OF OPTICS
tomic system), and those for which no such surface F can be
found (anorthotomic system). With the help of the preceding
principle the law of Malus can now be proved. This law is
stated thus : A n orthotomic system of rays remains orthotomic
after any number of reflections and refractions. From the
standpoint of the wave theory, which makes the rays the
normals to the wave front, the law is self-evident. But it can
also be deduced from the fundamental geometrical laws already
used.
Let (Fig. 5) ABCDE and A'B'C'D'E' be two rays infinitely
close together and let their initial direction be normal to a
surface F. If L represents the total
optical distance from A to E, then
it may be proved that every ray
whose total path, measured from its
origin A, A', etc., has the same
optical length Z, is normal to a sur-
face F' which is the locus of the ends
E, E', etc., of those paths. For
the purpose of the proof let A ' B and
E ' D be drawn.
According to the law of extreme
path stated above, the length of
must be equal to that of the infinitely
near path A'BCDE' ', i.e. equal to L, which is also the length
of the path ABCDE. If now from the two optical distances
A'BCDE' and ABCDE the common portion BCD be sub-
tracted, it follows that
= n-A'B
in which n represents the index of the medium between the
surfaces F and B, and n' that of the medium between D
and F 1 . But since AB = A' B, because AB is by hypothesis
normal to F, it follows that
DE = DE',
FIG. 5.
the path A'B'C'D'E
THE FUNDAMENTAL LAWS 13
i.e. DE is perpendicular to the surface F f . In like manner
it may be proved that any other ray D'E' is normal to F' .
Rays which are emitted by a luminous point are normal to
a surface F, which is the surface of any sphere described about
the luminous point as a centre. Since every source of light
may be looked upon as a complex of luminous points, it
follows that light rays always form an orthotomic system.
CHAPTER II
GEOMETRICAL THEORY OF OPTICAL IMAGES
i. The Concept of Optical Images. If in the neighbor-
hood of a luminous point P there are refracting and reflecting
bodies having any arbitrary arrangement, then, in general,
there passes through any point P' in space one and only one
ray of light, i.e. the direction which light takes from P to P r
is completely determined. Nevertheless certain points P' may
be found at which two or more of the rays emitted by P inter-
sect. If a large number of the rays emitted by P intersect in
a point P', then P' is called the optical image of P. The
intensity of the light at P' will clearly be a maximum. If the
actual intersection of the rays is at P' , the image is called real;
if P' is merely the intersection of the backward prolongation
of the rays, the image is called virtual. The simplest exam-
ple of a virtual image is found in the reflection of a luminous
point P in a plane mirror. The image P' lies at that point
which is placed symmetrically to P with respect to the mirror.
Real images may be distinguished from virtual by the direct
illumination which they produce upon a suitably placed rough
surface such as a piece of white paper. In the case of plane
mirrors, for instance, no light whatever reaches the point P 1 '.
Nevertheless virtual images may be transformed into real by
certain optical means. Thus a virtual image can be seen be-
cause it is transformed by the eye into a real image which
illumines a certain spot on the retina.
The cross-section of the bundle of rays which is brought
together in the image may have finite length and breadth or
may be infinitely narrow so as in the limit to have but one
GEOMETRICAL THEORY OF OPTICAL IMAGES 15
dimension. Consider, for example, the case of a single refrac-
tion. If the surface of the refracting body is the aplanatic
surface for the two points P and P ', then a beam of any size
which has its origin in P will be brought together in P' ; for
all rays which start from P and strike the aplanatic surface
must intersect in P', since for all of them the total optical dis-
tance from P to P' is the same.
If the surface of the refracting body has not the form of the
aplanatic surface, then the number of rays which intersect in
P is smaller the greater the difference in the form of the two
surfaces (which are necessarily tangent to each other, see
page 10). In order that an infinitely narrow, i.e. a plane,
beam may come to intersection in P', the curvature of the sur-
faces at the point of tangency must be the same at least in one
plane. If the curvature of the two surfaces is the same at
for two and therefore for all planes, then a solid elementary
beam will come to intersection in P '; and if, finally, a finite
section of the surface of the refracting body coincides with the
aplanatic surface, then a beam of finite cross-section will come
to intersection in P' .
Since the direction of light may be reversed, it is possible
to interchange the source P and its image P', i.e. a source at
P' has its image at P. On account of this reciprocal relation-
ship P and P' are called conjugate points.
2. General Formulae for Images. Assume that by means
of reflection or refraction all the points P of a given space are
imaged in points P' of a second space. The former space will
be called the object space ; the latter, the image space. From
the definition of an optical image it follows that for every ray
which passes through P there is a conjugate ray passing
through P . Two rays in the object space which intersect at
P must correspond to two conjugate rays which intersect in
the image space, the intersection being at the point P' which
is conjugate to P. For every point P there is then but one
conjugate point P' . If four points P^Pff^ of the object space
lie in a plane, then the rays which connect any two pairs of
1 6 THEORY OF OPTICS
these points intersect, e.g. the ray P 1 P 2 cuts the ray PjP^ in
the point A. Therefore the conjugate rays P\P' 2 and P f\
also intersect in a point, namely in A' the image of A. Hence
the four images P^P^P^P^ also lie in a plane. In other
words, to every point, ray, or plane in the one space there
corresponds one, and but one, point, ray, or plane in the
other. Such a relation of two spaces is called in geometry a
collinear relationship.
The analytical expression of the collinear relationship can
be easily obtained. Let x, y, z be the coordinates of a point
P of the object space referred to one rectangular system, and
x' , y f , z' the coordinates of the point P f referred to another
rectangular system chosen for the image space ; then to every
x, y y z there corresponds one and only one x' , y' , z' , and vice
versa. This is only possible if
+ b^y + c^z + d l
x =
ax -j- by -\- cz -j- d
+ b^y -f c^z -f d 2
ax -\- by -\- cz -\- d
z =
(0
ax -[- by -f- cz + d
in which a, b, c, d are constants. That is, for any given
x' , y' , z' , the values of x, y, z may be calculated from the
three linear equations (i); and inversely, given values of x, y,
z determine x' , y 1 ', z' . If the right-hand side of equations (i)
were not the quotient of two linear functions of x, y, z, then
for every x' , y' , z' there would be several values of x, y, z.
Furthermore the denominator of this quotient must be one and
the same linear function (ax -\- by -J- cz -f- d\ since otherwise
a plane in the image space
A'x' + B'y' + C'z' + D' = o
would not again correspond to a plane
in the object space.
GEOMETRICAL THEORY OF OPTICAL IMAGES 17
If the equations (i) be solved for x, y y and z, forms analo-
gous to (i) are obtained; thus
X =
/, etc. ... (2)
a'x' + b'y' + c'z' + d
From (i) it follows that for
ax -f- by + cz + d = o: x' = y' 2' = oo .
Similarly from (2) for
The plane ' of its points P lie at
infinity. Two rays which originate in a point P of this focal
plane correspond to two parallel rays in the image space.
The plane a'x' + b'y' -f c'z' + d' o is called the focal
plane g' of the image space. Parallel rays in the object space
correspond to conjugate rays in the image space which inter-
sect in some point of this focal plane g'.
In case a = b = c o, equations (i) show that to finite
values of x, y, z correspond finite values of x' , y' , z' ; and, in-
versely, since, when a, b, and c are zero, a' , b' ', c' are also
zero, to finite values of x' , y' ', z' correspond finite values of
x, y, z. In this case, which is realized in telescopes, there
are no focal planes at finite distances.
3. Images Formed by Coaxial Surfaces. In optical in-
struments it is often the case that the formation of the image
takes place symmetrically with respect to an axis; e.g. this
is true if the surfaces of the refracting or reflecting bodies are
surfaces of revolution having a common axis, in particular, sur-
faces of spheres whose centres lie in a straight line.
From symmetry the image P' of a point P must lie in the
plane which passes through the point P and the axis of the
system, and it is entirely sufficient, for the study of the image
formation, if the relations between the object and image in
such a meridian plane are known.
i8
THEORY OF OPTICS
If the xy plane of the object space and the x'y' plane of the
image space be made to coincide with this meridian plane, and
if the axis of symmetry be taken as both the x and the x' axis,
then the z and z' coordinates no longer appear in equations (i).
They then reduce to
" "" ax + by + d j ' '~ ax + by + d' ' ' ^
The coordinate axes of the xy and the x'y' systems are
then parallel and the x and x' axes lie in the same line. The
origin O' for the image space is in general distinct from the
origin O for the object space. The positive direction of x will
be taken as the direction of the incident light (from left to
O'
FIG. 6.
right); the positive direction of x' , the opposite, i.e. from
right to left. The positive direction of y and y' will be taken
upward (see Fig. 6).
From symmetry it is evident that x' does not change its
value when y changes sign. Therefore in equations (3)
b l = b o. It also follows from symmetry that a change in
sign of y produces merely a change in sign of y' . Hence
# 2 = d 2 = o and equations (3) reduce to
x'
(4)
Five constants thus remain, but their ratios alone are
sufficient to determine the formation of the image. Hence
GEOMETRICAL THEORY OF OPTICAL IMAGES 19
there are in general four characteristic constants which deter-
mine the formation of images by coaxial surfaces.
The solution of equations (4) for x and y gives
dx' d l a^d ad v y'
= 1 - ax" y ~~ b % ' ^ - a*T ' '
The equation of the focal plane of the object space is
ax -f- d o, that of the focal plane of the image space
ax' a l = o. The intersections F and F' of these planes
with the axis of the system are called the principal foci.
If the principal focus F of the object space be taken as the
origin of x, and likewise the principal focus F' of the image
space as the origin of x' , then, if X Q , x^ represent the coordi-
nates measured from the focal planes, ax^ will replace ax -\- d
and (P)
a* y ax^
Hence only two characteristic constants remain in the
equations. The other two were taken up in fixing the posi-
tions of the focal planes. For these two complex constants
simpler expressions will be introduced by writing (dropping
subscripts)
xx '=ff, y ~=^ = j, (7)
In this equation x and x' are the distances of the object and
the image from the principal focal planes g and g' respectively.
The ratio y' : y is called the magnification. It is I for
x f, i.e. x' =f. This relation defines two planes ^j and
)' which are at right angles to the axis of the system. These
planes are called the unit planes. Their points of intersection
//"and H 1 with the axis of the system are called unit points.
The unit planes are characterized by the fact that the dis-
tance from the axis of any point P in one unit plane is equal to
tJiat of the conjugate point P' in the other imit plane. The two
remaining constants /and/' of equation (7) denote, in accord-
20
THEORY OF OPTICS
ance with the above, the distance of the unit' planes , Q from
the focal planes g, g'. The constant / is called the focal
length of the object space; f, the focal length of the image
space. The direction of f is positive when the ray falls first
upon the focal plane g, then upon the unit plane ; for/"' the
case is the reverse. In Fig. 7 both focal lengths are positive.
The significance of the focal lengths can be made clear in
the following way: Parallel rays in the object space must have
conjugate rays in the image space which intersect in some
point in the focal plane g' distant, say, y' from the axis. The
value of ;/ evidently depends on the angle of inclination u of
the incident ray with respect to the axis. If u = o, it follows
from symmetry thaty = o, i.e. rays parallel to the axis have
conjugate rays which intersect in the principal focus F f . But
FIG. 7.
if u is not equal to zero, consider a ray PFA which passes
through the first principal focus F, and cuts the unit plane
in A (Fig. 7). The ray which is conjugate to it, A P' , must
evidently be parallel to the axis since the first ray passes
through F. Furthermore, from the property of the unit planes,
A and A' are equally distant from the axis. Consequently
the distance from the axis y' of the image which is formed by
a parallel beam incident at an angle u is, as appears at once
from Fig. 7,
y =/-tan u (8)
Hence the following law: The focal length of the object
space is equal to the ratio of the linear magnitude of an image
GEOMETRICAL THEORY OF OPTICAL IMAGES 21
formed in the focal plane of the image space to the apparent
(angular) magnitude of its infinitely distant object, A similar
definition holds of course for the focal length/' of the image
space, as is seen by conceiving the incident beam of parallel
rays to pass first through the image space and then to come
to a focus in the focal plane g.
If in Fig. 7 A'P' be conceived as the incident ray, so that
the functions of the image and object spaces are interchanged,
then the following may be given as the definition of the focal
length y, which will then mean the focal length of the image
space :
The focal length of the image space is equal to the distance
between the axis and any ray of the object space which is
parallel to the axis divided by the tangent of the inclination of
its conjugate ray.
Equation (8) may be obtained directly from (7) by making
tan u y\x and tan u = y' \ x' . Since x and x' are taken
positive in opposite directions and y and y' in the same direc-
tion, it follows that u and u' are positive in different directions.
The angle of inclination u of a ray in the object space is positive
if the ray goes upward from left to right; the angle of inclina-
tion u of a ray in the image space is positive if the ray goes
downward from left to right.
The magnification depends, as equation (7) shows, upon
x, the distance of the object from the principal focus F, and
upon /", the focal length. It is, however, independent of j/,
i.e. the image of a plane object which is perpendicular to the
axis of the system is similar to the object. On the other hand
the image of a solid object is not similar to the object, as is
evident at once from the dependence of the magnification
upon x. Furthermore it is easily shown from (7) that the
magnification in depth, i.e. the ratio of the increment dx' of
x to an increment dx of x, is proportional to the square of the
lateral magnification.
Let a ray in the object space intersect the unit plane in
22
THEORY OF OPTICS
A and the axis in P (Fig. 8). Its angle of inclination u with
respect to the axis is given by
AH AH
tan u =
PH
if x taken with the proper sign represents the distance of P
from F.
FIG. 8.
The angle of inclination u' of the conjugate ray with respect
to the axis is given by
A'H' A'H'
tan u' ~
P'H>
ff /
if x' represent the distance of P' from P , and P' and A' are
the points conjugate to P and A. On account of the property
of the unit planes AH = A'H \ then by combination of the
last two equations with (7),
tan u' f x x f
tan u ~~ f x 1 ~ f ' x''
(9)
The ratio of the tangents of inclination of conjugate rays is
called the convergence ratio or the angular magnification. It
is seen from equation (9) that it is independent of u and u! '.
The angular magnification = I for x = f or x' = f.
The two conjugate points K and K' thus determined are called
the nodal points of the system, They are characterized by the
GEOMETRICAL THEORY OF OPTICAL IMAGES 23
fact that a ray through one nodal point K is conjugate and
parallel to a ray through the other nodal point K' . The posi-
tion of the nodal points for positive focal lengths/ and f is
K
F
FIG. 9.
shown in Fig. 9. KA and K' A' are two conjugate rays. It
follows from the figure that the distance between the two nodal
points is the same as that between the tivo unit points. If
/=/', the nodal points coincide with the unit points.
Multiplication of the second of equations (7) by (9) gives
7
If e be the distance of an object P from the unit plane ,
and e' the distance of its image from the unit plane Q , e and
c' being positive if P lies in front of (to the left of) and P'
behind (to the right of) V, then
*=/-*, e '=f- X >.
Hence the first of equations (7) gives
The same equation holds if e and e' are the distances of P
and P' from any two conjugate planes which are perpendicular
to the axis, and /and/' the distances of the principal foci from
these planes. This result may be easily deduced from (7).
THEORY OF OPTICS
f-
Construction of Conjugate Points. A simple graphical
interpretation may be given to equation
(u). If ABCD (Fig. 10) is a rectangle
with the sides f and /', then any
straight line ECE' intersects the pro-
longations of /and/' at such distances
from A that the conditions AE = e and
AE' = e' satisfy equation (i i).
It is also possible to use the unit
plane and the principal focus to determine the point P f conju-
gate to P. Draw (Fig. 1 1) from P a ray PA parallel to the
axis and a ray PF passing through the principal focus F.
A'
B
FIG. 10.
FIG. ii.
A'F' is conjugate to PA, A' being at the same distance from
the axis as A ; also P'B' , parallel to the axis, is conjugate to
PFB, B' being at the same distance from the axis as B. The
intersection of these two rays is the conjugate point sought.
The nodal points may also be conveniently used for this con-
struction.
The construction shown in Fig. 1 1 cannot be used when P
and P' lie upon the axis. Let a ray from P intersect the focal
plane ^ at a distance g and the unit plane ^ at a distance //
from the axis (Fig. 12). Let the conjugate ray intersect '
and g at the distances k\-= h*) and g 1 ' . Then from the figure
g PF - x g' PF'
h ~ f -4- pf
f
f -
GEOMETRICAL THEORY OF OPTICAL IMAGES 25
and by addition, since from equation (7) xx' = ff,
g + g' 2xx'-fx'~f'x
h
ff'+xx -fx' -f'x
(12)
P' may then be found by laying off in the focal plane g' the
distance g 1 = h g, and in the unit plane ^)' the distance
FIG. 12.
h' h, and drawing a straight line through the two points thus
determined, g and g' are to be taken negative if they lie
below the axis.
5. Classification of the Different Kinds of Optical Sys-
tems. The different kinds of optical systems differ from one
another only in the signs of the focal lengths / and /'.
If the two focal lengths have the same sign, the system is
concurrent, i.e. if the object moves from left to right (x in-
creases), the image likewise moves from left to right
(x l decreases). This follows at once from equation (7) by
taking into account the directions in which x and x' are con-
sidered positive (see above, p. 18 ). It will be seen later that
this kind of image formation occurs if the image is due to
refraction alone or to an even number of reflections or to a
combination of the two. Since this kind of image formation is
most frequently produced by refraction alone, it is also called
dioptric.
26 THEORY OF OPTICS
If the two focal lengths have opposite signs the system is
contracurrent, i.e. if the object moves from left to right, the
image moves from right to left, as appears from the formula
xx 1 = ff. This case occurs if the image is produced by an odd
number of reflections or by a combination of an odd number of
such with refractions. This kind of image formation is called
katoptric. When it occurs the direction of propagation of the
light in the image space is opposite to that in the object space,
so that both cases may be included under the law : In all cases
of image formation if a point P be conceived to move along a ray
in the direction in which the light travels, the image P' of that
point moves along the conjugate ray in the direction in which
the light travels.
Among dioptric systems a distinction is made between those
having positive and those having negative focal lengths. The
former systems are called convergent, the latter divergent,
because a bundle of parallel rays, after passing the unit plane
V of the image space, is rendered convergent by the former,
divergent by the latter. No distinction between systems on
the ground that their foci are real or virtual can be made, for
it will be seen later that many divergent systems (e.g. the
microscope) have real foci.
By similar definition katoptric systems which have a nega-
tive focal length in the image space are called convergent,
for in reflection the direction of propagation of the light is
reversed.
There are therefore the four following kinds of optical
systems :
n . ,, . C a. Convergent:
Dioptric . . . \ j -,,. s , r
\b. Divergent: f,
T^ . , . C a. Convergent:
Katoptric . . < , ^. s .
( b. Divergent:
6. Telescopic Systems. Thus far it has been assumed
that the focal planes lie at finite distances. If they lie at
infinity the case is that of a telescopic system, and the coefB-
GEOMETRICAL THEORY OF OPTICAL IMAGES 27
cient a vanishes from equations (4), which then reduce by a
suitable choice cf the origin of the x coordinates to
#'= ax, / = Py (13)
Since x' = o when x = o, it is evident that any two conjugate
points may serve as origins from which x and x' are measured.
It follows from equation (13) that the magnification in breadth
and depth are constant. The angular magnification is also
constant, for, given any two conjugate rays OP and O'P', their
intersections with the axis of the system may serve as the
origins. If then a point P of the first ray has the coordinates
x, y, and its conjugate point P' the coordinates x' t y' ', the
tangents of the angles of inclination are
tan u = y : x, tan u' = y' : x f .
Hence by (13)
tan u' : tan u p \ a (14)
a must be positive for katoptric (contracurrent) systems, nega-
tive for dioptric (concurrent) systems. For the latter it is
evident from (14) and a consideration of the way in which u
and u' are taken positive (see above, p. 21) that for positive P
erect images of infinitely distant objects are formed, for nega-
tive /?, inverted images. There are therefore four different
kinds of telescopic systems depending upon the signs of a
and p.
Equations (14) and (13) give
y' tan 11 ft 2
y tan u a
(15)
A comparison of this equation with (10) (p. 23) shows that
for telescopic systems the two focal lengths, though both
infinite, have a finite ratio. Thus
/ P*
7=-T (' 6 )
If f = f, as is the case in telescopes and in all instru-
ments in which the index of refraction of the object space is
28
THEORY OF OPTICS
equal to that of the image space (cf. equation (9), Chapter III),
then a ~ {P. Hence from (14)
. tan u' : tan u = I : fi.
This convergence ratio (angular magnification) is called in the
case of telescopes merely the magnification /"". . From (13)
y-y'= - r > (14')
i.e. for telescopes the reciprocal of the lateral magnification is
numerically equal to the angular magnification.
7. Combinations of Systems. A series of several systems
must be equivalent to a single system. Here again attention
will be confined to coaxial systems. If/j and// are the focal
lengths of the first system alone, and / 2 and f 2 ' those of the
second, and /and/' those of the combination, then both the
focal lengths and the positions of the principal foci of the com-
bination can. be calculated or constructed if the distance
F^F 2 ^= A (Fig. 13) is known. This distance will be called
for brevity the separation of the two systems I and 2, and will
be considered positive if F^ lies to the left of F 2 , otherwise
negative.
A ray S (Fig. 13), which is parallel to the axis and at a
FIG. 13.
distance y from it, will be transformed by system I into the
ray S l , which passes through the principal focus F^ of that
system. S l will be transformed by system 2 into the ray S 7 .
GEOMETRICAL THEORY OF OPTICAL IMAGES 29
The point of intersection of this ray with the axis is the prin-
cipal focus of the image space of the combination. Its position
can be calculated from the fact that F^ and F' are conjugate
points of the second system, i.e. (cf. eq. 7)
in which F 2 F is positive if F' lies to the right of F 2 f . F' may
be determined graphically from the construction given above
on page 25, since the intersection of 5 L and S' with the focal
planes F 2 and F 2 f are at such distances g and g' from the axis
that + '= y r
The intersection A' of S' with 5 must lie in the unit plane
JQ' of the image space of the combination. Thus $g f is deter-
mined, and, in consequence, the focal length f of the com-
bination, which is the distance from $$ of the principal focus F'
of the combination. From the construction and the figure it
follows that/' is negative when A is positive.
f may be determined analytically from the angle of incli-
nation u' of the ray S'. For ^ the relation holds:
tan U L = y ://,
in which u v is to be taken with the opposite sign if S l is con-
sidered the object ray of the second system. Now by (9),
tan u' A
tan ^ ~~ //
or since tan u l y : //,
A
tan u' = - y ' 7777*
/1/2
Further, since (cf. the law, p. 21) y :f = tan u r , it follows
that
f'=-~T ....... OS)
A similar consideration of a ray parallel to the axis in the
image space and its conjugate ray in the object space gives
...... 09)
3 o THEORY OF OPTICS
and for the distance of the principal focus F of the combination
from the principal focus F^ ,
(20)
in which FF l is positive if F lies to the left of F r
Equations (17), (18), (19), and (20) contain the character-
istic constants of the combination calculated from those of the
systems which unite to form it.
Precisely the same process may be employed when the
combination contains more than two systems.
If the separation A of the two systems is zero, the focal
lengths f and f are infinitely great, i.e. the system is tele-
scopic. The ratio of the focal lengths, which remains finite,
is given by (18) and (19). Thus
From the consideration of an incident ray parallel to the axis
the lateral magnification y' : y is seen to be
/ -y = ft = -/, :/,' ..... (22)
By means of (21), (22), and (16) the constant a, which repre-
sents the magnification in depth (cf. equation (13)) is found.
Thus
<*>
Hence by (14) the angular magnification is
tan u : tan u = ft : a == / i : / a '. . . . ( 24 )
The above considerations as to the graphical or analytical
determination of the constants of a combination must be
somewhat modified if the combination contains one or more
telescopic systems. The result can, however, be easily
obtained by constructing or calculating the path through the
successive systems of an incident ray which is parallel to
the axis.
CHAPTER III
PHYSICAL CONDITIONS FOR IMAGE FORMATION
ABBE'S geometrical theory of the formation of optical
images, which overlooks entirely the question of their physical
realization, has been presented in the previous chapter, because
the general laws thus obtained must be used for every special
case of image formation no matter by what particular physical
means the images are produced. The concept of focal points
and focal lengths, for instance, is inherent in the concept of
an image no matter whether the latter is produced by lenses
or by mirrors or by any other means.
In this chapter it will appear that the formation of optical
images as described ideally and without limitations in the
previous chapter is physically impossible, e.g. the image of
an object of finite size cannot be formed when the rays have
too great a divergence.
It has already been shown on page I 5 that, whatever the
divergence of the beam, the image of one point may be pro-
duced by reflection or refraction at an aplanatic surface. Images
of other points are not produced by widely divergent rays, since
the form of the aplanatic surface depends upon the position of
the point. For this reason the more detailed treatment, of
special aplanatic surfaces has no particular physical interest.
In what follows only the formation of images by refracting and
reflecting spherical surfaces will be treated, since, on account
of the ease of manufacture, these alone are used in optical
instruments ; and since, in any case, for the reason mentioned
above, no other forms of reflecting or refracting surfaces furnish
ideal optical images.
31
3 2
THEORY OF OPTICS
It will appear that the formation of optical images can be
practically accomplished by means of refracting or reflecting
spherical surfaces if certain limitations are imposed, namely,
limitations either upon the size of the object, or upon the
divergence of the rays producing the image.
i. Refraction at a Spherical Surface, In a medium of
index n, let a ray PA fall upon a sphere of a more strongly
refractive substance of index ri (Fig. 14). Let the radius of
FIG. 14.
the sphere be r, its centre C. In order to find the path of thfe
refracted ray, construct about C two spheres I and 2 of radii
r l = r and r 2 = ,r (method of Weierstrass).
// ft
Let PA meet sphere I in B\ draw BC intersecting sphere
2 in D. Then AD is the refracted ray. This is at once
evident from the fact that the triangles ADC and BAG
are similar. For A C : CD = BC : CA = ri : n. Hence the
< DA C < ABC 0', the angle of refraction, and since
< BA C = 0, the angle of incidence, it follows that
sin : sin 0' = BC : AC = ri : n,
which is the law of refraction.
If in this way the paths of different rays from the point P
PHYSICAL CONDITIONS FOR IMAGE FORMATION 33
be constructed, it becomes evident from the figure that these
rays will not all intersect in the same point P '. Hence no
image is formed by widely divergent rays. Further it appears
from the above construction that all rays which intersect the
sphere at any point, and whose prolongations pass through
B y are refracted to the point D. Inversely all rays which
start from D have their virtual intersection in B. Hence upon
every straight line passing through the centre C of a sphere
of radius r> there are two points at distances from C of
r and r~ respectively which, for all rays, stand in the relation
of object and virtual (not real} image. These two points are
called the aplanatic points of the sphere.
If u and u' represent the angles of inclination with respect
to the axis BD of two rays which start from the aplanatic
points B and Z), i.e. if
ABC u, ADC = u',
then, as was shown above, ^AJ5C = ^DAC = u. From
a consideration of the triangle ADC it follows that
sin u' : sin u = AC : CD = n' : n. . . . (i)
In this case then the ratio of the sines of the angles of inclina-
tion of the conjugate rays is independent of u, not, as in equa-
tion (9) on page 22, the ratio of the tangents. The difference
between the two cases lies in this, that, before, the image of
a portion of space was assumed to be formed, while now only
the image of a surface formed by widely divergent rays is
under consideration. The two concentric spherical surfaces I
and 2 of Fig. 14 are the loci of all pairs of aplanatic points B
and D. To be sure, the relation of these two surfaces is not
collinear in the sense in which this term was used above,
because the surfaces are not planes. If s and s r represent the
areas of two conjugate elements of these surfaces, then, since
their ratio must be the same as that of the entire spherical
surfaces I and 2,
s':s= n* : ' 4 .
34 THEORY OF OPTICS
Hence equation (i) may be written:
sin 2 u-s-n 2 = sin 2 U'-S'-K'* (2)
It will be seen later that this equation always holds for two
surface elements s and s f which have the relation of object and
image no matter by what particular arrangement the image is
produced.
In order to obtain the image of a portion of space by means
of refraction at a spherical surface, the divergence of the rays
which form the image must be taken very small. Let PA
(Fig. 15) be an incident ray, AP' the refracted ray, and PCP'
FIG. 15.
the line joining P with the centre of the sphere C. Then from
the triangle PA C,
sin : sin a PH + r : PA ,
and from the triangle P'AC,
sin 0' : sin a P'H r : P'A.
Hence by division,
sin _ ' _ PH+r P'A
siiT^ ~ w Z ~ P'H- r' ~PA' ' (3)
Now assume that ^4 lies infinitely near to //, i.e. that the angle
APH is very small, so that /M may be considered equal to
r and P'A to P'//. Also let
PH = e, P'H = e'.
PHYSICAL CONDITIONS FOR IMAGE FORMATION 35
Then from (3)
or
n n
7+7-
In which r is to be taken positive if the sphere is convex
toward the incident light, i.e. if C lies to the right of H. e is
positive if P lies to the left of H\ e' is positive if P' lies to the
right of H. To every e there corresponds a definite e' which
is independent of the position of the ray PA, i.e. an image
of a portion of space which lies close to the axis PC is formed
by rays which lie close to PC.
A comparison of equation (4) with equation (n) on page
23 shows that the focal lengths of the system are
and that the two unit planes fa and $$ coincide and are tan-
gent to the sphere at the point H. Since /and/' have the
same sign, it follows, from the criterion on page 25 above,
that the system is dioptric or concurrent. If n' > n, a convex
curvature (positive r) means a convergent system. Real
images (e' > o) are formed so long as *>./". Such images
are also inverted.
Equation (10) on page 23 becomes
y' tan u ! n
v tan u n'
(6)
By the former convention the angles of inclination u and u' of
conjugate rays are taken positive in different ways. If they
are taken positive in the same way the notation 'u will be used
instead of u', i.e. 'u = u f . Hence the last equation may
be written:
ny tan u = n'y' tan 'u (7)
3 6 THEORY OF OPTICS
In this equation a quantity which is not changed by refrac-
tion appears, an optical invariant. This quantity remains
constant when refraction takes place at any number of coaxial
spherical surfaces. For such a case let n be the index of
refraction of the first medium, n' that of the last; then equa-
tion (7) holds. But since in general for every system, from
equation (10), page 23,
y' tan u' _ /
y tan u ~~ /"
there results from a combination with (7)
/:/'=:', (9)
i.e. In the formation of images by a system of coaxial refract-
ing spherical surfaces the ratio of the focal lengths of the
system is equal to the ratio of the indices of refraction of tJie
first and last media. If, for example, these two media are
air, as is the case with 'lenses, mirrors, and most optical instru-
ments, the two focal lengths are equal.
2. Reflection at a Spherical Surface. Let the radius r be
considered positive for a convex, negative for a concave mirror.
FIG. 16.
By the law of reflection (Fig. 16) ^ PAC = P 'AC.
Hence from geometry
PA :P'A = PC :P'C (10)
If the ray PA makes a large angle with the axis PC, then
the position of the point of intersection P' of the conjugate ray
PHYSICAL CONDITIONS FOR IMAGE FORMATION 3 7
with the axis varies with the angle. In that case no image of
the point P exists. But if the angle A PC is so small that the
angle itself may be used in place of its sine, then for every
point P there exists a definite conjugate point P', i.e. an image
is now formed. It is then permissible to set PA = PH,
P f A P'H, so that (10) becomes
PH:P'H= PC :P'C, . . . . (n)
or if PH = e, P'H = e', then, since r in the figure is nega-
tive,
<">
A comparison of this with equation (11) on page 23 shows
that the focal lengths of the system are
f=- l -r, f'=+ l ~r; .... (13)
that the two unit planes and ' coincide with the plane
tangent to the sphere at the vertex H\ that the two principal
foci coincide in the mid-point between C and H\ and that the
nodal points coincide at the centre C of the sphere. The
signs of e and e' are determined by the definition on page 23.
Since f and f have opposite signs, it follows, from the
criterion given on page 25, that the system is katoptric or con-
tracurrent. By the conventions on page 26 a negative r, i.e.
a concave mirror, corresponds to a convergent system ; on the
other hand a convex mirror corresponds to a divergent system.
A comparison of equations (13) and (5) shows that the
results here obtained for reflection at a spherical surface may
be deduced from the former results for refraction at such a sur-
face by writing ri \ n = i. In fact when n' \ n = i, the
law of refraction passes into the law of reflection. Use may
be made of this fact when a combination of several refracting
or reflecting surfaces is under consideration. Equation (9)
holds for all such cases and shows that a positive ratio/:/ 7
38 THEORY OF OPTICS
always results from a combination of an even number of reflec-
tions from spherical surfaces or from a combination of any
number of refractions, i.e. such systems are dioptric or concur-
rent (cf. page 25).
The relation between image and object may be clearly
brought out from Fig. 17, which relates to a concave mirror.
The numbers 7, 2, J, . . . 8 represent points of the object at a
constant height above the axis of the system. The numbers
7 and 8 which lie behind the mirror correspond to virtual
objects, i.e. the incident rays start toward these points, but fall
upon the mirror and are reflected before coming to an intersec-
tion at them. Real rays are represented in Fig. 17 by
FIG. 17.
continuous lines, virtual rays by dotted lines. The points
i 1 ', 2' t j f , . . . 8' are the images of the points /, 2, j, . . . 8.
Since the latter lie in a straight line parallel to the axis, the
former must also lie in a straight line which passes through the
principal focus F and through point 6, the intersection of the
object ray with the mirror, i.e. with the unit plane. The con-
tinuous line denotes real images ; the dotted line, virtual im-
ages. Any image point 2' may be constructed (cf. page 24)
by drawing through the object 2 and the principal focus F a
straight line which intersects the mirror, i.e. the unit plane, in
some point A^. If now through A 2 a line be drawn parallel
PHYSICAL CONDITIONS FOR IMAGE FORMATION 39
to the axis, this line will intersect the previously constructed
image line in the point sought, namely 2' . From the figure it
may be clearly seen that the images of distant objects are real
and inverted, those of objects which lie in front of the mirror
within the focal length are virtual and erect, and those of virtual
objects behind the mirror are real, erect, and lie in front of the
mirror.
Fig. 1 8 shows the relative positions of object and image
FIG. 18.
for a convex mirror. It is evident that the images of all real
objects are virtual, erect, and reduced; that for virtual objects
which lie within the focal length behind the mirror the images
are real, erect, and enlarged; and that for more distant virtual
objects the images are also virtual.
FIG. 19.
Equation (i i) asserts that PCP ' H are four harmonic points.
The image of an object P may, with the aid of a proposition
of synthetic geometry, be constructed in the following way:
40 THEORY OF OPTICS
From any point L (Fig. 19) draw two rays LC and LH, and
then draw any other ray PDB. Let O be the intersection of
DH with BC\ then LO intersects the straight line PH in a
point P' which is conjugate to P. For a convex mirror the
construction is precisely the same, but the physical meaning of
the points C and H is interchanged.
3. Lenses. The optical characteristics of systems com-
posed of two coaxial spherical surfaces (lenses) can be directly
deduced from 7 of Chapter II. The radii of curvature r l
and r 2 are taken positive in accordance with the conventions
given above ( i); i.e. the radius of a spherical surface is
considered positive if the surface is convex toward the inci-
dent ray (convex toward the left). Consider the case of a lens
of index n surrounded by air. Let the thickness of the lens,
i.e. the distance between its vertices S l and 5 2 (Fig. 20), be
/ .. A
G'
F F, 5,1 /V F* S*
\ "
F^ f"
FIG. 20.
denoted by d. If the focal lengths of the first refracting sur-
face are denoted by /j and//, those of the second surface by
/ 2 and//, then the separation A of the two systems (cf. page
28) is given by
4=d-fi-f t (14)
and, by (5),
PHYSICAL CONDITIONS FOR IMAGE FORMATION 41
Hence by equations (19) and (18) of Chapter II (page 29)
the focal lengths of the combination are
n -
/ T ^
while the positions of the principal foci F and F' of the com-
bination are given by equations (17) and (20) of Chapter II
(page 29). By these equations the distance a of the principal
focus F in front of the vertex S l , and the distance o, r 2 < o),
Plano-convex lenses (r l > o, r 2 = )
Concavo-convex lenses (r l > o, r 2 > o, r 2 > rj,
in short all lenses which are thicker in the middle than at the
edges.
PHYSICAL CONDITIONS FOR MAGE FORMATION 43
Lenses of negative focal length (divergent lenses) include
Double-concave lenses (r l < o, r 2 > o),
Plano-concave lenses (^ = 00, r 2 > o),
Convexo-concave lenses (r l > o, r 2 > o, r 2 < rj,
i.e. all lenses which are thinner in the middle than at the
edges.*
The relation between image and object is shown diagram-
matically in Figs. 21 and 22, which are to be interpreted in
FIG. 21.
the same way as Figs. 17 and 18. From these it appears that
whether convergent lenses produce real or virtual images of
FIG. 22.
real objects depends upon the distance of the object from the
lens ; but divergent lenses produce only virtual images of real
*The terms collective (dioptrics for systems of positive focal length, dispersive,
for those of negative focal length, have been chosen on account of this property of
lenses. A lens of positive focal length renders an incident beam more convergent.
one of negative focal length renders it more divergent. When images are formed
by a system of lenses, or, in general, when the unit planes do not coincide, say,
with the first refracting surface, the conclusion as to whether the system is con-
vergent or divergent cannot be so immediately drawn. Then recourse must be
had to the definition on page 26.
44 THEORY OF OPTICS
objects. However, divergent lenses produce real, upright,
and enlarged images of virtual objects which lie behind the
lens and inside of the principal focus.
If two thin lenses of focal lengths / t and / 2 are united to
form a coaxial system, then the separation A (cf. page 40) is
A = (/! -h/ 2 )- Hence, from equation (19) of Chapter II
(page 29), the focal length of the combination is
or
7 = 7+7, ....... (24)
It is customary to call the reciprocal of the focal length of
a lens its power. Hence the law: The power of a combination
of thin lenses is equal to the sum of the powers of the separate
lenses.
5. Experimental Determination of Focal Length, For
thin lenses, in which the two unit planes are to be considered
as practically coincident, it is sufficient to determine the posi-
tions of an object and its image in order to deduce the focal
length. For example, equation (11) of Chapter II, page 23,
reduces here, since / = /", to
7+7=7 .......
Since the positions of real images are most conveniently
determined by the aid of a screen, concave lenses, which
furnish only virtual images of real objects, are often combined
with a convex lens of known power so that the combination
furnishes a real image. The focal length of the concave lens
is then easily obtained from (24) when the focal length of the
combination has been experimentally determined. This pro-
cedure is not permissible for thick lenses nor for optical systems
generally. The positions of the principal foci are readily deter-
PHYSICAL CONDITIONS FOR IMAGE FORMATION 45
mined by means of an incident beam of parallel rays. If then
the positions of an object and its image with respect to the
principal foci be determined, equations (7), on page 19, or (9),
on page 22, give at once the focal length/ ( =/').
Upon the definition of the focal length given in Chapter II,
page 20 (cf. equation (8)), viz.,
(26)
it is easy to base a rigorous method for the determination of
focal length. Thus it is only necessary to measure the angular
magnitude u of an infinitely distant object, and the linear mag-
nitude y' of its image. This method is particularly convenient
to apply to the objectives of telescopes which are mounted
upon a graduated circle so that it is at once possible to read
off the visual angle u.
If the object of linear magnitude y is not at infinity, but is
at a distance e from the unit plane , while its image of linear
magnitude y' is at a distance e' from the unit plane ', then
(27)
because, when/ = /', the nodes coincide with the unit points,
i,e. object and image subtend equal angles at the unit points.
By eliminating e and e' from (25) and (27) it follows that
(28)
y y
Now if either e or e' are chosen large, then without appreci-
able error the one so chosen may be measured from the centre
of the optical system (e.g. the lens), at least unless the unit
planes are very far from it. Then either of equations (28)
may be used for the determination of the focal length /"when
e or e' and the magnification y'\y have been measured.
The location of the positions of the object or image may
be avoided by finding the magnification for two positions of
46 THEORY OF OPTICS
the object which are a measured distance / apart. For, from
(7), page 19,
hence
/
(29)
in which (y '.y'\ denotes the reciprocal of the magnification for
the position x of the object, (y : y'\ the reciprocal of the mag-
nification for a position x -{- / of the object. / is positive if, in
passing to its second position, the object has moved the dis-
tance /in the direction of the incident light (i.e. from left to
right).
Abbe's focometer, by means of which the focal lengths of
microscope objectives can be determined, is based upon this
principle. For the measurement of the size of the image y' a
second microscope is used. Such a microscope, or even a
simple magnifying-glass-, may of course be used for the meas-
urement of a real as well as of a virtual image, so that this
method is also applicable to divergent lenses, in short to all
cases.*
6. Astigmatic Systems. In the previous sections it has
been shown that elementary beams whose rays have but a
small inclination to the axis and which proceed from points
either on the axis or in its immediate neighborhood may be
brought to a focus by means of coaxial spherical surfaces.
In this case all the rays of the beam intersect in a single point
of the image space, or, in short, the beam is homocentric in
the image space. What occurs when one of the limitations
imposed above is dropped will now be considered, i.e. an
* A more detailed account of the focometer and of the determination of focal
lengths is given by Czapski in Winkelmann, Handbuch der Physik, Optik,
pp. .-85-296.
PHYSICAL CONDITIONS FOR IMAGE FORMATION 47
elementary beam having any inclination to the axis will now
be assumed to proceed from a point P.
In this case the beam is, in general, no longer homocentric
in the image space. An elementary beam which has started
from a luminous point P and has suffered reflections and re-
fractions upon surfaces of any arbitrary form is so constituted
that, by the law of Malus (cf. page 12), it must be classed
as an orthotomic beam, i.e. it may be conceived as made up
of the normals N to a certain elementary surface 2. These
normals, however, do not in general intersect in a point.
Nevertheless geometry shows that upon every surface 2 there
are two systems of curves which intersect at right angles (the
so-called lines of curvature) whose normals, which are also at
right angles to the surface 2, intersect.
If a plane elementary beam whose rays in the image space
are normal to an element / x of a line of curvature be alone
considered, it is evident that an image will be formed. The
image is located at the centre of curvature of this element / t ,
since its normals intersect at that point. Since every element
/ L of a line of curvature is intersected at right angles by some
other element / 2 of another line of curvature, a second elemen-
tary beam always exists which also produces an image, but
the positions of these two images do not coincide, since in
general the curvature of / t is different from that of / 2 .
What sort of an image of an object P will then in general
be formed by any elementary beam of three dimensions ? Let
/, 2, j, 4. (Fig. 23) represent the four intersections of the four
lines of curvature which bound the element d"2 of the sur-
face 2. Let the curves 1-2 and 34. be horizontal, 2-3 and
/ 4 vertical. Let the normals at the points I and 2 intersect
at 12, those at j and 4 at 34.. Since the curvature of the line
i2 differs by an infinitely small amount from that of the line
3 4, the points of intersection 12 and 34. lie at almost the same
distance from the surface 2. Hence the line p l which connects
the points 12 and 34 is also nearly perpendicular to the ray 5
which passes through the middle of d2 and is normal to it.
48 THEORY OF OPTICS
This ray is called the principal ray of that elementary beam
which is composed of the normals to d22. From the symmetry
of the figure it is also evident that the line p^ must be parallel
to the lines 2-3 and 1-4., i.e. it is vertical. The normals to
any horizontal line of curvature intersect at some point of the
line p r
FIG. 23.
Likewise the normals to any vertical line of curvature
intersect at some point of the line / 2 which connects 14. and = da + du = AB 1 ^- + ^^),
. . . (32)
But a differentiation of the equation of refraction sin =
n sin 0' gives
cos . d i.e. h = 10 cm. The
so-called lateral aberration C, i.e. the radius of the circle
which the rays passing through the edge of a lens form upon
* This minimum is never zero. A complete disappearance of the aberration
of the first order can only be attained by properly choosing the thickness of the
lens as well as the ratio of the radii.
fit follows at once that the form of the lens which gives minimum aberration
depends upon the position of the object.
THEORY OF OPTICS
a screen placed at the focal point /y, is obtained, as appears
at once from a construction of the paths of the rays, by multi-
plication of the longitudinal aberration by the relative aperture
h \f, i.e. in this case by T ^-. Thus the lateral aberration
determines the radius of the illuminated disc which the outside
rays from a luminous point P form upon a screen placed in the
plane in which P is sharply imaged by the axial rays.
f = i m. h 10 cm.
n = 1.5
n = 2
(T
;
cr
e
c
Front face plane
4e cm
4 5 mm
oo
2 cm
2 mm
1.67 "
i 67 "
I
i "
i "
o
1.17 "
1. 17 4<
o
0.5 "
o.? "
Most advantageous form
1
~ ff
1.07 "
1.07 "
+ 1
0.44 "
0.44"
That a plano-convex lens produces less aberration when its
convex side is turned toward a distant object than when the
sides are reversed seems probable from the fact that in the first
case the rays are refracted at both surfaces of the lens, in the
second only at one; and it is at least plausible that the dis-
tribution of the refraction between two surfaces is unfavorable
to aberration. The table further shows that the most favor-
able form of lens has but little advantage over a suitably placed
plano-convex lens. Hence, on account of the greater ease of
construction, the latter is generally used.
Finally the table shows that the aberration is very much
less if, for a given focal length, the index of refraction is made
large. This conclusion also holds when the aberration of a
higher order than the first is considered, i.e. when the remain-
ing terms of the power series in u are no longer neglected.
Likewise the aberration is appreciably diminished when a
single lens is replaced by an equivalent system of several
PHYSICAL CONDITIONS FOR IMAGE FORMATION 57
lenses.* By selecting for the compound system lenses of
different form, it is possible to cause the aberration not only
of the first but also of still higher orders to vanish, f One
system can be made to accomplish this for more than one
position of the object on the axis, but never for a finite length
of the axis.
When the angle of inclination u is large, as in microscope
objectives in which u sometimes reaches a value of 90, the
power series in u cannot be used for the determination of the
aberration. It is then more practicable to determine the paths
of several rays by trigonometrical calculation, and to find by
trial the best form and arrangement of lenses. There is, how-
ever, a way, depending upon the use of the aplanatic points of
a sphere mentioned on page 33, of diminishing the divergence
of rays proceeding from near objects without introducing aber-
ration, i.e. it is possible to produce virtual images of any size,
which are free from aberration.
Let lens i (Fig. 25) be plano-convex, for example, a hemi-
FIG. 25.
spherical lens of radius r l , and let its plane surface be turned
toward the object P. If the medium between P and this lens
has the same index n l as the lens, then refraction of the rays
* In this case, to be sure, the brightness of the image suffers somewhat on
account of the increased loss of light by reflection.
f Thus the aberration of the first order can be corrected by a suitable com-
bination of a convergent and a divergent lens.
58 THEORY OF OPTICS
proceeding from the object first takes place at the rear surface
of the lens; and if the distance of P from the centre of curva-
ture GI of the back surface is r l : n l , then the emergent rays
produce at a distance n^r^ from C l a virtual image P l free from
aberration. If now behind lens / there be placed a second
concavo-convex lens 2 whose front surface has its centre of
curvature in P l and whose rear surface has such a radius r z that
P l lies in the aplanatic point of this sphere r^ (the index of
lens 2 being ^ 2 ), then the rays are refracted only at this rear
surface, and indeed in such a way that they form a virtual
image P z which lies at a distance ;/ 2 r 2 from the centre of curva-
ture C 2 of the rear surface of lens 2, and which again is entirely
free from aberration. By addition of a third, fourth, etc.,
concavo-convex lens it is possible to produce successive virtual
images P 3 , P 4 , etc., lying farther and farther to the left, i.e.
it is possible to diminish successively the divergence of the
rays without introducing aberration.
This principle, due to Amici, is often actually employed in
the construction of microscope objectives. Nevertheless no
more than the first two lenses are constructed according to this
principle, since otherwise the chromatic errors which are intro-
duced are too large to be compensated (cf. below).
9. The Law of Sines. In general it does not follow that
if a widely divergent beam from a point P upon the axis gives
rise to an image P' which is free from aberration, a surface
element do" perpendicular to the axis at P will be imaged in
a surface element da' at P '. In order that this may be the
case the so-called sine law must also be fulfilled. This law
requires that if u and u' are the angles of inclination of any two
conjugate rays passing through P and P', sin u : sin u' = const.
According to Abbe systems which are free from aberra-
tion for two points P and P' on the axis and which fulfil the
sine law for these points are called aplanatic systems. The
points P and P' are called the aplanatic points of the system.
The aplanatic points of a sphere mentioned on page 33 fulfil
these conditions, since by equation (2), pa#e *A. the ratio of the
PHYSICAL CONDITIONS FOR IMAGE FORMATION 59
sines is constant. The two foci of a concave mirror whose
surface is an ellipsoid of revolution are not aplanatic points
although they are free from aberration.
It was shown above (page 22, equation (9), Chapter II)
that when the image of an object of any size is formed by a
collinear system, tan u : tan u' = const. Unless u and u' are
very small, this condition is incompatible with the sine law,
and, since the latter must always be fulfilled in the formation
of the image of a surface element, it follows that a point-for-
point imaging of objects of any size by widely divergent beams
is physically impossible.
Only when u and u' are very small can both conditions be
simultaneously fulfilled. In this case, whenever an image P'
is formed of P, an image do-' will be formed at P' of the surface
element do" at P. But if u is large, even though the spherical
aberration be entirely eliminated for points on the axis, unless
the sine condition is fulfilled the images of points which lie to
one side of the axis become discs of the same order of magni-
tude as the distances of the points from the axis. According
to Abbe this blurring of the images of points lying off the axis is
due to the fact that the different zones of a spherically corrected
system produce images of a surface element of different linear
magnifications.
The mathematical condition for the constancy of this linear
magnification is, according to Abbe, the sine law.* The same
conclusion was reached in different ways by Clausius t and v.
Helmholtz \. Their proofs, which rest upon considerations of
energy and photometry, will be presented in the third division
of the book. Here a simple proof due to Hockin will be
given which depends only on the law that the optical lengths
of all rays between two conjugate points must be equal (cf.
* Carl's Repert. f. Physik, 1881, 16, p. 303.
fR. Clausius, Mechanische Warmetheorie, 1887, 3d Ed. I, p. 315.
J v. Helmholtz, Pogg. Ann. Jubelbd. 1874, p. 557.
Hockin, Jour. Roy. Microsc. Soc. 1884, (2), 4. p. 337.
6o
THEORY OF OPTICS
page 9).* Let the image of P (Fig. 26) formed by an axial
ray PA and a ray PS of inclination u lie at the axial point P'.
Also let the image of the infinitely near point P l formed by a
ray P l A l parallel to the axis, and a ray P l S l parallel to PS,
lie at the point P^ . The ray F'P^ conjugate to P l A l must
evidently pass through the principal focus F' of the image
space. If now the optical distance between the points P and
P' along the path through A be represented by (PAP'), that
FIG. 26.
along the path through SS r by (PSS'P'), and if a similar
notation be used for the optical lengths of the rays proceeding
from P l , then the principle of extreme path gives
(PAP r ) = (PSS'P') ; (/V^/r'/V) = (/^S/P/),
and hence
(PAP') - (P,A,F'P{) = (PSS'P') - (P&SW. . (39)
Now since F' is conjugate to an infinitely distant object T on
the axis, (TPAF f ) = (TP^A^F'}. But evidently TP = TP l ,
since PP l is perpendicular to the axis. Hence by subtraction
") = (P 1 A 1 F') (40)
* According to Bruns (Abh. d. sachs. Ges. d. Wiss. Bd. 21, p. 325) the sine
law can be based upon still more general considerations, namely, upon the law of
Malus (cf. p. 12) and the existence of conjugate rays.
PHYSICAL CONDITIONS FOR IMAGE FORMATION 61
Further, since P'P^ is perpendicular to the axis, it follows
that when P'P{ is small F'P' = F'PJ. Hence by addition
(PAF'P') = (P
i.e. the left side of equation (39) vanishes. Thus
(PSS'P f )=(P l S l S l 'P l f ) ..... (41)
Now if Fj* is the intersection of the rays P'S' and P/5/, then
FI is conjugate to an infinitely distant object T l , the rays from
which make an angle u with the axis. Hence if a perpendic-
ular PN be dropped from P upon P 1 S 1 , an equation similar to
(40) is obtained; thus
(PSS'F l ') = (NS 1 S l 'F l ') ..... (42)
By subtraction of this equation from (41),
(F l 'P>)= -(NP l ) + (F l 'P l '). . . . (43)
If now n is the index of the object space, n' that of the image
space, then, if the unbracketed letters signify geometrical
lengths,
(NP^ = n - NP l = n . PP l . sin u. . . . (44)
Further, if P'N' be drawn perpendicular to F^P', then, since
P'Pi is infinitely small,
(/Y/Y) - (/V 7 *) = n'.N'PJ = n'.P'P^ sin '. . (45)
Equation (43) in connection with (44) and (45) then gives
n-PP^sin u n'.P'P^-sm u' .
If y denote the linear magnitude PP l of the object, and y r the
linear magnitude P'P^- of the image, then
sin u riy'
sin u' ny
(46)
Thus it is proved that if the linear magnification is con-
stant the ratio of the sines is constant, and, in addition, the
value of this constant is determined. This value agrees with
62
THEORY OF OPTICS
that obtained in equation (2), page 34, for the aplanatic points
of a sphere.
The sine law cannot be fulfilled for two different points on
the axis. For if P' and /Y (Fig. 27) are the images of P and
P lt then, by the principle of equal optical lengths,
(PAP')=(PSS'P'), (P 1 AP 1 ') = (P 1 S 1 S'P 1 '), . (47)
in which PS and P^ are any two parallel rays of inclina-
tion u.
JV"
FIG. 27.
Subtraction of the two equations (47) and a process ci*
reasoning exactly like the above gives
or
n-Pf(\ cos u) = n'-PJP' (i cos *'),
i.e.
n'-P'P{
(48)
This equation is then the condition for the formation, by a
beam of large divergence, of the image of two neighboring
points upon the axis, i.e. an image of an element of the axis.
However this condition and the sine law cannot be fulfilled
at the same time. Thus an optical system can be made
aplanatic for but one position of the object
PHYSICAL CONDITIONS FOR IMAGE FORMATION 63
The fulfilment of the sine law is especially important in the
case of microscope objectives. Although this was not known
from theory when the earlier microscopes were made, it can be
experimentally proved, as Abbe has shown, that these old
microscope objectives which furnish good images actually
satisfy the sine law although they were constructed from
purely empirical principles.
10. Images of Large Surfaces by Narrow Beams. It
is necessary in the first place to eliminate astigmatism (cf.
page 46). But no law can be deduced theoretically for accom-
plishing this, at least when the angle of inclination of the rays
with respect to the axis is large. Recourse must then be had
to practical experience and to trigonometric calculation. It is
to be remarked that the astigmatism is dependent not only
upon the form of the lenses, but also upon the position of the
stop.
Two further requirements, which are indeed not absolutely
essential but are nevertheless very desirable, are usually im-
FIG. 28.
posed upon the image. First it must be plane, i.e. free from
bulging, and second its separate parts must have the same
magnification, i.e. it must be free from distortion. The first
requirement is especially important for photographic objectives.
64 THEORY OF OPTICS
For a complete treatment of the analytical conditions for this
requirement cf. Czapski, in Winkelmann's Handbuch der
Physik, Optik, page 124.
The analytical condition for freedom from distortion may
be readily determined. Let PP 1 P 2 (Fig. 28) be an object
plane, P'P^P 2 the conjugate image plane. The beams from
the object are always limited by a stop of definite size
which may be either the rim of a lens or some specially intro-
duced diaphragm. This stop determines the position of a
virtual aperture B, the so-called entrance- pupil, which is so
situated that the principal rays of the beams from the objects
P lt P 2 , etc., pass through its centre. Likewise the beams in
the image space are limited by a similar aperture B' ', the
so-called exit-pupil, which is the image of the entrance-pupil.*
If /and /' are the distances of the entrance-pupil and the exit-
pupil from the object and image planes respectively, then, from
the figure,
tan ^ = PP l : /, tan u 2 = PP 2 : /,
tan / = />'/Y : /', tan * a ' = P'P 2 ' : /'.
If the magnification is to be constant, then the following rela-
tion must exist:
P'P{ : PP l = P'P 2 : PP 2 ,
hence
tan u' tan u'
= -7 - const (49)
tan u l tan u 2
Hence for constant magnification the ratio of the tangents of the
angles of inclination of the principal rays must be constant. In
this case it is customary to call the intersections of the prin-
cipal rays with the axis, i.e. the centres of the pupils, ortho-
scopic points. Hence it may be said that, if the image is to
be free from distortion, the centres of perspective of object and
image must be orthoscopic points. Hence the positions of the
pupils are of great importance.
* For further treatment see Chapter IV.
PHYSICAL CONDITIONS FOR IMAGE FORMATION 65
An example taken from photographic optics shows how the
condition of orthoscopy may be most simply fulfilled for the
case of a projecting lens. Let R (Fig. 29) be a stop on either
side of which two similar lens systems i and 2 are symmetrically
placed. The whole system is then called a symmetrical double
objective. Let 5 and S f represent two conjugate principal
rays. The optical image of the stop R with respect to the
system / is evidently the entrance-pupil, for, since all principal
rays must actually pass through the centre of the stop R, the
prolongations of the incident principal rays S must pass through
the centre of >, the optical image of R with respect to i.
Likewise B 1 ', the optical image of R with respect to 2, is the
exit-pupil. It follows at once from the symmetry of arrange-
ment that u is always equal to u' ', i.e. the condition of orthos-
copy is fulfilled.
FIG. 29.
Such symmetrical double objectives possess, by virtue of
their symmetry, two other advantages: On the one hand, the
meridional beams are brought to a sharper focus,* and, on the
other, chromatic errors, which will be more fully treated in the
next paragraph, are more easily avoided. The result u = u',
which means that conjugate principal rays are parallel, is
altogether independent of the index of refraction of the system,
* The elimination of the error of coma is here meant. Cf. Mtiller-Pouillet,
Optik, p. 774-
66 THEORY OF OPTICS
and hence also of the color of the light. If now each of the
two systems / and 2 is achromatic with respect to the position
of the image which it forms of the stop R, i.e. if the posi-
tions of the entrance- and exit-pupils are independent of the
color,* then the principal rays of one color coincide with those
of every other color. But this means that the images formed
in the image plane are the same size for all colors. To be
sure, the position of sharpest focus is, strictly speaking, some-
what different for the different colors, but if a screen be placed
in sharp focus for yellow, for instance, then the images of
other colors, which lie at the intersections of the principal
rays, are only slightly out of focus. If then the principal rays
coincide for all colors, the image will be nearly free from
chromatic error.
The astigmatism and the bulging of the image depend upon
the distance of the lenses / and 2 from the stop R. In
general, as the distance apart of the two lenses increases the
image becomes flatter, i.e. the bulging decreases, while the
astigmatism increases. Only by the use of the new kinds of
glass made by Schott in Jena, one of which combines large
dispersion with small index and another small dispersion with
'large index, have astigmatic flat images become possible.
This will be more fully considered in Chapter V under the head
of Optical Instruments.
ii. Chromatic Aberration of Dioptric Systems. Thus
far the index of refraction of a substance has been treated as
though it were a constant, but it is to be remembered that for
a given substance it is different for each of the different colors
contained in white light. For all transparent bodies the index
continuously increases as the color changes from the red to
the blue end of the spectrum. The following table contains
the indices for three colors and for two different kinds of glass.
n c is the index for the red light corresponding to the Fraun-
*As will be seen later, this achromatizing can be attained with sufficient accu-
racy; on the other hand it is not possible at the same time to make the sizes of the
different images of R independent of the color.
PHYSICAL CONDITIONS FOR IMAGE FORMATION 67
hofer line C of the solar spectrum (identical with the red
hydrogen line), n D that for the yellow sodium light, and n F that
for the blue hydrogen line.
Glass.
*c
n D
*p
_ *p ~ n C
n D - i
1.5153
I.5I79
1.5239
0.0166
Ordinary silicate-flint.
I 614^
I.62O2
1. 6^14
O.O276
The last column contains the so-called dispersive power v,
of the substance. It is defined by the relation
v =
(50)
It is practically immaterial whether n D or the index for any
other color be taken for the denominator, for such a change
can never affect the value of v by more than 2 per cent.
Since now the constants of a lens system depend upon the
index, an image of a white object must in general show colors,
i.e. the differently colored images of a white object differ from
one another in position and size.
In order to make the red and blue images coincide, i.e. in
order to make the system achromatic for red and blue, it is
necessary not only that the focal lengths, but also that the
unit planes, be identical for both colors. In many cases a
partial correction of the chromatic aberration is sufficient.
Thus a system may be achromatized either by making the focal
length, and hence the magnification, the same for all colors ;
or by making the rays of all colors come to a focus in the same
plane. In the former case, though the magnification is the
same, the images of all colors do not lie in one plane; in the
latter, though these images lie in one plane, they differ in size.
A system may be achromatized one way or the other according
to the purpose for which it is intended, the choice depending
upon whether the magnification or the position of the image is
most important.
68 THEORY OF OPTICS
A system which has been achromatized for two colors,
e.g. red and blue, is not in general achromatic for all other
colors, because the ratio of the dispersions of different sub-
stances in different parts of the spectrum is not constant.
The chromatic errors which remain because of this and which
give rise to the so-called secondary spectra are for the most
part unimportant for practical purposes. Their influence can
be still farther reduced either by choosing refracting bodies for
which the lack of proportionality between the dispersions is as
small as possible, or by achromatizing for three colors. The
chromatic errors which remain after this correction are called
spectra of the third order.
The choice of the colors which are to be used in practice
in the correction of the chromatic aberration depends upon the
use for which the optical instrument is designed. For a system
which is to be used for photography, in which the blue rays
are most effective, the two colors chosen will be nearer the
blue end of the spectrum than in the case of an instrument
which is to be used in connection with the human eye, for
which the yellow-green light is most effective. In the latter
case it is easy to decide experimentally what two colors can be
brought together with the best result. Thus two prisms of
different kinds of glass are so arranged upon the table of a
spectrometer that they furnish an almost achromatic image
of the slit; for instance, for a given position of the table
of the spectrometer, let them bring together the rays C
and F. If now the table be turned, the image of the slit will
in general appear colored ; but there will be one position in
which the image has least color. From this position of the
prism it is easy to calculate what two colors emerge from the
prism exactly parallel. These, then, are the two colors which
can be used with the best effect for achromatizing instruments
intended for eye observations.
Even a single thick lens may be achromatized either with
reference to the focal length or with reference to the position
of the focus. But in practice the cases in which thin lenses
PHYSICAL CONDITIONS FOR IMAGE FORMATION 69
are used are more important. When such lenses are com-
bined, the chromatic differences of the unit planes may be
neglected without appreciable error, since, in this case, these
planes always lie within the lens (cf. page 42). If then the
focal lengths be achromatized, the system is almost perfectly
achromatic, i.e. both for the position and magnitude of the
image.
Now the focal length f v of a thin lens whose index for a
given color is n^ is given by the equation (cf. eq. (22), page 42)
(, -0*k. (50
in which k^ is an abbreviation for the difference of the curva-
tures of the faces of the lens.
Also, by (24) on page 44, the focal length /of a combina-
tion of two thin lenses whose separate focal lengths are _/j and
/ 2 is given by
For an increment dn l of the index n^ corresponding to a
change of color, the increment of the reciprocal of the focal
length is, from (51),
".->. = -= ' ' 0*
in which r 1 represents the dispersive power of the material of
lens I between the two colors which are used. If the focal
ength/of the combination is to be the same for both colors,
it follows from (52) and (53) that
+"- <">
This equation contains the condition for achromatism. It
also shows, since r l and r 2 always have the same sign no
matter what materials are used for J and 2, that the separate
70 THEORY OF OPTICS
focal lengths of a thin double achromatic lens always Jiave
opposite signs.
From (54) and (52) it follows that the expressions for the
separate focal lengths are
_ 2 __ l
/i"/".-*! 1 7 2 ~~ fr 2 -ri
Hence in a combination of positive focal length the lens with
the smaller dispersive power has the positive, that with the
larger dispersive power the negative, focal length.
If /is given and the two kinds of glass have been chosen,
then there are four radii of curvature at our disposal to make
f l and/2 correspond to (55). Hence two of these still remain
arbitrary. If the two lenses are to fit together, r^ must be
equal to r 2 . Hence one radius of curvature remains at our
disposal. This may be so chosen as to make the spherical
aberration as small as possible.
In microscopic objectives achromatic pairs of this kind are
very generally used. Each pair consists of a plano-concave
lens of flint glass which is cemented to a double-convex lens
of crown glass. The plane surface is turned toward the
incident light.
Sometimes it is desirable to use two thin lenses at a greater
distance apart; then their optical separation is (cf. page 28)
Hence, from (19) on page 29, the focal length of the combina-
tion is given by
i i i a
~- = + - (56)
If the focal length is to be achromatic, then, from (56) and (53),
or
PHYSICAL CONDITIONS FOR IMAGE FORMATION 7'
ff the two lenses are of tJie same material
they are at the distance
= 7' 2 ), then, when
(58)
they form a system ivJiicJi is achromatic with respect to the focal
length. Since v l = v 2 , this achromatism holds for all colors.
If it is desired to achromatize the system not only with
reference to the focal length, but completely, i.e. in respect to
both position and magnification of the image, then it follows
from Fig. 30 that
y Ji
i.e. the ratio of the magnifications is
y
(59)
cc
Fir,. 30.
If, therefore, the image is to be achromatic both with
respect to magnitude and position, then, since ^ is constant
for all colors,
= o.
(60)
But since e{ -\- e 2 = a (distance between the lenses) is also
constant for all colors, it follows that de{ = de. 2 , while, from
(60), d(e^ /e^] = o. Hence de^ o and de 2 = o, i.e. each of
the two separate lenses must be for itself achromatized, i.e.
must consist of an achromatic pair.
Hence the following general conclusion may be drawn:
A combination which consists of several separated systems is
72 THEORY OF OPTICS
only perfectly achromatic (i.e. with respect to both position and
magnification of the image) when each system for itself is
achromatic.
When the divergence of the pencils which form the image
becomes greater, complete achromatism is not the only con-
dition for a good image even with monochromatic light. The
spherical aberration for two colors must also be corrected as
far as possible; and, when the image of a surface element is
to be formed, the aplanatic condition (the sine law) must be
fulfilled for the two colors. Abbe calls systems which are free
from secondary spectra and are also aplanatic for several
colors " apochromatic " systems. Even such systems have a
chromatic error with respect to magnification which may,
however, be rendered harmless by other means (cf. below
under the head Microscopes).
CHAPTER IV
APERTURES AND THE EFFECTS DEPENDING UPON THEM.
I. Entrance- and Exit-pupils. The beam which passes
through an optical system is of course limited either by the
dimensions of the lenses or mirrors or by specially introduced
diaphragms. Let P be a particular point of the object (Fig.
31); then, of the stops or lens rims which are present, that
one which most limits the divergence of the beam is found in
the following way: Construct for every stop B the optical
image B l formed by that part S l of the optical system which
lies between B and the object P. That one of these images
l which subtends the smallest angle at the object point P is
evidently the one which limits the divergence of the beam.
This image is called the entrance-pupil of the whole system.
The stop B is itself called the aperture or iris.* The angle
2U which the entrance-pupil subtends at the object, i.e. the
angle included between the two limiting rays in a meridian
plane, is called the angular aperture of the system.
The optical image B^ which is formed of the entrance-
pupil by the entire system is called the exit-pupil. This
evidently limits the size of the emergent beam which comes to
a focus in P f , the point conjugate to P. The angle 2U' which
the exit-pupil subtends at P' is called the angle of projection
of the system. Since object and image are interchangeable,
it follows at once that the exit-pupil B^ is the image of the
* If the iris lies in front of the front lens of the system, it is identical with the
entrance-pupil.
73
74
THEORY OF OPTICS
stop B formed by that part 5 2 of the optical system which lies
between B and the image space. In telescopes the rim of the
objective is often the stop, hence the image formed of this rim
by the eyepiece is the exit-pupil. The exit-pupil may be
seen, whether it be a real or a virtual image, by holding the
FIG. 31.
instrument at a distance from the eye and looking through it
at a bright background.
Under certain circumstances the iris of the eye of the
observer can be the stop. The so-called pupil of the eye is
merely the image of the iris formed by the lens system of the
eye. It is for this reason that the general terms entrance-
pupil and iris have been chosen.
As was seen on page 52, the position of the pupils is of
importance in the formation of images of extended objects by
beams of small divergence. If the image is to be similar to
the object, the entrance- and exit-pupils must be orthoscopic
points. Furthermore the position of the pupils is essential to
the determination of the principal rays, i.e. the central rays of
the pencils which form the image. If, as will be assumed, the
pupils are circles whose centres lie upon the axis of the
system, then the rays which proceed from any object point P
toward the centre of the entrance-pupil, or from the centre of
the exit-pupil toward the image point P ', are the principal
rays of the object and image pencils respectively. When the
APERTURES AND THEIR EFFECTS
75
paths of the rays in any system are mentioned it will be
understood that the paths of the principal rays are meant.
2. Telecentric Systems. Certain positions of the iris can
be chosen for which the entrance- or the exit-pupils lie at
infinity (in telescopic systems both lie at infinity). To attain
this it is only necessary to place the iris behind Sj at its
principal focus or in front of S 2 at its principal focus (Fig. 31).
The system is then called telecentric, in the first case, tele-
centric on the side of the object; in the second, telecentric on the
side of the image. In the former all the principal rays in the
object space are parallel to the axis, in the latter all those
of the image space. Fig. 32 represents a system which is
telecentric on the side of the image. The iris B lies in front
of and at the principal focus of the lens 5 which forms the
real image P^P^ of the object P l and P y The principal rays
FIG. 32.
from the points P l and P 2 are drawn heavier than the limiting
rays. This position of the stop is especially advantageous when
the image /\'/Y ' 1S to ^ e measured by any sort of a micrometer.
Thus the image P^P^ always has the same size whether it
coincides with the plane of the cross- hairs or not. For even
with imperfect focussing it is the intersection of the principal
rays with the plane of the cross-hairs which determines for the
observer the position of the (blurred) image. If then the prin-
cipal rays of the image space are parallel to the axis, even
with improper focussing the image must have the same size as
if it lay exactly in the plane of the cross-hairs. But when the
principal rays are not parallel in the image space, the apparent
76 THEORY OF OPTICS
size of the image changes rapidly with a change in the position
of the image with respect to the plane of the cross-hairs.
If the system be made telecentric on the side of the object,
then, for a similar reason, the size of the image is not depen-
dent upon an exact focussing upon the object. This arrange-
ment is therefore advantageous for micrometer microscopes,
while the former is to be used for telescopes, in which the
distance of the object is always given (infinitely great) and the
adjustment must be made with the eyepiece.
3. Field of View. In addition to the stop B (the iris), the
images of which form the entrance- and exit-pupils, there are
always present other stops or lens rims which limit the size of
the object whose image can be formed, i.e. which limit t\\e field
of view. That stop which determines the size of the field of view
may be found by constructing, as before, for all the stops the
optical images which are formed of them by that part S l of the
entire lens system which lies between the object and each stop.
Of these images, that one G t which subtends the smallest angle
2w at the centre of the entrance-pupil is the one which deter-
mines the size of the field of view. 2w is called the angular
field of view. The correctness of this assertion is evident at
once from a drawing like Fig. 31 . In this figure the iris B, the
rims of the lenses S^ and S 2 , and the diaphragm G are all
pictured as actual stops. The image of G formed by S^ is
G^\ and since it will be assumed that G l subtends at the centre
of the entrance-pupil a smaller angle than the rim of 5^ or the
image which S l forms of the rim of the lens 5 2 , it is evident that
G acts as the field-of-view stop. The optical image G{ which
the entire system S^ -[~ S 2 forms of G l bounds the field of view
in the image space. The angle 2w' which G^ subtends at the
centre of the exit-pupil is called the angle of the image.
In Fig. 31 it is assumed that the image G l of the field-of-
view stop lies in the plane of the object. This case is charac-
terized by the fact that the limits of the field of view are
perfectly sharp, for the reason that every object point P can
either completely fill the entrance-pupil with rays or else can
APERTURES AND THEIR EFFECTS 77
send none to it because of the presence of the stop G r If the
plane of the object does not coincide with the image 6^, the
boundary of the field of view is not sharp, but is a zone of con-
tinuously diminishing brightness. For in this case it is evident
that there are object points about the edge of the field whose
rays only partially fill the entrance-pupil.
In instruments which are intended for eye observation it is
of advantage to have the pupil of the eye coincide with the
exit-pupil of the instrument, because then the field of view is
wholly utilized. For if the pupil of the eye is at some distance
from the exit-pupil, it itself acts as the field-of-view stop, and
the size of the field is thus sometimes greatly diminished. For
this reason the exit-pupil is often called the eye-ring, and its
centre is called the position of the eye.
Thus far the stops have been discussed only with reference
to their influence upon the geometrical configuration of the
rays, but in addition they have a very large effect upon the
brightness of the image. The consideration of this subject is
beyond the domain of geometrical optics ; nevertheless it will be
introduced here, since without it the description of the action
of the different optical instruments would be too imperfect.
4. The Fundamental Laws of Photometry. By the total
quantity of light M which is emitted by a source Q is meant
the quantity which falls from Q upon any closed surface 5 com-
pletely surrounding Q. S may have any form whatever, since
the assumption, or better the definition, is made that the total
quantity of light is neither diminished nor increased by propa-
gation through a perfectly transparent medium.*
It is likewise assumed that the quantity of light remains
constant for every cross-section of a tube whose sides are
made up of light rays (tube of light). t If Q be assumed
* In what follows perfect transparency of the medium is always assumed.
f The definitions here presented appear as necessary as soon as light quantity
is conceived as the energy which passes through a cross-section of a tube in unit
time. Such essentially physical concepts will here be avoided in order not to for-
sake entirely the dorrain of geometrical optics.
78 THEORY OF OPTICS
to be a point source, then the light-rays are straight lines
radiating from the point Q. A tube of light is then a cone
whose vertex lies at Q. By angle of aperture (or solid angle)
1 of the cone is meant the area of the surface which the cone
cuts out upon a sphere of radius i (i cm.) described about its
apex as centre.
If an elementary cone of small solid angle dQ, be consid-
ered, the quantity of light contained in it is
dL = Kdl ....... (61)
The quantity K is called the candle-power of the source Q in
the direction of the axis of the cone. It signifies physically
that quantity of light which falls from Q upon unit surface at
unit distance when this surface is normal to the rays, for in
this case dO. = i.
The candle-power will in general depend upon the direction
of the rays. Hence the expression for the total quantity of
light is, by (61),
M= TK-dfl, ..... (62)
in which the integral is to be taken over the entire solid angle
about Q. If A" were independent of the direction of the rays,
it would follow that
M =
since the integral of dO. taken over the entire solid angle about
Q is equal to the surface of the unit sphere described about Q
as a centre, i.e. is equal to 471. The mean candle-power K m
is defined by the equation
(63)
If now the elementary cone dl cuts from an arbitrary sur-
face 5 an element dS, whose normal makes an angle with
the axis of the cone, and whose distance from the apex Q of
APERTURES AND THEIR EFFECTS 79
the cone, i.e. from the source of light, is r, then a simple
geometrical consideration gives the relation
dD,.r*= aTS-cos 9 (64)
Then, by (61), the quantity of light which falls upon dS is
4L = K<*^ (65)
The quantity which falls upon unit surface is called the
intensity of illumination B. From (65) this intensity is
B=K^ (66)
i.e. the intensity of illumination is inversely proportional to the
square of the distance from the point source and directly pro-
portional to the cosine of the angle which the normal to the
illuminated surface makes with the direction of the incident rays.
If the definitions here set up are to be of any practical
value, it is necessary that all parts of a screen appear to the eye
equally bright when they are illuminated with equal intensities.
Experiment shows that this is actually the case. Thus it is
found that one candle placed at a distance of I m. from a screen
produces the same intensity of illumination as four similar
candles placed close together at a distance of 2 m.
Hence a simple method is at hand for comparing light
intensities. Let two sources Q l and Q 2 illuminate a screen
from such distances r^ and r 2 ( being the same for both) that
the intensity of the two illuminations is the same. Then the
candle-powers K l and K 2 of the two sources are to each other
as the squares of the distances r l and r v A photometer is used
for making such comparisons accurately. The most perfect
form of this instrument is that constructed by Lummer and
Brodhun.*
* A complete treatment of this instrument, as well as of all the laws of pho-
tometry, is given by Brodhun in Winkelmann's Handbuch der Physik, Optik, p.
45 sq-
8o
THEORY OF OPTICS
The most essential part of this instrument is a glass cube
which consists of two right-angled prisms A and B (Fig.
33) whose hypothenuses are polished so as to fit accurately
together. After the hypothenuse of prism A has been ground
upon a concave spherical surface until its polished surface has
been reduced to a sharply defined circle, the two prisms are
pressed so tightly together that no air-film remains between
them. An eye at O, which with the help of a lens w looks
FIG. 33-
perpendicularly upon one of the other surfaces of the prism B,
receives transmitted and totally reflected light from immedi-
ately adjoining portions of the field of view. Between the two
sources Q l and Q 2 which are to be compared is placed a screen
5 of white plaster of Paris, whose opposite sides are exactly
alike. The light diffused by 5 is reflected by the two mirrors
S l and 5 2 to the glass cube AB. If the intensities of illumina-
tion of the two sides of 5 are exactly equal, the eye at O sees
the glass cube uniformly illuminated, i.e. the figure which dis-
tinguishes the transmitted from the reflected light vanishes.
The sources Q l and Q 2 are then brought to such distances ^
and r* from the screen 5 that this vanishing of the figure takes
APERTURES AND THEIR EFFECTS 81
place. In order to eliminate any error which might arise from
a possible inequality in the two sides of S y it is desirable to
make a second measurement with the positions of the two
sources Q l and Q 2 interchanged. The screen S, together with
the mirrors S 1 and S 2 and the glass cube, are rigidly held in
place in the case KK.
As unit of candle-power it is customary to use the flame of a
standard paraffine candle burning 50 mm. high, or, better still,
because reproducible with greater accuracy, the Hefner light.
This light was introduced by v. Hefner-Alteneck and is pro-
duced by a lamp which burns amyl-acetate and is regulated
to give a flame 40 mm. high.
When the candle-power of any source has been measured,
the intensity at any distance can be calculated by (66). The
unit of intensity is called the candle-meter. It is the in-
tensity of illumination produced by a unit candle upon a
screen standing I m. distant and at right angles to the direc-
tion of the rays. Thus, for example, an intensity of 50 candle-
meters, such as is desirable for reading purposes, is the
intensity of illumination produced by 50 candles upon a book
held at right angles to the rays at a distance of I m., or that
produced by 12^- candles at a distance of ^ m., or that pro-
duced by one candle at a distance of 1 m.
Photometric measurements upon lights of different colors
are attended with great difficulties. According to Purkinje
the difference in brightness of differently colored surfaces varies
with the intensity of the illumination.*
If the source Q must be looked upon as a surface rather
than as a point, the amount of light emitted depends not only
upon the size of the surface, but also upon the inclination of the
rays.
A glowing metal ball appears to the eye uniformly bright.
Hence the same quantity of light must be contained in all ele-
* Even when the two sources appear colorless, if they are composed of different
colors physiological effects render the measurement uncertain. Cf. A. Tschermak,
Arch. f. ges. Physiologic, 70, p. 297, 1898.
82 THEORY OF OPTICS
mentary cones of equal solid angle doo whose vertices lie at the
eye and which intersect the sphere. But since these cones
cut out upon the metal sphere (cf. eq. (64) ) surface elements
ds such that
cos
(67)
in which $ is the angle of inclination of ds with the axis of the
cone, it follows that the surface elements which send a given
quantity of light to the eye increase in size as the angle
included between the normal and the direction of the rays to
the eye increases, i.e. the surfaces are proportional to I : cos 0.
Hence (cf. eq. (65)) the quantity of light dL which a sur-
face element ds sends to another surface element dS is
$-cos
-,*... (68)
in which r represents the distance between the surface elements,
and 8 and represent the inclinations of the normals at ds
and dS to the line joining the elements, i is called the inten-
sity of radiation of the surface ds. It is the quantity which unit
surface radiates to another unit surface at unit distance when
both surfaces are at right angles to the line joining them.
The symmetry of eq. (68) with respect to the surface
element which sends forth the radiations and that upon which
they fall is to be noted. This symmetry can be expressed in
the following words : The quantity of light which a surface
element radiating with an intensity i sends to another surface
element is the same as the former would receive from the latter
if it were radiating with the intensity i.
Equation (68; can be brought into a simpler form by intro-
ducing the solid angle dfl which dS subtends at ds. The
* This equation, which is often called the cosine law of radiation, is only approxi-
mately correct. Strictly speaking, i always varies with 0, and this variation is
different for different substances. The subject will be treated more fully when
considering Kirchhoft's law (Part III, Chapter II). This approximate equation will,
however, be used here, i.e. i will be regarded as constant.
APERTURES AND THEIR EFFECTS 83
relation existing bewteen dl and dS is expressed in equation
(64). Hence (68) may be written
ttL = t-ds-cos&-ttn, (69)
On the other hand it is possible to introduce the solid angle
doo which ds subtends at dS. A substitution in (68) of its
value taken from (67) gives
dL = i-dS-cos -dco (70)
The relation which the intensity of radiation i bears to the
total quantity M which is emitted by ds is easily obtained.
Thus a comparison of equations (61) and (69) shows that
the candle-power K of the surface ds in a direction which
makes an angle $ with its normal has the value
K ids cos $ (71)
Let now the quantity of light be calculated which is con-
tained between two cones whose generating lines make the
angles $ and $ -j- d$ respectively with the normal to the sur-
face ds. The volume enclosed between the two cones is a.
conical shell whose aperture is
dl 2n sin (3)
96 THEORY OF OPTICS
in which x 1 denotes the distance of the image from the second
principal focus, and a that of the eye. Generally a may be
neglected in comparison with d, in which case the magnifi-
cation produced, by the lens is
(4)
Thus it is inversely proportional to the focal length of the
lens.
If the diameter of the magnifying-glass is greater than that
of the image which it forms of the pupil of the eye, then the
latter is the aperture stop, the former the field-of-view stop.
In order to obtain the largest possible field of view it is neces-
sary to bring the eye as near as possible to the lens. As the
distance of the lens from the eye is increased, not only does
the field of view become smaller, but also the configuration of
the rays changes in that the images of points off the axis are
formed by portions of the lens which lie to one side of the axis.
This is evident at once from a graphical construction of the
entrance-pupil of the system, i.e. a construction of the image
of the pupil of the eye formed by the lens. The orthoscopy
is in this way generally spoiled, i.e. the image appears blurred
at the edges.
A simple plano-convex lens gives good images for mag-
nifications of less than eight diameters, i.e. for focal lengths
greater than 3 cm. The plane side of the lens must be turned
toward the eye. Although this position gives a relatively
large spherical aberration on the axis (cf. page 55), because
the object lies near its principal focus of the lens, nevertheless
it is more satisfactory than the inverse position on account of
the smaller aberration off the axis.
The image may be decidedly improved by the use of two
simple lenses because the distribution of the refraction over
several lenses greatly diminishes the spherical aberration on
the axis. Figs. 35 and 36 show the well-known Fraunhofer
and Wilson magnifying-glasses. In the latter the distance
OPTICAL INSTRUMENTS 97
between the lenses is much greater than in the former. In
this way the advantage is gained that the differences in the
magnifications for the different colors is diminished, although
at the cost of the distance of the object from the lens.*
Achromatization is attained in Steinheil's so-called apla-
natic magnifying-glass by a choice of different kinds of glass
(Fig: 37). In this a double-convex lens of crown glass is
cemented between two convexo-concave lenses of flint glass.
FIG. 35. FIG. 36. t FIG. 37.
The Briicke magnifying-glass, which consists of a conver-
gent achromatic front lens and at some distance from it a
simple divergent lens, is characterized by the fact that the
object lies at a considerable distance. The divergent lens
produces inverted, enlarged, virtual images of virtual objects
which lie behind its second principal focus (cf. Fig. 22, page
43). The arrangement of the lenses may be the same as in
the teleobjective (Fig. 34), i.e. the optical separation of the
convergent and the divergent lenses may be positive. Never-
theless, if the object is sufficiently close, the image formed by
the convergent lens may lie behind the second focus of the
divergent lens. Like the simple magnifying-glass this com-
bination furnishes erect images, for the image formed by the
convergent lens alone would be inverted were another inver-
sion not produced by the divergent lens. The objectionable
feature of this instrument is the smallness of the field of view.
3. The Microscope. a. General Considerations. In order
to obtain greater magnification it is advantageous to replace
* The effect of the distance between the lenses upon achromatism has been
treated above, p. 71. The subject will come up again when the eyepieces of
telescopes and microscopes are under consideration.
98 THEORY OF OPTICS
the magnifying-glass of short focal length by a microscope.
This consists of two convergent systems relatively far apart.
The first system (the objective) produces a real, inverted, en-
larged image of an object which lies just beyond its first
principal focus. This image is again enlarged by the second
system (the eyepiece) which acts as a magnifying-glass. Apart
from the fact that, on account of the greater distance apart of
the two systems of the microscope, a greater magnification
can be produced than with a single system used as a simple
magnifier, the chief advantage of the instrument lies in this,
that the problem of forming the image is divided into two
parts which can be solved separately by the objective and the
eyepiece. This division of labor is made as follows: the
objective, which has the greatest possible numerical aperture,*
forms an image of a surface element, while the eyepiece, like
any magnifying-glass, forms the image of a large field of view
by means of pencils which must be of small divergence, since
they are limited by the pupil of the eye. It has been shown
above (Chapter III, 8, 9, 10) that these two problems may
be separately solved.
b. The Objective. The principal requirements which an
objective must fulfil are as follows :
1. That with a large numerical aperture the spherical
aberration upon the axis be eliminated and the aplanatic
condition, i.e. the sine law, be fulfilled.
2. That chromatic errors be corrected. This requires that
the aplanatic condition be fulfilled for at least two colors, and
that a real achromatic image of the object be formed by the
objective. If only partial achromatism is required it is suf-
ficient to make the objective achromatic with respect to the
first principal focus ; for the position of the image of an object
which lies near this focal point F would vary rapidly with the
color if the position of F depended upon the color. If a system
has been achromatized thus with respect to the focus F, i.e.
*This requirement is introduced not only for the sake of increased brightness
but also of increased resolving power. Cf. above, pp. 90, 92.
OPTICAL INSTRUMENTS
99
with respect to the position of the image, it is not achromatic
with respect to the focal length. The different colors, there-
fore, produce images of different sizes, i.e. chromatic differences
in magnification still remain. These must be corrected by
means of the eyepiece.
It is customary to distinguish between dry and immersion
systems. In the latter the space between the front lens of the
objective and the cover-glass under which the object lies is
filled with a liquid. The advantages of this method of increas-
ing the numerical aperture are evident. Furthermore, by the
use of the so-called homogeneous immersions, in which the
liquid has the same index and dispersion as the cover-glass
and the front lens, the formation of aplanatic images by a
hemispherical front lens may be attained in accordance with
the principle of Amici (cf. page
58). Fig. 38 shows, in double the
natural size, an objective designed
by Abbe, called an aprochromat, in
which the above conditions are ful-
filled by a combination of ten
different lenses used with a homo-
geneous immersion. The apro-
chromat, being achromatic for three
colors, is free from secondary spec- (
tra, and the aplanatic conditions
are fulfilled for two colors.
2 mm. and its numerical
FIG. 38.
The focal length of the system is
aperture a= 1.40. The light-
collecting and dioptric excellence of this objective is such that
the limit of resolving power of a microscope (equation (87),
page 92) may be considered as actually attained by it.
c. The Eyepiece. The chief requirements for the eyepiece
are those for the formation of the image of an extended object
by means of narrow pencils, namely:
1. The elimination of astigmatism in the oblique pencils.
2. The formation of orthoscopic images.
3. The formation of achromatic images.
too THEORY OF OPTICS
The first two points have been discussed in Chapter III,
10, page 63; as to the last, partial achromatization is
sufficient. Consider the case in which the image formed by
an objective is free from chromatic errors. On account of the
length of the microscope tube, i.e. on account of the relatively
large distance between the real image formed by the objective
and the exit-pupil of the objective, the principal rays which
fall upon the eyepiece have but a small inclination to the axis
of the instrument. If now the eyepiece is made achromatic
with respect to its focal length, then it is evident from the
construction of conjugate rays given on page 24, as well as
from the property of the focal length given on page 20, that a
ray of white light which falls upon the eyepiece is split up into
colored rays all of which emerge from the eyepiece with the
same, inclination to the axis. Hence an eye focussed for
parallel rays sees a colorless image. Even when the image
lies at the distance of most distinct vision (25 cm.) an eyepiece
which has been made achromatic with respect to its focal
length nearly fulfils the conditions 71 for a colorless image.
Now it was shown on page 71 that two simple lenses of
focal lengths /j a.ndf 2 , made of the same kind of glass, when
placed at a distance apart a = ~, have a resultant focal
length /"which is the same for all colors. Since, in addition,
the construction of an eyepiece from tw r o lenses produces an
improvement of the image in the matter of astigmatism, eye-
pieces are usually made according to this principle. The lens
which is nearer the objective is called the field-lens , that next
the eye the eye- lens.
The two most familiar forms of achromatic eyepiece are the
following :
I. The Ramsden eyepiece (cf. Fig. 40, page 109). This
consists of two equal plano-convex lenses which have their
curved sides turned toward each other. Since /j =J^> the
distance a between the lenses is a =f l =/ 2 . But this arrange-
ment has the disadvantage that the field-lens lies at the prin-
OPTICAL INSTRUMENTS 101
cipal focus of the eye-lens, and hence any dust-particles or
scratches upon the former are seen magnified, by the, latter.
Hence the field-lens is placed somewhat nearer' tk^'ehe eye-
lens, for instance, a |/ r In this way a further: aciVarrta$fe
is obtained. When a = f /j , the optical separation of the
two lenses (cf. page 28) A = f f r Hence, by equation (20)
on page 30, the focal length F of the combination lies
at a distance J^ before the field-lens ; while, when a = /j ,
i.e. A = fij it would fall in the objective lens itself. Since
the real image formed by the objective of the microscope lies
near the principal focus F of the eyepiece, if a f/j , it is
still in front of the field-lens ; hence the image in the micro-
scope may be measured by introducing in front of the field-
lens, at the position of the real image formed by the objective,
a micrometer consisting of fine graduations upon glass or a
cross-hair movable by means of a screw.
2. The Huygens eyepiece (Fig. 39). In this the focal
length /j of the field-lens is larger than that f 2 of the eye-
lens. Generally/! = 3/ 2 . Then from a = l ~^ it follows
that a = f/J 2/2- The optical separation has the value
A f/i , hence by (20) on page 30 the focal length F of the
combination lies a distance /j behind the field-lens. The
real image formed by the objective must, therefore, fall behind
the field-lens as a virtual object, and a micrometrical measure-
ment of it is not easily made since both the lenses in the eye-
piece take part in the formation of the image of the object,
while the image of the micrometer is formed by the eye-lens
alone. This eyepiece also consists of two plano-convex lenses
but their curved surfaces are both turned toward the object. The
advantage of the combination of a weak field-lens with an eye-
lens three times as powerful lies in the fact that the bending
of the rays at the two lenses is uniformly distributed between
them.*
*For this calculation cf. Heath, Geometrical Optics, Cambr., 1895.
102 THEORY OF OPTICS
If chromatic errors exist in the image formed by the objec-
tive, they m'ay be eliminated by constructing the eyepiece to
have -chromatic errors of opposite sign. It was shown above
i(pag. 99) that the chromatic errors of magnification are not
eliminated in the aprochromat objective, the blue image being
larger than the red. Abbe then combines with such objectives
the so-called compensating eyepieces which are not achrome-
tized with respect to focal length, i.e. with respect to mag-
nification, but which produce larger red images than blue.
d. The Condenser. In order that full advantage may be
taken of the large numerical aperture of the objective, the rays
incident upon it must be given a large divergence. To obtain
such divergence there is introduced under the stage of the
microscope a condenser which consists of one or more conver-
gent lenses of short focal length arranged as in an objective,
but in the inverse order. From the discussion above on page
85 it is evident that such a condensation of the light does not
increase the intensity of the source but merely has the effect
of bringing it very close to the objective.
e. Geometrical Configuration of the Rays. If the normal
magnification (cf. page 90) has not been reached, the pupil of
the eye is the exit-pupil of the entire microscope, and the image
of the pupil of the eye formed by the instrument is the
entrance-pupil. If the normal magnification is exceeded, a
stop or the rim of a lens in the microscope is the aperture stop.
This stop always lies in the objective, not in the eyepiece. Fig.
39 shows a case of very frequent occurrence in which the rim
B^B 2 of the hemispherical front lens of the objective is both
aperture stop and entrance-pupil. The image B^B 2 of B 1 B 2
formed by the whole microscope is the exit-pupil. If the
length of the tube is not too small, this image lies almost at
the principal focus of the eyepiece. The eyepiece shown in
Fig. 39 is a Huygens eyepiece. The real image of the object
P^P 2 formed by the objective and the field-lens of the eyepiece
is /Y-^Y- The field-of-view stop GG is placed at /Y^Y- In
this way the edge of the field of view becomes sharply defined,
OPTICAL INSTRUMENTS
103
because the image of G formed by the field-lens and the objec-
tive lies in the plane of the object P^P 2 (cf. remark on page
76). The points P^P 2 must ^ e on tne edge of the field-of-
view stop. Then P-f^ is the size of the field of view on the
side of the object. The virtual image P^'P^' formed by the
eye-lens of the real image P^P^ is the image seen by the
observer. If this image is at a distance d from the exit-pupil,
then the observer, the pupil of whose eye ought to be coin-
cident with the exit-pupil B^B 2 (cf. page 77), must focus his
eye for this distance tf. By a slight raising or lowering of the
whole microscope with respect to the object P^P 2 the image
/Y'/y may easily be brought to any desired distance d. It
is usually assumed that d is the distance of most distinct vision,
namely, 25 cm.
In Fig. 39 the principal and the limiting rays which proceed
from P l are shown. From P 2 the principal ray only is drawn,
the limiting rays being introduced behind the eye-lens.
io 4 THEORY OF OPTICS
f. The Magnification. Let the object have the linear
magnitude y. By equation (7) on page 19, the objective
forms a real image of size y' =y- t in which/j' is the second
/i
focal length of the objective,* and / the distance of the image
from the second principal focus. Since, as was shown above,
this image y' lies immediately in front of or behind the field-
lens of the ocular, /may with sufficient accuracy be taken as the
length of the microscope tube. Likewise, by equation (7), the
<\
virtual image formed by the eyepiece has the size y" y'>~^
-/2
in which f 2 represents the focal length of the eyepiece and 6
the distance of the virtual image from its second principal
focus. Since, as was above remarked, this eyepiece lies close
to the exit-pupil, i.e. to the pupil of the eye, d may be taken
as the distance of the image from the eye. The magnification
V produced by the whole microscope is then
v _y^_sj_
' y -/,'/,-
Since the second principal focal length f of the entire
microscope is, by equation (18) on page 29, t
ff _ -A/2 xx-v
J i ...... W
J, the optical separation between the objective and the eye-
piece being almost equal to /, it follows that, disregarding the
s ig" n > (5) mav be written
(7)
Thus the magnification depends upon three factors which
are entirely arbitrary, namely, upon//, / 2 , and /. The length
* A distinction between first and second principal foci is only necessary for
immersion systems.
f For the eyepiece / 2 =/ 2 '.
OPTICAL INSTRUMENTS 105
/ of the tube cannot be increased beyond a certain limit with-
out making the instrument cumbrous. It is more practicable to
obta'in the effect of a longer / by increasing the power of the
eyepiece. Furthermore the focal length of the objective is
always made smaller than that of the eyepiece. In this way
not only may the lenses in the objective be made relatively
small even for high numerical aperture, but also a certain
quality of image (near the axis) may be more easily obtained
for a given magnification the smaller the focal length of the
objective. But since, with the diminution of the focal length
of the objective, the errors in the final image formed by the
eyepiece increase for points off the axis, the shortening of f^
cannot be carried advantageously beyond a certain limit (1.5-2
mm. in immersion systems).
g. The Resolving Power. This is not to be confused with
magnification, for, under certain circumstances, a microscope
of smaller magnifying power may have the larger resolving
power, i.e. it may reveal to the eye more detail in the object
than a more powerfully magnifying instrument. The resolving
power depends essentially upon the construction of the objec-
tive : the detail of the image formed by it depends (cf. page
92) on the one hand upon the numerical aperture of the
objective, on the other upon the size of the discs which arise
because the focussing is not rigorously homocentric. If two
points P l and P 2 of an object be considered such that the discs
to which they give rise in the image formed by the objective
do not overlap, they may be distinguished as two distinct
points or round spots in case the eyepiece has magnified the
image formed by the objective to such an extent that the visual
angle is at least I ' '. But if these discs in the image formed by
the objective overlap, then the most powerful eyepiece cannot
separate the points P l and P 2 . For every objective there' is
then a particular ocular magnification, which will, just suffice
to bring out completely the detail in the image formed by the
objective. A stronger magnification may indeed be con-
veniently used in bringing out this detail, but it adds no new
106 THEORY OF OPTICS
element to the picture. From the focal length of the objective,
the length of the tube, and the focal length of the eyepiece
which is just sufficient to bring out the detail in the image, it
is possible to calculate from (5) the smallest permissible mag-
nification for complete resolution. This magnification is
greater the greater the resolving power of the objective.
Assuming a perfect objective, the necessary magnification of
the whole instrument depends only upon the numerical aper-
ture. This has not yet been pushed beyond the limit (for
immersion systems) a 1.6. Hence, by equation (87) on
page 92 , the smallest interval d which can be optically resolved
is
A. 0.00053 mm.
d = = - = 0.00016 mm.
2a 3.2
if A be the wave-length of green light. Now at a distance
d 25 cm. from the eye an interval d' = 0.145 mm. has a
visual angle of 2', which is the smallest angle which can be
easily distinguished. Since d' \ d 905, the limit of resolution
of the microscope is attained when the total magnification is
about poo. Imperfections in the objective reduce this required
magnification somewhat. By equation (85) on page 89 the
ratio of the brightness of the image to the normal brightness
is for this case
/- \ 2-900 20
the radius / of the pupil of the eye being assumed as 2 mm.
h. Experimental Determination of the Magnification and the
Numerical Aperture. The magnification maybe determined
by using as an object a fine glass scale and drawing with the
help of a camera lucida its image upon a piece of paper placed
at a distance of 25 cm. from the eye. The simplest form of
camera lucida consists of a little mirror mounted obliquely to
the axis of the instrument, from the middle of which the silver-
ing has been removed so as to leave a small hole of about 2
mm. diameter. The image in the microscope is seen through the
OPTICAL INSTRUMENTS 107
hole, while the drawing-paper is at the same time visible in the
mirror.* The ratio of the distances between the divisions in
the drawing to those upon the glass scale is the magnification
of the instrument.
From the magnification and a measurement of the exit-
pupil of the microscope its numerical aperture may be easily
found. Since, according to the discussion on page 88, the
ratio of the brightness of the image to the normal brightness is
equal to the ratio of the exit-pupil to the pupil of the eye, it
follows, from (85) on page 89, that
H P _ a v
H,-~f- fV*
in which b represents the radius of the exit-pupil. Hence the
numerical aperture is
bV .
A substitution of the value of V from (7) gives
* = :/', ...... (10)
i.e. the numerical aperture is equal to the ratio of the radius of
the exit-pupil to the second focal length of the whole microscope.
Abbe has constructed an apertometer which permits the
determination of the numerical aperture of the objective
directly, t
4. The Astronomical Telescope. This consists, like the
microscope, of two convergent systems, the objective and the
eyepiece. The former produces at its principal focus a real
inverted image of a very distant object This image is enlarged
by the eyepiece, which acts as a simple magnifier. If the eye
of the observer is focussed for parallel rays, the first focal plane
of the eyepiece coincides with the second focal plane of the
* Other forms of camera lucida are described in Mtiller-Pouillet, Optik, p. 839.
f A description of it will be found in the texts referred to at the beginning of
this chapter.
io8 THEORY OF OPTICS
objective, and the image formation is telescopic in the sense
used above (page 26), i.e. both the object and the image lie
at infinity. The magnification F means then the ratio of the
convergence of the image rays to the convergence of the object
rays. But, by (24) on page 30,
F= tan u' : tan u = /j :/ 2 , . . . . (n)
in which f v is the focal length of the objective and f 2 that of
the eyepiece. Hence for a powerful magnification /j must be
large and/ 2 small.
The magnification may be experimentally determined by
measuring the ratio of the entrance-pupil to the exit-pupil of
the instrument. For when the image formation is telescopic,
the lateral magnification is constant (cf. page 26), i.e. it is
independent of the position of the object and, by (14') on page
28, is equal to the reciprocal of the angular magnification.
Now (without reference to the eye of the observer, cf. below)
the entrance-pupil is the rim of the objective, hence the exit-
pupil is the real image (eye-ring) of this rim formed by the
eyepiece. Hence if the diameter of this eye-ring be measured
with a micrometer, the ratio between it and the diameter of
the objective is the reciprocal of the angular magnification of
the telescope.
Fig. 40 shows the configuration of the rays when a Rams-
den eyepiece is used (cf. page 100). B^B 2 is the entrance-
pupil (the rim of the objective), B^B^ the exit-pupil, and P l
is the real image formed by the objective of an infinitely dis-
tant point P. The principal ray is drawn heavy, the limiting
ray light. P l lies somewhat in front of the field-lens of the
eyepiece. The field-of-view stop GG is placed at this point.
Since its image on the side of the object lies at infinity, the
limits of the field of view are sharp when distant objects are
observed. P' is the infinitely distant image which the eyepiece
forms of P r When the eye of the observer is taken into con-
sideration, it is necessary to distinguish between the case in
which the exit-pupil of the instrument is smaller than the
OPTICAL INSTRUMENTS
109
pupil of the eye and that in which it is greater. Only in the
first case do the conclusions reached above hold, while in the
second the pupil of the eye is the exit-pupil for the whole
system of rays, and the image of the pupil of the eye formed
by the telescope is the entrance-pupil.
The objective is an achromatic lens which is corrected for
spherical aberration. In making the eyepiece achromatic the
P .
>:
FIG. 40.
same conditions must be fulfilled which were considered in the
case of the microscope. Since the principal rays which fall
upon the eyepiece are almost parallel to the axis, it is sufficient
if it be achromatized with respect to the focal length. Hence
the same eyepiece may be used for both microscope and tele-
scope, but the Ramsden eyepiece is more frequently employed
in the latter because it lends itself more readily to micrometric
measurements.
Here, as in the microscope, in order to bring out all the
detail, the magnification must reach a certain limit beyond
which no advantage is obtained in the matter of resolving
power. In telescopes the aperture of the objective corresponds
to the numerical aperture in microscopes.
$. The Opera-glass. If the convergent eyepiece of the
astronomical telescope be replaced by a divergent one, the
instrument becomes an opera-glass. In order that the image
formation may be telescopic, the second principal focus of the
eyepiece must coincide with the second principal focus of the
no
THEORY OF OPTICS
objective. Thus the length of the telescope is not equal to
the sum, as in the astronomical form, but rather to the differ-
ence of the focal lengths of the eyepiece and the objective.
Since equation (11) of this chapter holds for all cases of
telescopic image formation, the angular magnification T of the
opera-glass may be obtained from it. This instrument, how-
ever, unlike the astronomical telescope, produces erect images,
for the inverted image formed by the objective is again inverted
by the dispersive eyepiece.
Without reference to the eye of the observer, the rim of
the objective is always the entrance-pupil of the instrument.
The eyepiece forms directly in front of itself a virtual diminished
image of this rim (the exit-pupil). The radius of this image is
(12)
in which h is the radius of the objective.
Since this exit-pupil lies before rather than behind the eye-
piece, the pupil of the eye of the observer cannot be brought
into coincidence with it; consequently the pupil of the eye acts
as a field-of-view stop in case the quantity b determined by
-0*
FIG. 41.
equation (12), i.e. the exit-pupil of the instrument, is smaller
than the eye, which means that the normal magnification is
exceeded. Hence for large magnifications the field of view is
very limited. Fig. 41 shows the geometrical configuration of
the rays for such a case. /, / represents the pupil of the eye,
w' the angular field of view of the image. Since the image of
OPTICAL INSTRUMENTS
in
the field -of- view stop (the pupil of the eye), formed by the
whole telescope lies at a finite distance, i.e. since it is not at
infinity with the object, the edge of the field of view is not
sharp (cf. page 76).
But if the exit-pupil B{B 2b of the instrument is larger
than the pupil of the eye, i.e. if the normal magnification has
not been reached, then, taking into account the eye of the
observer, the pupil of his eye is the exit- pupil for all the rays,
and the rim of the objective acts as the field-of-view stop.
The field of view on the side of the image is bounded by the
image 2b of the rim of the objective (in Fig. 42 this is repre-
sented by B^B). Hence in this case the field of view may
be enlarged by the use of a large objective. But again, for
the same reason as above, the limits of the field of view are
not sharp. Fig. 42 shows this case, w' being the angular field
of view on the side of the image.
FIG. 42.
If the radius of the pupil of the eye is assumed as 2 mm ,
then the paths of the rays will be those shown in 41 or 42,
according as *
h ^ 2 F mm ;
* The difference between these cases may be experimentally recognized by
shading part of the objective with an opaque screen and observing whether the
brightness of the image or the size of the field is diminished.
112
THEORY OF OPTICS
for example, for a magnification of eight diameters, 2/1 = 32
mm. is the critical size of the objective.
6, The Terrestrial Telescope. For observation of objects
on the earth it is advantageous to have the telescope produce an
erect image. If the magnification need not be large, an opera-
glass may be used. But since for large magnifications this
has a small field of view, the so-called terrestrial telescope is
often better. This latter consists of an astronomical telescope
with an inverting eyepiece. The image is then formed as fol-
lows: the objective produces a real inverted image of the
object; this image is then inverted without essential change in
size by a convergent system consisting of two lenses. The
erect image thus formed is magnified either by a Ramsden or
a Huygens eyepiece.
7. The Zeiss Binocular. The terrestrial eyepiece has an
inconvenient length. This difficulty may be avoided by invert-
ing the image formed by the objective by means of four total
reflections within two right-angled prisms placed as shown in
Fig. 43. The emergent beam is parallel to the incident, but
FIG. 43.
has experienced a lateral displacement. Otherwise the con-
struction is the same as that of the astronomical telescope.
The telescope may be appreciably shortened by separating
the two prisms I and II, since the ray of light traverses the
distance between the prisms three times. By a suitable division
and arrangement of the prisms the lateral displacement
between the incident and the emergent rays may be made as
large as desired. In this way a binocular may be constructed
OPTICAL INSTRUMENTS 113
in which the exit-pupils (the lenses of the objective) are much
farther apart than the pupils of the eyes. Thus the stereo-
scopic effect due to binocular vision is greatly increased.
8. The Reflecting Telescope. This differs from the refract-
ing telescope in that a concave mirror instead of a lens is used
to produce the real image of the object. For observing this
image various arrangements of the eyepiece are used.*
Reflecting telescopes were of great importance before achro-
matic objectives were invented, for it is evident that concave
mirrors are free from chromatic errors.
To obtain the greatest possible magnification large mirrors
with large radii of curvature must be used. Herschel built an
enormous concave mirror of 16 m. radius of curvature. Since
the visual angle of the sun is about 32', the image of the sun
formed by it was 7 cm. in diameter.
. *For further details cf. Heath, Geometrical Optics, Cambr., 1895.
PART II
PHYSICAL OPTICS
SECTION I
GENERAL PROPERTIES OF LIGHT
CHAPTER I
THE VELOCITY OF LIGHT
I. Romer's Method. Whether light is propagated with
finite velocity or not is a question of great theoretical impor-
tance. On account of the enormous velocity with which light
actually travels, a method depending on terrestrial distances
which was first tried by Galileo, gave a negative result. For
the small distances which must be used in terrestrial methods
the instruments employed must be extremely delicate.
Better success was attained by astronomical methods, which
permit of the observation of the propagation of light over very
great distances. The first determination of the velocity of
light was made by Olaf Romer in 1675. He observed that
the intervals of time between the eclipses of one of Jupiter's
satellites increased as the earth receded from Jupiter and
decreased as it approached that planet. This change in the
interval between eclipses can be very accurately determined
by observing a large number of consecutive eclipses. Romer
114
THE VELOCITY OF LIGHT 115
found that the sum of these intervals taken over a period
extending from the opposition to the conjunction of the earth
and Jupiter differed by 996 seconds from the product of the
number of eclipses and the mean interval between eclipses
taken throughout the whole year. He ascribed this difference
to the finite velocity of light. According to this view, then, light
requires 996 seconds to traverse the earth's diameter. Glase-
napp's more recent observations make the correct value of this
interval 1002 seconds. The diameter of the earth's orbit may
be obtained from the radius of the earth and the solar parallax,
i.e. the angle which the radius of the earth subtends at the sun.
According to the most recent observations the most probable
value of the solar parallax is 8. 85". The radius of the earth
is 6378 km., so that the diameter, d, of its orbit is
2-6378 180-60.60
sec .
Hence the velocity of light V is
V = 296 700 km -/sec. = 2.967 - IO M cm -/
On account of errors in the determination of the solar parallax
this value is uncertain by from ^ to I per cent.
2. Bradley's Method. Imagine that a ray of light from
a distant source P reaches the eye of an observer after passing
successively through two holes 5 X and S 2 which lie upon the
axis of a tube R. If the tube R moves with a velocity v
in a direction at right angles to its axis, while the source P
remains at rest, then if the light requires a finite time to trav-
erse the length of the tube R a ray of light which has passed
through the first hole S l will no longer fall upon the hole S 2 .
Therefore the observer no longer sees the source P. In order
to see it again he must turn the tube R through an angle a.
Thus the line of sight to P appears inclined in the direction of
the motion of the observer an angle such that
tan C = z/: F, ...... (i)
in which V represents the velocity of light.
n6 THEORY OF OPTICS
This consideration furnished the explanation of the aberra-
tion of the fixed stars, a phenomenon discovered in 1727 by
Bradley. He found that if the line of sight and the motion of
the earth are at right angles, the line of sight is displaced a
small angle in the direction of the earth's motion. According
to the most recent observations the value of this angle is 20. 5".
Since the velocity v of the earth in its orbit is known from the
size of the orbit, equation (i) gives as the velocity of light
V 2.982.io locm -/ sec-
This method, like Romer's or any astronomical method,
is subject to the uncertainty which arises from the imperfect
knowledge of the solar parallax and hence of the size of the
earth's orbit.
The result agrees well with that obtained by Romer, a fact
which justifies the assumption made in both calculations, that
the rays, in passing through the atmosphere which is moving
with the earth, receive from it no lateral velocity. Never-
theless aberration cannot be completely explained in this
simple way. From the considerations here given it would be
expected that when a fixed star is viewed through a telescope
rilled with water the aberration would be greater, since, as will
be shown later, the velocity of light in water is less than in
air. As a matter of fact, however, the aberration is indepen-
dent of the medium in the tube. In order to explain this a
more complete investigation of the effect of the motion of a
body upon the propagation of light within it is necessary.
This will be given farther on. It is sufficient here to note
that the phenomenon of aberration is capable of giving the
velocity of light in space, i.e. in vacuo.
3. Fizeau's Method. The first successful determination
of the velocity of light by a method employing terrestrial dis-
tances was made by Fizeau in the year 1849. An image of a
source of light P is formed at^by means of a convergent lens
and a glass plate / inclined to the direction of the rays (Fig.
44). The rays are then made parallel by a lens L l and pass
THE VELOCITY OF LIGHT
117
to the second lens L 2 distant from L l 8.6 km. A real image
is formed upon a concave mirror s whose centre of curvature
lies in the middle of the lens L 2 . The mirror s returns the
light back over the same path so that the reflected rays also
form a real image at f. This image is observed through the
obliquely inclined plate / by means of the eyepiece o. At f t
FIG. 44.
where the real image is formed, the rim of a toothed wheel is
so placed that the light passes freely through an opening, but
is cut off by a tooth. If the wheel is rotated with small
velocity, the image alternately appears and disappears. When
the velocity is increased, the image is seen continuously on
account of the persistence of vision. As the velocity of the
wheel is still further increased, a point is reached at which the
image slowly disappears. This occurs when, in the time re-
quired by the light to travel from/" to s and back, the wheel has
turned so that a tooth is in the position before occupied by an
opening. When the velocity is twice as great the light again
appears, when it is three times as great it disappears, etc. From
the velocity of rotation of the wheel, the number of teeth, and
the distance between /and s, the velocity of light can easily be
calculated. Fizeau used a wheel having 720 teeth. The first
disappearance occurred when the rate of rotation was 12,6
u8
THEORY OF OPTICS
revolutions per second. Since the distance between Z x and
was 8.633 km., the velocity of light was calculated as
The principal difficulty in the method lies in the production
and measurement of a uniform velocity of rotation. By using
more refined methods of measurement Cornu obtained the
value
Young and Forbes the value
V 3-Oi3-i >
4. Foucault's Method. This method does not require so
large distances as the above and is in several respects of great
importance in optical work.
Rays from a source P pass
through an inclined plate /
(Fig. 45) and fall upon the
rotating mirror m. When this
mirror m is in a certain position,
the rays are reflected through
the lens Z,* which is close to m
FlG> 45< and so placed that a real image
of the source P is formed at a distance D upon a concave mir-
ror s whose centre of curvature is at m. The mirror s reflects
the rays back over the same path provided the mirror m has
not appreciably changed its position in the time required for
the light to travel the distance 2D. An image P' of the source
P is then formed by the rays reflected from m, s, and /. But
if, in the time required for the light to travel the distance 2D,
the rotating mirror has turned through an angle <*, then the
ray returning from m to p makes an angle 2oc with the original
ray and a displaced image P" is produced after reflection at /.
* In Foucault's experiment the lens L was actually between the source P and
the mirror m, instead of between m and s; but the discussion is essentially the
same for either arrangement so long as L is close to w,-*- TR.
THE VELOCITY OF LIGHT 119
From the displacement P'P'' ', the velocity of rotation of the
mirror m, and the distances D and <4, the velocity of light may
be easily obtained.
If A = i m., D = 4 m., and the mirror m makes 1000
revolutions a second, then the displacement P'P" is 0.34 mm.
By reflecting the light back and forth between five mirrors
slightly inclined to one another, Foucault made the distance
D 20 m. instead of 4.
Theoretically this method is not so good as Fizeau's, since
it is necessary to measure not only the number of revolutions
but also the small displacement P ! P" . However, by increas-
ing the distance D to 600 m. Michelson materially improved
the method, since in this way he obtained a displacement P'P"
of 13 cm. without using a rate of revolution greater than 200
a second. With Foucault's arrangement it was not possible
to materially increase D, because the light returned would be
too faint unless the concave mirror s were of enormous dimen-
sions. Michelson avoided this difficulty by placing the lens L
so that m lay at its principal focus. In this way the principal
rays of all beams which are reflected by m to the lens L are
made parallel after passage through L, so that D can be taken
as large as desired and a plane mirror s perpendicular to the
axis of L used for reflection. Thus the mirror need be no larger
than the lens. From a large number of measurements Michel-
son obtained
V= 2.999- io- c + 4^. e tc., the
latter when A = -J- TT, -(- 371-, etc. Entire darkness must
result at a minimum if A l = A 2 .
These conditions are realized in the Fresnel-mirror experi-
ment in which two virtual sources
Q l and Q 2 (Fig. 47) are produced
by reflecting light from a single
source Q upon two mirrors 5 and
S' which are slightly inclined to
one another. In the space illumi-
nated by both of the sources interference occurs.* From the
calculation above there will be darkness at a point P if
A 3*.
r l r a =-> , etc. . . . (12)
Considering only such positions of the point P as lie on a line
parallel to Q^ 2 (Fig. 47), then if d represent the distance
* This space will be considerably diminished if the mirror S projects in front
of the mirror S'. Hence care must be taken that the common edge of the mirrors
coincides with their line of intersection.
132
THEORY OF OPTICS
between Q l and Q 2 , a the distance of the line d from the line
P P, and / the distance of a point P from the point P Q , which
lies on the perpendicular erected at the middle of d>
.e.
r 2 )fa
or since r^ -f- r 2 is approximately equal to 2 a when p and d
are small in comparison with a, it follows that
r l r 2 dp : a,
i.e. darkness occurs at the points
a A a 3 A. # 51
> = JT iJ'T- J'T' etc ' ^ I3 )
Hence, if the light be monochromatic, interference fringes
will appear on a screen held at a distance a from the line d,
and the constant distance between these fringes will be a\ : d.
-
FIG. 47.
If white light is used, colored fringes will appear upon the
screen since the different colors contained in white light, on
account of their different wave lengths, produce points of maxi-
mum and minimum brightness at different places upon the
screen. But at the point P Q there will be no color, since there
all the colors have a maximum brightness (r l r 2 o).
The distance d between the virtual sources may be calcu-
lated from the position of the actual source Q with respect to
the mirrors and the angle between the mirrors. This angle
must be very small (only a few minutes) in order that d may
INTERFERENCE OF LIGHT 133
be small enough to permit of the separation of the interference
fringes. Since (13) contains only the ratio a : d, it is merely
necessary to measure the angle subtended at P Q by the two
images Q l and Q 2 .
Instead of receiving the interference pattern upon a screen,
it is possible to observe it by means of a lens or by the eye
itself, if it be placed in the path of the rays coming from Q l
and <2 2 an d focussed upon a point P at a distance a from those
sources.* Fig. 48 shows an arrangement for making quanti-
tative measurements such as the determination of wave lengths.
A cylindrical lens / brings to a line focus the rays from a lamp.
This, acting as a source Q, sends rays to both mirrors S and
FIG. 48.
S', whose line of intersection is made parallel to the axis of the
cylindrical lens. The direct light from Q is cut off by a screen
attached to the mirrors and at right angles to them. The
* If the eye be focussed with or without a lens upon P, the two interfering
beams reach the image of P upon the retina with the same difference of phase
which they have at P itself, since the optical paths between P and the retinal
image are the same for all the rays. Hence the intensity upon the retina is zero
if it would be zero upon the corresponding point of a screen placed at P.
134 THEORY OF OPTICS
interference fringes are observed by means of a micrometer
eyepiece L which is movable by the micrometer screw K.
The question arises whether interference fringes might not
be more simply produced by using as sources not the two
virtual images of a real source, but two small adjacent open-
ings in a screen placed before a luminous surface.
In this case no interference phenomena are obtained even
with monochromatic light such as a sodium flame. For if two
sources are to produce interference, their phases must always
be either exactly the same or else must have a constant dif-
ference. Such sources are called coherent. They may always
be obtained by dividing a single source into two by any sort of
optical arrangement. With incoherent sources, however, like
two different points of a flame, although the difference of phase
is constant for a large number of periods, since, as will be
shown later, a monochromatic source emits a large number
of vibrations of constant period, yet irregularities in these
vibrations occur within so short intervals of time that separate
impressions are not produced in the eye. Thus incoherent
sources change their difference of phase at intervals which are
extremely short although they include many millions of vibra-
tions. This prevents the appearance of interference.
As was remarked on page 124, diffraction is not entirely-
excluded from this simple interference experiment. All the
boundaries of the mirrors can give rise to diffraction, but
especially the edge in which the two touch. In order to avoid
this effect it is desirable that the incident light have a consider-
able inclination to the mirrors (say 45), and that the point of
observation be at a considerable distance from them. Also
the angle between the mirrors must not be made too small.
In this way it is possible to arrange the experiment so that the
extreme rays which proceed from Q l and Q 2 to the common
edge of the mirrors are removed as far as possible from the
point of observation P.
4, Modifications of the Fresnel Mirrors. The considera-
tions advanced in paragraph 3 are typical of all cases in which
INTERFERENCE OF LIGHT
'35
interference is produced by the division of a single source into
two coherent sources <2i an d Q r This division may be brought
about in several other ways. The Fresnel bi-prism, shown in
cross-section in Fig. 49, is particularly convenient. The light
FIG. 49.
from a line source Q which is parallel to the edge B is refracted
by the prism in such a way that two coherent line sources Q l
and <2 2 are produced.
If such a prism be placed upon the table of a spectrometer
so that the edge B is vertical, and if the vertical slit of the
collimator focussed for parallel rays be used for the source, then
two separate images of the slit appear in the telescope of the
spectrometer. The angle a between these images may be
read off upon the graduated circle of the spectrometer when
the cross-hairs have been set successively upon the two images.
This angle a is the supplement of the angle ABC (Fig. 49)
which the two refracted wave fronts AB and BC make with
each other after passage through the prism. If the telescope
be removed, dark fringes may be observed at any point P for
which (cf. 12) r t r 2 = i^> f^ etc., in which r l and r 2 are
the distances of the point P from the wave fronts AB and BC.
From the figure it is evident that
hence
^ b sin (ABP), r 2 = b sin (CBP),
, ABC .
^ r 2 = 20 cos sin
136 THEORY OF OPTICS
The angle is very small so that sin = tan = / : a.
Furthermore ABC n a, and since b = a approximately,
and sin a = a, it follows finally that
Thus the relative distance between the fringes is k : <*, i.e.
it is independent of a. Since has been measured by the
telescope, the measurement of the distance between the fringes
furnishes a convenient method of determining A.
Billet's half-lenses (Fig. 50), which produce two real or
virtual images of a source Q t are similar in principle to the
FIG. 50.
Fresnel bi-prism. The space within which interference occurs
is shaded in the figure.
5. Newton's Rings and the Colors of Thin Plates. Suf-
ficiently thin films of all transparent bodies show brilliant colors.
These may be most easily observed in soap-bubbles, or in thin
films of oil upon, water, or in the oxidation films formed upon
the heated surfaces of polished metals.
The explanation of these phenomena is at once evident as
soon as they are attributed to interference taking place between
the light reflected from the front and the rear surface of the
film.
Consider a ray AB of homogeneous light (Fig. 51) incident
at an angle upon a thin plane parallel plate of thickness d.
At the front surface of the plate AB divides into a reflected
ray BC and a refracted ray BD. At the rear surface the latter
is partially reflected to B' and passes out of the plate as the
ray B'C' . The essential elements of the phenomena can be
presented by discussing the interference between the two rays
INTERFERENCE OF LIGHT 137
BC and B'C' only. If these two rays are brought together at
a point on the retina, as is done when the eye is focussed for
parallel rays, the impression produced is a minimum if the
phase of the ray BC differs from that of B'C' by TT, 3?r, 577-,
etc.
Of course for a complete calculation of the intensity of the
reflected light all the successive reflections which take place
between the two surfaces must be taken into account. This
FIG. 51.
rigorous discussion will be given in Section II, Chapter II,
11. It is at once apparent that the introduction of these
repeated reflections will not essentially modify the result, since
the intensity of these rays is much smaller than that of BC and
B'C' ', which have experienced but one reflection.
If a perpendicular B'E be dropped from B' upon BC, the
two rays BC and B'C' would have no difference of phase if the
phase at B' were the same as that at E. The two rays would
then come together at a point upon the retina in the same
phase. The difference of phase between the points E and B'
is identical with the difference of phase between the rays BC
and B'C'.
138 THEORY OF OPTICS
But the difference of phase between B' and E is
BD DB 1 BE
provided A' represents the wave length of the light within the
plate, A its wave length in the surrounding medium. If now
the angle of refraction be denoted by X, then
BD B'D = d : cos x, BE BB' sin = 2,d tan % sin ;
further, A : A' = n (index of the plate with respect to the sur-
rounding medium). Hence
cos
or, since from the law of refraction sin 9 .
The difference of phase which is thus introduced between the
two rays depends upon the inclination of the plate p l to AB.*
With Jamin's instrument it is not possible to produce a
separation between the two rays of more than 2 cm. A much
larger separation may be obtained if, as in Zehnder's instru-
ment^ four nearly parallel plates be used. According to
Mach t it is advantageous to replace two of these plates by
metal mirrors S x and S r Fig. 55 shows Mach's arrangement.
He also introduced a device for increasing the intensity of the
* For the more rigorous calculation cf. F. Neumann, Vorles. uber theor.
Optik (Leipzig, 1885), p. 286.
f Cf. Zehnder, Ztschr. Instrkd. 1891, p. 275.
\ Mach, Wien. Ber. 101 (II. A.), p. 5. 1892. Ztschr. Instrkd. 1892, p. 89.
INTERFERENCE OF LIGHT
light. In the arrangements shown in Figs. 54 and 55, the rays
coming to the eye at E are of small intensity because they have
undergone one reflection at a glass surface and have thus been
materially weakened. In Fig. 55 the rays from S which
FIG. 55-
FIG. 56.
pass through PP 2 are much more intense than those which are
reflected from PP 2 to E. This difficulty can be diminished
by increasing the reflecting power of the glass surface. This
is done by depositing a thin film of silver or gold upon the sur-
face, the most favorable thickness of such a film being that for
which the intensity of the reflected light is equal to that of the
transmitted. But with the arrangement shown in Fig. 55 it is
not necessary to use two plates P l and P 2 of finite thickness in
order to produce interference; it is sufficient if, instead, the
division of the ray into a reflected ray and a transmitted ray is
accomplished by means of a thin film of metal. This may be
done by pressing together the partially silvered hypothenuse sur-
faces of two right-angled glass prisms. The reflections upon the
mirrors 5 t and S 2 may be replaced by total reflections upon the
unsilvered surfaces of right-angled glass prisms. Finally these
latter prisms may be united with the prisms which divide the
j
i 4 8 THEORY OF OPTICS
light so as to form single pieces of glass. Thus Fig. 56 shows
Mach's construction of the interferometer, in which to the two
equal glass rhombs K^ and K 2 the two prisms K{ and K^ are
cemented with linseed oil, the surfaces of contact P l and P 2
being coated with a thin film of gold. The rays are totally
reflected at the inclined surfaces S l and S 2 . When the two
rhombs K l and K 2 are set up so as to be nearly parallel to each
other, an eye at E sees interference fringes.
8. Interference with Large Difference of Path. If the
Newton ring apparatus be viewed in monochromatic light, such
as is furnished by a sodium flame, the interference rings are
seen to extend over the whole surface of the glass. This is a
proof that light retains its capacity for interference when the
difference of path is as much as several hundred wave lengths.
How far this difference of path can be increased before the
interference disappears is a question of the greatest importance.
This question cannot be answered by simply separating the
two plates of the Newton ring apparatus farther and farther
and focussing the eye or the lens upon the surface O l of one of
the plates, for, in accordance with the note on page 141, the
interference fringes would soon become indistinct on account
of the changing inclination of the coherent pairs of rays which
intersect at a point of the surface O r It is necessary, therefore,
to provide that all coherent pairs of rays which are brought
together in the same point upon the retina of the observer have
the same difference of phase.
This condition is fulfilled when the interference arises from
reflections at two parallel surfaces O l and O 2 , and the eye or
the observing telescope is focussed for parallel rays. All the
interfering coherent pairs of rays which are brought together
at a point of the retina then traverse the interval of thickness
d between the two surfaces at the same inclination to the
common normal N to these two surfaces and hence have the
same difference of phase, provided the distance d is constant.
This difference of phase changes with the angle of inclination
to N, so that the interference figure consists of concentric
INTERFERENCE OF LIGHT 149
circles whose centres lie upon the perpendicular from the eye
to the plates.* The interference rings thus produced are curves
of equal inclination, rather than ciirves of equal thickness, such
as are seen in a thin wedge or the Newton ring apparatus.
Such curves of equal inclination may be observed in mono-
chromatic light in plane parallel plates several millimeters
thick, so that interference takes place when the difference of
path amounts to several thousand wave lengths. In order to
be able to vary continuously the difference in path Michelson
devised the following arrangement: f
The ray QA (Fig. 57) falls at an angle of 45 upon the
half-silvered front face of a plane parallel glass plate, where it
is divided into a transmitted ray,
which passes on to the plane
mirror D, and a reflected ray,
which passes to the mirror C.
These two mirrors return the ray
to the point A, where the first is
reflected, the second transmitted j)\
to E.
A second plane parallel glass
plate B, of the same thickness
as A, makes the difference in the E
paths of the two rays which come FlG - 57-
to interference at E equal to zero, provided the two mirrors D
and C are symmetrically placed with respect to the plate A.
It is evident that, as far as interference is concerned, this
arrangement is equivalent to a film of air between two plane
surfaces O l and O 2 , O l being the mirror C, and O 2 the image
* Lummer uses this phenomenon (cf. Muller-Pouillet, Optik, pp. 916-924) to
test glass plates for parallelism. The curves of equal inclination vary from their
circular form as soon as the distance d between the two reflecting surfaces O l and
O z is not absolutely constant.
f A. A. Michelson, Am. J. Sci. (3) 34, p. 42?> l88 7- Travaux et Mem. du
Bureau International d. Poids et Mes. n, 1895, pp. 1-237. In this second work
Michelson determined the value of the metre in wave lengths of light by the use of
his interferometer.
1 50 THEORY OF OPTICS
of D in the plate A. This image O z must also be parallel to
C if the interference curves of equal inclination are to be seen
clearly when the difference of path is large. In order to vary
the difference of path, one of the mirrors C is made movable
in the direction AB by means of a micrometer-screw. With
this apparatus, using as a source of light the red cadmium line
from a Geissler tube, Michelson was able to obtain interference
when the difference of path in air was 20 cm., a distance equal
to about 300,000 wave lengths. Interference was obtained
with the green mercury radiation when the difference of path
was 540,000 wave lengths.*
These experiments are particularly instructive because
observations upon the change of visibility of the interference
fringes with variations of the difference of path furnish data for
more accurate conclusions as to the homogeneity of a source of
light than can be drawn from spectroscopic experiments.
Fizeau had already observed that a continuous change of
the thickness d of the air film produced a periodic appearance
and disappearance of the fringes produced by sodium light.
The fringes first disappear when the thickness d is o. 1445 mm. ;
when d = 0.289 they are again clear; when d 0.4335 they
reach another minimum of clearness; etc. The conclusion
may be drawn from this that the sodium line consists of two
lines close together. The visibility of the fringes reaches a
minimum when a bright fringe due to one line falls upon a dark
fringe due to the other. Since the mean wave length of sodium
light is 0.000589 mm., the thickness ^ 0.289 mm. corre-
sponds to 491 wave lengths. If the difference between the
wave lengths of the two sodium lines be represented by
A _ A , it follows that
(\ ~~ ^2) "49 l ~ 0-0002 94 mm.,
2
i.e.
A X A 2 = o.ooo 0006 mm.
* A. Perot and Ch. Fabry (see C. R. 128, p. 1221, 1899), using a Geissler
tube fed by a high-voltage battery, obtained interference for a difference of path of
790,000 wave lengths.
INTERFERENCE OF LIGHT 151
Michelson has given a more general solution of the
problem.*
According to equation (n) on page 131 the intensity of
illumination produced by two equally bright coherent rays
whose difference of path is 2/ is
Instead of the wave length A of light in air, its reciprocal
\= m (20)
will be introduced. Then m denotes the number of waves in
unit length.
If now the light is not strictly homogeneous, i.e. if it con-
tains several wave lengths A, or wave numbers m, then if the
wave numbers lie between m and m -f- dm, the factor A 2 in
equation (19) maybe represented by *p(m)*dm. The intensity
J obtained when interference is produced by an air film of
thickness / is
J = 2 / if>(m)[i +cos 4?r lm\dm, . . (21)
(*m*
= 2 / />(
*J m l
in which the limits of integration are those wave numbers
between which $(m) differs appreciably from zero.
Assuming first that the source consists of a single spectral
line of small width, and setting
m = m-\- x, m^ = m a, m 2 = m -\- a, . (22)
(21) becomes
C +a
J= 2 / $(X)\1 +COS 4 7T/(/ +#)]<*;
J - a
* This development is found in Phil. Mag. 5th Sei.,, VoL 31, p. 338, 1891;
Vol. 34, pp. 380 and 407 (Rayleigh), 1892.
152 THEORY OF OPTICS
or setting
= fl, CM*)** = P, )
(22/)
sn
r r
/ $(x) cos (^rtlx)-dx = C, I i
cos -Ssin S. . . . (23)
If the thickness of the air-plate be slightly altered, J varies
because $ does. On the other hand, C and S may be con-
sidered independent of small changes in /, provided the width
of the spectral line, i.e. the quantity a, is small.
Hence, by (23), maxima and minima of the intensity J
occur when
5
tanfl:= -, (24)
the maxima being given by
the minima by
i/min. = P - t'C+S ..... (25')
Hence no interference is visible when C = S = o. But
also when these two expressions are small there will be no
perceptible interference. The visibility of the interference
fringes is conveniently defined by
......
w/max. "I J min.
Hence, from (25) and (25'),
. f 2 + 5 2
*^= -- /5^ ...... (27)
This equation shows how the visibility of the fringes varies
with the difference of path 2/ of the two interfering beams
when / is changed by the micrometer-screw.
INTERFERENCE OF LIGHT 153
If the distribution of brightness of the spectral line is sym-
metrical with respect to the middle, S o and (27) becomes
If it be assumed that ty(x) = constant = c, then
ic sin A.7tla sin
. (28)
Thus the interference fringes vanish when ^la = I, 2, 3,
etc., and the fringes are most distinct (V = i) when / = o.
As / increases, the fringes, even for the most favorable values
of /, become less and less distinct, e.g. for 4/0 = f
V =. 2 : 3?r = 0.212.
Likewise a periodic vanishing and continual diminution in
the distinctness of the maxima occur if, instead of ^(x) = con-
stant,
ty(x\ = COS^TT .
rv ; 2a
The smallest value of / for which the fringes vanish is given
by 4/jtf = -- f- I ; they vanish again when 4/ 2 # = -|- 2,
4/ 3 # = |- 3, etc. Hence from the distances / L , / 2 , / 3 , at
which the visibility curve becomes zero, the width a of the
line, as well as the exponent /, which gives its distribution of
brightness, may be determined.
If * = *
there is a gradual diminution of the visibility without periodic
maxima and minima.
In like manner, when the source consists of several narrow
spectral lines, the visibility curve may be deduced from (21).
Thus, for example, two equally intense lines produce periodic
* This intensity law would follow from Maxwell's law of the distribution of
velocities of the molecules as given in the kinetic theory of gases.
i 5 4 THEORY OF OPTICS
zero values of V. If the two lines are not equally intense, the
visibility does not actually become zero, but passes through
maxima and minima. This is the case of the double sodium
line.
This discussion shows how, from any assumed intensity
law i/>(m), the visibility V of the fringes may be deduced.
The inverse problem of determining $(m) from V is much
more difficult. Apart from the fact that the numerical values
of V can only be obtained from the appearance of the fringes
by a somewhat arbitrary process,* the problem is really not
solvable, since, as follows from (27), only C 2 -j- S 2 can be de-
termined from F, and not C and 5 separately, t Under the
assumption that the distribution of brightness in the several
spectral lines is symmetical with respect to the middle, a solu-
tion may indeed be obtained, since then, for a single line,
5=o, and for several lines similar simplifications may be made.
Michelson actually observed the visibility curves V of numer-
ous spectral lines and found them to differ widely .\ He then
found by trial what intensity law fy(m) best satisfied the ob-
served forms of V. It must be admitted, however, that the
resulting i/>(m) is not necessarily the correct one, even though
the distribution of intensity and the width of the several spectral
lines are obtained from this valuable investigation of Michelson 's
with a greater degree of approximation than is possible with a
spectroscope or a diffraction grating. In any case it is of great
interest to have established the fact that lines exist which are
so homogeneous that interference is possible when the differ-
ence of path is as much as 500,000 wave lengths.
9. Stationary Waves. In the interference phenomena
which have thus far been considered, the two interfering
* F" might be determined rigorously ify m ax. andy m i n . were measured with a
photometer or a bolometer.
f From Fourrier's theorem i/}(m) could be completely determined if C and S
were separately known for all values of /.
\ Ebert has shown in Wied. Ann. 43, p. 790, 1891, that these visibility curves
vary greatly with varying conditions of the source.
INTERFERENCE OF LIGHT 155
beams have had the same direction of propagation. But inter-
ference can also be detected when the two rays travel in
opposite directions. If upon the train of plane waves
s l = A sin 27Z"
// M
(T ~ I''
which is travelling in the positive direction of the .s-axis, there
be superposed the train of plane waves
(f+S-
s z = A sin 27r(
which is travelling in the negative direction of the ^r-axis, there
results
t 2
s = s l -f- s 2 = 2A sin 27t cos 27Tj. . . (29)
This equation represents a light vibration whose amplitude
2 A cos 27iz/^ is a periodic function of z. For ~- = J, |, J, etc.,
the amplitude is zero, and the corresponding points are called
nodes. For ^- = o, J, f, etc., the amplitude is a maximum,
and the corresponding points are called loops. The distance
between successive nodes or successive loops is therefore %h.
This kind of interference gives rise to waves called stationary,
because the nodes and loops have fixed positions in space.
Wiener * proved the existence of such stationary waves by
letting light fall perpendicularly upon a metallic mirror of high
reflecting power. In this way stationary waves are produced
by the interference of the reflected with the incident light.
In order to be able to prove the existence of the nodes and
loops Wiener coated a plate of glass with an extremely thin
film of sensitized collodion, whose thickness was only -fa of a
light- wave = 20 millionths of a mm., and placed it nearly
parallel to the front of the mirror upon which a beam of light
from an electric arc was allowed to fall. The sensitized film
* O. Wiener, Wied. Ann. 40, p. 203, 1890.
156 THEORY OF OPTICS
then intersects the planes of the nodes and loops in a system
of equidistant straight lines, whose distance apart is greater
the smaller the angle between the mirror and the collodion
film. Photographic development of the film actually shows
this system of straight lines. This proves not only that photo-
graphic action maybe obtained upon such a thin film, but also
that such action is different at the nodes and the loops. These
interesting interference phenomena may also be conveniently
demonstrated by means of the fluorescent effects which take
place in thin gelatine films containing fluorescin.* Such a film
shows a system of equidistant green bands. It is a fact of
great theoretical importance, as will be seen later, that the
mirror itself lies at a node.
10. Photography in Natural Colors. Lippmann has made
use of these stationary light-waves in obtaining photographs in
color. As a sensitive film he chose a transparent uniform
layer of a mixture of collodion and albumen containing iodide
and bromide of silver. This he laid upon mercury, which
served as the mirror. When this plate has been exposed to
the spectrum, developed, and fixed, it reproduces approxi-
mately the spectrum colors. The simplest explanation is that
in that part of the film which was exposed to light whose
wave length within the film was A, thin layers of silver have
been deposited at a distance apart of tjrA. If now these parts
of the film be observed in reflected white light, the light- waves
are reflected from each layer of silver with a given intensity.
But these reflected rays agree in phase, and hence give maxi-
mum intensity only for those waves whose wave lengths are
equal to either A, or -JA, or ^A, etc. Hence a spot which was
exposed to green light, for instance, appears in white light
essentially green, for the wave length JA lies outside the visible
spectrum. But under some circumstances a part of the plate
exposed to deep red appears violet, because in this case the
wave length -JA falls within the visible spectrum.
If such a photograph be breathed upon, the colors are dis-
* Drude and Nernst, Wied. Ann. 45, p. 460, 1892,
INTERFERENCE OF LIGHT 157
placed toward the red end of the spectrum, because the
moisture thickens the collodion film, and the reflecting layers
are a greater distance apart. If the plate be observed with
light of more oblique incidence, the colors are displaced toward
the violet end of the spectrum, for the same reason that the
Newton's rings shift toward the lower orders as the incidence
is more oblique. For, as is evident from (14) on page 138,
the difference of phase A between two rays reflected from two
surfaces a distance d apart is proportional to cos x, in which x
is the angle of inclination of the rays between the two surfaces
to the normal to the surfaces. When the angle of incidence
increases A decreases; but in Newton's rings this effect is
much more marked than in Lippmann's photographs, since, in
the former, within the film of air which gives rise to the inter-
ference, x varies much more rapidly with the incidence than it
does in the collodion film, whose index is at least as much as
1.5.
Although the facts presented prove beyond a doubt that
the colors are due to interference, yet the explanation of these
colors by periodically arranged layers of silver is found, upon
closer investigation, to be probably untenable. For Schutt*
has made microscopic measurements upon the size of the par-
ticles of silver deposited in such photographic films, and found
them to have a diameter of from 0.0007 to 0.0009 mm., which
is much larger than a half wave length. According to Schiitt,
the stationary waves and the fixing of the sensitive film pro-
duce layers of periodically varying index of refraction, due to
a periodic change in the arrangement of the silver molecules.
This theory does not alter the principle underlying the expla-
nation of the colors, for it also ascribes to the collodion film a
variable reflecting power whose period is ^A.
This theory makes it possible to calculate the intensity of
any color after reflection. The complete discussion will be
omitted, especially as the calculation is complicated by the
fact that it is not permissible to assume the number of periods
* F. Schiitt, Wied. Ann. 57, 533, 1896.
158 THEORY OF OPTICS
in the photographic film as large.* The best color photo-
graphs are obtained when the thickness of the photographic
film does not exceed o.ooi mm. This thickness corresponds to
3-5 half wave lengths. But without calculation it may be seen
at once that the reflected colors are a mixture and not pure
spectral colors, a fact which can be verified by an analysis of
the reflected light by the spectroscope. t For even if that
color whose wave length is the same as that of the light
to which the plate was exposed must predominate in the
reflected light, yet the neighboring colors, and, for that matter,
all the colors, must be present in greater or less intensity.
According to an experiment of Neuhauss, J the gradual
reduction of the thickness of the film by friction causes the
reflected colors to undergo certain periodic changes. This
effect follows from theory if the small number of periods in the
photographic film be taken into consideration.
A further peculiarity of these photographs is that, in
reflected light, they do not show the same color when viewed
from the front as from the back. Apart from the fact that
the glass back gives rise to certain differences between the two
sides, it is probable that the periodic variations in the optical
character of the film are greater in amplitude on the side of
the film which lay next to the metal mirror. On account of a
slight absorption of the light, the stationary waves which, in
the exposure of the plate, lie nearest the metal mirror are most
sharply formed.
If this assumption be introduced into the theory, both the
result of Neuhauss and the difference in the colors shown by
the opposite sides of the plate are accounted for.
* The only calculations thus far made, namely those published by Meslin
(Ann. de chim. et de phys. (6) 27, p. 369, 1892) and Lippmann (Jour, de phys.
(3) 3' P- 97> x ^94) n t on ly ma ke this untenable assumption, but they also lead to
the impossible conclusion that under certain circumstances the reflected intensity
can be greater than the incident.
f Cl, for instance, the above-mentioned article by Schutt.
\ R. Neuhauss, Photogr. Rundsch. 8, p. 301, 1894. Cf. also the article by
Schutt.
Cf. Wiener, Wied. Ann. 69, p. 488, 1899.
CHAPTER III
HUYGENS' PRINCIPLE
i. Huygens' Principle as first Conceived. The fact has
already been mentioned on page 127 that the explanation of
the rectilinear propagation of light from the standpoint of the
wave theory presents difficulties. To overcome these difficulties
Huygens made the supposition that every point P which is
reached by a light-wave may be conceived as the source of
elementary light-waves, but that these elementary waves
produce an appreciable effect only upon the surface of their
envelope. If the spreading of the rays from a point source Q
is hindered by a screen S X S 2 containing an opening A } A.,,
then the wave surface at which the disturbance has arrived
after the lapse of the time t may be constructed in the follow-
ing way:
Consider all the points A. 6 in the plane o r the opening A^A^
as new centres of disturbance which send out their elementary
waves into the space on both sides of the screen. These
elementary wave surfaces are spheres described* about the
points A. These spheres have radii of different lengths, if they
are drawn so as to touch the points at which the light from Q
has arrived in the time t. Since, for instance, the disturbance
from Q has reached A 3 sooner than A lt the elementary wave
about A must be drawn larger than that about A l in proportion
to the difference between these two times. It is evident that
the radii of all the elementary waves, plus the distance from Q
to their respective centres, have the same value. But in this
way there is obtained, as the enveloping surface of these ele-
159
160 THEORY OF OPTICS
mentary waves, a spherical surface (drawn heavier in Fig. 58)
whose centre is at Q, and which is limited by the points B l ,
B 2 , i.e. which lies altogether within the cone drawn from Q
to the edge of the aperture S { S 2 . Inside this cone the light
from Q is propagated as though the screen were not present,
but outside of the cone no light disturbance exists.
Though the rectilinear propagation of light is thus actually
obtained from this principle, yet its application in this form is
subject to serious objection. First, it is evident from Fig. 58
FIG. 58.
that the elementary waves from the points A have also an
envelope C^ 2 in the space between the screen and the source.
Hence some light must also travel backward; but, as a matter
of fact, in a perfectly homogeneous space, no such reflection
takes place. Furthermore, the construction here given for the
rectilinear propagation of light ought always to hold how-
ever small be the opening A^A Z in the screen. But it was
shown on page I that, with very small apertures, light no
longer travels in straight lines, but suffers so-called diffraction.
Again, why do not these considerations hold also for sound,
which is always diffracted, or, at least, never produces sharp
shadows ?
HUYGENS' PRINCIPLE 161
Before considering Fresnel's improvements upon Huygens'
work, the latter 's explanation of reflection and refraction will
be presented. Let A^A 2 be the bounding surface between two
media I and II in which the velocities of light are respectively
V l and V 2 y and let a wave whose wave front at any time to
JL
FIG. 59.
occupies the position A^B fall obliquely upon the surface A^A^
What then is the position of the wave surface in medium II at
the time / -f- / ? Conceive the points A of the bounding sur-
face as centres of elementary waves which, as above, have
different radii, since the points A are reached at different times
by the wave front AB. Since the disturbance at A l begins at
the time t , the elementary wave about A l must have a radius
represented by the line A^C = V<. Let the position of the
point A 2 b& so chosen that the disturbance reaches it at the
time / -(- /. This will be the case if the perpendicular dropped
from A 2 upon the wave front has the length Vj> since, accord-
ing to Huygens' construction, in a homogeneous medium such
as I any element of a plane wave is propagated in a straight
line in the direction of the wave normal. The elementary
wave about A 2 has then the radius zero. For any point A
between A l and A 2 the elementary wave has a radius which
diminishes from V 2 t to zero proportionally to the distance
A^A. The envelope of the elementary waves in medium II
is, therefore, the plane through A 2 tangent to the sphere
162
THEORY OF OPTICS
about A r The angle A 2 CA 1 is then a right angle. Since
now sin =
sn
CA l :
: A 1 A 2J it follows that
sin V,
= ~ = const.
sin x ~ V*.
But since and x ar e the angles of incidence and refraction
respectively, this is the well-known law of refraction. Hence,
as was remarked though not deduced on page 129, the
index of refraction n is equal to the ratio of the velocities of
propagation of light in the two media.
By constructing in the same way the elementary waves
reflected back into medium I the law of reflection is at once
obtained.
2. FresnePs Improvement of Huygens' Principle. Fres-
nel replaced Huygens' arbitrary assumption that only the
envelope of the elementary waves
produces appreciable light effects
by the principle that the elementary
waves in their criss-crossing influ-
ence one another in accordance with
the principle of interference. Light
ought then to appear not only upon
the enveloping surface, but every-
where where the elementary waves
reinforce one another ; on the other
hand, there should be darkness
wherever they destroy one another.
Now as a matter of fact it is possi-
ble to deduce from this Fresnel-
Huygens principle not only the
laws of diffraction, but also those of straight-line propagation,
reflection, and refraction.
Consider the disturbance at a point P caused by light from
a source <2, and at first assume that no screen is interposed
between P and Q. A sphere of radius a described about Q
HUYGENS' PRINCIPLE 163
(Fig. 60) may be considered as the wave surface, and the dis-
turbance which exists in the elements of this sphere may be
expressed by (cf. page 127)
A t t a
(I)
in which A represents the amplitude of the light at a distance
a = I from the source Q. Fresnel now conceives the spherical
surface to be divided in the following way into circular zones
whose centres lie upon the straight line QP: The central zone
reaches to the point M l , at which the distance M^P =. r l is
|A greater than the distance MJP. Calling the latter b y
M^P = r l = b + A. The second zone reaches from M l to
M 2 , where Mf = r 2 = r l + 4^ The third zone reaches from
M 2 to M 3 , where MJP = r B = r 2 -\- -JA, etc. Consider now in
any zone, say the third, an elementary ring which lies
between the points M and M ' . Let the distances MP r,
M'P = r + dr, and MQP = u, ^M'QP = u+ du. The
area of this elementary zone is
do = 27fa 2 sin udu ...... (2)
Also, since
r 2 = c? + (a + Vf 2a(a + b) cos u,
it follows by differentiation that
2r dr = 2a(a -f- <) sin u du,
so that equation (2) may be written
do = 27t - r dr. ... (3)
a b VJ/
The disturbance ds' which is produced at P by this ele-
mentary zone must be proportional directly to do and inversely
to r, since (cf. page 126) the amplitude of the disturbance due
to an infinitely small source varies inversely as the distance
from it. Hence, from (i),
kA It
(4)
i6 4 THEORY OF OPTICS
or, in consideration of (3),
In this equation k is a factor of proportionality which can
depend only upon the inclination between the element do and
the direction of r. Fresnel assumes that this factor k is smaller
the greater the inclination between do and r. If this inclination
be assumed to be constant over an entire Fresnel zone, i.e.
between M M _ l and M ni an assumption which is allowable if a
and b are large in comparison with the wave length A, it follows
from (4') that the effect of this nth zone is (k n denoting the
constant k under these circumstances)
or
,_ k n \A
But since
it follows that
2k M \A ft a 4-
= (-')"' jipj sin 2*( r - --- - (6)
From this it is evident that the successive zones give alter-
nately positive and negative values for s' . If the absolute
value of s n ' be represented by s n , then by the principle of in-
terference the whole effect s' at P due to the first n zones is
given by the series
s' = s l -s 2 + s 3 -s,+ ... + (~- !)" + *. . (7)
\ik n were assumed equal for all zones, s lt s 2 , s s , etc., would
all be equal, and the value of the series (7) would vary with the
number of terms n. But k n and hence s n diminish continuously
HUYGENS' PRINCIPLE 165
as n increases, since the greater the value of n the greater the
inclination between r and do. In this case the value of the
series may be obtained in the following way : * If n is odd, the
series may be written in the form :
Y ..... (8)
or in the form :
** -
H-i- + ... (9)
If now every s p is greater than the arithmetical mean of the
two adjacent quantities s^ and s p+I , the conclusion may be
drawn from (8) that
while it follows from (9) that
y > ,, _ + ,
These two limits between which s' is in this way contained
are, however, equal to one another when, as is here the case,
every s p differs by an infinitely small amount both from s p _ l
and s p+l . Hence
A similar conclusion may be drawn when each s p is smaller
than the arithmetical mean between the two adjacent quantities
s p _ t and s f + l . In this latter case if at equal distances along an
axis of abscissae the s^s be erected as successive ordinates,
*A. Schuster, Phil. Mag. (5), 31, p. 85, 1891.
X 66 THEORY OF OPTICS
the line connecting the ends of these ordinates is a curve which
is convex toward the axis of abscissae. In the former case
this curve is concave toward this axis. These same conclu-
sions may be drawn, i.e. equation (10) obtained, if the s p curve
consists of a finite number of concave and convex elements.
Only when this number becomes infinitely large does equation
(10) cease to hold. On account of the presence of the factor
k n this case can never occur.
If n is even, a similar argument, with a somewhat different
arrangement of the terms of series (7), gives
According to Fresnel these zones are to be drawn until the
radius vector r from P becomes tangent to the wave surface
about Q. For the last zone r is perpendicular to QM and
both k n and S H become zero. Hence the values of (10) and
(10') are identical and the light disturbance at P is
s. k^A It a + b
sin
It a + b \
U=- \ 1. . . (u)
\T A /
2 a -f- b
Thus it may be looked upon as due solely to the effect of
the elementary waves of half the central zone.
The effect at P of introducing any sort of a screen will
depend upon whether the central zone and those immediately
adjacent to it are covered or not. It might be expected that
the effect at P would be completely cut off by a circular screen
whose centre lies at J/ and which covers half of the central
zone. But this is not the case. For when a circular screen
is introduced perpendicular to PQ with its centre at M , the
construction of the Fresnel zones may begin at the edge of
this screen. Then half of this first zone is still effective at P,
i.e. equation (11) still holds, but b now represents the distance
between P and the edge of the screen, and k l refers to the first
zone about the edge of the screen. Hence there can be dark-
ness at no point along the central line M Q P. This surprising
conclusion is actually verified by experiment. However, for
HUYGENS' PRINCIPLE 167
screens which are large in comparison with the wave length
as well as in comparison with the distance b, the effect at P
is small, because the factor k n in equation (5) is then small.
Likewise the effect at P is small if the screen 5 is not exactly
circular. For, consider that the screen 5 is bounded by
infinitely small circular arcs of varying radii drawn about M Q
as a centre. Let the angle subtended at the centre by the
first arc be d^ , the distance of this arc from the point P be & lt
and from (7, a v Then, by (n) and the above considerations,
the effect of the entire opening which lies between the two
radii vectores drawn from M Q through the ends of this first arc is
t * \
ds' =
b l
sm 2
kj^A which is not covered by the screen is
It
sin 27
2?t
etc. All these effects must be summed in order to obtain the
value of s f at P after the introduction of the irregular screen
at M . If the screen is not too large, it is possible to set
k v = k 2 = / 3 , etc. Likewise the differences between the various
rt's and //s in the denominator may be neglected so that
2 sn 2^
In the argument of the sin it is not permissible to set
a^ + b l a 2 + b z , etc., since these quantities are divided by
the small quantity A. For if the screen S is many wave
lengths in diameter (it need be but a few mm.), the differences
between the quantities a -\- b amount to many wave lengths.
Hence with an irregular screen the different terms of equation
(n) are irregularly positive and negative so that in general
the whole sum is small. Only when the screen has a regular
168 THEORY OF OPTICS
form, for instance when all the a's and ^'s are exactly equal,
is the sum s' finite. Hence it is possible to speak of rectilinear
propagation of light, since the result of interposing a screen of
sufficient size and irregular form upon the line QP is darkness
at P.
If between Q and P a screen with a circular opening whose
centre is at M be introduced, then the effect at P varies greatly
with the size of this opening. If the opening has the same size
as half of the central zone, the effect at P is the same as though
no screen were present, i.e. the light at P has the natural
brightness. If the opening corresponds to the whole central
zone, s' at P is twice as great as before, i.e. the intensity at
P is four times the natural brightness. If the size of the open-
ing be doubled, so that the first two central zones are free,
then, according to (7), s' ^ s 2 , an expression whose
value is nearly zero; etc. This conclusion also has been veri-
fied by experiment. Instead of using screens or apertures of
various sizes, it is only necessary to move the point of observa-
tion along the line QM .
Although Fresnel's modification of Huygens' principle not
only accounts for the straight-line propagation of light, show-
ing this law to be but a limiting case,* but also explains the
departures from this law shown in diffraction phenomena in a
way which is in agreement with experiment, nevertheless his
considerations are deficient in two respects. For, in the first
place, according to his theory, light ought to spread out from
any wave surface not only forward, but backward toward the
source. This difficulty was contained in the original concep-
tion of the Huygens' principle (cf. page 161). In the second
place, Fresnel's calculation gives the wrong phase to the light
disturbance s' at P. For, according to equation (i) on page
163, in the case of direct propagation s' ought to be
A It a-\-b
s = FT cos 27rU=r-. y
a -j- o \2 A
* That this is not true for sound is due to the fact that the sound-waves are so
long that the obstacles interposed are not large in comparison.
HUYGENS' PRINCIPLE 169
while by (ii) on page 166, s f , as determined by the considera-
tion of the elementary waves upon a wave surface, is
k.\A it a + b
s = ; , sin 2n\-7F> --- 1
a -\- b \T A
In order to obtain agreement between the amplitudes in
the two expressions for s', k l may be assumed equal to -^ , but
the phases in the two expressions cannot be made to agree.
These difficulties disappear as soon as Huygens' principle is
placed upon a more rigorous analytical basis. This was first
done by Kirchhoff.* The simpler deduction which follows is
due to Voigt.t
3. The Differential Equation of the Light Disturbance.
It would have been possible to find the analytical expression for
the light disturbance s at any point Pin space if all waves were
either spherical or plane. But when light strikes an obstacle
the wave surfaces often assume complicated forms. In order
to obtain the analytical expression for s in such cases, it is
necessary to base the argument upon more general considera-
tions, i.e. to start with the differential equation which s
satisfies.
Every theory of light, and, for that matter, every theory of
the propagation of wave-like disturbances, leads to the differ-
ential equation
in which / represents the time, x, y, z the coordinates of a
rectangular system, and V the velocity of propagation of the
waves. This result of theory may for the present be assumed ;
a deduction of the equation from the standpoint of the electro-
magnetic theory will be given later (Section II, Chapter I).
* G. Kirchhoff, Ges. Abh. or Vorles. tiber math Optik.
| W. Voiet, Kompendium d. theor. Physik, II, p. 776. Leipzig, 1896.
1 70 THEORY OF OPTICS
It will first be shown how the analytical forms of s given
above for plane and spherical waves are obtained from (12).
For plane waves let the ;r-axis be taken in the direction of
the normal to the wave front, i.e. in the direction of propaga-
tion; then s can depend only upon x and /, since in every
plane x = const, which is a wave-front, the condition of vibra-
tion for a given value of / is everywhere the same. Equation
(12) then reduces to
&s T72 3 2 s
3? = *"a? ....... 03)
The general integral of this equation is
in which /j is any function whatever of the argument / -- - ,
x
and f 2 any function of the argument / -j~ T7- For if the first
derivatives of the functions /j and f 2 with respect to their argu-
ments be denoted by /j' and / 2 ', the second derivatives by
fi'ifz"' respectively, then
. .f> + f . +./ 4. - f
'dx ~ V l V 2 ' 3^r 2 ~ V* l ~ YV*
i.e. equation (13) is satisfied. If now the variation of s with the
time is of the simple harmonic form, i.e. if it is proportional to
cos 2?r , as is the case for homogeneous light, then, by (14),
(15)
in which A lt A 2 , d lt d t are constants. This corresponds to
our former equation for a plane wave of wave length A VT.
A l is the amplitude of the waves propagated in the positive
HUYGENS' PRINCIPLE 171
direction of the ;t>axis, A 2 the amplitude of those propagated
in the negative direction of the ^-axis.
For spherical waves whose centre is at the origin, s can
depend only upon / and the distance r from the origin. Hence
d s __ d* dr_ _ 9f
dy dr dy dr r j
dz dr ds dr r
For since r* = x* -f- y 2 -f- z*, partial differentiation gives
r-dr = x-dx, i.e. = - cos (rx),
and similarly
dy~~r' dz~~ r'
Also,
r* dr* dr \r
and similarly
I
dv*~~ ~r*' dr** dr r ' r 3 ''
Equation (12) becomes, therefore, for this case
which may also be written in the form
0..2 ~ V 2 ^l2~ ( T 7)
172 THEORY OF OPTICS
This equation has the same form as (13) save that rs
replaces s, and r replaces x. The integral of (17) is therefore,
by (14).
If, again, homogeneous light of period T be used, it follows that
s = cos 27T L-. _ +d+ 4> cos 2
This is our former equation for spherical waves. One train of
waves moves from the origin, the other moves toward it. The
amplitudes, for example , are inversely proportional to r.
This result, which was used above on page 126 in defining the
measure of intensity, follows from equation (12).
Before deducing Huygens' principle from equation (12) the
following principle must be presented.
4. A Mathematical Theorem. Let dr be an element of
volume and F a function which is everywhere finite, continuous,
and single-valued within a closed surface 5. Consider the
following integral, which is to be taken over the entire volume
contained within 5:
C-dF
f
J
First perform a partial integration with respect to x, i.e. make
dF
a summation of all the elements -^~dr which lie upon any
straight line @ parallel to the axis of x. The result is
dy d *J^ dx = dy dz( ~ F * + F * ~ F + F * etc -)-
in which F IJ F 2 , etc. , represent the values of the function F
at those points upon the surface 5 where the straight line
intersects it. For the sake of generality it will be assumed
that this line intersects the surface several times; since, how-
HUYGENS' PRINCIPLE 173
ever, 5 is a closed surface, the number of such intersections
will always be even. In moving along the line ($ in the direc-
tion of increasing x, F IJ F B , etc., which have odd indices,
represent the values of F at the points of entrance into the
space enclosed by S\ while F 2 , F, etc., which have even
indices, represent the values of F at the points of exit. Con-
struct now upon the rectangular base dy dz a column whose
axis is parallel to the ;r-axis. This column will then cut from
the surface S, at the points of entrance and exit, the elements
dS^ , dS z , etc., whose area is given by
dy dz dS-cos(nx),
in which (nx) represents the angle between the ^r-axis and the
normal to the surface 5 at each particular point of intersection.
The sign must be taken so that the right-hand side is positive,
since the elements of surface dS are necessarily positive, n
will be taken positive toward the interior of the space enclosed
by S. Then, at the points of entrance,
dy dz = -f-^Sj- cos (n^x) = -\-dS 3 -cos ( 3 ^), etc.,
and at the points of exit
dy dz = d
Hence
f
I
J
dy dz I dr = F 1 cos (# r r) dS l F 2 cos (n 2 x) dS 2 etc.
J x
If now the integration be performed with respect to y and
z in order to obtain the total space integral, i.e. if the summa-
tion of the products F cos (nx)dS over the whole surface be
made, there results
/ -dr I F cos (nx)-dS, .
(20)
in which on the right-hand side F represents the value of the
function at the surface element dS.
Thus by means of this theorem the original integral, which
i 7 4 THEORY OF OPTICS
was to be extended over the whole volume, is transformed into
one which is taken over the surface which encloses the volume.
From the method of proof it is evident that F must be finite,
continuous, and single-valued within the space considered,
since otherwise in the partial integration not only would there
appear values F lt F 2 , etc., of F corresponding to points on
the surface, but also values for points inside.
5. Two General Equations. Let U be a function which
contains explicitly x, y, z, and r. Let r represent the dis-
tance from the origin, i.e. r* = x* -\- y* + z*. Let - - repre-
QX
sent a differentiation with respect to the variable x as it
explicitly appears, so that jr, #, and r are in this differentiation
considered constants. On the other hand let -= represent
the differential coefficient of U, which arises from a motion dx
along the jr-axis ; in which it is to be remembered that in this
case r varies with x. Then
-r- = -^ h ^-- = ^- 4- ^ cos (rx}. (2i\
fL3C c)3C i\f ^ y 7^3? (\y *
9?" x
But (cf. page 171) = - = cos (rx). Hence
d ii dU\ 9 /i dU\ 9 /i ?>U
9 /i dU\
= ^\r ^>
or, since in the differentiation - - the radius r is constant.
'
d
dx\r
d
C S
, (22)
HUYGENS' PRINCIPLE 175
Now let r- represent the ratio of the total change in U to a
change in r, which arises from a motion dr along the fixed
direction r. This change in U is a combination of several
partial changes : First, U varies with r as it explicitly occurs,
the amount of this variation being. Second, it varies
because x, y, z, which occur explicitly in U, are functions
of r. Further a simple geometrical consideration shows that
dx dr cos (rx\ dy dr cos (ry), dz dr cos (rz\ hence
If in this equation /be replaced by , the result is
d foU\
-r- ^r- =
r '
or> _. (24)
dr \ dr 2 } ' ^ J '
Addition of the three equations (22) gives, in consideration of
(23) and (24),
,, r ox , + -t.L <^. / + ^lr
>*7
r*\dr~~ dr
But
If equation (25) be multiplied by the volume element dr =
dxdydz and integrated over a space within which - , - ,
are finite, continuous, and single-valued, and if theorem
r oz
i 7 6 THEORY OF OPTICS
(20) on page 173 be applied three times,* there results, in
consideration of (26),
- / i ] - cos (nx) + -^ cos (ny) + -^- cos (nz) \ dS
J r ( 9;tr 3y 3^ ' )
The space over which the integration is extended evidently
cannot contain the origin, since there ; becomes infinite.
Now two cases are to be distinguished: I. The space over
which the integration is extended is bounded by a surface S
which does not include the origin ; II. The outer surface 5 of
that space does include the origin.
CASE II. In this case, which will be first considered, con-
ceive the origin to be excluded from the space over which the
integration is extended by means of a sphere A" of small radius
p about the origin as a centre. The region of integration has
then two boundaries, the outer one the surface 5, the inner
one the surface K of the sphere. The surface integral of
equation (27) is therefore to be extended over both these sur-
faces. The value of the integral over the surface K is, how-
ever, not finite when p is infinitely small, since this surface is
an infinitesimal of the second order with respect to p, and r
appears in the denominator of the left-hand side of (27) in the
first power only. Further,
_ C os (nx) + ^ cos (ny) + cos (m) =-- , (28)
in which 9 7: dn is the differential coefficient which arises from
a motion *dn in the positive direction along the normal n to 5
* The symbol ~ which appears in equation (20) has the same meaning as
here. That equation is also to be applied in this case when the differentiation is
taken with respect toy and z.
HUYGENS' PRINCIPLE 177
when r is treated as a constant. Hence the left-hand side of
equation (27) becomes
and this integral is to be taken over the outer surface only,
not over the small spherical surface K.
The last term on the right-hand side of (27) will now be
transformed by writing
dr = r*d= +dS-cos (nr),
while at the point of exit
r*d = dS-cos (nr).
HUYGENS' PRINCIPLE 179
Hence the volume integral (30') may be written as the surface
integral
Hence for this case (27) becomes
C ( : 9/ t ^ ^ I U \} ^
I \ - ^ -- cos (nr) I dS =
J \ r dn ^ ' dr\ r I }
- ' ' (34)
6. Rigorous Formulation of Huygens' Principle. The
following application will be made of (34) and (34') : Let s be
the light disturbance at any point, S Q the value of s at the
origin, s satisfies the differential equation (12) on page 169.
U will now be understood to be that function which is obtained
by replacing in s the argument t (time) by / r /v. This
will be expressed by
U=s(f-'lJ).
It is then evident that /" = ^ , since at the origin r = o.
Furthermore, from (12),
but since U is a function of t r / v , (cf. equations (17) and
(18), page 171) the following relation also holds:
Hence, from the last two equations,
Hence (34) gives, for the case in which the origin lies within
the surface 5,
tfV - 'M
= Tr r _,__, i-Mf-'/r)
* u.v* V ( jr / ftt .\
cos (^) - - -^ - - J 5. (35)
i8o THEORY OF OPTICS
This equation may be interpreted in the following way:
The light disturbance s at any point P (which has been taken
as origin) may be looked upon as the superposition of disturb-
ances which are propagated with a velocity V toward P from
the surface elements dS of any closed surface which includes the
point P Q . For, since the elements of the surface integral (35)
are functions of the argument / r I v, any given phase of the
elementary disturbance will exist at P Qt r /V seconds after it
has existed at dS.
In this interpretation of (35) it is easy to recognize the
foundation of the original Huygens' principle, but the condition
of vibration of the separate sources dS is much more compli-
cated than was required by the earlier conceptions, according
to which the elements of the integration were simply propor-
tional to s(t r '/V) (cf. (4) on page 163).
Further, it is possible to calculate from equation (35) the
disturbance S Q at the point P if the disturbances s and are
known over any closed surface 5. In certain cases these are
known, as, for instance, when the source is a point and the
spreading of the light is not disturbed by screens or changes
in the homogeneity of the space. In this case, to be sure, s
can be determined directly; nevertheless, for the sake of what
follows, it will be useful to calculate it from (35).
Let the source <2 ne outside of the closed surface S. Let
the disturbance at any point P which lies upon S and is
distant r l from the source <2 be represented by
-. ..... (36)
Then
*ds Vs
= x cos (nr.),
or
2nA
(37)
HUYGENS' PRINCIPLE 181
Now r^ must be large in comparison with A, hence the first
term is negligible in comparison with the second, so that
t r
Further, from (36),
*- A
COS
If this expression be differentiated with respect to r, a term
may again be neglected as in (37), since r also is large in
comparison with A; hence
sin 27r^~ ^ LJ. . . (39)
Substitution of the values (38) and (39) in (35) gives
A Ci It r+r\
S Q = Y / - - sin 27r I r i 1 cos (nr) cos (nr, )]dS. (40)
2 A, / T'/' \ _/ A ' / y
/
This equation contains the principle of Fresnel stated above
on page 163, but with the following improvements:
i. Fresnel' s factor k is here determined directly from the
differential equation for s, which constitutes the basis of the
theory. Consider, for example, an element dS which lies at
the point M^ (Fig. 61) along the line QP Q ; then for this ele-
FIG. 61.
ment cos (nr) = cos (nr^j since the positive directions of r
and r l are opposite. Hence Fresnel's radiation factor k is
cos (nr]
i8 2 THEORY OF OPTICS
If dS is perpendicular to QP Q , then cos (nr) = I, and,
save for the sign, the factor k l (cf. page 169) of the central zone
has been deduced in an indirect way.
2. For an element dS, which lies at J/ ' (Fig. 61), the
positive directions of r and r^ are the same, i.e.
cos (nr) cos (nr^) = o.
Hence the influence of this element upon the value of S Q dis-
appears, i.e. the elementary waves are not propagated back-
ward as they should be according to Huygens' and Fresnel's
conceptions of the principle. It is at once evident that this
disappearance of the waves which travel backward is a conse-
quence of the fact that in (35) every elementary effect appears
as the difference of two quantities.
3. The phase at P is determined correctly, being the same
as that due to the direct propagation from Q to P Q . For
surface elements dS which lie at M perpendicular to QP Q are
multiplied in (40) by the factor
~ sm 2n ~
and hence the effect is the same as though these surface ele^
ments vibrated in a phase which is - ahead * of that of the di-
rect wave from Q to dS, which, in accordance with (36), would
/ / r -4- r \
lead to the expression cos 2n\ - . L U. When the inte-
gration is performed over the surface 5 there is again obtained
for the point P Q : + cos 27r(~ _ a "|" j, not, as in Fresnel's
* If the light disturbance be assumed to exist not as a convex, but as a con-
cave, spherical wave, which travels toward a point Q outside of S, the considera-
tions are somewhat modified, as may be seen from (35). (In Mascart, Traited'Op-
tique, I, p. 260, Pans, 1889, this case is worked out.) Under some circumstan-
ces this case is of great importance for interference phenomena. Cf. Gouy, C. R.
no, p. 1251; in, p. 33, 1890. AlsoWied. Beibl. 14, p. 969.
HUYGENS' PRINCIPLE 183
calculation, sin 27t f j~"j ( cf - P a g e l6 9)- Thus this
contradiction in Fresnel's theory is also removed.
Now if any screen be introduced, the problem of rigorously
determining s is extremely complicated, since, on account of
the presence of the screen, the light disturbance s at a given
point P is -different from the disturbance ~s which would be
produced by the sources alone if the screen were absent. In
order to obtain an approximate solution of the problem, the
assumption may be made that, if the screen is perfectly opaque
and does not reflect light, both s and vanish at points which
lie close to that side of the screen which is turned away from
the source; while, for points which are not protected from the
sources by the screen, the disturbance s has the value s~ which
it would have in free space.
In fact this was the method of procedure in the above
presentation of Fresnel's theory. Then, starting from equa-
tion (40), by constructing the surface S so that as much as
possible lies on the side of the screen remote from the source,
a very approximate calculation of the disturbance ^ at any
point P Q may be made. Only the unprotected elements
appear in (40). It is immaterial what particular form be given
to this unprotected surface, provided only that it be bounded
by the openings in the screen. This result can be deduced
from equation (34') on page 179, which shows that the right-
hand side of (40) becomes zero for this case, if the closed
surface 5 excludes the point P (and also the source 0, for
which s is to be calculated. Hence if the integral s of equa-
tion (40) be taken over an unclosed surface 5 which is bounded
by a curve C, and if another surface S' be constructed which
is likewise bounded by C, then 5+5' may be looked upon as
one single closed surface which does not include the origin P .
(34') shows that the sum s + s ' of the two integrals extended
over 5 and S' vanishes. But in this n is always drawn toward
the interior of the closed surface formed by 5 and S', so that,
1 84 THEORY OF OPTICS
if the positive direction of the normal to S points toward the
side upon which P Q lies, then the positive direction of the
normal to S' points away from this side. If then the positive
direction of the normal to S' be taken toward the side upon
which P Q lies, the sign of the integral S Q ' becomes reversed.
Hence it follows that S Q s f = o, or s = s ', or, expressed
in words: The integral s , defined by equation (^o), has the
same value for all unclosed surfaces S of any form which are
bounded by a curve C, provided tJie normal be always reckoned
positive in the same direction {from the side upon which the
source lies to that upon which P Q lies) , and provided these
different surfaces S do not enclose either the source <2 or the
point P Q for which S Q is to be calculated.
How, now, from equation (40) the rectilinear propagation
of light, and certain departures from the same, may be
deduced has already been shown in 2 with the aid of Fres-
nel's zones. In the following chapter these departures from
the law of rectilinear propagation, the so-called diffraction
phenomena, will be more completely treated.
CHAPTER IV
DIFFRACTION OF LIGHT
As is evident from the discussion in 2 of the preceding
chapter, diffraction phenomena always appear when the screens
or the apertures are not too large in comparison with the
wave length. But, as will be seen later, diffraction phe-
nomena may appear under certain circumstances even if the
screen is large, for example at the edge of the geometrical
shadow cast by a large object. If now, starting with equation
(40), the diffraction phenomena be calculated in accordance
with the considerations on page 182, it must not be forgotten
that the theoretical results thus obtained are only approximate;
since, on the one hand, when screens are present, the value of
s is not exactly the same at unprotected points as it would be
with undisturbed propagation, and, on the other hand, at pro-
tected points s and do not entirely vanish. The approxi-
mation is more and more close
as the size of the apertures in the
screens is increased ; in fact the
approximate results obtained
from theory agree well with ex-
periment if the apertures are not
unusually small. The rigorous
theory of diffraction will be pre-
sented in 7 of this chapter.
i . General Treatment of Dif-
fraction Phenomena. Assume
that between the source Q and
the point P Q there is introduced a plane screen 5 which is of
185
1 86 THEORY OF OPTICS
infinite extent and contains an opening cr of any form. Let
this opening be small in comparison with its distance j\ from
the source Q, and also in comparison with its distance r from
the point P at which the disturbance s is to be calculated by
equation (40) of the preceding chapter. In performing the
integration over cr the angles (nr) and (nr t ) are, on account of
the smallness of cr, to be considered constant; likewise the
quantities r and r l whenever they are not divided by A ; hence
A cos (nr\ cos (nr.) C It r -I-
f ~
It r -I- r\
sm27t(~ --- 4 n
J \T A /
(i)
Assume now a rectangular coordinate system x y y, z.
Let the ;rj/-plane coincide with the screen 5, and let some
point P in the opening cr have the coordinates x and y. Let
x \ ' y\ z \ ^ e * ne coordinates of the source, z^ being positive;
and ;r , j/ , ^ those of P . Z Q is then negative. Then
Let the distances of Q and P Q from the origin be p l and P Q
respectively; then
Then the following relations hold:
- (4)
The dimensions of the opening cr and its distance from the
origin are to be small with respect to p l and p . Hence, in
the integration over cr, x and y are small with respect to p.
If now the expression (4) be expanded in a series with increas-
ing powers of x/p^ y/p l and x/p QJ y/p^ and if powers
higher than the second be neglected, there results, since
( l + e )* = l +i e ~ i e2 provided e is small in comparison
with I,
DIFFRACTION OF LIGHT
187
Denoting the direction cosines of p l and /9 by a l , /^ , ^
and #0 , /? , r respectively, in which the positive directions
of p l and P point away from the origin, then
Hence the addition of (5) and (6) gives
'1+ r = Pi+Po-^K + "<>)-
Substituting this value in (i) and writing for brevity
T~ ~ l A. - T '
A cos (nr) - cos (r t ) _ ^^
(i) becomes
i/( *'
s A' \ sm 27r-^
COS 27T -
(9)
-, cos
. (10)
J may therefore be conceived as due to the superposition
of two waves whose amplitudes are proportional to
S =
C 11 )
This change displaces the origin of time.
i88 THEORY OF OPTICS
and whose difference of phase is - . Hence, from the law on
2
page 131 [cf. equation (n)], the intensity of illumination of
the light at the point P Q is
J =**(&+&) ...... (12)
Now two cases are to be distinguished : I . That in which
both the source and the point P^ lie at finite distances (FresneT s
diffraction phenomena)] and 2. That in which the source and
P Q are infinitely far apart (Frannhofer 's diffraction phenomena].
2. Fresnel's Diffraction Phenomena. Let the origin lie
upon the line QP and in the plane of the screen. Then p l
and p Q lie in the same straight line, but have opposite signs,
hence
<*\ = a o > ft\ Ar
A comparison of equations (8) with equations (9), which
define f(x, y), gives
+ |)[^+> a -(*"i+M) s ]- ('3)
I TV
This equation may be still further simplified by choosing as
the .r-axis the projection of QP Q upon the screen. Then
/? t = o. Also if the angle which p l makes with the ^-axis be
represented by 0, then
In order to avoid the necessity of interrupting the discussion
later by lengthy calculations, a few mathematical considera-
tions will be introduced here.
3. Fresnel's Integrals. The characteristics of the func-
tions which are known as Fresnel'j> integrals will here be dis-
cussed geometrically.* There are two of these integrals,
namely,
C V TtV* C* TtV*
% = / cos dui *? = I sin dv. . . . (15)
* This method was proposed by Cornu in Jour, de Phys. 3, 1874.
DIFFRACTION OF LIGHT 189
The Z and rj which correspond to each particular value of
the parameter v may be thought of as the rectangular coordi-
nates of a point E. Then, as v changes continuously, E
describes a curve whose form will be here determined.
Since, when v = o, = V = o, the curve passes through
the origin. When v changes to v, the expression under the
integral is not altered, but the upper limit of the integral, and
hence also and rj, change sign. Hence the origin is a centre
of symmetry for the curve, for to every point + , + 77, there
corresponds a point , 17. The projections of an element
of arc ds of the curve upon the axes are, by (15),
d = dv-cos , drj dv-sm - . . . (16)
Hence
ds = Vd& + drj* = dv,
or, if the length s be measured from the origin,
s = v ........ (17)
The angle r which is included between the tangent to the
curve at any point E and the -axis is given by
drf ntf . it ,
tan r = ^ = tan , i.e. r = -z/ 2 . . . (18)
U&, 2 2
Hence at the origin the curve is parallel to the -axis ; when
z/=i, i.e. when the arc s = I, it is parallel to the ?;-axis;
when s 2 2 it is parallel to the -axis ; when s 2 = 3 it is
parallel to the ?;-axis; etc.
The radius of curvature p of the curve at any point E is
given by [cf. (17) and (18)]
ds i i
Hence at the origin, where v o, there is a point of inflec-
tion. As v increases, i.e. as the arc increases, p continually
diminishes. Hence the curve is a double spiral, without
double points, which winds itself about the two asymptotic
1 90 THEORY OF OPTICS
points F and F f , whose position is determined by v = -|- oo
and v = oo . The coordinates of these points will now be
calculated. For F,
r Ttv* r. *
Z F = I cos dv, t} F =: I sin dv. . (20)
To obtain the value of this definite integral set
If y is the variable, then also
e-*dy = M.
The product of these two definite integrals is
y = M> ..... (22)
If now x and y be conceived as the rectangular coordinates
of a point P, then x z -j- j/ 2 = r 2 , in which r is the distance of P
from the origin. Furthermore dx dy may be looked upon as a
surface element in the ;rj/-plane. But if a surface element be
bounded by two infinitely small arcs which have the origin as
centre, subtend the angle d ....... (23)
Hence, since the integration is to be taken over one quad-
rant of the coordinate plane, (22) may be written
/JT/2 x>00
dJ Q e-'*r dr ..... (24)
But now
Hence
7T
=Vx (25)
DIFFRACTION OF LIGHT 191
Writing in (21) for x
in which * represents the imaginary, there results from (21)
and (25)
or, because
4/7 = ' + *'
- i 1/2 '
i 4- i
e^dv = -^
o
But since
= cos + *sin , . . . (28)
it follows, by equating the real and the imaginary parts of (27),
that
-" j l I . nv* . i , N
cos dv = , I sin dv = . . (29)
Hence, in accordance with (20), the asymptotic point F has the
coordinates % F = ri p = \. The form of the curve is therefore
that given in Fig. 63. The curve may be constructed in the
following way: Move from o along the -axis a distance
I 10
s = o. i. Construct a circle of radius p = = which
ns n
passes through the point o and whose centre lies upon a line
which passes through the point s = o. i and makes with the
?;-axis the angle r = - - = o.oi [cf. (18)]. On the circle
2 2
thus constructed lay off from o the arc s = o. I . Through its end
I 9 2
THEORY OF OPTICS
point draw another circular arc of radius p = -- =
7i s n . o. 2
whose centre lies upon a line which passes through the point
0.1 O.X 0.3 O.t 0.5 0.0 0,7 0,9
',*
FIG. 63.
s = O. I on the curve and which makes with the ?/-axis an
angle f =
n
0.04 . Proceeding in this way, tiie entire
curve may be constructed.
4. Diffraction by a Straight Edge. Resume the notation
of 2. Let thejj/-axis be parallel to the edge of the screen,
and let the screen extend from x -\- oo to x = x' (the edge
of the screen, cf. Fig. 64). In the figure x' is positive, i.e.
P Q lies outside of the geometrical shadow of the screen. Con-
sider the intensity of the light in a plane which passes through
the source Q and is perpendicular to the edge of the screen.
QP then lies in the ^-^-plane. Equation (14) is here appli-
DIFFRACTION OF LIGHT
cable, and gives, in combination with (n), the following ex-
pressions to be evaluated:
5 =
/?
X cot/ oo
dy
- -* cos
i ' o
-. (30)
It is necessary first to justify the extension in this case of the
integration over the whole portion of the ;rj/-plane not covered
by the screen, for it will be remembered that in the preceding
discussion (cf. page 186) the integral was extended only over
an opening all of whose points lay at distances from the origin
which were small in comparison with p x and p Q . As a matter
FIG. 64.
of fact such a limited region of integration is in itself determina-
tive of the intensity J of the light at the point P , since it
includes the central zones, and indeed a large number of them.
An extension of the integration over a larger region adds
nothing to J, since, as was previously shown, the edge of the
screen exerts no further influence upon the intensity at the
point P when it is many zones distant from the line connect-
i 9 4
THEORY OF OPTICS
ing P and Q. Hence in (30) the result is not altered when
the integration is taken over the entire portion of the xy-
not covered by the screen.
Substitution in (30) of
S = (30
gives
7t
dv du cos ~(v* -\- u 2 ),
+ 00
J ' n
> du sin
S =
in which
If in (32) the following substitution be made,
(32)
TT V ' (33)
it. .
cos -(v* +
2
nv nu TTV . KU
cos cos -- sin - sin
2222
and for sin (z^-j- & 2 ) the analogous expression, the integration
with respect to u may be immediately performed and there re-
sults, in consideration of (29),
", (34)
C=f-\ I cos dv /sin a
( r. nit r * )
S =/ i \ sin dv + i cos dv \ ,
(y-co fc/-oo j^
1
.... (35)
2 COS
DIFFRACTION OF LIGHT 195
Hence it follows from (12) that
J == 2A'*,f>. cos *d^\ + lm . . (36)
The value of A' is given in (9), page 187. Since, according to
the observations on the preceding page, only those portions of
the ;try-plane which lie near the origin are in the integration
determinative of the intensity J at the point P Q , it is possible
to set in the expression for A'
r P , r^ = p l , cos (nr) = cos (nr^ = cos 0.
Hence
(37)
The two Fresnel integrals which occur in (36) will be inter-
preted geometrically as in 3. If the coordinates of a point
E of the curve of Fig. 63 be represented by the above
equations (15), i.e. by
/V 9 /*Z>
itir I
cos dv, rj /sin
*y o
and the coordinates of another point E' on the curve, corre-
sponding to the parameter v' , by
%' I cos dv y rf / sin dv t
2 12
y o c/ o
then evidently
cos n dv =' %, / sin
VJ - (38)
From the form of the curve in Fig. 63 it is evident that J has
maxima and minima for positive values of ' v 1 ', i.e. for cases in
which P Q lies outside the geometrical shadow of the screen. But
when P Q lies inside the shadow, the intensity of the light
decreases continuously as P moves back into the shadow; for
in this case v' is negative and the point E' continuously
approaches the point F' .
If v' = + then ( oo , -f- oo ) 2 = 2, since each of the
points F and F' has the coordinates if = : . In this case
P Q lies far outside of the geometrical shadow, and by (38) the
intensity is the same as though no screen were present. For
v' = o, P Q lies at the edge of the geometrical shadow, in which
case ( oo , o) 2 = , and, by (38), the intensity is one fourth
the natural intensity.
The rigorous calculation of the maxima and minima of
intensity when P Q lies outside the shadow will not be given
here.* It is evident from Fig. 63 that these maxima and
minima lie approximately at the intersections of the line FF'
with the curve. Since this line cuts the curve nearly at right
angles, it is evident that at the maxima the angle of inclination
r of the curve with the -axis is (f + 2^)?r, at the minima
t (J + 2/i)n, in which // = o, I, 2, etc. Hence at the
maxima, cf. equation (18) on page 189, v' = l/f +4^, at the
minima, v' V% + 4^. Now in order to determine the
position of the diffraction fringes, conceive the screen so
*Cf. Fresnel, CEuvr. compl. I, p. 322. For a development in series of
Fresnel's integrals, cf. F. Neumann, Vorles. u. theor. Optik. herausgeg. von
Dorn, Leipzig, 1885, p. 62. Lommel in the Abhandl. d. bayr. Akad., Vol. 15,
p. 229, 529, treats very fully, both theoretically and experimentally, the diffraction
produced by circles and straight edges.
DIFFRACTION OF LIGHT 197
rotated* about its edge that it stands perpendicular to the
shortest line a which can be drawn from Q to the edge
(cf. Fig. 64). Then p l = a : cos 0. Further, draw through
P Q a line parallel to the jr-axis, and let the distance of P Q from
the geometrical shadow of the screen measured along this line
be represented by d. Then x' : d = a : a -\- b. Hence d
denotes the distance of the point P from the geometrical
shadow, in a plane which lies a distance b behind the screen.
Introducing now in (33) the quantity d in place of x' , and set-
ting p l a, P = b, which is allowable since cos does not
differ appreciably from I provided P Q be taken in the neigh-
borhood of the shadow, there results
= d:p, ... (39)
in which p is an abbreviation for
P=\I-^T-' (4)
There are therefore maxima of intensity when d = p V\ +
i.e. when
dv -f- / cos dv,
/ v \ 2 /* +
cos dv -\- I cc
oo i/ z/ a
N = /sin dv + / si
*J - oo U I'-i
71V*
dv.
Now the first term of M \s equal (cf. the analogous develop-
ment on page 195) to the ^-component of the line which con-
nects F' and the point E l which corresponds to the parameter
(cf. Fig. 67). The second term of M is equal to the -
FIG. 67.
component of the line (E 2 F) in which the point E 2 corresponds
to the parameter v r The two terms in N have similar signifi-
* A straight wire may be conveniently used as such a screen.
202 THEORY OF OPTICS
cations. If the and rj components of the lines (F'EJ and
(E 2 F) be denoted by ^ , 2 , % , ?7 2 , then
M* + N* = (^ + ,) + (V, + ?7 2 ) 2 .
If at the end of the line (F'E^ the line (,/?"), having the
same length and direction as the line (E 2 F), be drawn, then
the line (F'F"} has the components ^ + ^2' ^i + ^2- The
intensity y at the point /^ is then proportional to the square
of the line (F'F"), which is the geometrical sum of the two
lines (F'Ei) and (E 2 F), i.e.
(46)
_
From this it appears that the central line (d = o) is always
bright, although it lies farthest inside the geometrical shadow ;
for along it the values of i\ and v 2 are equal and of opposite
sign, so that the two points E l and E 2 in Fig. 67 are sym-
metrically placed with respect to the origin, and hence the
lines F'E l and E 2 F are equal and have the same direction, so
that their sum can never be zero. The broader the screen, the
smaller is the intensity along the middle line.
If the screen is sufficiently broad so that z^ and ^ 2 are large,
the points E l and E 2 lie close to F' and F. The lines (F'E^)
and (EyF) are then approximately equal, and complete dark-
ness results, provided (F'E^) and (E Z F) are parallel and oppo-
site in direction.
Since, for large values of v l and ?/ 2 , the lines (F'E^) and
(FE^ are almost perpendicular to the curve in Fig. 67, it fol-
lows that if these lines have the same direction, the tangents
which are drawn to the curve at E l and E 2 are approximately
parallel to each other; and their positive directions, which are
taken in the direction of increasing arc, are opposite. Hence
the difference between the angles which the tangents make
with the -axis, i.e. T I r 2 , is an odd multiple of TT, or since,
by (18), r = z/ 2 , dark fringes occur when
L( VI I _ ^ = i, 3, 5, etc.
DIFFRACTION OF LIGHT 203
This becomes, in consideration of (43),
2dd = hU, h = i, 3, 5, etc. . . . (47)
These fringes become less dlack as 4 increases. they are
equidistant and independent of the distance a of the source
from the screen. These results hold only inside the geometri-
cal shadow, i.e. only so long as d < \S , and only then
with close approximation provided the values of v l and v z which
correspond to the edges of the screen are sufficiently large,
i.e. provided the screen is broad enough and the point P Q is
sufficiently near to it and to the middle line of the shadow.
As P Q moves toward the edge of the geometrical shadow
or passes outside of it, maxima and minima occur at different
positions of P Q which can be determined for every special case
by the construction given in Fig. 67. The law determining
the positions of these fringes is, however, not a simple one.
These examples will suffice to show the utility of the
geometrical method used by Cornu.* Observation verifies all
the consequences here deduced.
7. Rigorous Treatment of Diffraction by a Straight Edge.
As was remarked at the beginning of this chapter (page 185),
the foregoing treatment of diffraction phenomena, based upon
Huygens' principle, is only approximately correct. Now it
is important to notice that in at least one case, namely, that
of diffraction by a straight edge, the problem can be solved
rigorously, as has been shown by Sommerfeld.t This solution
both furnishes a test of the accuracy of the approximate solu-
tion, and also makes it possible to discuss theoretically the
phenomena when the angle of diffraction is large, i.e. when P Q
lies far within the limits of the geometrical shadow, a discus-
sion which was not possible with the other method, at least
without making important extensions.
* Complicated cases are treated by this method by Mascart, Traite d'Optique,
Paris, 1889, Vol. I, p. 283.
f A. Sommerfeld, Math. Annalen, Vol. XLVII, p. 317, 1895.
204
THEORY OF OPTICS
In the rigorous treatment of the diffraction phenomena the
differential equation (12) on page 159,
for the light disturbance must be integrated so as to satisfy
certain boundary conditions which must be fulfilled at the sur-
face of the diffraction screen. The form of these conditions
will be deduced in Section II, Chapters, I, II, and IV; here
the results of that deduction will be assumed. In the first
place, to simplify the discussion, assume that the source is an
infinitely long line parallel to the j-axis. Also let the edge
of the screen be chosen as the j-axis, and let the ;r-axis be
positive on the side of the screen, and the ^-axis positive
toward the source (cf. Fig. 68). In this case it is evident that
Incident light
FIG. 68.
s cannot depend upon the coordinate
equation reduces to
so that the above
. (48)
Let the screen be infinitely thin and have an infinite absorp-
tion coefficient. It can then transmit no light, but can reflect
perfectly, as will be shown in Section II. A very thin, highly
DIFFRACTION OF LIGHT 205
polished film of silver may constitute such a screen. It is then
not a "perfectly black" screen, but rather one "perfectly
white. ' ' * The boundary conditions at such a screen are :
(49) s = o, if the incident light is polarized in a plane per-
pendicular to the edge of the screen,
(50) = o, if the light is polarized in a plane parallel to the
edge of the screen, t
The meaning of these symbols and of the word polarized
will not be explained until the next chapter. Here it is suffi-
cient to know that the solution of the differential equation (48)
must satisfy either (49) or (50). The boundary conditions
hold upon the surface of the screen, i.e. for z = o, x > o;
or if polar coordinates are introduced by means of the equa-
tions
x r cos 0, z = r sin 0, . . . . (51)
for = o or = 2?r.
If these polar coordinates be introduced into the differential
equation (48), there results
' ' ' (52)
Now a solution of this differential equation, which satisfies
the boundary condition (49) or (50), gives for the particular
* A perfectly black screen, i.e. one which neither transmits nor reflects light, is
realized when the index of refraction of the substance constituting it changes
gradually at the surface to that of the surrounding medium, and the coefficient of
absorption at the surface changes gradually to the value zero. Every discontinuity
in the properties of an optical medium produces necessarily reflection of light.
Hence an ideal black screen, consisting of a thin body, with sharp boundaries, at
which definite boundary conditions can be set up, is inconceivable.
f As will be seen later in the discussion of the electro-magnetic theory, s has
not the same meaning in the two equations. In (49) s represents the electric force
vibrating parallel to the edge of the screen, in (50) the magnetic force vibrating
parallel to the edge of the screen. The intensity is calculated in both cases ir^the
same way, at least for the side of the screen which is turned away from the
source.
206 THEORY OF OPTICS
case in which the source lies at infinity and the incident rays
make an angle -'), r' = -y- cos (0 + '), . (54)
X sin (0-0')- ^=-^sin(0 + 0')- (55)
In (53) the sign is minus or plus according as it is the con-
dition (49) or (50) which must be fulfilled. The letter i denotes
the imaginary V i. Thus the solution of s appears as a
-complex quantity. In order to obtain its physical significance,
it is only necessary to take into account the real part of this
quantity. Thus setting
the physical meaning of s is the real part, i.e.
s = A cos 27Tyr B sin 2n . . . . (57)
The intensity of the light would in this case be (cf. similar
conclusion on page 188)
(58)
This result could have been obtained from (56) directly by
multiplying s by the conjugate complex quantity, i.e. by that
quantity which differs from the right-hand side of (56) only in
. t_
the sign of i, namely, by (A Bt)e~ l T ' . For the sake of
later use this result may be here stated in the following form :
When the expression for the light disturbance s is a complex
quantity (in which s signifies physically only the real part of
DIFFRACTION OF LIGHT 207
this complex quantity), tJie intensity of the light is obtained by
multiplication by the conjugate complex quantity.
That equations (53), (54), and (55) are a real solution of
the differential equation (52) can be shown by taking the
differential coefficients with respect to r and 0.* Also the
boundary condition (49) is fulfilled when the minus sign is used
in (53), since for = o and = 2?r, y = y', ' <
The real part of this expression corresponds to plane waves
which have amplitude A, and whose direction of propagation
makes the angle 0' with the .r-axis. The solution actually
corresponds then, for large values of r, to the incident light
from an infinitely distant source Q which lies in the direction 0'.
3. The region of reflection: 2tt 0' < < 2?r. & and < ?r, and it is to be noticed that, on account
of the small denominator A. (wave length), 27t 0')
equation (61) assumes values of considerable size.
Hence if it is desired to deduce a general rigorous equation
for the intensity of the light, integral (61) cannot be neglected
in comparison with (60). This is true, both for the region of
reflection and for the other regions, when r is very small or
when the angle of diffraction <+>' is large.
This rigorous equation for the intensity J is obtained by
multiplying the right-hand side of (53) by the conjugate com-
plex expression. Using the notation of (60) and (61), the
following is thus obtained:
2 sin(r - y'
or
, (64)
in which X denotes the angle included between the lines (F'E)
and (F'E'\ x is taken positive when the rotation which leads
most directly from F'E to F'E' takes place in the same direc-
tion as a rotation from the q- to the -axis. By (54),
Y y' -jj- sin sin 0' ..... (65)
By (64) J is proportional to the square of the geometrical
difference or sum of the two lines of length (F'E} and (F'E')
which include the angle x+ Y y f . The geometrical differ-
DIFFRACTION OF LIGHT 211
ence is to be taken when the incident light is polarized in a
plane perpendicular to the edge of the screen, the geometrical
sum when it is polarized in the plane parallel to that edge.
The expression (64) may still be much simplified when the
intensity J"is reckoned for points which are not in the neigh-
borhood of the edge of the shadow, i.e. when the difference
between and E/ = ^F'* = f < - ''*>
Now, from (55) and (65), y y' -\-x = o, and hence, from
If the values of cr and cr f given in (55) be introduced here,
then, when the sign is negative, i.e. when the incident light is
polarized in a plane perpendicular to the edge of the screen,
r (cos - cos 0'f
^ 7 }
while when the sign is positive, i.e. when the incident light is
polarized in a plane parallel to the edge of the screen,
A* \ cos'^-si ,,._.
U \) J - - rf' r - (cos _ cos /a-
212 THEORY OF OPTICS
These equations for the region of the shadow hold only so
long as is very small and the difference between and 0' is
large. Thus they do not hold at the edge of the shadow.
The equations show that, at the screen itself (
f '. But then equation (69) no longer
holds, and for points close to the boundary of the region of
reflection the result must be obtained from (64) and the curve
of Fig. 63, since in this case F'E' is larger.
In the region of reflection, at a sufficient distance from its
boundary = 2^ 0', both F'E and F'E' are approximately
equal to t/2 and x - Hence, from (64) and (65), the in-
tensity changes periodically from perfect darkness to four times
2;'
the intensity of the incident light according as -j- sin sin 0'
is a whole number or half of an odd number. Hence the
phenomenon of stationary waves, discussed above on page 155,
is again encountered. Such stationary waves always occur
when the incident and the reflected light are superposed. But
it is important to remark that the significance of s depends
upon the condition of polarization of the incident light (cf.
foot-note, p. 205). This matter will be discussed in a later
chapter.
8. Fraunhofer's Diffraction Phenomena. As was re-
marked on page 188, Fraunhofer's diffraction phenomena are
those in which the source Q lies at an infinite distance from the
point P of observation. These phenomena may be observed
by placing a point source Q at the focus of a convergent lens,
so as to render the emergent rays parallel, and observing by
means of a telescope placed behind the diffraction screen and
focussed for parallel rays.
The discussion will be based, as in i, on Huygens'
principle; and hence the treatment will not be altogether
rigorous. But, as has already been seen, this principle gives a
2i 4 THEORY OF OPTICS
very close approximation when the angle of diffraction is not
too large. In accordance with equations (8) and (9) on page
187, when p t = P Q oo ,
fa y) = - *(! + " ) + X/*i + fit) > (70)
in which ar l , /? x , tf , /? denote the direction cosines with
respect to the x- and j/-axes of the lines drawn from the origin
to the source Q and the point of observation P respectively.
(Cf. Fig. 62, page 185.)
Hence, from equations (n) and (12) on pages 187 and
1 88, using the abbreviations
xK + o) == * r(/. + /*.) = *-. (70
there results for the intensity of the light at the point P ,
j = A >*(C* + s*) ...... (72)
in which
C = /"cos (px + vy)dv, S = fs'm (x + vy)d. (76)
in a coordinate system x'y' whose origin lies at the focus F of
the objective, and whose axes are parallel to the sides of the
rectangle, /"represents the focal length of the objective. In
(76) it is assumed that <* , /? are small quantities, i.e. the
angle of diffraction is small.
Now, from (71),
2nx' 2 ny'
A/
r =
(77)
Hence complete darkness occurs when
pa =
and when
.e.
= *y-, h = i, 2 , 3
=
.e.
= ^, * = I, 2, 3 .
216 THEORY OF OPTICS
Hence in the focal plane of the objective there is produced,
when monochromatic light is used, a pattern crossed by dark
lines as shown in Fig. 69. The lines are a constant distance
FIG. 69.
apart save in the middle of the pattern, where their distance is
twice as great. The aperture which produced this pattern is
shown in the upper left-hand corner of the figure. Hence the
fringes are rectangles which are similar to the aperture but lie
inversely to it.
At the focus of the objective the intensity reaches its
greatest value J = J' ; for when /* o, the limiting value of
l*a l*a
the quotient sin : = i . J has other but weaker maxima
approximately in the middle points of the rectangles bounded
by the diffraction fringes in Fig. 69. For these points
i), vb = 7t(2k+ i), //, k= i, 2, 3 . . .
But for the middle points of those rectangles upon the ;tr'-axis
pa = n(2h + i), v = o, h = i, 2, 3 . . .
Hence the intensities in the maxima upon the jr'-axis (or the
jj/-axis) are
' 4
7t\2h + I) 2 '
DIFFRACTION OF LIGHT 217
while the intensities at the middle points of other rectangles
for which neither x' nor y' vanish are
J
J
(2k + l)\2k + If
Thus the intensities y z are much smaller than the intensi-
ties y x ; so that the figure viewed as a whole gives the im-
pression of a cross which grows brighter toward the centre and
whose arms lie parallel to the sides of the rectangle. In Fig.
69 the distribution of the light is indicated by the shading.
10. Diffraction through a Rhomboid. This case may be
immediately deduced from the former by noting that in (73)
the integrals C and S, and consequently the intensity J,
remain unchanged if the coordinates x, y of the diffraction
aperture are multiplied by the factors /, q, while at the same
time the /*, Y, i.e. the cordinates x' , y' of the diffraction
pattern, are divided by the same factors /, q. Thus a rectan-
gular parallelogram whose sides are not parallel to the coordi-
nate axes x, y may be reduced to a rhomboid by the use of
two factors /, q, and in this case the diffraction fringes will
also be rhomboids whose sides are perpendicular to the sides
of the diffracting opening.
11. Diffraction through a Slit. A slit may be looked
upon as a rectangle one of whose sides b is very large. Hence
the diffraction pattern reduces to a narrow strip of light along
the ;r'-axis. This is crossed by dark spots corresponding to
the equation
pa- 2
(78)
in which, when the incident light is perpendicular to the plane
of the slit,
/i = ^ sin 0, (78')
2l8
THEORY OF OPTICS
where denotes the angle of diffraction, i.e. the angle included
between the diffracted and the incident rays. If Q is a line
source which is parallel to the slit, the diffraction pattern
becomes a broad band of light which is crossed by parallel
fringes at the places determined by pa = 2hn. Between the
limits pa = 27t the intensity is much greater than elsewhere.
The position of the dark fringes can also be determined directly
from the following considerations :
In order to find the intensity for a given angle of diffraction
* (cf. Fig. 70) conceive the slit
AB divided into such portions
AA lf A 1 A 2J A 2 A 3 , etc., that the
distances from A, A lt A 2 , . . . to
the infinitely distant point P differ
from each other successively by
JA. The combined effect of any
two neighboring zones is zero.
Hence there is darkness if AB can
be divided into an even number
of such zones, i.e. if the side BC
k. w.here
A,A 2 A,
FIG. 70.
of the right-angled triangle ACB is equal to
h i, 2, 3, etc. Since now BC = a sin 0, in which a is the
width of the slit, there is darkness when the angle of diffraction
is such that
sin = &-.
(79)
But from (78') this is identical with the condition pa = 2hn.
Hence it follows that when a < A there is darkness for no angle
of diffraction, i.e. diffusion takes place (cf. page 199).
If the incident light is white, and if the intensity J* which
corresponds to a given color, i.e. a given wave-length A, be
denoted by J' K , and if the abbreviation na sin a' be intro-
duced, then for a given value of a' the whole intensity is
,2 a'/
A (79)
sn
DIFFRACTION OF LIGHT 219
If a' is not very small, e.g. if it is about 3^, then
in (79') sin y varies much more rapidly with A than does - .
A A,
If y be considered approximately constant, (79') assumes the
form given for the intensity of light reflected from a thin plate
(cf. Section II, Chapter II, i). Hence at some distance from
the centre of the field of view colors appear which resemble
closely those of Newton's rings.
12. Diffraction Openings of any Form. With any sort
of unsymmetrical opening, the integrals C and S have in
general a value different from zero. At positions of zero
intensity in the diffraction pattern the two conditions C = o
and 5=o must be simultaneously fulfilled. Hence in general
such positions are discrete points, not, as with a rectangular
opening, continuous lines. For the theoretical discussion of
special forms of diffraction apertures cf. Schwerd, "Die
Beugungserscheinungen," Mannheim, 1835.
13. Several Diffraction Openings of like Form and Orien-
tation. Let the coordinates of any point of a diffraction open-
ing referred to a point A lying within that opening be and
T;, and let the point A in all the openings be similarly placed.
Let the coordinates of the points A referred to any arbitrary
coordinate system xy lying in the diffraction screen be x^y v ,
x z y 2 , x^y z , etc. Then for any point in any opening, for
instance the third,
x = x z + , y = y^ + ?7,
and, from (73),
/-
(80)
sin
The and rj vary in all the openings within the same limits.
Hence denoting the integrals C and 5 when they are extended
over a single opening by c and s, that is, setting
c = /"cos (/* + vrf)d$dri t s = fs'm (p% + vrj)d$dri, (8l)
220 THEORY OF OPTICS
and, for the sake of brevity, writing
c' = 2 cos (MXg + ^-), s' = 2 sin (j*x + ry t ), . (82)
then, from (80),
C = c f -c - s'-s, S = s'-c + c'-s,
and hence, from (72),
/ = A"\C'* + 5")( + S*) ..... (8 3 )
From this equation it appears that those places in the
diffraction pattern which in the case of a single opening are
dark remain dark in the case of several similar openings.
The intensity at any point is c' 2 -f- s' 2 times that due to a
single opening. This quantity c' 2 -\- s' 2 may have various
values. It may be written in the form
+ 2 sin 2 OUT, + v
or ^ 2 + ^ /2 = w+22cos[^ / -^) + ^ / -^)], . (84)
/,
in which w denotes the number of openings. In the case of a
large number of openings irregularly arranged, the second
term of the right-hand side of (84) may be neglected in com-
parison with the first, because the values of the separate terms
under the sign 2 vary irregularly between I and + I-
Hence the intensity in the diffraction pattern is everywhere m
times greater than when there is but one opening. This
phenomenon may be studied by using as a diffraction screen a
piece of tin-foil in which holes of equal size have been pierced
at random by a needle. The diffraction pattern consists of a
system of concentric rings which differ from those produced by
a single hole only in that they are more intense.
The result is entirely different when the holes are regularly
arranged or are few in number. Consider, for example, the
case of two openings, and set
DIFFRACTION OF LIGHT 221
then
The diffraction pattern which is produced by a single open-
ing is now crossed by dark fringes corresponding to the equa-
tion vd (2k + i) 7 *, i.e. by fringes which are perpendicular
to the line connecting two corresponding points of the openings
and which are, in the focal plane of the objective, a distance
A/": d apart.
14. Babinet's Theorem. Before passing to the discussion
of the grating, which consists of a large number of regularly
arranged diffraction openings, the case of two complementary
diffraction screens will be considered. If a diffraction screen
<7j has any openings whatever, while a second screen cr 2 has
exactly those places covered which are open in & l , while the
places in o" 2 are open which are covered in = o). From
this equation it appears that the diffraction pattern is the same
as that of a single slit (which is represented by the first two fac-
tors) save that it is crossed by a series of dark fringes which are
very close together and correspond to the equation - - = hit.
These fringes are closer together the greater the number m
of the slits. Between the fringes the intensity J reaches
maxima which are, however, at most equal to the intensities
produced at the same points by a single slit. But much
tut
stronger maxima occur when sin vanishes, i.e. when
\
i.e. sin /z-, . . . . (86)
in which denotes the angle of diffraction. (The light is
assumed to fall perpendicularly upon the grating.)
For the diffraction angles thus determined
.
sin 2 -
2
so that the intensity is m* times as great as it is at the same
point when there is but one slit. When m is very great, it
is these maxima only which are perceptible.* One of these
maxima may be wanting if a minimum of the diffraction pattern
due to a single slit falls at the same place, i.e. if both (86) and
are at the same time fulfilled.
* If the constant of the grating is less than A., no maxima appear, since, by (86\
sin > i. Hence transparent bodies may be conceived as made up of ponderable
opaque particles embedded in transparent ether. If the distance between the
particles is less than a wave length, only the undiffracted light passes through.
224 THEORY OF OPTICS
This is only possible if the width of the slit a is an exact
multiple of the constant of the grating d. Close-line gratings
are produced by scratching fine lines upon glass or metal by
means of a diamond. The furrows made by the diamond act
as opaque or non-reflecting places. According to Babinet's
theorem the width of the furrow may also be looked upon as
the width a of the slit. This latter then is much smaller than
the constant d of the grating, so that, in any case, the first
maxima, which in (86) correspond to small values of h, do not
vanish. These maxima have a nearly constant intensity, since
for small values of the width a of the slit the diffraction figure
which is produced by a single slit illuminates the larger portion
of the field with a nearly constant intensity.
Hence, when the number m of the slits is sufficiently large,
the diffraction pattern in monochromatic light, which proceeds
from a line source Q, consists of a series of fine bright lines which
appear at the diffraction angles , 0j , 2 , etc., determined by
A. 2*. 3 A,
= o, sin 0j = ^, sin 2 = -^-, sin 3 = , etc.
If the grating is illuminated by white light from a line
source Q, pure spectra must be produced, since the different
colors appear at different angles. These grating spectra are
called normal spectra, to distinguish them from the dispersion
spectra produced by prisms, because the deviation of each
color from the direction of the incident light is proportional to
its wave length, at least so long as is so small that it is
permissible to write sin = 0. Since each color correspond-
ing to the different values of h in (86) appears many times,
many spectra are also produced. The spectrum corresponding
to h = i is called that of the first order; that to h = 2, the
spectrum of the second order, etc. In the first spectrum the
violet is deviated least; the other colors follow in order to the
red. After an interval of darkness the violet of the second
order follows. But the red of the second spectrum and the
blue of the third overlap, since 3^ < 2^ r , in which A^ and k r
DIFFRACTION OF LIGHT 225
denote the wave lengths of the visible violet and red rays
contained in white light. This overlapping of several colors
increases rapidly with the angle of diffraction.
That pure spectral colors are produced by a grating and
not by a slit, which gives approximately the colors of Newton's
rings (cf. page 219), is due to the fact that in the case of the
grating it is the positions of the maxima, while in the case of
a slit it is the positions of the minima, which are sharply
defined.
The grating furnishes the best means of measuring wave
lengths. The measurement consists in a determination of d
and and is more accurate the smaller d is, since then the
diffraction angles are large. Rutherford made gratings upon
glass which have as many as 700 lines to the millimetre. The
quality of a grating depends primarily upon the ruling engine
which makes the scratches. The lines must be exactly
parallel and a constant distance apart. Rowland now pro-
duces faultless gratings with a machine which is able to rule
1700 lines to the millimetre.
16. The Concave Grating. A further advance was made
by Rowland in that he ruled gratings upon concave spherical
mirrors of speculum metal, the distance between the lines
measured along a chord being
equal. These gratings produce a
real image P of a line source Q
without the help of lenses; the
diffraction maxima P lt P 2 , etc.,
are also real images. In order to
locate these images, construct a
circle tangent to the grating (Fig.
71) upon the radius of curvature
of the grating as its diameter. If
the line source Q lies upon the FlG - n-
circle, an undiffracted image is produced upon the same circle
at P by direct reflection, in such a way that P and Q are sym-
metrical to C, C being the centre of curvature of the grating
226 THEORY OF OPTICS
GG. For the line CB is the normal to the mirror at the point
B, hence the angle of incidence QBC is equal to the angle of
reflection PBC. But a ray reflected from any point B' of the
mirror must also pass through P because CB' is the normal to
the mirror at B' ', since C is the centre of curvature of the
mirror and since approximately < QB'C = <^C PB'C, and
therefore B' P is the direction of the reflected ray. The angles
QB'C and PB'C would be rigorously equal if B' lay upon the
circle itself, since then they would be inscribed angles sub-
tended by equal arcs. P is then the position of the undiffracted
image which is formed by reflection by the mirror of the light
from Q*
The position of the diffraction image P l is at the intersection
of two rays BP V and B'P^ which make equal angles with BP
and B'P. Hence it is evident that P l also lies upon the circle
passing through PCQB, since the angles PB'P l and PBP^
would be rigorously equal if B' lay upon the circle.
If the real diffraction spectrum at P l were to be received
upon a screen 5, it would be necessary to place the screen very
, r obliquely to the rays. Since it
is better that the rays fall per-
pendicularly upon the screen S,
the latter is placed at the point
C parallel to the grating. The
source Q must also lie upon the
circle whose diameter is CB,
i.e. the angle CQB must always
be a right angle. In practice,
in order to find the positions of Q which throw diffraction
spectra upon 5, the grating G and the screen 5 are mounted
upon a beam of length r (radius of curvature of the grating)
which slides along the right-angled ways QM, QN, as shown
* This would follow from the second of equations (34), page 51, which apply
to the formation of astigmatic images by reflection. For this case ^ CBQ = (p,
CB = r, and hence QB = s = r cos 0. Hence s l = j, i.e. the point />,
symmetrical to Q with respect to C, must be the image of Q upon the circle.
DIFFRACTION OF LIGHT 227
in Fig. 72. The source is placed at Q. As S is moved away
from Q the spectra of higher order fall successively upon the
screen.
17. Focal Properties of a Plane Grating.- If the distance
d between the lines of a grating is not constant, then the
diffraction angle which corresponds to a maximum, for
instance the first which is given by sin = A : d, is different
for different parts of the grating, d may be made to vary in
such a way that these directions which correspond to a maxi-
mum all intersect in a point F. This point is then a focal
point of the grating, since it has the same properties as the
focus of a lens.*
18. Resolving Power of a Grating. The power of a grat-
ing to separate two adjacent spectral lines must be proportional
to its number of lines ;^, since it has been already shown that
the diffraction maxima which correspond to a given wave
length A become narrower as m increases. By equation (86)
on page 223, the maximum of the order h is determined by
// = 2h7t : d, i.e. sin = h\ : d.
If A* rises above or falls below this value, then, by (85), the
first position of zero intensity occurs when /* has changed in
such a way that m^d/2 has altered its value by it, i.e. when
the change in p amounts to
dp 2K : md.
Hence the corresponding change in the diffraction angle
0, whose dependence upon p is given in equation (/S 7 ), is
d$ = A : m d cos ...... (87)
Hence this quantity d(f> is half the angular width of the diffrac-
tion image.
* For the law of distribution of the lines cf. Cornu, C. R. 80, p. 645, 1875 ;
Fogg. Ann. 156, p. 114, 1875 ; Soret, Arch. d. Scienc. Phys. 52, p. 320, 1875 ;
Fogg. Ann. 156, p. 99, 1875 I Winkelmann's Handbuch, II, p. 622.
228 THEORY OF OPTICS
For an adjacent spectral line of wave length A -|- d\ the
position of the diffraction maximum of order h is given by
sin + d f must be greater than half the breadth of the diffrac-
tion image of one of the lines, i.e.
d d, h-d^ > A. : m y -^- > T . . . (88)
Thus the resolving power of a grating is proportional to
the total number of lines m and to the order h of the spectrum,
but is independent of the constant d of the grating. To be
sure, if d is too large, it may be necessary to use a special
magnifying device in order to separate the lines, but the sep-
aration may always be effected if only the resolving power
defined by (88) has not been exceeded.
In order to separate the double D line of sodium for which
d^ : A. = o.ooi, a grating must have at least 500 lines if the
observation is made in the second spectrum.
19. Michelson's Echelon.* From the above it is evident
that the resolving power may be increased by using a spectrum
of high order. With the gratings thus far considered it is not
practicable to use an order of spectrum higher than the third,
on account of the lack of intensity of the light in the higher
orders. But even when the angle of diffraction is very small,
if the light be made to pass through different thicknesses of
glass, a large difference of phase may be introduced between
the interfering rays, i.e. the same effect may be obtained as
with an ordinary grating if the spectra of higher orders could
be used. Consider, for instance, two parallel slits, and let a
* A. A. Michelson, Astrophysical Journal, 1898, Vol. 8, p. 37.
DIFFRACTION OF LIGHT
229
glass plate several millimetres thick be placed in front of one
of the slits; then at very small angles of diffraction rays
come to interference which have a difference of path of several
thousand wave lengths. This is the fundamental idea in
Michelson's echelon spectroscope, m plates of thickness 6
are arranged in steps as in Fig. 73. Let the width of the
FIG. 73-
steps be a, and let the light fall from above perpendicularly
upon the plates. The difference in path between the two
parallel rays A A' and CC', which make an angle with the
incident light, is, if CD is \_AA' and if n denote the index of
refraction of the glass plates,
n-BC AD = nS 6 cos $ + a sin 0,
since AD = DE AE and DE = # cos 0, AE = a sin 0.
If this difference of path is an exact multiple of a wave length,
i.e. if
^A, = n$ ^ cos -f- a sin 0, . . . (89)
then a maximum effect must take place in the direction 0,
since all the rays emerging from AB are reinforced by the
parallel rays emerging from CF. Hence equation (89) gives
the directions of the diffraction maxima.
The change d in the position of the diffraction maxima
2 3 o THEORY OF OPTICS
corresponding to a small change dh in A, is large, since it fol-
lows from (89) by differentiation that
hdk S'dn -f- (3 sin + a cos 0)dT0',
i.e. if be taken small,
h-d^ $ - dn
dV = -- - - ..... (90)
Since, by (89), when is small M (n i)tf, (90) may be
written
^=(-i)-h; (90')
Hence d(f> f is large when d : a is large. It is to be observed
that it is in reality a summation and not a difference which
occurs in this equation, since in glass, and, for that matter, all
transparent substances, n decreases as A. increases.
One difficulty of this arrangement arises from the fact that
the maxima of different orders, which yet correspond to the
same A, He very close together. For, by (89), the following
relation exists between the diffraction angle + dty" of order
h -f- I and the wave-length A. :
A = (d sin -|- a cos 0)^/0",
i.e. when is small,
d" > d) correspond-
ing to those zero positions which are immediately adjacent to
the maxima determined by (89). In order to find these posi-
tions of zero intensity, consider the m plates of the echelon
divided into two equal portions I and II. Darkness occurs for
those angles of diffraction -f- d(f> for which the difference of
path of any two rays, one of which passes through any point
of portion I, the other through the corresponding point of
portion II, is an odd multiple of A. Just as the right side of
(89) gives the difference of path of two rays, one of which has
passed through one more plate than the other, so the difference
of path in this case, in which one wave has passed through -
more plates than the other, may be obtained by multiplying
the right-hand side of (89) by .
Hence, at a position of zero intensity which corresponds to
the angle of diffraction -\- d} + a sin (0 may be as small as possible, i.e. in order to
obtain the two positions of zero intensity which are closest to
the maxima determined by (89), it is necessary, as a compari-
/ yyi
son with (89) shows, to make in this equation k = k. Hence
from these two equations
= (d sin + a cos 0) 0.61.-
If A be assumed to be 0.00056 mm., and if be expressed in
minutes of arc, then
> ' , (96)
in which h must be expressed in mm. A telescope whose
objective is 20 cm. in diameter is then able to resolve two stars
whose angular distance apart is = o.oi 17' = 0.7".
22. The Limit of Resolution of the Human Eye. The
above considerations may be applied to the human eye with
the single difference that the wave length A of the light in the
lens of the eye, whose index is 1.4, is i: 1.4 times smaller
than in air. The radius of the pupil takes the place of Ji. If
h be assumed to be 2 mm., then the smallest visual angle
which two luminous points can subtend if they are to be
resolved by the eye is
= 0.42'.
The actual limit is about = i'.
23. The Limit of Resolution of the Microscope. The
images formed by microscopes are of illuminated, not of self-
luminous, objects, t The importance of this distinction was first
pointed out by Abbe. From the standpoint of pure geometri-
cal optics, which deals with rays, the exact similarity of object
and image follows from the principles laid down in the first
part of this book. From the standpoint of physical optics,
which does not deal with rays of light as independent geometri-
cal directions, since this is not rigorously permissible, but which
is based upon deformations of the wave front, the similarity of
* On account of the smallness of 0, may be written for sin 0.
f Objects which are visible by diffusely reflected light may be approximately
treated as self-luminous objects.
DIFFRACTION OF LIGHT 237
object and image is not only not self-evident, but is, strictly
speaking, unattainable. For the incident light, assumed in the
first case to be parallel, will, after passing through the object
which it illuminates, form a diffraction pattern in that focal plane
g ' of the objective which is nearest the eyepiece. The question
now is, what light effect will this diffraction figure produce in
the plane ^' which is conjugate with respect to the objective
to the object plane ^ ? The image formed in this plane is the
one observed by the eyepiece. The formation of the image of
an illuminated object is therefore not direct (primary) but
indirect (secondary), since it depends upon the effect of the
diffraction pattern formed by the object.
It is at once clear that a given diffraction pattern in the focal
plane g' gives rise always to the same image in the plane ty'
upon which the eyepiece is focussed. Now in general different
objects produce different diffraction patterns in the plane $'.*
But if the aperture of the objective of the microscope is very
small, so that only the small and nearly uniformly illuminated
spot of the diffraction pattern produced by two different objects
is operative, then these objects must give rise to the same light
effects in the plane *|3', i.e. they look alike when seen in the
microscope. Now in this case there is seen in the microscope
only a uniformly illuminated field, and no evidence of the
structure of the object. In order to bring out the structure,
the numerical aperture of the microscope must be so great that
not only the effect of the central bright spot of the diffraction
pattern appears, but also that of at least one of the other
maxima. When this is so, the distribution of light in the plane
$' is no longer uniform, i.e. some sort of an image appears
*By the introduction of suitable stops in the plane %' the same diffraction
pattern may be produced by different objects. In this case the same image is also
seen at the eyepiece in the plane )', although the objects are quite different.
Thus if the object is a grating whose constant is J, and if all the diffraction images
of odd order be cut out by the stop, then the object seems in the image to have a
grating constant . Cf. Muller-Pouillet (Lummer), Optik, p. 713. The house of
C. Zeiss in Jena constructs apparatus to verify these conclusions.
23*
THEORY OF OPTICS
which has a rough similarity to the object. As more maxima
of the diffraction pattern are admitted to the microscope tube,
i.e. as more of the diffraction pattern is utilized, the image in
the microscope becomes more and more similar to the object.
But perfect similarity can only be attained when all the rays
diffracted by the object, which are of sufficient intensity to be
able to produce appreciable effects in the focal plane g' of the
objective, are received by the objective, i.e. are not cut off by
stops. This shows the great importance of using an objective
of large numerical aperture. The greater the aperture the
sooner will an image be formed which approximately repro-
duces the fine detail in the object. Perfect similarity is an
impossibility even theoretically. A microscope reproduces the
detail of an object up to a certain limit only.
To illustrate this by an example, assume that the object P
is a grating whose constant is d, and that the incident beam is
parallel and falls perpendicularly upon the grating. The first
maximum from the centre of the field lies in a direction deter-
mined by sin = A : d. Let the real image of this maximum
in the focal plane g' of the objective be C l , while C is that
of the centre of the field (Fig. 75). Let the distance between
FIG. 75.
these two images be e. Now the two images C and C l have
approximately the same intensity and send out coherent waves,
i.e. waves capable of producing interference. Hence there is
formed at a distance x' behind the focal plane g' a system of
fringes whose distance apart is d' = x'\ : e. If now the objec-
tive is aplanatic, i.e. fulfils the sine law (cf. page 58), then
sin = e-sin 0',
DIFFRACTION OF LIGHT 239
in which e denotes a constant. Setting sin 0' = e : x' , which
is permissible since ' is always small (while may be large),
and remembering that sin = A : d, it follows that
A e
~d~~ V'
i.e. the distance d' between the fringes is
,, *'\ .
a = = ea
e
or, the distance between the fringes is proportional to the con-
stant of the grating and independent of the color of the light
used.
Hence in order that the grating lines may be perceptible
in the image, the objective must receive rays whose inclination
is at least as great as that determined by sin = A : d. In
the case of an immersion system A denotes the wave length in
the immersion fluid, i.e. it is equal to A : n when A denotes
the wave length in air and n the index of the fluid with respect
to air. Hence
n sin = A : d.
Now n sin U = a is the numerical aperture of the microscope
(cf. equation (80) on page 86), provided U is the angle
included between the limiting ray and the axis. Hence the
smallest distance d which can be resolved by a microscope of
aperture a is
d = A : a (97)
This equation holds for perpendicular illumination of the object.
With oblique illumination the resolving power may be in-
creased, for, if the central spot of the diffraction pattern does
not lie in the middle but is displaced to one side, the first
diffraction maximum appears at a smaller angle of inclination
to the axis. The conditions are most favorable when the inci-
dent light has the same inclination to the axis as the diffracted
light of the first maximum, and both just get in to the objec-
tive.
2 4 o THEOR Y OF OP TICS
If the incident and the diffracted light make the same angle
U with the normal to the grating, then, by (71) on page 214,
/* = j--2 sin U. Since, further, by (86) on page 223, the
first diffraction maximum appears when ^ = -, it follows that
in this case
Hence the smallest distance d which the microscope objective
is able to resolve with the most favorable illumination is
(98)
in which a is the numerical aperture of the microscope and A.
the wave length of light in air. This is the equation given on
page 92 for the limit of resolution of the microscope.
In order to increase the amount of light in the microscope,
the object is illuminated with strongly convergent light (with
the aid of an Abbe condenser, cf. page 102). The above
considerations hold in this case for each direction of the incident
light; but in the resolution of the object only those directions
are actually useful for which not only the central image but
also at least the first maximum of the diffraction pattern falls
within the field of view of the eyepiece. The diffraction
maxima corresponding to the different directions of the inci-
dent light lie at different places in the focal plane of the
objective, but they exert no influence whatever upon one
another, since they correspond to incoherent rays ; for the light
in each direction comas from 1 a different point of the source, for
example the sky.
If, instead of a grating, a single slit of width d were used,
no detail whatever would be recognizable unless the diffraction
pattern were effective at least, to the first minimum. Since,
according to equation (79) on page 218, for perpendicularly
incident light this first minimum lies at the diffraction angle
DIFFRACTION OF LIGHT 241
determined by sin h \ d* the result for one slit is the same
as for a grating. Only in this case a real similarity between
the image and the slit, i.e. a correct recognition of the width
of the slit, is not obtained if the diffraction pattern is effective
only up to the first minimum.
If only an approximate similarity between object and image
is sufficient, for example if it is only desired to detect the
existence of a small opaque body, its dimensions may lie con-
siderably within the limit of resolution d as here deduced; for
so long as the diffraction pattern formed by the object causes
an appreciable variation in the uniform illumination in the
image plane which is conjugate to the object, its existence
may be detected.
From the above considerations it is evident that the limit
of resolution d is smaller the shorter the wave length of the
light used. Hence microphotography, in which ultraviolet
light is used, is advantageous, although no very great increase
in the resolving power is in this way obtained. But the
advantages of an immersion system become in this case very
marked, since by an immersion fluid of high index the wave
length is considerably shortened. This result appears at once
from equations (97) and (98), since the numerical aperture a
is proportional to the index of refraction of the immersion fluid.
* d here has the same signification as a there.
CHAPTER V
POLARIZATION
i. Polarization by Double Refraction. A ray of light is
said to be polarized when its properties are not symmetrical
with respect to its direction of propagation. This lack of
symmetry is proved by the fact that a rotation of the ray about
the direction of propagation as axis produces a change in the
observed optical phenomena. This was first observed by
Huygens * in the passage of light through Iceland spar. Polar-
ization is always present when there is double refraction.
Those crystals which do not belong to the regular system
always show double refraction, i.e. an incident ray is divided
within the crystal into two rays which have different directions.
The phenomenon is especially easy to observe in calc-spar,
which belongs to the hexagonal system and cleaves beautifully
in planes corresponding to the three faces of a rhombohedron.
In six of the corners of the rhombohedron the three intersect-
ing edges include one obtuse and two acute angles, but in the
two remaining corners A, A' , which lie opposite one another
(cf. Fig. 76), the three intersecting edges enclose three equal
obtuse angles of 101 53'. A line drawn through the obtuse
corner A so as to make equal angles with the edges intersect-
ing at A lies in the direction of the principal crystallographic
axis.^ If a rhombohedron be so split out that all of its edges
are equal, this principal axis lies in the direction of the line
connecting the two obtuse angles A, A'. Fig. 76 represents
such a crystal.
* Huygens, Traite de la Lumiere, Leyden, 1690.
f- The principal axis, like the normal to a surface, is merely a direction, not a
definite line.
242
POLARIZATION
243
If now a ray of light LL be incident perpendicularly upon
the upper surface of the rhombohedron, it splits up into two
rays LO and LE of equal intensity
which emerge from the crystal
as parallel rays OL' and EL"
perpendicular to the lower face.
Of these rays LO is the direct
prolongation of the incident ray
and hence follows the ordinary
law of refraction in isotropic
bodies, in accordance with which
no change in direction occurs
when the incidence is normal.
This ray LO together with its
prolongation L 'O is therefore
called the ordinary ray. But the
second ray LE, with its prolonga-
tion L"E, which follows a law of refraction altogther different
from that of isotropic bodies, is called the extraordinary ray.
Also the plane defined by the two rays is parallel to the direc-
tion of the crystallographic axis. A section of the crystal by
a plane which includes the normal to the surface and the axis
is called a principal section. Hence the extraordinary ray lies
in the principal section; it rotates about the ordinary ray as the
crystal is turned about LL as an axis.
The intensities of the ordinary and extraordinary rays are
equal. But if one of these rays, for instance the extraordinary,
is cut off, and the ordinary ray is allowed to fall upon a second
crystal of calc-spar, it undergoes in general a second division
into two rays, which have not, however, in general the same
intensity. These intensities depend upon the orientation of the
two rhombohedrons with respect to each other, i.e. upon the
angle included between their principal sections. If this angle
is o or 1 80, there appears in the second crystal an ordinary
but no extraordinary ray ; but if it is 90, there appears only
an extraordinary ray. Two rays of equal intensity are pro-
244 THEORY OF OPTICS
duced if the angle between the principal sections is 45.
Hence the appearance continually changes when the second
crystal is held stationary and the first rotated, i.e. when the
ordinary ray turns about its own direction as an axis. Hence
the ray is said to be polarized. This experiment can also be
performed with the extraordinary ray, i.e. it too is polarized.
Also if the first rhombohedron is rotated through 90 about
the normal as an axis, the extraordinary ray produces in the
second crystal the same effects as were before produced by the
ordinary ray. Hence the ordinary and extraordinary rays
are said to be polarized in planes at right angles to each other.
The two rays produced by all other doubly refracting
crystals are polarized in planes at right angles to each other.
The principal section is conveniently chosen as a plane of
reference when it is desired to distinguish between the direc-
tions of polarization of the two rays. Since these phenomena
produced by two crystals of calc-spar depend only upon the
absolute size of the angle included between their principal sec-
tions and not upon its sign, the properties of the ordinary and
extraordinary rays must be symmetrical with respect to the
principal section.
The principal section is called the plane of polarization of
the ordinary ray, an expression which asserts nothing save
that this ray is not symmetrical with respect to the direction
of propagation, but that the variations in symmetry in different
directions are symmetrical with respect to this plane of polar-
ization, the principal section.
Since, as was observed above, the ordinary ray is polarized
at right angles to the extraordinary ray, it is necessary to call
the plane which is perpendicular to the principal section the
plane of polarization of the extraordinary ray. These relations
may also be expressed as follows: The ordinary ray is polar-
ized in the principal section, the extraordinary perpendicular
to the principal section.
2. The Nicol Prism. In order to obtain light polarized in
but one plane, it is necessary to cut off or remove one of the
POLARIZATION 245
two rays produced by double refraction. In the year 1828
Nicol devised the following method of accomplishing this end :
By suitable cleavage a crystal of calc-spar is obtained which is
fully three times as long as broad. The end surfaces, which
make an angle of 72 with the edges of the side, are ground
off until this angle (ABA' in Fig. 77) is 68. The crystal is
L"
FIG. 77.
then sawed in two along a plane A A', which passes through
the corners A A' and is perpendicular both to the end faces and
to a plane defined by the crystallographic axis and the long axis
of the rhombohedron. These two cut faces of the two halves
of the prism are then cemented together with Canada balsam.
This balsam has an index of refraction which is smaller than
that of the ordinary but larger than that of the extraordinary
ray. If now a ray of light LL enters parallel to the long axis
of the rhombohedron, the ordinary ray LO is totally reflected
at the surface of the Canada balsam and absorbed by the
blackened surface BA' ', while the extraordinary ray alone
passes through the prism. The plane of polarization of the
emergent light EL" is then perpendicular to the principal
section, i.e. parallel to the long diagonal of the surfaces AB
or A'B'.
The angle of aperture of the cone of rays which can enter
the prism in such a way that the ordinary ray is totally reflected
amounts to about 30. Furthermore a convergent incident
beam is not rigorously polarized in one plane, since the plane
of polarization varies somewhat with the inclination of the
incident ray ; for the plane of polarization of the extraordinary
ray is always perpendicular to the plane defined by the ray and
the crystallographic axis (principal plane). The principal plane
and the principal section are identical for normal incidence.
2 4 6
THEORY OF OPTICS
3. Other Means of Producing Polarized Light. Apart
from polarization prisms* constructed in other ways, tourmaline
plates may be used for obtaining light polarized in one plane,
provided they are cut parallel to the crystallographic axis and
are from one to two millimetres thick. For under these con-
ditions the ordinary ray is completely absorbed within the
crystal. Also, polarized light may be obtained by reflection at
the surface of any transparent body if the angle of reflection
fulfils the condition (Brewster's law) tan = n, in which n is
the index of refraction of the body. This angle is called
the polarizing angle. For crown
glass it is 57. The reflected
light is polarized in the plane of
incidence, as may be shown by
passing the reflected light through
a crystal of calc-spar.
If light reflected at the polar-
izing angle from a glass plate
be allowed to fall at the same
angle upon a second glass plate,
the final intensity depends upon
the angle a included between the
planes of incidence upon the two
surfaces and is proportional to
cos 2 a. This case can be studied
by means of the Norrenberg
polariscope. The ray a is polar-
ized by reflection upon the glass
plate A and then falls perpendic-
ularly upon a silvered mirror at c.
This mirror reflects it to the black
glass mirror 5 which turns upon a
FlG< 78> vertical axis. The ray cb falls
also at the polarizing angle upon 5 and, after reflection upon
* Cf. W. Grosse, Die gebrauchlichen Polarisationsprismen, etc.
1889 ; Winkelmann's Handbuch d. Physik, Optik. p. 629.
Klaustahl,
POLARIZATION 247
5, has an intensity which varies as 5 is turned about a vertical
axis. Between A and 5 a movable glass stage is introduced
in order to make it convenient to study transparent objects at
different orientations in polarized light. But since the intensity
of light after but one reflection is comparatively small, this
means of producing polarized light is little used; the same
difficulty is met with in the use of tourmaline plates (not to
mention a color effect).
A somewhat imperfect polarization is also produced by the
oblique passage of light through a bund e of parallel glass
plates. This case will be treated in Section II, Chapter II.
That polarization is also produced by diffraction was mentioned
on page 212.
4. Interference of Polarized Light. The interference
phenomena described above may all be produced by light
polarized in one plane. But two rays which are polarized at
right angles never interfere. This can be proved by placing
a tourmaline plate before each of the openings of a pair of slits.
The diffraction fringes which are produced by the slits are seen
when the axes of the plates are parallel, but they vanish com-
pletely when one of the plates is turned through 90.
Fresnel and Arago investigated completely the conditions
of interference of two rays polarized at right angles to each
other after they had been brought back to the same plane of
polarization by passing them through a crystal of calc-spar
whose principal section made an angle of 45 with the planes
of polarization of each of the two rays. They found the fol-
lowing laws:
1 . Two rays polarized at right angles to each other, which
have come from an unpolarized ray, do not interfere even when
they are brought into the same plane of polarization.
2. Two rays polarized at right angles, which have come
from a polarized ray, interfere when they are brought back to
the same plane of polarization.
5. Mathematical Discussion of Polarized Light. It has
been already shown that the phenomena of interference lead
248 THEORY OF OPTICS
to the wave theory of light, in accordance with which the light
disturbance at a given point in space is represented by
s = A sin27r -f d ..... (i)
It is now possible to make further assertions concerning
the properties of this disturbance. For in polarized light these
properties must be directed quantities, i.e. vectors, as are lines,
velocities, forces, etc. Undirected quantities like density and
temperature are called scalars to distinguish them from vectors.
If the properties of polarized light were not vectors, they could
not exhibit differences in different azimuths. For the same
reason these vectors cannot be parallel to the direction of
propagation of the light. Hence s will now be called a light
vector. Now a vector may be resolved into three components
along the rectangular axes x, y, z. These components of s
will be denoted by u, v, w. Hence the most general repre-
sentation of the light disturbance at a point P is
u = A sin (zn-f + A v = sin \27r
( t \
w = C sin \27t- 4- rj.
The meaning of these equations can be brought out by
representing by a straight line through the origin the magni-
tude and direction of the light vector at any time. The end
@ of this line can be located by considering u, v, w, as its
rectangular coordinates. The path which this point &
describes as the time changes is called the vibration form and
is obtained from equations (2) by elimination of t. (2) may
be written
u t t
r- = sin 2 7t cos / -\- cos 27^-^ sin /,
yjt ./ y
v t t
-= = sin 27T - -cos ^ 4- cos 27f-= -sin ^,
n 1 1
w t t
^ = sin 2 TT- c os r + cos 2 TT-^ sin r.
(^ - .J[. ^ .-..--.*,. J_
. (3)
POLARIZATION 249
Multiplying these equations by sin (q r), sin (r /),
and sin (p q) respectively, and adding them, there results
u v w
-j sin (f-r) + -g sin (r -/) + ^ sin (/ - q) = o, (4)
i.e. since a linear equation connects the quantities u, v, w, the
vibration form is always a plane curve.
The equations of its projections upon the coordinate planes
may be obtained by eliminating t from any two of equations (3).
Thus, for instance, from the first two of these equations
/ u v
sin 27t- (cos /sin q cos q sin/) = -^ sin q sin/,
t u v
cos 2 n (cos / sin q cos q sm /) = -^- cos ^ + -g- cos /.
Squaring and adding these two equations gives
u 2 v* 2uv
-?). . . (5)
But this is the equation of an ellipse whose principal axes
coincide with the coordinate axes when / q = -. Hence,
in the most general case, the vibration form is a plane elliptical
curve. This corresponds to so-called elliptically polarized
light. "When the vibration form becomes a circle, the light
is said to be circularly polarized. This occurs, for instance,
when w = o, A B, and / q = -, so that either the
relation
u = A sin 27t~, v = A cos 2?r , ... (6)
or the relation
u = A sin 27t , v = A cos 27T . . (6')
holds. These two cases are distinguished as right-handed and
left-handed circular polarization. The polarization is right-
handed when, to an observer looking in a direction opposite
2 5 o THEORY OF OPTICS
to that of propagation, the rotation corresponds to that of the
hands of a watch. When the vibration ellipse becomes a
straight line, the light is said to be plane-polarized. This
occurs when w = o, and / q = o or n. The equation of
the path is then, by (5),
The intensity of the disturbance has already been set equal
to the square of the amplitude A of the light vector. This
point of view must now be maintained, and it must be remem-
bered that the square of the amplitude is equal to the sum of
the squares of the amplitudes of the three components. The
intensity J is then, in accordance with the notation in (2),
J~A* + JP+C* ...... (8)
An investigation will now be made of the vibration form
which corresponds to the light which in the previous paragraph
was merely said to be polarized, i.e. the light which has suffered
double refraction or reflection at the polarizing angle. The
principal characteristic of this light is that two rays which are
polarized at right angles never interfere, but give always an in-
tensity equal to the sum of the intensities of the separate rays.
If there be superposed upon ray (2), which is assumed to
be travelling along the -s'-axis, a ray of equal intensity, which
is polarized at right angles to it and whose components are u' ',
v' , w f , and which differs from it in phase by any arbitrary
amount #, then
(9)
V 1 = A sin ( 27T-^r
?v r = C sin UTT^T 4- r +
For, save for the difference in phase #, these equations become
equations (2) if the coordinate system be rotated through 90
about the ^-axis.
By superposition of the two rays (2) and (9), i.e. by taking
POLARIZATION 25 ,
the sums u + u r , v -f v r , w + w', there results, according to
the rule given above [equation (n) page 131], for the squares
of the amplitudes of the three components
A' 2 = A 2 + B* + 2AB cos (8 + ^ - /),
" = ^2 B* - 2AB cos d -
i cos .
Addition of these three equations gives, in consideration of (8),
J' = 2j + 2C 2 cos tf - 4^^ sin tf sin (q /).
Since now experiment shows that J' is equal simply to the
sum of the intensities of the separate rays and is wholly inde-
pendent of tf, it follows that = o, i.e. the light vector is
perpendicular to the direction of propagation, or the wave is
transverse; it also follows that sin (p q] = o, i.e., from (5)
or (7), the vibration form is a straight line.
Hence rays which have suffered double refraction or reflec-
tion at the polarizing angle are plane-polarized transverse
waves.
Since, as was shown on page 244, the properties of a
polarized ray must be symmetrical with respect to its plane of
polarization, it follows that the light vector must lie either in
the plane of polarization or in the plane perpendicular to it.
Whether it lies in the first or the second of these planes is a
question upon which light is thrown by the following experi-
ment.
6. Stationary Waves produced by Obliquely Incident
Polarized Light. Wiener investigated the formation of sta-
tionary waves by polarized light which was incident at an
angle of 45 (cf. page 155), and found that such waves were
distinctly formed when the plane of polarization coincided with
the plane of incidence, but that they vanished completely when
the plane of polarization was at right angles to the plane of
incidence. The conclusion is inevitable that the light vector
which produces the photographic effect * is perpendicular to the
* The same holds for the fluorescent effect produced by stationary waves.
Cf. foot-note, p. 156 above,
252 THEORY OF OPTICS
plane of polarization; for stationary waves can be formed only
when the light vectors of the incident and reflected rays are
parallel. When they are perpendicular to each other every
trace of interference vanishes.
It will be seen later that, from the standpoint of the elec-
tromagnetic theory, the above question has no meaning if merely
the direction of the vector be taken into account. For in that
theory, and in fact in any other, two vectors which are at
right angles to each other (the electric and the magnetic force)
are necessarily involved. However, the question may well be
asked, which of these two vectors is determinative of the light
phenomena, or whether, in fact, both are. If both were
determinative of the photographic effect, then in Wiener's
experiment no stationary waves could have been obtained even
with perpendicular incidence, since the nodes of one vector
coincide with the loops of the other, and inversely, as will be
proved in the later development of the theory of light. But
the fact that stationary waves are actually observed proves
that, for the photo-chemical as well as for the fluorescent
effects, only one light vector is determinative; and indeed that
it is the one which is perpendicular to the plane of polarization
is shown by the experiments in polarized light mentioned
above.
The phenomena shown by pleochroic crystals like tourma-
line lead also to the same conclusions.
7. Position of the Determinative Vector in Crystals. In
crystals the velocity depends upon the direction of the wave
normal and upon the plane of polarization. Similarly in the
pleochroic crystals the absorption of the light depends upon
the same quantities. Now it appears * that these relations are
most easily understood upon the assumption that the light vector
is perpendicular to the plane of polarization. For then the
velocity and the absorption t of the wave depend only upon the
* This is more fully treated in Section II, Chap. II, 7.
f The fluorescence phenomena in crystals lead also to the same conclusion.
Cf. Lommel, Wied. Ann. 44, p. 311.
POLARIZATION 253
direction of the light vector with respect to the optical axis of
the crystal. The following example will illustrate: A plate
of tourmaline cut parallel to the principal axis does not change
color or brightness when rotated about that axis, i.e. when the
light is made to pass through obliquely, but its direction is kept
perpendicular to the axis. But the brightness of the plate
changes markedly if it be rotated about an axis perpendicular
to the principal axis of the crystal. The plane of polarization
of the emergent ray is in the first case perpendicular to the
principal axis, i.e. to the axis of rotation of the plate; in the
second case it is parallel to this axis. The vector which is
perpendicular to the plane of polarization is, therefore, in the
first case continually parallel to the principal axis of the plate,
but in the second it changes its position with respect to this
axis.
Thus far no case has been observed in which a light vector
which lies in the plane of polarization is alone determinative
of the effects, i.e. furnishes the simplest explanation of the
phenomena. Hence in view of what precedes it may be said :
The light vector is perpendicular to the plane of polarization *
8. Natural and Partially Polarized Light. It has been
shown above that two plane-polarized beams may be obtained
by double refraction from a single beam of natural light.
Superposition of two plane -polarized rays which have the same
direction but different phases and azimuths produces, as is
shown by equation (5), elliptically polarized light. The vibra-
tion in such a ray is, however, wholly transverse, since the
plane of the ellipse is perpendicular to the direction of propa-
gation.
As will be fully shown later, elliptically polarized light is
produced by the passage of a plane-polarized beam through a
doubly refracting crystal whenever the two beams produced
by the double refraction are not separated from each other.
* At least this assumption gives a simpler presentation of optical phenomena
than the other (which is also possible) which makes the light vector parallel to the
plane of polarization.
254 THEORY OF OPTICS
Also the most general case, represented by equations (2), of
elliptically polarized light which is not transverse can be
realized by means of total reflection or absorption, as will be
shown later.
The question now arises, What is the nature of natural
light ? Since it does not show different properties in different
azimuths, and yet is not identical with circularly polarized light,
because, unlike circularly polarized light, it shows no one-sided-
ness after passing through a thin doubly refracting crystal, the
only assumption which can be made is that natural light is
plane or elliptically polarized for a small interval of time tf/,
but that, in the course of a longer interval, the vibration form
changes in such a way that the mean effect is that of a ray
which is perfectly symmetrical about the direction of propa-
gation.
Since Michelson has observed interference in natural light
for a difference of path of 540,000/1 (cf. page 150), it is
evident that in this case light must execute 540,000 vibrations
at least before it changes its vibration form. But since a
million vibrations are performed in a very short time, namely,
in 20. io~ 10 seconds, the human eye could never recognize a
ray of natural light as polarized even though several million
vibrations were performed before a change occurred in the
vibration form. For, in the shortest interval which is neces-
sary to give the impression of light, the vibration form would
have changed several thousand times.
As regards the two laws announced by Fresnel and Arago
(cf. page 247), the second, namely, that two rays polarized at
right angles interfere when they are brought into the same
plane of polarization provided they originated in a polarized
ray, is easily understood; for in this case the original ray has
but one vibration form, hence the two reuniting rays must be
in the same condition of polarization, i.e. must be capable of
interfering. This is the case also when the original ray is
natural light so long as the vibration form does not change,
i.e. within the above-mentioned interval 6t. But for another
POLARIZATION 255
interval #/', although interference fringes must be produced,
the position of these fringes is not the same as that of the
fringes corresponding to the first interval 6t. For a change in
the vibration form of the original ray is equivalent to a change
of phase. Hence the mean intensity, taken over a large num-
ber of elements 8t, is equivalent to a uniform intensity, i.e.
two rays polarized at right angles to each other, which origi-
nated in natural light, do not interfere even though they are
brought together in the same azimuth. This is the first of the
Fresnel-Arago laws.
The term partially polarized light is used to denote the
effect produced by a superposition of natural light and light
polarized in some particular way. Partially polarized light has
different properties in different directions, yet it can never be
reduced to plane polarized light, as can be done with light
which has a fixed vibration form (cf. below).
9. Experimental Investigation of Elliptically Polarized
Light. In order to obtain the vibration form of an elliptically
polarized ray, it is changed into a plane-polarized ray by means
of a doubly refracting crystalline plate. For, as was remarked
upon page 242, the passage of plane-polarized light through
a doubly refracting crystal decomposes it into two waves
polarized at right angles to each other. The directions of the
light vectors in the two waves are called the principal direc-
tions of vibration. These have fixed positions within the
crystal and are perpendicular to each other. Since now the
two rays are propagated with different velocities within the
crystal, they acquire a difference of phase which depends upon
the nature and thickness of the plate. An incident light vector
which is parallel to one of these two principal directions of
vibration within the crystal is not decomposed into two waves.
Two methods of procedure are now possible : first, the
plate of crystal may be of such thickness that it introduces a
difference of phase of (difference of path JA) between the
two waves propagated through it. This is called a quarter- wave
256 THEORY OF OPTICS
plate (Senarmonfs compensator}. If the quarter- wave plate is
rotated until its principal directions are parallel to the principal
axes of the elliptical vibration form of the incident light, the
emergent light must evidently be plane-polarized, and the
position of its plane of polarization must depend upon the ratio
of the principal axes of the incident ellipse. For the two light
vectors which lie in the directions of the principal axes of this
ellipse have, after passage through the plate, a difference of
phase of o or TT, and in this case there results (cf. page 250)
plane-polarized light in which the direction of the light vector
is given by equation (7). Hence if the emergent light is
observed through a nicol, entire darkness is obtained when the
nicol is in the proper azimuth. Hence this method of investi-
gation requires a rotation both of the crystalline plate about
its normal and of the nicol about its axis until complete dark-
ness is obtained. The position of the crystal then gives the
position of the principal axes of the incident ellipse; that of
the nicol, the ratio of these axes.
Second, a fixed plate of variable thickness, such as a quartz
wedge, may be used in order to give those two components of
the incident light which are in the principal directions of vibra-
tion of the plate such a difference of phase that, after passage
through the crystal, they combine to form plane-polarized
light. A nicol is used to test whether or not this has been
accomplished. The position of the nicol gives the ratio of the
components u, v, of the incident light, while their original
difference of phase is calculated from the thickness of the plate
which has been used to change the incident light into plane-
polarized light.
In order that the crystal may produce a difference of phase
zero, it is convenient to so combine two quartz wedges, whose
optical axes lie in different directions, that they produce differ-
ences of phase of different sign. Thus,
for example, in Fig. 79, A is a wedge
FIG. 79. . of quartz whose crystal! ographic axis
is parallel to the edge of the wedge, while B is another plate
POLARIZATION 257
whose principal axis is perpendicular to the edge but parallel
to the surface (Babinefs compensator). Only the difference
in the thickness of the two wedges is effective. Hence, if the
incident light is homogeneous and elliptically polarized, a suit-
able setting of the analyzing nicol brings out dark bands which
run parallel to the axis of the wedge. These bands move
across the compensator if one wedge is displaced with reference
to the other. A micrometer screw effects this displacement.
After the instrument has been calibrated by means of plane-
polarized light, it is easy from the reading on the micrometer
when a given band has been brought into a definite position
to calculate the difference of phase of those two components
u, v, which are parallel to the two principal axes of the quartz
wedges.
The construction must be somewhat altered if it is desired
to obtain a large uniform field of plane-polarized light. Then,
in place of a quartz wedge, a plane parallel plate of quartz
must be used as a compensator.
Such a plate is produced by com-
bining two adjustable quartz wedges
whose axes lie in the same direc-
FIG. 80.
tion (Fig. 80). In order to make
it possible to introduce a difference of phase zero, the two
wedges are again combined with a plane parallel plate of
quartz B whose principal axis is at right angles to the axes of
A and A' \ so that the effective thickness is the difference
between the thickness of B and the sum of the thicknesses of
the wedges A and A 1 '. This construction, that of the Soleil-
Babinet compensator, is shown in Fig. 80. In the wedges A,
A' the principal axis is parallel to the edges of the wedges; in
the plate B the principal axis is perpendicular to the edge and
parallel to the surface. It is convenient to have one plate, for
example A', cemented to B, while A is micrometrically adjust-
able. For a suitable setting of the micrometer and the
analyzing nicol the whole field is dark.
This construction of the compensator is particularly con-
258 THEORY OF OPTICS
venient for studying the modifications which plane-polarized
light undergoes upon reflection or refraction. In a spectrom-
eter (Fig. 81) the collimator K and the telescope F are fur-
nished with nicol prisms whose orientations may be read off
on the graduated circles /, /'. The Soleil-Babinet compen-
FIG. 81.
sator C is attached to the telescope. Its principal directions
of vibration (the principal axes) are parallel and perpendicular
to the plane of incidence of the light. 5 is the reflecting or
refracting body. Thus the light is parallel in passing through
the nicols and the compensator.*
* Since the telescope must be focussed for infinity, the simple Babinet compen-
sator cannot be used.
SECTION II
OPTICAL PROPERTIES OF BODIES
CHAPTER I
THEORY OF LIGHT
I. Mechanical Theory. The aim of a theory of light is to
deduce mathematically from some particular hypothesis the
differential equation which the light vector satisfies, and the
boundary conditions which must be fulfilled when light crosses
the boundary between two different media. Now the differen-
tial equation (12) on page 169 of the light vector is also the
general equation of motion in an elastic medium, and hence it
was natural at first to base a theory of light upon the theory
of elasticity. According to this mechanical conception, a light
vector / must be a displacement of the ether particles from their
positions of equilibrium, and the ether, i.e. the medium in
which the light vibrations are able to be propagated, must be
an elastic material of very small density.
But a difficulty arises at once from the fact that light-waves
are transverse. In general both transverse and longitudinal
vibrations are propagated in an elastic medium ; but fluids which
have no rigidity are capable of transmitting longitudinal vibra-
tions only, while solids which are perfectly incompressible can
transmit transverse vibrations only. The fact that the heavenly
bodies move without friction through free space would point
strongly to the conclusion that the ether is a fluid, not an in-
259
260 THEORY OF OPTICS
compressible solid. Nevertheless this difficulty may be met
by the consideration that, with respect to such slowly acting
forces as are manifested in the motions of the heavenly bodies,
the ether acts like a frictionless fluid; while, with respect to
the rapidly changing forces such as are present in the vibra-
tions of light, a slight trace of friction causes it to act like a
rigid body.
But a second difficulty arises in setting up the boundary
conditions for the light vector. The theory of elasticity fur-
nishes six conditions for the passage of a motion through the
bounding surface between two elastic media, namely, the
equality on both sides of the boundary of the components of
the displacements of the particles, and the equality of the com-
ponents of the elastic forces. But in order to satisfy these
six conditions both transverse and longitudinal waves must be
present. How the various mechanical theories attempt to
meet this difficulty will not be considered here : * suffice it to
say that most of these theories retain only four of the boundary
conditions.
In order to bring theory into agreement with the observa-
tions upon the properties of reflected light, for instance to
deduce Brewster's law as to the polarizing angle (cf. page
246), it is necessary to assume either that the density or that
the elasticity of the ether is the same in all bodies. The
former standpoint was taken by F. Neumann, the latter by
Fresnel. Neumann's assumption leads to the conclusion that
the displacement of the ether particles in a plane-polarized ray
lies in the plane of polarization, while Fresnel's makes it per-
pendicular to this plane.
2. Electromagnetic Theory. The fundamental hypothe-
sis of this theory, first announced by Faraday, and afterwards
mathematically developed by Maxwell, is that the velocity of
light in a non-absorbing medium is identical with the velocity of
* For complete presentation cf. Winkelmann's Handbuch, Optik, pp.
641-674.
THEORY OF LIGHT 261
an electromagnetic wave in the same medium. Either the elec-
tric or the magnetic force may be looked upon as the light
vector; both are continually vibrating and, in a plane-polarized
ray, are perpendicular to each other. This two-sidedness of
the theory leaves open the question as to the position of
the light vector with respect to the plane of polarization;
nevertheless, for the reasons stated on page 252, it is simpler
to interpret the electric force, which lies perpendicular to the
plane of polarization, as the light vector. This leads to the
results of Fresnel's mechanical theory, while Neumann's re-
sults are obtained when the magnetic force is interpreted as the
light vector.
The following are the essential advantages of the electro-
magnetic theory:
1. That the waves are transverse follows at once from
Maxwell's simple conception of electromagnetic action,
according to which there exist only closed electrical circuits.
2. The boundary conditions hold for every electromag-
netic field. It is not necessary, as in the case of the mechan-
ical theories, to make special assumptions for the light
vibrations.
3. The velocity of light in space, and in many cases in
ponderable bodies also, can be determined from pure electromag-
netic experiments. This latter is an especial advantage of this
theory over the mechanical theory, and it was this point which
immediately gained adherents for the electromagnetic concep-
tion of the nature of light. In fact it is an epoch-making
advance in natural science when in this way two originally
distinct fields of investigation, like optics and electricity, are
brought into relations which can be made the subject of quan-
titative measurements.
Henceforth the electromagnetic point of view will be main-
tained. But it may be remarked that the conclusions reached
in the preceding chapters are altogether independent of any
particular theory, i.e. independent of what is understood by a
light vector.
362 THEORY OF OPTICS
3. The Definition of the Electric and of the Magnetic
Force. Two very long thin magnets exert forces upon each
other which appear to emanate from the ends or poles of the
magnets. The strengths of two magnet-poles m and m^ are
defined by the fact that in a vacuum, at a distance apart
r, they exert upon each other a mechanical force (which can be
measured in C. G. S. units)
CD
In accordance with this equation a unit magnetic pole (in = i)
is defined as one which, placed at unit distance from a like
pole, exerts upon it unit force.
The strength $ of a magnetic field in any medium* is the
force which the field exerts upon unit magnetic pole. The
components of along the rectangular axes x, y y z will be
denoted by a, /?, y.
The direction of the magnetic lines of force determines the
direction of the magnetic field; the density of the lines, the
strength of the field, since in a vacuum the strength of field is
represented by the number of lines of force which pass per-
pendicularly through unit surface. A correct conception of the
law offeree (i) is obtained if a pole of strength m be conceived
as the origin of ^nm lines of force. For then the density of
the lines upon a sphere of radius r described about the pole as
centre is equal to m : r 2 , i.e. is equal to the strength of field
, according to law (i).
Similar definitions hold in the electrostatic system for the
electric field.
The quantities of two electric charges e and e l are defined
by the fact that in a vacuum, at a distance apart r, they exert
upon each other a measurable mechanical force
The definition of unit charge is then similar to that of unit pole
above.
* This medium can be filled with matter or be totally devoid of it.
THEORY OF LIGHT 263
The strength g of any electric field in any medium is the
force which it exerts upon unit charge. The components of g
along the three rectangular axes will be denoted by X, F, Z.
The direction of the electric lines of force determines the
direction of the electric field, and the number of lines which
intersect perpendicularly unit surface in a vacuum determines
the strength g of the field. Hence, since law (2) holds, 47^
lines offeree originate in a charge whose quantity is e.
4. Definition of the Electric Current in the Electrostatic
and in the Electromagnetic Systems. In the electrostatic sys-
tem the electric current i which is passing through any cross-
section q is defined as the number of electrostatic units of quan-
tity which pass through q in unit time. Thus if, in the element
of time dt, the quantity de passes through q, the current is
de
If the cross-section q is unity, i is equal to the current
density/. The components of the current density, namely,
Jxt Jy> Jz> are obtained by choosing q perpendicular to the
x-, y-, or ^-axis respectively.
In the electromagnetic system, the current i' is defined by
means of its magnetic effect. A continuous current is obtained
in a wire when the ends of the wire are connected to the poles
of a galvanic cell. In this case also definite quantities of elec-
tricity are driven along the wire, for the isolated poles of the
cell are actually electrically charged bodies. A magnetic pole
placed in the neighborhood of an electric current is acted upon
by a magnetic force. In the electromagnetic system the current
i' is defined by the fact that it requires ^.Tti' [ units of work
to carry unit magnetic pole once around the current. *
Take, for example, a rectangle whose sides are dx, dy
(Fig. 82), and through which a current i' = j' z -dxdy flows in a
* The work 51 is independent of both the path of the magnet pole and the
nature of the medium surrounding the current. Cf. Drude, Physik des Aethers,
PP- 77, 83.
264 THEORY OF OPTICS
direction perpendicular to its plane. j' z is the #- component of
the current density in the electromagnetic system. If the cur-
rent flows toward the reader (Fig. 82), and the positive direc-
tion of the coordinates is that shown in the figure, then, accord-
ing to Ampere's rule, a positive magnetic pole is deflected in
the direction of the arrow. The whole work 5( done in mov-
ing a magnet pole m = -f- I around the circuit from A through
B, C, D, and back to A is
$ = a-dx -\-ft'-dy a'-dx fi-dy, ... (4)
if a and /? denote the components of the magnetic force which
act along AB and AD, while a' and ft' denote the components
which act along DC and BC. ' differs from a only in that it
acts along a line whose j-coordinate is dy greater than the
j/-coordinate of the line AB along which a acts. When dy is
sufficiently small (a 1 #) : dy is the differential coefficient
3<*:d so that
Similarly
so that, from (4),
Since now by the definition of the current i' this work is
equal to 4^2' = ^nj'^dx dy, it follows that
and in the same way the two other differential equations may
be deduced, namely,
(5)
THEORY OF LIGHT 265
These are Maxwell's differential equations of the electro-
magnetic field. In order to use them with the signs given in
(5), the coordinate system must be chosen in accordance with
Fig. 82. In these equations the current density j' defined
electromagnetically may be replaced by the current density j
defined electrostatically by introducing c, the ratio of the elec-
tromagnetic to the electrostatic unit. Thus
i : i' = c, j x \ j' x = c, etc (6)
Hence, by (5),
These equations are independent of the nature of the
medium in which the electromagnetic phenomena occur (cf.
note i, page 263), and hence they hold also in non-homogeneous
and crystalline media.
The value of the ratio c can be obtained by observing the
magnetic effect which is produced by the discharge of a quan-
tity e of electricity measured in electrostatic units. It may be
shown that c has the dimensions of a velocity. Its value is
c 3 io 10 cm. /sec.
5. Definition of the Magnetic Current. Following the
analogy of the electric current, the magnetic current which
passes through any cross-section q is defined as the number of
units of magnetism which pass through q in unit time. The
magnetic current divided by the area of the surface q is called
the density of the current, and its components are represented
by**, V s *'
Equations (7) express the fact that an electric current is
always surrounded by circular lines of magnetic force. But on
the other hand a magnetic 3[
current must always be sur- j> c
z
rounded by circular lines of
electric force. This follows
at once from an application A B
of the principle of energy.
Imagine the rectangle ABCD of Fig. 82 traversed by an elec-
266 THEORY OF OPTICS
trie current of intensity i (measured in electrostatic units) flow-
ing in the direction of the arrows. Then a positive magnetic
pole would be driven through the rectangle toward the reader,
i.e. in the positive direction of the ^-axis, and would continually
revolve about one side of the rectangle. The work thus per-
formed must be done at the expense of the amount of energy which
is required to maintain the current at the constant intensity i
while it is doing the work ; or, in other words, the motion of
the pole must create a certain counter-electromotive force which
must be overcome if the current is to remain constant. The
expression for the work done when a unit charge is carried
once about the rectangle in the direction of the arrows is
analogous to that given in (4) and (4'), i.e.
In order to maintain the current at intensity i during the time
/, this work must be multiplied by the number of unit charges
which traverse the circuit in the time /, i.e. by/'-/. The prin-
ciple of energy requires that this work ty.it be equal to the
work which is done upon a magnet pole of strength m in
carrying it once around a side of the rectangle in the time /.
Since (cf. page 263) this work is equal to ^irmi' 4^mi\ c, it
follows that
^{'i-t = 47rmi : c, i.e. 91 ^m : ct. . . (9)
But m\ t is the strength of the magnetic current which passes
through the rectangle, and m/t-dx dy is equal to the ^-com-
ponent of the magnetic density. Hence from (8) and (9) it
follows that
47t 3F *dX
s z - -- -^ ...... (10)
c dx 9/
And similarly two other equations for s x and s y are obtained.
In (10) X and Y represent the electric forces which must
be called into play in order to keep the current constant. But
THEORY OF LIGHT 267
if X and Y denote the opposite forces produced by the mag-
netic current by induction, they are of the same magnitude but
opposite in sign. Hence
47T dY dZ 47f QZ dX 47T
These equations are perfectly general and hold in all media,
even in those which are non-homogeneous and crystalline.
The general equations (7) and (11) may be called the
fundamental equations of Maxwell' s theory. In all extensions
of the original theory of Maxwell to bodies possessing
peculiar optical properties, such as dispersion, absorption,
natural and magnetic rotation of the plane of polarization,
these fundamental equations remain unchanged. But the
equations which connect^ and s x , etc., with the electric and
magnetic forces have different forms for particular cases.
6. The Ether. Constant electric currents can only be
produced in conductors like the metals, not in dielectrics.
Nevertheless a change in an electric charge produces in the
latter currents which are called displacement currents to dis-
tinguish them from the conduction currents, and the corner-
stone of Maxwell's theory is the assumption that these dis-
placement currents have the same magnetic effects as the
conduction currents. This assumption gives to Maxwell's
theory the greatest simplicity in comparison with the other
electrical theories. Constant magnetic currents cannot be
produced, since there are no magnetic conductors.
It is first necessary to determine how the electric and
magnetic current densities in the free ether depend upon the
electric and magnetic forces. In the free ether there are no
charges e or poles m concentrated at given points, but there
are lines of force. Now, in accordance with the convention
adopted on pages 262 and 263, namely, that every charge e or
pole m sends out ^ne or ^nm lines of force, it may be said
that 47r multiplied by the current density is equal to the change
in the density of the lines of force in unit time, i.e.
268
THEORY OF OPTICS
in which N x , N y , N z , M x , M y , M 2 are the components of the
densities of the electric and magnetic lines offeree. But now,
in accordance with the definitions on pages 262 and 263, in a
vacuum the density of the electric or magnetic lines of force is
numerically equal to the electric or magnetic force, so that, for
a vacuum, equations (12) become
dX dY dZ 1
4;;== 37, 4% = -^, 4%==-gj, i
}. . (13)
da dp dy \
4 7r ^x == 7> 4 7f ^v = ~ ~cw~> 4 7 *s = T^T.
Q c d* J
Hence for the free ether the equations (7) and (n) of the
electromagnetic field take the form
da
dY
dz
i
c
dP
dz
dX
~ 3*'
i
c
dr
dt
dX
I"Y
dt
~ dz
~ ,-. >
dy
dt '
dx
dy
'die' j
7. Isotropic Dielectrics. For a space filled with insulat-
ing matter laws (i) and (2) must be modified. For if the
electric charges e and e l are brought from empty space into a
dielectric, for example a fluid, they exert a weaker influence
upon each other than in empty space, so that it is necessary
to write
The constant e is called the dielectric constant. The definition
holds also for solid bodies, only in them the attracting or
repelling forces cannot be observed so conveniently as in fluids.
But there are other methods of determining the dielectric con-
stant of solid bodies for which the reader is referred to texts
THEORY OF LIGHT 269
upon electricity. The dielectric constant of all material bodies
is greater than I.
Similarly the forces between magnetic poles are altered
somewhat when the poles are brought from a vacuum into a
material substance, so that it is necessary to write
i mm,
The constant fit is called the permeability of the substance.
It is sometimes greater than i {paramagnetic bodies), some-
times less than I (diamagnetic bodies). It differs appreciably
from i only in the paramagnetic metals iron, nickel, and
cobalt. At present dielectrics only are important since it is
desired to consider first perfectly transparent substances,
namely, those which transmit the energy of the electromagnetic
waves without absorption, i.e. without becoming heated. In
dielectrics /* differs so little from I (generally only a few
thousandths of i per cent) that in what follows it will always
be considered equal to i .*
Because of the change of the law (2) into (15) a change
must also be made in equations (13), since with the same cur-
rents the electric force in the dielectric is - weaker than in the
e
free ether. Hence (13) become
-da
etc., ns x = /^--, etc. . . (17)
For an isotropic dielectric, since equations (7) and (n) are
applicable to this case also, the following equations hold when
(18)
edX 'dy dfi e dY _d<* dy e dZ _ dfi
c a/ dy dz' c dt dz dx j c dt dx
i dot dY dZ i d/3 dZ dX i dy _ dX
cdt dz dy'c dt dx dz' c dt dy
* In the discussion of the optical properties of magnetized bodies it will be
shown why it is justifiable to assume for light vibrations /< = i for all bodies.
The reason for this is not that the magnetization of a body cannot follow the rapid
changes of field which occur in light vibrations, but is far more complicated.
270 THEORY OF OPTICS
These equations completely determine all the properties of
the electromagnetic field in a dielectric.
If equations (12) be considered general, i.e. if the number
of lines of force which originate in a charge be considered
independent of the nature of the medium, then a comparison
of (17) with (12) shows that within the body
i.e. only in the ether (e = i, /* i) is the density of the lines
of force numerically equal to the electric ', or the magnetic, force.
4?r e lines of force must be sent out from the entire surface
of an elementary cube which contains the charge e and has the
dimensions dx dy dz. But the number of emitted lines can
also be calculated from the surface of the cube; thus the two
sides which lie perpendicular to the ^r-axis emit the number
(N x \(fy dz -\- (N^^dy dz, in which the indices i and 2 relate
to the opposite faces which are dx apart. Now evidently, from
the definition of a derivative,
so in this way the whole number of lines passing out of the
surface is found to be
f- -zr- \dx-dy dz.
oz I
If this expression be placed equal to ^.ne, then it follows, in
consideration of (19), if e : dx dy dz = p be called the density
of the charge (charge of unit volume),
It is evident from its derivation that this equation holds also
for isotropic non-homogeneous bodies, i.e. for bodies in which e
varies with x, y, z. An analogous equation may be deduced
for the density of the magnetization.
THEORY OF LIGHT 271
8. The Boundary Conditions. If two different media are
in contact, there are certain conditions which the electric and
magnetic forces must fulfil in passing from one medium into
the other. These conditions may be obtained from the equa-
tions (18) by the following consideration: In the passage from
a medium of dielectric constant e l to one of dielectric constant
e 2 the change in the electric and magnetic forces is not
abrupt, as would be the case if the surface of separation were
a mathematical plane, but gradual, so that within the transi-
tion layer the dielectric constant varies continuously from the
value j to the value e 2 . Also within this transition layer the
equations (7), (u), and (17), and hence also (18), must hold,
i.e. all the differential coefficients which appear in them must
remain finite. Assume now, for example, that the plane of
contact between the two media is the ;rj/-plane. Since the
9/5f -da
differential coefficients -^ , , , must remain finite
oz oz oz oz
within the transition layer, it follows that, if the thickness of
this layer, i.e. dz> is infinitely small, the changes in Y, X,
/?, a in the transition layer are infinitely small. In other
words, the components of the electric and magnetic forces parallel
to the surface must vary continuously in passing through the
transition layer, assumed to be infinitely thin. That is,
X 1 = X 2 , Y l 3= F 2 , ! = ,, A = ft a for * = o, (21)
in which the subscripts refer to the two different media.
Since in equations (18) the differential coefficients ^ and ^~
^z cz
do not appear, the same conclusions do not hold for Z and y
which held for X, F, /?, ex. Nevertheless it is evident from the
'oy
last of equations (18) that , and hence also y, has the same
value on both sides of the transition layer, because, for all
values of x and y, X and Y have the same values on both sides
of that layer. Hence there is no discontinuity in y in passing
through the infinitely thin boundary layer. In the same way
the conclusion may be drawn from the third of equations (18)
272 THEORY OF OPTICS
that the product eZ is continuous and hence that Z is discon-
tinuous. To the boundary conditions (21) there are then also
to be added
But on account of the existence of the principal equations
(18) only four of the six equations (21) and (21') are independent
of one another.
Equation (19) in connection with (21) shows that the lines
of force do not have free ends at the boundary between tzvo media.
(N.B in (21') /* is assumed equal to I, otherwise it would be
necessary to write J^ l y l = /* 2 X 2 -
9. The Energy of the Electromagnetic Field. If equa-
tions (18) be multiplied by the factors Xdr, Ydr, Zdr, adr,
fidr, ydr, in which dr represents an element of volume, and
then integrated over any region, there results, after adding and
setting
C T
(22)
(23)
The application of theorem (20) on page 173 gives
ds -
in which dS denotes an element of the surface which bounds
the region over which the integration is taken, and n the inner
normal to dS. When this transformation is applied to the first
three integrals which appear on the right-hand side of (23) the
volume integrals disappear, and there results
^.(dr=^ [_(yY- ftZ) cos (nx) + (aZ - yX) cos (ny)
+ (fiX - F
^ 3/ 2 " = a* 2 H ~ c)/ " a* 2 ~M^ r + "^r
Also differentiation of the first three of the equations (18)
with respect to ;r, y, z, and addition of them gives
= 0<
Since in what follows we are only concerned with periodic
changes in the electric and magnetic forces, and since for
these the differential coefficient with respect to the time is
proportional to the changes themselves (when the phase
has been added), the conclusion may be drawn from the
last equation that
Hence equation (i) becomes
e -&X
'
Similar equations hold for Y and Z, so that the following
system of equations is obtained :
For the components of the magnetic force similar equations
hold, thus
(20
276 THEORY OF OPTICS
Now it has been shown on page 1 70 that differential equa-
tions of the form of (3) and (3') represent waves which are
propagated with a velocity
This is then, according to the electromagnetic view of the
nature of light, the velocity of light, and it is immaterial
whether the electric or the magnetic force be interpreted as the
light vector, for the two are inseparably connected and have
the same velocity.
Applying equation (4) to the case of the free ether, it fol-
lows that the velocity of light in ether is equal to the ratio of
the electromagnetic to the electrostatic units. This conclusion
has actually been strikingly verified, for (cf. page 119) the
mean of the best determinations of the velocity of light was
seen to be V =. 2.9989-10 cm. /sec., a number which agrees
within the observational error with that given for the ratio of
the units, namely, c = 3- io 10 cm. /sec.
This is the first b+ ^liam success of the electromagnetic
theory.
According to (4) the velocity in ponderable bodies musf
be I Ve smaller than in the free ether, or, since the index of
refraction n Q of a body with respect to the ether is the ratio of
the velocities in ether and in the body,
(5)
i.e. the square of the index of refraction is equal to the dielectric
constant.
Evidently this relation cannot be rigorously fulfilled, for
the reason that the index depends for all bodies upon the color,
i.e. upon the period of oscillation, while from its definition e is
independent of the period of oscillation.
But in case of the gases, in which the dependence of the
index upon the color is small, the relation (5) is well satisfied,
as is shown by the following table, in which the values of the
TRANSPARENT ISOTROPIC MEDIA
277
dielectric constants are due to Boltzmann,* while the indices
are those for yellow light :
V~e
Air
OOO 2Q4.
I OOO 2QC
ooo 138
I OOO 172
Carbon dioxide . ...
OOO 4.J.O
I OOO J.77
ooo 74.6
I OOO 7d?
Nitrous oxide
.000503
i . ooo 497
Relation (5) also holds well for the liquid hydrocarbons; for
example, for benzole n (yellow) = 1.482, Ve = 1.49.
On the other hand many of the solid bodies, such as the
glasses, as well as some liquids, like water and alcohol, show
a marked departure from equation (5). For these substances
e is always larger than ;/ 2 , as the following table shows :
iT*
Water
I. -5 -2
90
1.74
c.7
Ethyl alcohol
1.76
e O
In order to explain these departures, the fundamental
equations of the electric theory must be extended. This
extension will be made in Chapter V of this section. In this
extension the quantity e which is here considered as constant
will be found to depend upon the period of oscillation.
But first an investigation will be made from the standpoint
of the electric theory of those optical properties of bodies which
do not depend upon dispersion. In what follows it will be
assumed that the light is monochromatic, and that the extension
to be given in Chapter V has already been made, so that the
constant e appearing ir the fundamental equations is equal to
the square of the index of refraction for the given color.
*L. Boltzmann, Wien. Ber. 69, p. 795, 1874. Fogg. Ann. 155, p. 407, 1873,
278 THEORY OF OPTICS
2. The Transverse Nature of Plane Waves. A plane
wave is represented by the equations
/
X = A X -COS -Tfrlt
V
27ft mx -\-ny-\- pz
Y = A y - cos -jr\t -
-r- , J mx -\-ny-\- pz
(6)
For the phase is the same in the planes
mx _|_ n y _|_ p z const (7)
which is then the equation of the wave fronts, m, n, and/ are
the direction cosines of the normal to the wave front, provided
the further condition be imposed that
rfj rf p+p =l (8)
A x , A y , A z are the components of the amplitude of the
resultant electrical force. They are then proportional to the
direction cosines of the amplitude A . In consequence of equa-
tion (2) on page 275,
A x *m + A,.n + A,-p = o, . . . . (9)
an equation which expresses the fact that the resulting ampli-
tude A is perpendicular to the normal to the wave front, i.e.
to the direction of propagation; or in other words, that the
wave is transverse. This conclusion holds for the magnetic
force also. That plane waves are transverse follows from equa-
tions (2) or (2'), i.e. from the form of the fundamental equa-
tions of the theory.
3. Reflection and Refraction at the Boundary between
two Transparent Isotropic Media. Let two media i and 2
having the dielectric constants e l and e 2 meet in a plane which
will be taken as the ;rj/-plane. Let the positive ^-axis extend
from medium i to medium 2 (Fig. 83). Let a plane wave fall
from the former upon the latter at an angle of incidence 0, and
let the .r^-plane be the plane of incidence. The direction
cosines of the direction of propagation of the incident wave are
then
m = sin 0, n =5 p, / = cos 0. . . . do]
TRANSPARENT ISOTROP1C MEDIA
279
Let the incident electric force be resolved into two com-
ponents, one perpendicular to the plane of incidence and of
amplitude E t , and one in the plane of incidence and of ampli-
tude E p . The first component is parallel to the jj/-axis so that,
in consideration of (6) and (10), the /-component of the incident
force may be written
27t x sin -\- z cos 0\
e *" - T* \ 77 /' * \* U
\ K 1
in which V l is the velocity of light in the first medium.
By (4),
V, = c: Ve t (12)
Since the wave is transverse, the component E p of the elec-
trical force, which lies in the plane of incidence, is perpendic-
ular to the ray, i.e. the components A x and A z , along the
x- and ^-axes, of the amplitude E p must have the values
A x = ^-cos 0, A 2 = jE^-sin 0,
if, as shown in Fig. 83, the positive direction of E p is taken
downward, i.e. into the second medium.
The x- and ^-components of the electric force of the inci-
dent wave are, therefore,
27t f x sin -f- z cos 0\
v,
2 7t I x sin -4- z cos \
+'7\t- X_ -j.
(13)
Now a magnetic force is necessarily connected with the
electric force in the incident wave, and from the fundamental
equations (18) on page 269, and (12) above, the components
of this force are found to be
04- ^cos
27t
a, = .,. cos yejcos-^
27t / x sin + z cos
27t f x sin -f- z cos
y e =-\-E f > sin i / e 1 cos - (/ ^
* * i
(H)
280
THEORY OF OPTICS
If E, = o,
then a f =y e = o, and /?, differs from
zero, i.e. the amplitude E p of the
electric force, which lies in the
plane of incidence, gives rise to a
component fi e of the magnetic force
which is perpendicular to the plane
of incidence. Conversely, the
component E s of the electric force,
which is perpendicular to the plane
of incidence, gives rise to a mag-
netic force which lies in the plane
of incidence. This conclusion that the electric and magnetic
forces which are inseparably connected are always perpendic-
ular to each other follows from the considerations already given
on page 274.
When the incident electromagnetic wave reaches the
boundary it is divided into a reflected and a refracted wave.
The electric forces in the reflected wave can be represented by
expressions analogous to those in (11) and (13), namely, by
* sin $' * COS
--
v
X
x sin 0' -f z cos
~^~
2?r / ;r sin 0' + # cos
Z r = Rj'sm cos \t y
The corresponding equations for the refracted wave are
x sin x + 2 cos x\
(15)
cos
*(t
rV~
x sn
^ 2
COS J
Z 2 =
27T
x sin X + z cos X
(16)
In these equations R p , R s , D p , D s denote amplitudes, 7
the angle of reflection, i.e. the angle between the -\- ,0-axis
TRANSPARENT 1SOTROPIC MEDIA 281
and the direction of propagation of the reflected wave, x tne
angle of refraction .
The corresponding magnetic forces are, cf. (14),
^
0'
27r L -^ sin 0' -(- cos 0'
y r = + R f .sin 0' Videos (t ....).
- 2nl x sin x + si cos x\
= - A -cos x - Ve 2 -cos -=[t - - -A_E
-(18)
On account of the boundary conditions (21) of the previous
chapter, there must exist between the electric (or the magnetic)
forces certain relations for all values of the time and of the
coordinates x and y. Such conditions can only be fulfilled if,
for z = o, all forces become proportional to the same function
of/, ;r, y, i.e. the following relations must hold:
sin sin 0' sin x
~y ~ V ~ y ( J 9)
v\ v\ v i
From the first of these equations it follows immediately that
sin 0= sin 0'; i.e., since the direction of the reflected ray
cannot coincide with that of the incident ray,
cos cos 0', i e. 0' = TT 0. . . (20)
This is the law of reflection, in accordance with which the
incident and reflected rays lie symmetrically with respect to
the normal at the point of incidence.
The second of equations (19) contains the law of refrac-
tion, since from this equation
sin : sin x V\ : ^2 == n > ( 2I )
282
THEORY OF OPTICS
in which n is the index of refraction of medium 2 with respect
to medium i.
The laws of reflection and refraction follow, then, from the
fact of the existence of boundary conditions and are altogether
independent of the particular form of these conditions.
As to the form of these conditions it is to be noted that
here X^ = X e + X r , with similar expressions for the other
components, since the electric force in medium I is due to a
superposition of the incident and reflected forces. Hence the
boundary conditions (21) on page 272 give, in connection with
(20),
(E p R p ) cos = D p cos x>
(22}
(E s - R s ) Ve l cos = D. 4/e 2 cos x,
From this the reflected and refracted amplitudes can be
calculated in terms of the incident amplitude. Thus:
cos
/ ^ e i cos \ n ( Ve. cos . \
\-7 --i) = R s (--- - + i),
\ V 2 COS X / \ *' 2 COS X /
Ve.
cos
+ ^
(23)
'cos
cos x
If the ratio i/e 2 : Ve v which, according to (4), is the index
of refraction n of medium 2 with respect to I, be replaced by
sin : sin x [cf. (21)], then (23) may be written in the form
sin (0 x) ^ tan (0 x)
I
(24)
*,= -
D E 2 S1 ' n ^ cos ^ r; r
J sin (0 + ^) ' * *
2 sin j cos
cos(0-j)'
TRANSPARENT ISOTROPIC MEDIA 283
These are Fresnel 's reflection equations, from which the
phase and the intensity of the reflected light can be calculated
in terms of the characteristics of the incident light.
It is seen from (24) that R s never vanishes, but that R p
becomes zero when
tan (0 + x) = oo, + =-, . . . (25)
z
i.e. when the reflected ray is perpendicular to the refracted ray.
In this case it follows from (25) that
sin x = sin ~ = cos 0> or > cf. (21),
tan = n ....... (25')
When, then, the angle of incidence has this value, the
electric amplitude in the reflected wave has no component
which lies in the plane of incidence, no matter what the nature
of the incident light, i.e. no matter what ratio exists between E
and E p . Thus if natural light is incident at an angle which
corresponds to (25'), the electric force in the reflected wave
has but one component, namely, that perpendicular to the
plane of incidence ; in other words, it is plane-polarized. Now
this angle actually corresponds to Brewster's law given
above on page 246. At the same time it now appears, since
the plane of incidence was called the plane of polarization, that
in a plane-polarized wave the light vector is perpendicular to
the plane of polarization, provided this vector be identified with
the electric force.
On the other hand the light vector would lie in the plane
of polarization if it were identified with the magnetic force, since,
by equation (17) (cf. also page 280), R p signifies the amplitude
of the component of the magnetic force which is perpendicular
to the plane of incidence. Neumann s reflection equations
would follow .from the assumption that the magnetic force is
the light vector.
The intensities of the reflected electric and magnetic waves
are equal. For, given incident light polarized in the plane of
284 THEORY OF OPTICS
incidence, in order to calculate the reflected intensity it is
necessary to apply only the first of equations (24), no matter
whether the electric or the magnetic force be interpreted as the
light vector. For, by (14) on page 279, E t is in every case
the amplitude of the incident light.
On the other hand the signs of the reflected electric and
magnetic amplitudes are opposite. This difference does not
affect the intensity, which depends upon the square of the
amplitude only, but it does affect the phase of the wave. This
will be more fully discussed for a particular case.
4. Perpendicular Incidence. Stationary Waves. Equa-
tions (24) become indeterminate when I, the
reflected electric amplitude is of opposite sign to the incident
amplitude. But the second equation asserts the same thing,
for, when = o, like signs of R p and E p actually denote oppo-
site directions of these amplitudes, as appears from the way in
which R p and E p are taken positive in Fig. 83 on page 280.
The stationary waves (cf. page 155) produced by the interfer-
ence of the incident and reflected waves must have a node at
the reflecting surface, which, to be sure, would be a point of
complete rest only if R s were exactly as large as E tJ i.e. if
n = oo . For finite n only a minimum occurs at the mirror,
since the reflected amplitude only partially neutralizes the
effect of the incident amplitude.
For the magnetic forces, however, E p and R p represent the
components of the amplitude which are perpendicular to the
plane of incidence, i.e. parallel to the j^-axis. Like signs of these
amplitudes represent actually like directions, so that it follows
from the second of equations (26) (also from the first, if the
proper interpretation be put upon the direction of the amplitudes
TRANSPARENT ISOTROPIC MEDIA 285
in space) that the reflected magnetic amplitude has the same
direction as the incident magnetic amplitude. Hence stationary
magnetic waves have a loop at the mirror itself if n > I .
Wiener's photographic investigation showed that at the
bounding surface between glass and metal a node was formed
at the surface of the mirror. This indicates that the electric
force is the determinative vector for photographic effects, as
was even more convincingly proved by the investigation of
stationary waves formed in polarized light at oblique incidence
(cf..page 251).
5. Polarization of Natural Light by Passage through a
Pile of Plates. From equation (24) it is seen that R s : E s
continually increases as increases from zero to -. On the
other hand Rp : E p first decreases, until it reaches a zero value
at the polarizing angle, and then increases to the maximum
7t
value i when = (grazing incidence). But for all angles
of incidence if E s E p , R s > R p . For, from (24),
*, Ep cos (0 + X )
R s ' E s 'cos (0 -
Hence at every angle of incidence natural light is partially (or
completely) polarized in the plane of incidence. And since
by assumption no light is lost, the refracted light must be
partially polarized in a plane perpendicular to the plane of
incidence. This explains the polarizing effect of a pile of
plates.
Also an application of the last two of equations (24) to the
two surfaces of a glass plate gives directly, for the passage of
the light through the plate,
&L = .L cos* (0 - *), .... (28)
in which D' s , D' p denote the amplitudes of the ray emerging
from the plate. Hence when E s ,, it follows from (28)
286 THEORY OF OPTICS
that D' t < D' pj i.e. incident natural light becomes by passage
through the plate partially polarized in a plane perpendicular
to the plane of incidence. To be sure, there is no angle
by
(28), when E s = E p ,
D' p ' ~ (i + *f
Hence when n = 1.5, D' s : D' p = 0.85, and the ratio of the
intensities Z/ 2 : D'f = 0.73. After passage through five plates
this ratio sinks to O.73 5 = 0.20, i.e. the light would still differ
considerably from plane-polarized light.
6. Experimental Verification of the Theory. Equations
(24) may be experimentally verified either by comparing the
intensities of the reflected and incident light, or more con-
veniently by measuring the rotation which the plane of polariza-
tion of the incident light undergoes at reflection or refraction.
The amount of this rotation may be calculated from equations
(27) or (28).
If the incident light is plane-polarized, the quantity a con-
tained in the expression for the ratio of the components,
namely, E p : E s = tan a, is the azimuth of the plane of polariza-
tion of the incident light. The reflected and refracted light is
likewise plane-polarized and the azimuth ^ of its plane of polar-
ization is determined by (27) and (28). Thus tan ^ = R p : J? s .
For the measurement of this angle it is convenient to use the
apparatus shown on page 258 without the Babinet compen-
sator. The incident light is polarized by means of the Nicol
/ (the polarizer), and the Nicol /' (the analyzer] is then turned
until the light is extinguished. The value of ?/> which corre-
sponds to any particular a can thus be observed.
* At this angle the transmitted light is by no means completely polarized.
TRANSPARENT ISOTROPIC MEDIA 287
Both methods furnish satisfactory verification of the laws of
reflection ; but Jamin found by very careful investigation that,
in the neighborhood of the polarizing angle, there is always a
departure from those laws, in that the polarization of the
reflected light is not strictly plane but somewhat elliptical.
Hence it cannot be entirely extinguished by the analyzer
unless the compensator is used. The explanation of thh
phenomenon follows.
7. Elliptic Polarization of the Reflected Light and the
Surface or Transition Layer. The above developments make
application of the boundary conditions (21) on page 271 and
rest upon the assumption that when light passes from medium
I to medium 2 there is a discontinuity at the bounding sur-
face. But strictly speaking there is no discontinuity in Nature.
Between two media I and 2 there must always exist a tran-
sition layer within which the dielectric constant varies continu-
ously from e l to e 2 . This transition layer is indeed very thin,
but whether its thickness may be neglected, as has hitherto
been done, when so short electromagnetic waves as are the
light-waves are under consideration, is very doubtful. Further-
more the thickness of this transition layer between two media
is generally increased by polishing the surface.
In any case the actual relations can be better represented
if a transition layer be taken into account.
Nevertheless, in order not to unnecessarily complicate the
calculation, it may be assumed that the thickness / of this
transition layer is so small that all terms of higher order than
the first in / may be neglected.
First the boundary conditions which hold for the electric
and magnetic forces at the two boundaries of the transition
layer will be deduced. These boundaries are defined as the
loci of those points at which the dielectric constant first attains
the values e l and e 2 respectively.
According to the remark of page 267 equations (18) on
page 269 hold within the transition layer also.
If the fourth and fifth of these equations (18) be multiplied
288 THEORY OF OPTICS
by an element dz of the thickness of the transition layer, and
integrated between the two boundaries I and 2, there results,
since the quantities involved do not depend upon y, provided y
be taken perpendicular to the plane of incidence,
-*2 M
(29)
Now, by (21) and (21') on pages 271 and 272, a, /3, and eZ
are approximately constant within the transition layer, so that
a, (3, and eZ may be placed before the sign of integration in
the above equations and replaced by a 2 , /3 2 , e 2 Z 2 (or by ^ ,
ft, e^). Thus
c ; C; r* z ; ^ z * r <**
J-*=*J*> 1 &*=&! ^
Introducing the abbreviation
/2 /*2 /*2 r
dz = /, j'dg = f, I -^ = q, . . (30)
in which / denotes the thickness of the transition layer and e
its dielectric constant at the point corresponding to the element
dz of the thickness, equations (29) become
I'd/3 3Z 2 / fiat
*> = * + 71? -*aF* F '=^-737' (30
Likewise the first two of equations (18) give, after multipli-
cation by dz, integration, and treatment as above,
Equations (31) and (32) take the place of the previous
boundary conditions (21) on page 271.
To determine the electric and magnetic forces in media I
and 2, equations (11), (13), (14), (15), (16), (17), (18) of this
chapter may be used, but with the limitation that the forces in
TRANSPARENT ISOTROPIC MEDIA 289
the reflected and refracted wave must differ in phase from the
incident wave by an amount which must be deduced from
equations (31) and (32). Without such a difference of phase
these equations cannot be satisfied.
Now these differences of phase may be most simply taken
into account in the following way: Write, for instance [cf.
equations (15), page 280],
Y r = R. cos , -
then Y r is the real part of the complex quantity
Writing now
R t -e* = R s , (33)
then
f .aw / _ x sin ^' -f- * cos <'\ )
F,= 9?JR S .* T v, ; j f .
(34)
in which the symbol 9^ means that the real part of the complex
quantity which follows it is to be taken. This complex
quantity within the brackets contains the amplitude R s which
is also complex, so that an advance in phase 6 which occurs in
Y r may be represented by setting Y r equal to the real part of
an exponential function containing a complex factor (complex
amplitude). The other electric and magnetic forces may be
treated in the same way.
Instead of performing the calculations with the real parts
only of the complex quantities, it is possible, when only linear
equations (or linear differential equations) are involved, to first
set the electric and magnetic forces equal to the complex
quantities and, at the end of the calculation, to take the real
parts only into consideration in determining the physical
meaning.
Thus in the previous equations (11), (13), (14), (15), (16),
(17), (18) for the electric and magnetic forces, the real ampli-
tudes E s , E p , R s , R p , etc., will be replaced by the complex
2 9 o THEORY OF OPTICS
amplitudes E s , E p , R s , R p , etc., and the cosines by the
exponential expression (cf. equation (34) ). Then the boundary
conditions (31) and (32) give, since they are to hold for z = o,
and since X l X e -)- X r , a l = a e -J- ct r , etc.,
(E p - R p ) cos = D p cos
s + R s =D s [i +i
-(35)
(E p + R p ) V = D p +i- cos X
From these equations R s , R p , D s , D p may be calculated
in terms of E s and E p . It is the reflected light only which is
here of interest. If the product Tc be replaced by A, the wave
length in vacuo of the light considered, and if F" 2 be replaced
by c : Ve 2 , then, from (35),
_ 27C
R cos \/e 2 -cos xV^+i ^ [> cos cos x -(l-q
W P
. 27T r
* L/o
> (36)
s cos 4/ei-fcos itf f\-i \l cos cos xV e \ 2-\-p? t sin * # 1
A L -*
Now it is to be remembered that the terms which contain
the factor i are very small correction terms, since they are
proportional to the thickness / of the transition layer. Hence
if the expressions (36) be developed to terms of the first power
only of the ratio / : A, there results
cos
Rp = cosftj/^-cosx^j j , ? -47T cQs ,-p cos 2 x le*+qeS sin 2 X )
* cos2 e i cos 2 x j
Rs cos
cos
cos ' 2
TRANSPARENT ISOTROPIC MEDIA 291
The denominator of the correction term which appears in
the second of these equations can never vanish, i.e. e t cos 2
can never be equal to e 2 cos 2 , for if e > e lt then always
> J, and hence cos < cos X- But the denominator of the
correction term of the first of equations (37) does vanish if
cos Ve 2 = cos x ^ ..... (38)
A simple transformation of (38) shows, since Ve 2 : Ve, = n,
that this condition is fulfilled for the polarizing angle 0, which,
according to Brewster's law, is determined by tan ~ n.
Hence for this angle of incidence it follows from (37), or also
directly from (36), that
P = cos (39)
EP A 1 (cos Ve 2 + cos x Vetf
Equations (37) can be further simplified by consideration
of the law of refraction, namely,
sin : sin x = n = ^ 2 : Ve r (40)
For from this it follows that
e l cos 2 e 2 cos 2 x = e !
2
6 2 cos 2 - e l cos 2 x = e 2 ( e i sin 2 - e 2 cos 2 0) j
Now the nature of the reflected light is completely deter-
mined by the ratio R p : R s . Assume that the incident light is
plane-polarized at an azimuth of 45 to the plane of incidence
(cf. page 286). Then E p = E s , and from (37) it follows, in
consideration of (40) and (41), that
Rp_ cos (0+x) ( ATT ee l cos sin 2 )
R s ~ ~cos(0-z) ( l * A 6l -e 2 '6 1 sin 2 0-6 2 cos 2 77 ) '
in which rj is an abbreviation for
rf = p - /( 6l + e 2 ) + qe^. . . . (43)
At the polarizing angle (tan = n) (42) assumes the value
R P .* ^ + 6,
=VI ..... (44)
292 THEORY OF OPTICS
as is seen most easily from (39) by dividing it by the second
of equations (37) and retaining terms of the first order only in
n- A.
In order now to recognize the physical significance of (42)
and (44) it must be borne in mind that, according to (33),
R P =* / .A R. = *,/', . . . (45)
in which R p and R s are the components which are respectively
parallel and perpendicular to the plane of incidence of the
amplitude of the reflected electric force, and d f and d s are the
advances in phase of these components with respect to the in-
cident wave. Hence
r>
'*)= p.e^, .... (46)
in which p is the ratio of the amplitudes and A the difference in
phase of the two components. Hence, from (44), it follows that
at the polarizing angle
/: i .
A = x/2, . . . (47)
i.e. the reflected light is not plane-polarized in the plane of
incidence as it was above shown to be when the transition
layer was not considered, but it is elliptically polarized. The
principal axes of the ellipse are parallel and perpendicular to
the plane of incidence (cf. page 249) and their ratio is p. p will
be called the coefficient of ellipticity. By (43), (47), and (30)
this may be written
_ __ Tt VX + 2 Ae - 6 t )(e - 6 2 )
in which the integration is to be extended through the transi-
tion layer between the two media.
According to (48) ~p is positive if the value of the dielectric
constant e of the transition layer varies continuously between
the limiting values e l and e 2 , and if e 2 > e r But if at any
point within the transition layer e > e, and also e > e, , then p
(48)
TRANSPARENT ISOTROPIC MEDIA 293
is negative when e 2 > e r The relations are inverted when
e t > e 2 , i.e. when the medium producing the reflection has the
smaller refractive index. In consideration of the way in which
the amplitude R p is taken positive (cf. Fig. 83, page 280), it is
evident that, if the coefficient of ellipticity p is positive, the
direction of rotation of the reflected light in its elliptical
vibration form is counter-clockwise to an observer standing in
the plane of incidence and looking toward the reflecting sur-
face, provided the incident electrical force makes an angle of
45 with the plane of incidence and is directed from upper left
to lower right. But if p is negative, then when the same con-
ditions exist for the incident electrical force, the direction of
rotation of the reflected electrical force is clockwise.
Also for any other angle of incidence the reflected light is
always elliptically polarized, even though the incident light is
plane-polarized, for there is always a difference of phase A
between the /- and ^-components, which, according to (42)
and (46), has the value
n e 2 Ve, cos sin 2
tan A = 4 ~r~ *1 ~ L ~ - - ^ ( 4Q)
A ! ~ e 2 e, sin 2 - e 2 cos 2 0'
while the ratio p of the amplitudes does not depart appreciably
from the normal value
cos (0 + *)
which is obtained without the consideration of a surface layer.
In consideration of (47), (49) may be written
# 2 sin tan
tan A = 4p . 9 , 2 . . . (51)
\/i _|_ #2 tan 2 n 2
On account of the smallness of p the difference of phase is
appreciable only in the neighborhood of the polarizing angle,
for which tan = n.
These theoretical conclusions have been completely verified
by experiment. For, in the first place, it is observed that
294 THEORY OF OPTICS
when the angle of incidence is that determined by Brewster's
law, the reflected light is not completely (though very nearly)
plane-polarized, since it is not possible to entirely extinguish it
with an analyzing Nicol. The results of the investigation of
the elliptic polarization of reflected light by means of the
analyzer and compensator (cf. page 255) are in good agreement
with equations (50) and (51).
It is further found that the coefficient of ellipticity is smaller
the less the reflecting surface has been contaminated by con-
tact with foreign substances. Thus, for example, it is very
small at the fresh surfaces of cleavage of crystals, and at the
surfaces of liquids which are continually renewed by allowing
the liquid to overflow. For polished mirrors p is considerable.
The change in the sign of p when the relations of the two
media are interchanged is in accord with the theory. The
theory is also confirmed by the fact that, in the case of reflec-
tion from polished surfaces, ~p is in general positive. Only in
the case of media which have relatively small indices of refrac-
tion, like fluor-spar (n = 1.44) and hyalite (n = 1.42), has /)
been observed to be negative. This also might be expected
from the theory, provided the index of refraction of the
polished transition layer were greater than that of the
medium.
For well-cleaned polished glass surfaces, when the reflec-
tion takes place in air, the value of p lies between 0.03 (for
heavy flint glass of index n 1.75) and 0.007.
For liquids in contact with air the value of ~p does not
exceed o.oi. Water has a negative coefficient of ellipticity
which, when the surface is thoroughly cleaned, may be as
small as 0.00035. There are also so-called neutral liquids
like glycerine which produce no elliptic polarization by reflec-
tion. According to the theoretical equation given above for
the coefficient of ellipticity it is not necessary that these liquids
have no transition layer, i.e. that an actual discontinuity occur
in the dielectric constants in passing from the air to the liquid.
Rather, layers which have intermediate values of the dielectric
TRANSPARENT ISOTROPIC MEDIA 295
constant may exist, provided only other layers whose dielectric
constant is greater than that of the liquid are also present.
When the coefficient of ellipticity is positive (for reflection
in air) it is possible to determine a lower limit for the thickness
of the transition layer. For evidently, for a given positive
value of p, the smallest thickness which the transition layer
can have is attained when its dielectric constant is assumed to
be a constant whose value is determined by making the factor
(e e i)(e e 2 ) . / \ T-I ^
in equation (48) a maximum. This is the case
when e = Ve^, i.e. when the dielectric constant of the transi-
tion layer is a geometrical mean of the dielectric constants of
the two media. Hence, from (48), the lower limit / for the
thickness of the transition layer is given by
L- g . V ** + ^ = - _j "+ 1 (52)
A. ~ xVe^e^ Ve 2 - Ve l ~ n V I + ri> n - I
in which n denotes the index of refraction of the medium 2
with respect to the medium I (air). Thus for flint glass, for
which 0=1.75, 7> = -3 ( cf - P a g e 2 94)> T: A = 0.0175.
Hence the assumption of a transition layer of very small thick-
ness is sufficient to account for a very strong elliptic polarization
in reflected light.
8. Total Reflection. Consider again the case in which the
light incident in medium I is reflected from the surface of
medium 2. If the index n of 2 with respect to I is less than
i , the angle of refraction j which corresponds to the angle of
incidence
i (53)
At this angle of incidence there is then no refracted
light, but all of the incident light is reflected (total reflection).
In order to determine in this case the relation between the
nature of the reflected light and that of the incident light, the
method used in 3 of this chapter must be followed. The
discussion and the conclusions there given are applicable. In
29 6 THEORY OF OPTICS
order to avoid the use of the angle of refraction x in equations
(22), (23), and (24), sin X may be regarded as an abbreviation
for sin : n, so that cos X may be replaced by
cos =
If sin > n, this quantity is imaginary. In order to bring
this out clearly the imaginary unit V i i will be introduced,
thus:
/sin 2
cos X = -H/--3--I.* (54)
Equations (23) must hold under all circumstances, t for they are
deduced from the general boundary conditions for the passage
of light through the surface between two isotropic media, and
these conditions always hold, whether total reflection occurs
or not. But when (54) is substituted in (23) the amplitudes in
the reflected light become complex, even when those of the
incident light are real. From the physical meaning of a com-
plex amplitude which was brought out on page 289, it is
evident that in total reflection the reflected light has suffered a
change of phase with respect to the incident light.
In order to calculate this change of phase, write, in accord-
ance with (45), for the reflected amplitudes which appear in
(23) the complex quantities R p e*p, R s e { ^, so that from (23) and
(54), since Ve 2 : Ve l = n,
i cos ^ r> a / * cos ^ \ 1
,\ r- (55)
t'sin 2 /> 2 Vsin 2 -
=
;>
* Cos x must be an imaginary with a negative sign. According to the condk
tions which are to be fulfilled, either a positive or a negative value of cos x would
be possible. This could be physically realized only if the medium 2 were a plate
upon both sides of which light were incident at the same angle 0, which must also
be greater than the critical angle. This appears from the considerations in 9.
f The transition layers will here be neglected. They have but a small influence
upon total reflection; cf. Drude, Wied. Ann. 43, p. 146, 1891.
TRANSPARENT ISOTROPIC MEDIA 297
In order to obtain the intensities of the reflected light, i.e.
the values of R\ and R* p , it is only necessary to multiply equa-
tions (55) by the conjugate complex equations, i.e. by those
equations which are obtained from (55) by substituting i
for i.* The result is
Z72 /? 2 772 _ > 2
E' ** ** *
i.e. the intensity of the reflected light is equal to that of the
incident light (total reflection). This holds also for each of
the components (the s and /) separately.
The absolute differences of phase ti g and ^ will not be dis-
cussed, but the relative difference A = 6^ 6 s is of interest
because, according to page 292, the vibration form of the
reflected light is obtained from it. Division of the first of
equations (55) by the second gives, when E s = E py i.e. when
the incident light is plane-polarized at an azimuth of 45 with
respect to the plane of incidence, since then, according to
/cos Vsin 2 .0 n* ^ _ g /cos 0+ Vsin* n 2
icos0-n -- 4/sin 2 ri* 2cos0-/z-| I/sin 2 n 2
From this it follows that
(57)
sin 2 -f- i cos 4/sin 2 ri*
sin 2 / cos I sin 2 n*
Hence
i e*A / cos t/sin 2 ;/ 2
i + e*A sin 2 '
If this equation be multiplied by the conjugate complex
expression, there results, since j
' (66)
Now repeated reflections and refractions take place at the
surfaces of the plate (cf. above, page 137); but it is not neces-
sary to follow out each one of these separately, since their
total effect can be easily brought into the calculation.* This
effect consists in the propagation of waves within the plate
along both the positive and the negative directions of the
.s'-axis. For the former the following equations hold:
-- t ft ' = 0t r ' = .}
while for the latter
t Z" = o\
a" = D")/f'+* ft" = o, y" = o. )
Let the total effect of all the waves which have passed
through the plate be
'"'<-
It is now necessary to apply at both sides of the plate
(z = o, z = d ) the boundary conditions (21) on page 271,
which here take the form
F,+ F r = F'+F", , + =' + " for^-o, . (70)
a" = c< d forz = d. . (71)
The conditions (70) give
~ (700
* Equations (66) are to represent the total effect of all the separate waves which
are propagated in medium I along the negative 2-axis.
3 o 4 THEORY OF OPTICS
and the conditions (71)
D'e-# + D"e + # = De-*' 9
(D'e- > - D"e+ '>) V^ = Zfc - '>
in which / and q are abbreviations for
2.7t d d in d
. .__ . .
From (71') follows at once
(D'e ~V + D"e + V) Ve[ = (D'e -** D"e + *) VT % ,
from which is deduced
D'e ~ '>( i/F 2 - V^) = /?' V + <>( VF 2 + i/^). . . (73)
From (70'),
E + R _ Z? 7 + D" V7,
E -HR" Z> r - U'' ^
i.e.
R^ _ D'( V^ - V7 2 ) 4- D"( 4/^ + ^F 2 )
J '' = D'( V^ + i/^) + /?"( vT t - t/F 2 )'
In consideration of (73) this last may be written
R +-^-6-6
* sin /(e 1 + e 2 ) + 2 Vefr cos /
In order to obtain the intensity J r of the reflected light,
this equation must be multiplied by the conjugate complex
equation (cf. page 297). Thus, when J e denotes the intensity
of the incident light, there results
sin/(6 1 -6,y sin* /(i -tff
provided e 2 : e^ = n*, so that n is the index of the plate 2 with
respect to medium I .
TRANSPARENT ISOTROPIC MEDIA 305
From (70') and (7 1 ') it is easy to deduce the equation
+
i sin /( e i + e 2 ) + 2 1/ e^. cos /
So that the intensity J A of the transmitted light is
(75)
Hence the relation holds
Jd + Jr = J.> (76)
as was to be expected, since the plate absorbs no light
According to (74) the reflected light vanishes completely
when/ o, n, 2n, etc., i.e. when the thickness of the plate
d = o, JA 2 , A 2 , fA 2 , etc. This is in agreement with the results
deduced from equation (17) on page 139. A maximum of
/i _ ; /V
intensity occurs when sin / = i. Then J r = yJ -J .
[In the case of normal reflection at one surface only, equation
(26) on page 284 gives J r = /,(f^|) ]
If media I and 2 are air and glass, n = 1.5. In the case
of Newton's rings these media are glass and air, so that
n = i : 1.5. In both cases equation (74) becomes
sin 2 / i. 56
^ r ~~ 'sin 1 /- 1. 56 +9'
Hence, for an approximation, the term sin 2 X r ^ 2 ) 2 1>n tne
denominator of (74) may be neglected in comparison with 4// 2 ,
so that at a point in the Newton ring apparatus at which the
thickness of the air film is d,
f **'/* (77)
306 THEORY OF OPTICS
A denotes the wave length in air. If the incident light is
white, and if J x denotes the intensity in the incident beam of
light of wave length , then the intensity of the reflected light
is, provided dispersion or the dependence of n upon A be
neglected,
*'i* . . (78)
The colors of thin plates are then a mixture composed of
pure colors in a manner easily evident from (78).
12. Non-Homogeneous Media : Curved Rays. The opti-
cal properties of a non-homogeneous medium, in which the
dielectric constant e is a function of the coordinates x t y, z, will
be briefly considered. The most logical way of doing this
would be to integrate the differential equations (18) on page
269; for these hold for non-homogeneous media also. To do
this e must be given as a function of x, y, and z. This method
would give both the paths of the rays and the intensities of the
reflections necessarily taking place inside of a non-homogene-
ous medium. But even with the simplest possible assumption
for e this method is complicated and has never yet been carried
out. Investigation has been limited to the determination of
the form of the rays from Snail's law or Huygens' principle
a process which succeeds at once if the medium be conceived
to be composed of thin homogeneous layers having different
indices. When the index varies continuously, the ray must of
course be curved. Heath * has deduced for its radius of curva-
ture p at a point P the equation
I _ d log n
~p = ~~dv~' ( 79 )
in which v denotes the direction of most rapid change (decreas-
ing) of the index n.
This equation explains the phenomenon of mirage, which
is observed when the distribution of the density of the air over
* Heath, Geometric il Optics. Cambridge, 1897.
TRANSPARENT ISOTROPIC MEDIA 307
the earth's surface is abnormal, as is the case over heated
deserts. At a certain height above the earth the index n of
the air is then a maximum. But in this case, by (79), P = oo ,
i.e. at this height the ray has a point of inflection. Hence two
different rays can come from an object to the eye of an ob-
server, who then sees two images of the object, one erect, the
other inverted.*
An interesting application of the theory of curved rays has
been made by A. Schmidt. t He explains the appearance of
the sun by showing that a luminous sphere of gas of the dimen-
sions of the sun, whose density increases continuously from
without towards the interior, would have sharp limits, as the
sun appears to have. For a ray of light which travels towards
such a sphere of gas so as to make an angle less than a certain
angle with the line drawn from the observer to the centre of
the sphere is deflected toward the centre of the sphere and
passes many times around that centre. It thus attains depths
from which a continuous spectrum is emitted, for an incan-
descent gas emits such a spectrum when the pressure is suffi-
cient. But a ray which makes an angle greater than with
a line drawn to the centre of the sphere must again leave the
sphere without having traversed intensely luminous layers.
Although there is no discontinuity in the sun's density yet it
appears as a sharply bounded disc which subtends a visual
angle 20.
For the experimental presentation of curved rays cf.
J. Mace de Lepinay and A. Perot (Ann. d. chim. et d. phys.
(6) 27, page 94, 1892); also O. Wiener (Wied. Ann. 49, page
105, 1893). The latter has made use of the curved rays in
investigations upon diffusion and upon the conduction of heat.
* A more complete discussion of these interesting phenomena with the refer-
ences is given in Winkelmann's Handb., Optik, pp. 344-384.
f A. Schmidt, Die Strahlenbrechung auf der Sonne. Stuttgart, 1891.
. CHAPTER III
OPTICAL PROPERTIES OF TRANSPARENT CRYSTALS
i. Differential Equations and Boundary Conditions. A
crystal differs from an isotropic substance in that its properties
are different in different directions. Now in the electromag-
netic theory the specific properties of a substance depend solely
upon its dielectric constant, provided the standpoint taken on
page 269, that the permeability of all substances is equal to
unity, be maintained.
Now an inspection of the deduction of the differential
equations for an isotropic body as given upon pages 269 sq.
shows that equations (17) contain only the specific properties
of the body, i.e. its dielectric constants. But equations (7)
and (n) are also applicable to crystals, as has been already
remarked. Hence only equations (17) need to be extended,
since in a crystal the dielectric constant depends upon the
direction of the electric lines of force. The most general
equations for the extension of ( 1 7) are
(0
4*7,- 6 21 ^ + 6 22 3, + e 23~37
3Z
4*7. = e si+ ^32- + 6
33 ,
since the components of the current must always remain linear
'dX dY 3Z
functions of -= , , . Equations (i) assert that in general
in a crystal the direction of a line of current flow does not
308
PROPERTIES OF TRANSPARENT CRYSTALS 309
coincide with the direction of a line of force, since if, for
example, Y and Z vanish while X remains finite, j and j t do
not vanish.
Equation (23) on page 272 for the flow of energy may be
deduced by multiplying the general equations (9) and (n),
namely,
_ -
~c jx " : "" 3*' ' ' ' ~c s * ~
by Xdr, . . . adr, and integrating with respect to r. (dr repre-
sents element of volume.) The result is
in which ( represents the energy in the volume element d T.
This equation may also be applied to crystals, since the specific
properties of the medium do not appear in it. Hence the
change in the electromagnetic energy in unit volume with
respect to the time is
Since the last three of equations (17) on page 269 hold in
this case also (when /i = i) the last three terms of this equation
are a differential coefficient with respect to the time, i.e.
Consequently j^X -\-j y Y-{-j z Z must also be a differential
coefficient with respect to the time. In order that this may be
possible in consideration of (i), the following conditions must
be fulfilled:
3 io THEORY OF OPTICS
and in this case the part (5^ of the energy which depends upon
the electric forces is
i
By means of a transformation of coordinates (j may always
be reduced to the canonical form
(4)
When the coordinates have been thus chosen the factors
vanish and equations (i) take the simplified form
_ __ _ __ _ _ _ c
J * ~ 47T -dt' Jy ~ 4?t Vt ' Jt 47T *dt '
These coordinate axes are characterized by the fact that
along their direction the electric current coincides with the
direction of the electric force. These rectangular axes will be
called axes of electric symmetry, since the crystal is symmetrical
in its electrical properties with respect to them, or also with
respect to the three coordinate planes which they define.
e i 6 2 e s signify the dielectric constants corresponding to the
three axes of symmetry. They will be called the principal
dielectric constants.
As was remarked above, the assumption will be made that
the permeability of the crystal is the same in all directions.
Although this is not rigorously true, as is evident from the
tendency shown by a sphere of crystal when hung in a power-
ful magnetic field to set itself in a particular direction, yet
experiment justifies the assumption in the case of light vibra-
tions.*
Hence in the differential equations (18) on page 269, which
apply to isotropic media, only such modifications are necessary
* The theoretical reason for setting n = I in the case of the light vibrations will
be given later, in Chapter VII.
PROPERTIES OF TRANSPARENT CRYSTALS 311
as are due to the fact that the dielectric constant has different
values in different directions. The dielectric constant appears
in only the first three of equations (18). These equations assert
that the components of the current are proportional to the quan-
tities -- ~~, etc. Since the components of the current in a
^y 02
crystal are given by equations (i) and (5), the general differ-
ential equations (7) and (n)of the electromagnetic field on
pages 265 and 267 become for a crystal, when its axes of
electric symmetry have been chosen as coordinate axes,
_ .2 = _ -_
> 3 ~~ a* ' ~c a/ "" a* fa' c ^t " fa ^z *
I2fL -- L'M -. l -^L = - i_i ( 7 )
c ~a7 a# aj ' c a/ " ~ fa BS ' c a/ BJK a*
When referred to any arbitrary system of coordinates,
equations (6) must be replaced by
O ** \ O / C A-' / x- / \
=, 3 ) = -~ ^~ etc. . (6)
The conditions which must be fulfilled at the bounding sur-
face between two crystals, or between a crystal and an isotropic
medium, for example air, may be obtained from the considera-
tions which were presented in 8 of Chapter I, page 271.
They demand that, in passing through the boundary, the com-
ponents of the electric and magnetic forces parallel to the
boundary be continuous.
2. Light-vectors and Light-rays. In the discussion of
isotropic media on page 283 it was shown that different
interpretations of optical phenomena are obtained accord-
ing as the light-vector fs identified with the electric or with
the magnetic force. Both courses accord with the results of
experiment if the phenomena of stationary waves be left out
of account. The case is similar in the optics of crystals, save
that there is here a third possibility, namely, that of choosing
the electric current as the light-vector. Its components are
3 i2 THEORY OF OPTICS
then proportional to 6 1 -^r, 2~^7> e s?J7' Thus in the optics
of crystals there are three possible theories which differ from
one another both as regards the position of the light-vector
with respect to the plane of polarization, and also as regards
its position with respect to the wave normal in the case of
plane waves. As to the latter difference it appears from page
278 that the light-vector is perpendicular to the wave normal
in the case of plane waves (i.e. plane waves are transverse), if
its components, which will here be represented by u, v> and zu,
satisfy the differential equation
Differentiation of equations (7) with respect to x, y, and
and addition of them gives, as above on page 275,
i.e. the waves are transverse if the magnetic force is taken as
the light- vector.
If the same operation be performed upon the three equations
(6), there results
?>X\
i.e. the waves are likewise transverse if the electric current be
interpreted as the light- vector.
But the waves are not transverse if the electric force is
taken as the light-vector, since, in consequence of the last
equation, because of the differences between e t , e 2 , and e 3 ,
the following inequality exists: *
BX BF tZ
The plane of polarization is defined by the direction of the
wave normal and the magnetic force, as was shown on page
283 to be the case for isotropic media.
PROPERTIES Of TRANSPARENT CRYSTALS 313
Thus the characteristics of the three possible theories of
the optics of crystals are the following :
1 . The magnetic force is the light-vector. Plane waves are
transverse; the light-vector lies in the plane of polarization.
(Mechanical theory of F. Neumann, G. KirchhofT, W. Voigt,
and others.)
2. The electric force is the light-vector. Plane waves are
not strictly transverse; the light-vector is almost perpendicular
to the plane of polarization. (Mechanical theory of Ketteler,
Boussinesq, Lord Rayleigh, and others.)
3 . The electric current is the light-vector. Plane waves are
transverse; the light-vector lies perpendicular to the plane of
polarization. (Mechanical theory of Fresnel.)
These differences in the theory cannot lead to observable
differences in phenomena so long as the observations of the
final light effect are made in an isotropic medium upon ad-
vancing, not stationary, waves. No other kinds of observations
are possible in the case of crystals. Hence nothing more can
be done than to solve each particular problem rigorously, i.e.
in consideration of its special boundary conditions.
The system of differential equations and boundary condi-
tions to be treated are then completely determined, and there
results one definite value for the electric force in the outer
isotropic medium no matter what is interpreted as the light-
vector in the crystal. The results which can be tested by
experiment are the same whether the magnetic force or the
electric force is taken as the light-vector in the outer medium.
For, according to the fundamental equations, the intensity of
the advancing magnetic wave is always the same as the
intensity of the advancing electric wave.
The electromagnetic theory has then the advantage that it
includes a number of analytically different theories and shows
why they must lead to the same result.
A ray of light was defined on page 273 as the path of the
energy flow. According to the equation given on page 310
for the electromagnetic energy in crystals, equation (23) on
3 T 4 THEORY OF OPTICS
page 272 for the flow of energy holds for crystals also. The
direction cosines of the ray of light are then also in the crystal
proportional to the quantities f x , f y , f z , defined in equation
(2 5) on page 273.
The ray of light is then perpendicular both to the electric,
and to the magnetic force. In general it does not coincide
with the normal to a plane wave, since from the inequality (i i)
this normal is not perpendicular to the electric force.
3. FresnePs Law for the Velocity of Light. In order to
find the velocity of light in crystals, it is necessary to deduce
from equations (6) and (7) such differential equations as
contain either the electric force alone or the magnetic force
alone. The former are obtained by differentiating the three
equations (6) with respect to t and substituting for , , -^-,
Of Ot Of
which appear upon the right-hand side, their values taken
from (7). Thus from the first of equations (6)
e^X 3 i^X VY\ 3 I'bZ t*X\
~&~W '"^byVSy" ' ~^TJ~ aAl)F~ 3* /'
The right-hand side of this equation can be written in the
more symmetrical form
3Z
Similarly, from the two other equations of (6),
. (12)
'bx Tty 3^
From the discussion of the preceding paragraph it appears
that only analytical differences result from differences in the
choice of the light-vector. In order to bring the discussion
into accord with Fresnel's theory, the light-vector will be
assumed to be proportional to the electric current. Let u. r,
PROPERTIES OF TRANSPARENT CRYSTALS 315
Wj be the components of the light-vector for plane waves,
thus:
* = eX = A cos t -
in which it is assumed that
9ft 2 + Sft 2 +
b > c, the
THEORY OF OPTICS
sections of the wave surface by the planes of symmetry are
shown in Fig. 85. In the ^-plane, for two directions of the
wave normal, which are denoted by A l and A 2 , the two roots
V v and V 2 of necessity coincide, since the two sheets of the
normal surface intersect. It can be shown that this occurs for
IG. 85.
no other directions of the wave normal ; for the quadratic equa-
tion in V 2 is, by (18),
<*) + V 4- a 1 } + / V + />') i
=0. ... (22)
(23)
- F 2 -!
If the following abbreviations be introduced :
M= m*(b* - r 2 ), N = n^ - a 2 ), P = f(d* -
the solution of (22) is
-f
- 2MN
^WP^MP. \ {24]
PROPERTIES OF TRANSPARENT CRYSTALS 319
Now since a > b > <:, M and P are positive, N negative.
Since the quantity under the radical may be put in the form
(M+N PJ* - 4MN,
it is made up of two positive terms. Hence when the two
roots in V* are equal, the two following conditions must be
satisfied:
M+N- P = o, MN=o.
Now M cannot be zero, since in that case N= P, which is
impossible, for N is negative and P positive. Consequently
the expression under the radical vanishes only when
N = o, M = P,
i.e. when
n = O, m\P - **) = / 2 (0 2 - 2 ), . . (25)
or since m -J- n* + / 2 = J > when
- d*
These equations determine the two directions of the wave
normals for which the two velocities are the same. These
directions are called the optic axes. The axes of electric
symmetry x and z which bisect the angles between the optic
axes are called the median lines of the crystal.
The value of the common velocity of the two waves when
the wave normal coincides with an optic axis is V v = F 2 = b.
This is evident from Fig. 85 as well as from equation (24)
taken in connection with (26). Hence, from (19), the direction
of vibration of these waves is indeterminate, since an indeter-
minate expression, namely, n : ft f 72 = o : o, occurs in these
equations. Hence along the optic axis any kind of light may
be propagated, i.e. light polarized in any way, or even natural
light.
The velocity V can be calculated more conveniently by
introducing the angles g^ and g 2 which the wave normal
makes with the optic axes than by the use of (24). Let the
320
THEORY OF OPTICS
positive direction of one of the optic axes A l be so taken that
it makes acute angles with the positive directions of the x- and
2-axes. The direction cosines of this axis are then, by (26),
_ 2
Let the positive direction of the other optic axis A 2 be so taken
that it makes an acute angle with the ^-axis but an obtuse
angle with the .r-axis. Its direction cosines are then
7~^~P
Hence the cosines of the angles ^ and g 2 between the
wave normal and the positive directions of A l and A 2 are
cos =
nn
x ,
.e.
+ P
A 2 -
= m A / -=
y ^
- fr
7 2
(27)
+t
In consequence of the relation n 2 = I m 2 p 2 it is easy to
deduce the following:
cos g 2 , (28)
= a 2 + <* + (a* c*} cos
2NP - 2MP
Hence, from (24),
- COS -
6. Geometrical Construction of the Wave Surface and of
the Direction of Vibration. Fresnel gives the following geo-
metrical construction for finding, with the aid of a surface called
an ovaloid, the velocity and the direction of vibration: Let
PROPERTIES OF TRANSPARENT CRYSTALS 321
the direction cosines of the radius vector of the ovaloid be
^i ^2 > ^3- The equation of the ovaloid is then
P 2 = a *$2 + t>*$? + r># 3 2 , .... (30)
a, b, and c being its principal axes. In order to obtain the
velocity of propagation of a wave front, pass a plane through
the centre of the ovaloid parallel to the wave front, and deter-
mine the largest and the smallest radii vectores p l and p 2 of
the oval section thus obtained. These are equal to the veloci-
ties of the two waves, and the directions of p l and p 2 are the
directions of vibration in the waves, the directions p } and p 2
corresponding to the velocities p l and p 2 respectively.
In order to prove that this construction is correct, account
must be taken of the fact that $ lt $ 2 , $ 3 must also satisfy both
of the conditions
i = V + V+'V, (31)
o = m& l + fl a +/&,.... (32)
The last equation is an expression of the fact that the oval
section is perpendicular to the wave normal. In order to
determine those directions & l , $ 2 , $ 3 for which p has a maxi-
mum or a minimum value, $j, $ 2 , $ 3 may, in accordance
with the rules of differential calculus, be regarded as indepen-
dent variables provided equations (31) and (32) be multiplied
by the indeterminate Lagrangian factors & l and cr 2 , and added
to equation (30). By setting the separate differential coeffi-
cients of p 2 with respect to fy, 2 , $ 3 equal to zero, there
results
o = 2<>2 + o-j)^ + mo- 2 , \
o = 2 (/ 2 + **= -&*&=& (34)
If these equations be multiplied by m, n, and / respec-
tively and added, then it follows from (32) that
^ ^ __ _
^ .- p* -r p _ p2 -t- ^ _ p * -
i.e. p actually satisfies the same equation as the velocity V
[cf. equation (18), page 316].
From (34) it follows that fy , 2 , # 3 stand in the same
ratio to one another as 9ft, $1, and ^ in (19), i.e. the direction
of the light-vector is that of the maximum or minimum radius
vector of the oval section.
Since, by 5, the direction of vibration is indeterminate in
the case in which the wave normal coincides with one of the
optic axes, the oval section has in this case no maximum or
minimum radius vector, i.e. the intersections with tJie ovaloid
of planes which are perpendicular to the optic axes are circles.
The radii of these two circles are the same and equal to b.
Any arbitrary oval section of a plane wave whose normal is N
cuts the two circular sections of the ovaloid in two radii
vectores 1\ and r 2 which have the same length b. These radii
r l and r 2 are perpendicular to the planes which are defined by
the wave normal N and the one or the other of the optic axes
A l and A 2 \ since, e.g., r^ is perpendicular to N as well as
to A r Hence these planes (NA^) or (NA 2 ) also cut the oval
section of the ovaloid by the plane wave in two equal radii r^
and r 2 , since r^ is perpendicular to r lt and r 2 to r 2 . Also,
since r^ = r 2 , it follows, from the symmetry of the oval section,
that r/ = r 2 ', and that the principal axes p l and P 2 of this sec-
tion bisect the angles between r l and r 2 , 1\ and r% . The
directions of vibration of the light-vectors (which coincide with
p l and P 2 ) lie in the two planes which bisect the angles formed
by the planes (NA^ and (NA 2 ). Thus the directions of the
PROPERTIES OF TRANSPARENT CRYSTALS 323
vibrations are determined, since they are also perpendicular to
the wave normals N. The direction of vibration which corre-
sponds to F 2 [defined by (29)] lies in the plane which bisects
the angle (A l , N, A 2 ), in which A l and A 2 denote the positive
directions of the optic axes defined by (26') ; the direction of
vibration corresponding to V l is perpendicular to this plane,
i.e. in the plane which bisects the angle (A lt A 7 , A 2 ).
7. Uniaxial Crystals. When two of the principal veloci-
ties a, b, c are equal, for example when a = b, the equations
become much simpler. From (26) on page 319 it follows that
both optic axes coincide with the -axis. Hence these crystals
are called uniaxial. From (29) it follows, since g l = g' 2y that
V? = a\ V} = d> cos 2 g + c* sin 2 g, . . (35)
in which g denotes the angle included between the wave
normal and the optic axis. One wave has then a constant
velocity; it is called the ordinary wave. The direction of
vibration of the extraordinary wave lies, according to the con-
struction of the preceding page, in the principal plane of the
crystal, i.e. in the plane defined by the principal axis and the
normal to the wave. The direction of vibration of the ordinary
wave is therefore perpendicular to the principal plane of the
wave. Since the principal plane of the wave was defined above
(page 244) as the plane of polarization of the ordinary wave,
the direction of vibration is perpendicular to the plane of polar-
ization, as is the case from Fresnel's standpoint for isotropic
media. When the angle g which the wave normal makes with
the optic axis varies, N remaining always in the same principal
section, the direction of vibration of the ordinary wave remains
fixed, while that of the extraordinary wave changes. Hence,
as was mentioned on page 252, 7, Fresnel's standpoint has
the advantage of simplicity in that the direction of vibration is
alone determinative of the characteristics of the wave. If this
is unchanged, the velocity of the wave is unchanged even
though the direction of the wave normal varies.
Uniaxial crystals belong to those crystallographic systems
3 2 4 THEORY OF OPTICS
which have one principal axis and perpendicular to it two or
three secondary axes, i.e. to the tetragonal or hexagonal
systems. The optic axis coincides with the principal crystal-
lographic axis. The crystals of the regular system do not
differ optically from isotropic substances, since from their
crystallographic symmetry a = b = c.
Rhombic, monoclinic, and triclinic crystals can be optically
biaxial. In the first the axes of crystallographic symmetry
coincide necessarily with the axes of electric symmetry, since
in all its physical properties a crystal has at least that sym-
metry which is peculiar to its crystalline form. In monoclinic
crystals the crystalline form determines the position of but one
of the axes of electric symmetry, since this latter is perpendic-
ular to the one plane of crystallographic symmetry. In
triclinic crystals the axes of electric symmetry have no fixed
relation to the crystalline form.
In the case of uniaxial crystals (a = b) the ovaloid becomes,
according to (30), the surface of revolution
/# = a* + (^ - **)& ..... (36)
According as this surface is flattened or elongated in the direc-
tion of the axis, the crystal is said to be positively or negatively
uniaxial. Thus in the former a > , in the latter a < c.
According to (35), in positive crystals the ordinary wave
travels faster, i.e. is less refracted, while in negative crystals
the ordinary wave is more strongly refracted than the extraor-
dinary. Quartz is positively, calc-spar negatively, uniaxial.
8. Determination of the Direction of the Ray from the
Direction of the Wave Normal. Let the direction cosines of
the ray be m, n, p. From the considerations presented on
page 313 and equation (25) on page 273,
m : n : - p = y Y - PZ : aZ - yX : pX - aY. . (37)
But from equations (13) and (16) on page 315,
(38)
PROPERTIES OF TRANSPARENT CRYSTALS 325
Also, from equations (7), page 311, and (13), it is easy to
deduce
a : ft : y = b*p$l c*n^> : c*m% - a*pW : a^n^R - frm$i. (39)
Substitution of the values (38) and (39) in (37) gives
m . n . = _
+ Wa\a*mm + Prftl + c^} :...:... (40)
The terms denoted thus . . . can be obtained from the
written terms by a cyclical interchange of letters.
If now the abbreviation (16') on page 315 be introduced,
i.e. if
tfrnW + Pribl + c*p<$ = &, ... (41)
it follows from (17) that
p F 2 + pG\
If these three equations be squared and added, then, since
(cf. page 315)
9ft2 +'SR2 + $p 2 = _|_ n * _|_ ^ __ If
mm + Mn + p/ = o,
it follows that
am* + &*W + c^ = F 4 + G*. . . . (42)
Squaring and adding equations (17') gives
If now the value of 9Jte 2 obtained from (17') be introduced,
namely,
then, in consideration of (41) and (42), (40) becomes
<7 2
m : n : p = w( F 4 + ^ 4 ) +
326 THEORY OF OPTICS
or
G* \ / &
m : n : to =
This equation gives the direction of the ray in terms of the
direction of the wave normal, for V 2 is expressed in terms of
;/z, ft, and / in Fresnel's law (18), and G* [cf. (43)] in terms
of m, ft, />, and F 2 .
In order to determine the absolute values of m, n, p, not
their ratios merely, it is possible to write
= "(*"'+ T*?^)' n = na (
(45)
in which cr is a factor of proportionality which can be deter-
mined by squaring and adding these three equations. This
gives, in consideration of (18) and (43),
I = = C7 2 (F 4 + G*) ( 4 6)
9. The Ray Surface. If a wave front has travelled parallel
to itself in unit time a distance F, then V is called the velocity
along the normal. The ray is oblique to the normal, making
with it an angle which is given by
cos C = mm + \\n + p/. . . . . (47)
The ray has then in unit time travelled a distance $ such
that
58 cos C =-. V (48)
55 is called the velocity of the ray: it is larger than the
velocity along the normal.
If the three equations (45) be multiplied by m, n, p, respec-
tively, and added, it follows that cos = o-F 2 , or, in con-
sideration of (48),
(49)
PROPERTIES OF TRANSPARENT CRYSTALS 327
Hence, from (46),
or, in consideration of (48),
G 2 = V* tan C ....... (51)
If the value of 4 from (50) be substituted in (45), then, in
consideration of (49), there results, after a simple transforma-
tion,
mSB mV 11% nV
a2 F 2 # 2 ' $ 2 - 2 ~ F 2 -
If these three equations be multiplied by ma 2 , n^ 2 ,
respectively, and added, then, in consideration of (17'),
But the light-ray is perpendicular to the electric force.
Hence the right-hand side of the last equation vanishes, since
the components of the electric force satisfy (38). Hence
. . (53)
which may also be written in the form
m 2 n 2 p 2
T- T + r "z + rT = * (53)
^ 2 ~ SS 2 2 SS 2 ^ 2 $B 2
The addition to (53) of m 2 + n 2 + p 2 = I gives
. (53 r/ )
This equation expresses the velocity 35 of the ray as a function
of the direction of the ray. If in every direction m, n, $ the
corresponding 35 be laid off from a fixed point, the so-called
ray surface is obtained. This surface, like the normal surface,
consists of two sheets. These two surfaces are very similar to
each other, since equation (53') of the former is obtained from
(18) of the latter by substituting for all lengths which appear
328 THEORY OF OPTICS
in (18) their reciprocal values. Each of the planes of symmetry
intersects the ray surface in a circle and an ellipse.
Hence, in order to apply the geometrical construction given
in 6 to this case, it is necessary to start from the surface
[cf- (30)]
1 V, V, V
p*~ a 2 "" J 8 " 1 " <*'
i.e. from an ellipsoid whose axes are a, b, c. The velocities
%$ of the ray in a direction m, n, p are given by the principal
axes p i and p 2 of that ellipse which is cut from the ellipsoid by
a plane perpendicular to the ray.
In this case also there must be two directions, ^ and 51 2 , for
which the two roots %$ 2 of the quadratic equation (S3 7 ) are the
same. These directions are obtained from the equations for
the optic axes, namely, (26') and (26"), by substituting in them
for all lengths the reciprocal values. Thus
tn= / _, n = o, p =
or
n = o, p = fy^r-J. (54)
These two directions are called the r#jj/ #;re.r.
The ray surface can be looked upon as that surface at which
the light disturbance originating in a point P has arrived at the
end of unit time. For this reason it is commonly called the
wave surface.
If, in accordance with Huygens' principle, the separate
points P of a wave front are looked upon as centres of disturb-
ance and if the wave surfaces are constructed about these points,
the envelope of these surfaces represents the wave front at the
end of unit time (cf. page 159). According to this construe-
PROPERTIES OF TRANSPARENT CRYSTALS 329
tion the wave front corresponding to a ray PS is a plane tangent
to the wave surface at the point S.
This result can also be deduced from the equations. If
the rectangular coordinates of a point 5 of the wave surface
are denoted by x, y, and z, then m35 = x, etc., and $ 2 =
** ~t~ y* + ^ ! > an d> fro m (SS'O*
*2
~ * = o. - (55)
If this equation be written in the general form F(x, y, z) = o,
the direction cosines of the normal to the tangent plane at the
point x, y, z are proportional to , , . Hence it is
necessary to prove that
- : 77 : -TT = m : n : p. . . .
dx dy dz
Now, from (55),
From (52), ^ : S5 2 - a = m V : V* a*, etc. Hence, in con
sideration of (43) and (50),
i.e., in consideration of (52),
dF F 3
(57)
From this equation , may be written out by a simple
interchange of letters. Hence equation (56) immediately
results, i.e. the construction found from Huygens' principle is
verified.
From these considerations it is evident that the direction
m, n, $ of the ray can be determined from the direction m, n,
330 THEORY OF OPTICS
p of the normal in the following way: Suppose a light disturb-
ance to start at any instant from a point P ; the ray surface is
then tangent to all the wavs fronts, i.e. it is the envelope of
the wave fronts. Consider three elementary wave fronts the
directions of whose normals are infinitely near to the direction
of the line PN. Their intersection must then be infinitely near
to the end point S of the ray PS which corresponds to the
normal PN, since 5 is common to all three waves. The cor-
rectness of this construction will now be analytically proved.
The equation of a wave front is
mx + ny + pz = V. ..... (58)
If the point x, y, z is to lie upon an infinitely near wave front,
the equation obtained by differentiating (58) with respect to
m t n, and / will also hold. But these quantities are not inde-
pendent of one another, since m* + # 2 + / 2 = i. According
to the theorem of Lagrange (cf. above, page 321) there can
be added to (58) the identity
so that there results
mx + ny + pz +f(m* + n 2 + /*) = V + f. . (59)
/is an unknown constant. Since this constant has been intro-
duced into the equation, ??/, n, and p in (59) may be looked
upon as independent variables, and the partial differential
coefficients of (59) with respect to m, n, and / may be formed,
namely,
(60)
y
But, from (18) and (43),
m G 4
Similar expressions hold for , . If the three equations
(60) be multiplied by m, n, and /, respectively, and added, it
PROPERTIES OF TRANSPARENT CRYSTALS 331
is evident from ( 1 8) and (6 1 ) that the right-hand side of the *
resulting equation reduces to zero, while the left-hand side is,
by (5 8 )> V + 2 /> so that the constant 2/ is determined as
2/ V. Hence, in consideration of (61), the first of equa-
tions (60) becomes
* =
and similarly
Hence the radius vector drawn from the origin to the point of
intersection x, y, z of the three infinitely near wave fronts
coincides in fact with the direction of the ray as calculated on
page 326, since x :y : z = m : n : p. Further, the velocity of
the ray Vx* -\-y* -f- #* is found to have the same value as that
given above in (45) and (49).
For other geometrical relations between the ray, the wave
normal, the optic axes, and the ray axes, cf. Winkelmann's
Handbuch der Physik, Optik, p. 699.
10. Conical Refraction. Corresponding to any given
direction of a wave normal there are, in general, according
to equation (44), two different rays, since for a given value
of m, n, and p there are two different values of F 2 . , But it
may happen that these equations assume the indeterminate
form o : o. Thus this occurs when one of the quantities m, n,
or/ is equal to zero. If, for example, m = o, then, from (21)
on page 317, V? = a z . In this case, by (43) and (44),
4 = (V* a^ : m*,
G* V - *
The value of this expression, which is of the form o : o, is easily
332 THEORY OF OPTICS
determined, since, by Fresnel's equation (18) on page 316, the
expression m 2 : V* a 2 has a finite, determinate value, namely,
V* - a 2 6* - V*
The right-hand side of this equation can never be zero, since
for a > b > c and V* = a 2 both terms of the right-hand side
are negative. Hence, by (62), m = o when m o, i.e. the
light-ray is in the j/^-plane when the wave normal is in the
j/^-plane. When / = o the conclusion is similar. But the
case in which n o requires special consideration. For then,
when V = b, equations similar to (62) and (63) are obtained,
namely,
V 2 - b 2 n* m* p 2
* ' V 2 - P ~ a 2 - V*
The right-hand side of this equation which corresponds to the
case V = b may become zero, namely, when
m\c 2 - &) + /V a - ^ 2 ) = -
Now this relation is actually fulfilled when the wave normal
coincides with an optic axis [cf. (25), page 319]. In this case,
by (64), n still retains the indeterminate form o : o, i.e. to this
particular wave normal there correspond not two single deter-
minate rays, but an infinite number of them, since n always
remains indeterminate. The locus of the rays in this case can
be most simply determined from the equation
mm tin
which is deduced from (52) by multiplying by m, n, and /,
respectively, adding, and taking account of (18). If the wave
normal coincides with an optic axis, then n o, but n is not
necessarily zero and $$ is therefore in this case different from b.
Hence
mm ,
2 ~r
Further, from (47) and (48), since V = b,
+ p/) = b ...... (67)
PROPERTIES OF TRANSPARENT CRYSTALS 333
Elimination of SS 2 from these two equations gives
(mmc* + Wa*)(mm + p/) = 2 . . . . (68)
If the coordinates of the end points of a ray are denoted by
x, y, z, so that m = x : Vx* -f-j/ 2 -\- z 2 , etc., it follows that
(xm* + *l>rf)(xm + jsf) = t\x t +y> + j3*). . . (69)
This equation of the second degree represents a cone whose
vertex lies at the origin. Hence when the wave normal coin-
cides with the optic axis there are an infinite number of rays
which lie upon the cone defined by equation (6p). This cone
intersects the wave front
xm -f- zp = const (70)
in a circle, since when (70) is substituted in (69) the latter
becomes
(xnu* + zpa*)- const. = b\x? + y* + z*\
which is the equation of a sphere.
Hence from the discussion on page 328 it follows that the
wave surface has two tangent planes which are perpendicular
to the optic axis and tangent to the wave surface in a circle.
The axis of the cone coincides with the optic axis ; it is there-
fore perpendicular to the plane of the circle. The aperture j
of the cone is determined from (69) as
- -
tan;r=- ^- -1 (71)
This phenomenon is known as internal conical refraction, for
the following reason : If a ray of light is incident upon a crystal
in such a direction that the refracted wave normal coincides
with the optic axis of the crystal, then the light-rays within
the crystal lie upon the surface of a cone. The rays which
emerge from the plate lie therefore upon the surface of an
elliptical cylinder whose axis is parallel to the incident light
in case the plate of crystal is plane parallel.* Aragonite is
* For the direction of the rays in the outer medium depends only upon the
position of the wave front within the crystal, not upon the direction of the internal
rays. The law of refraction will be more fully discussed in the next paragraph.
334 THEORY OF OPTICS
especially suited for observation of this phenomenon, since in it
the angle of aperture of the cone is comparatively large
(X = i 52').* The arrangement
of the experiment is shown in
Fig. 86. A parallel beam so is
incident through a small opening
0upon one side of a plane-parallel
FlG> 86 plate of aragonite which is cut
perpendicular to the line bisecting the acute angle between the
optic axes. When the plate is turned into the proper position
by rotating it about an axis perpendicular to the plane of the
optic axes, an elliptical ring appears upon the screen S6\
A microscope or a magnifying-glass focussed upon o may
be used instead of a screen for observation.
The equation representing the dependence of the direction
of the wave normal upon the direction of the ray may be easily
deduced from (52) taken in connection with (47) and (48).
The result shows that in general for each particular value of
nt, n, p there are two values of m, n, /. Only when n = o
and ^ = b*, i.e. when the ray coincides with the ray axis,f
does n become indeterminate, as can be shown by a method
similar to that used above. Hence when the ray coincides with
the ray axis, then at the point of exit of the ray the ray surface
does not have merely two definite tangent plane 's, but a cone of
tangent planes. The corresponding wave normals lie upon a
cone of aperture ^ such that
This equation is obtained from (71) by substituting in it
for all the lengths their reciprocal values.
* Sulphur is still better, since its angle of aperture is 7; but its preparation is
much more difficult. The use of a sphere of sulphur for demonstrating conical
refraction is described by Schrauf, Wied. Ann. 37, p. 127.
f The ray axis is the axis of the cone of rays to which a single ray SO([g. 86)
gives rise when SO has the direction which corresponds to internal conical
refraction. TR.
PROPERTIES OF TRANSPARENT CRYSTALS 335
This phenomenon is called external conical refraction, for
the reason that a ray which inside the crystal coincides with
the ray axis becomes, upon emergence from the crystal, a cone
of rays. For the rays after refraction into the outer medium
have different directions corresponding to the different posi-
tions of the wave front in the crystal (cf. note, page 333).
Fig. 87 represents an arrangement for demonstrating
experimentally external conical refraction. A beam of light
is concentrated by a lens L upon a small opening o in front of
FIG. 87.
an aragonite plate. A second screen with an opening o' is
placed on the other side of the plate. If the line oo' coincides
with the direction of a ray axis, a ring appears upon the
screen 55. The diameter of this ring increases as the distance
from o 1 to the screen increases. In this arrangement only
those rays are effective which travel in the direction oo ', the
others are cut off by the second screen. The effective incident
rays are parallel to the rays of the emergent cone.
The phenomena of conical refraction were not observed
until after Hamilton had proved theoretically that they must
exist.
ii. Passage of Light through Plates and Prisms of
Crystal. The same analytical condition holds for the passage
of light from air into a crystal as was shown on page 280 to
hold for the refraction of light by an isotropic medium. If the
incident wave is proportional to
27T
mx -\-ny-\-
336 THEORY OF OPTICS
while the refracted wave is proportional to
m'x + n'y + p'z
and if the boundary surface is the plane s- = o, then the fact
that boundary conditions exist requires, without reference to
their form, the equations
m _ m' n n'
F" : "F' V ~~~~ F"'
This is the common law of refraction, i.e. the refracted ray lies
in the plane of incidence, and the relation between the angle of
incidence and the angle of refraction 0' is
sin : sin 0' = V : V, . . . . (73)
in which V and V are the velocities in air and in the crystal
respectively. But in the case of crystals this relation does not
in general give the direct construction of the refracted wave
normal, since in general V depends upon the direction of this
normal.
But the application of Huygens' principle, in accordance
with the same fundamental laws which were stated on page
161 for isotropic bodies, does give directly not only the rela-
tion (73), but also the construction of both the refracted wave
normal and the refracted ray. For let A^B (Fig. 88) be the
intersection of an incident wave front with the plane of inci-
dence (plane of the paper), and let the angle A^BA^ = , and
BA 2 = V, and construct about A l the ray surface 2 within the
crystal, this surface being the locus of the points to which the
disturbance originating at A r has been propagated in unit time.
Draw through A 2 a line perpendicular to the plane of incidence,
and pass through it two planes A 2 T^ and A 2 T 2 tangent respec-
tively to the two sheets of the ray surface. According to
Huygens' principle these tangent planes are the wave fronts of
the refracted waves. The lines drawn from A l to the two points
of tangency C l and C 2 of the planes with the ray surface give
PROPERTIES OF TRANSPARENT CRYSTALS 337
the directions of the refracted rays. In general these do not
lie in the plane of incidence.
Hence for perpendicular incidence the wave normal is not
doubly refracted, but there are two different rays whose direc-
tions may be determined by finding the points C l and C 2 in
which the two sheets of the wave surface constructed about a
point A of the bounding surface are tangent to two planes
Crystal
FIG. 88.
parallel to the bounding surface G. The directions of the rays
are A \ and A C 2 respectively.
When the light passes from the crystal into air a similar
construction is applicable. Hence in the passage of light
through a plane-parallel plate of crystal there is never a
double refraction of the wave normal, but only of the ray. In
order to observe the phenomena of double refraction it is
necessary to view a point on the remote side of the crystal.
This point appears double, since its apparent position depends
upon the paths of the rays.* But the introduction of a crystal-
line plate between collimator and telescope produces no dis-
placement of the image, since in this case the wave normal is
determinative of the position of the image. In order to detect
double refraction in this case, which occurs in all observations
* The apparent position is displaced not only laterally but also vertically. Cf.
Winkelmann's Handbuch d. Physik, Optik, p. 705.
33 8 THEORY OF OPTICS
with the spectrometer, it is necessary to introduce a prism of
the crystal.
With the help of such a prism it is possible to find the prin-
cipal indices of refraction, i.e. the quantities
! = V : a, ?* 2 = V : b, n^ = V : c. . . (74)
If, for example, a prism of uniaxial crystal (a = b) be used
whose edge is parallel to the optic axis, then the velocity V
of the wa,ves whose normals are perpendicular to the edge of
the prism has the two constant values a and c. n^ and n B can
therefore be found by the method of minimum deviation exactly
as in the case of prisms of isotropic substances. The different
directions of polarization of the emergent rays make it possible
to recognize at once which index corresponds to n^ and which
to n y
In the same way one of the principal indices of refraction
of a prism of a biaxial crystal whose edge is parallel to one of
the axes of optic symmetry may be found. In order to find
the other two indices it is necessary to observe the deviation
of a wave polarized parallel to the edge of the prism for at
least two different angles of incidence.
From the meaning which the electromagnetic theory gives
to the principal velocities a, b, c, it is evident from equations
(16) on page 315 and (74) that
*i = n ?, 2 = n ^ 3 = 3 2 > (75)
at least if C, the velocity in vacuo, be identified with V, the
velocity in air. The error involved in this assumption may be
neglected in view of the uncertainty which attends measure-
ment of the dielectric constant.
The relation (75) cannot be rigorously fulfilled, if for no
other reason, because the index depends upon the color, i.e.
upon the period of the electric force, while the dielectric con-
stant of a homogeneous dielectric is, at least within wide limits,
independent of the period. It is, however, natural to test (75)
' PROPERTIES OF TRANSPARENT CRYSTALS 339
under the assumption that ;z 2 is the index of infinitely long
waves, i.e. the A of the Cauchy dispersion equation
30
= A
Relation (75) is approximately verified in the case of ortho-
rhombic sulphur, whose dielectric constants have been deter-
mined by Boltzmann.* Its indices were measured by Schrauf.t
In the following table n 1 denotes the index for yellow light and
A the constant of (76) :
n? = 3.80; A* = 3.59; e 1 = 3-81
^ = 4.16; A* = 3.89; e 2 = 3.97
n*=$.02', A/ = 4.60; e 3 -4.77
Thus the dielectric constants have the same sequence as
the principal indices of refraction when both are arranged in
the order of their magnitudes, but are uniformly larger than
the A 's. With some other crystals this difference is even
greater. The departure from the requirements of the electro-
magnetic theory is of the same kind as that shown by isotropic
bodies (cf. page 277). Its explanation will be given in the
treatment of the phenomena of dispersion.
Thus the electromagnetic theory is analytically in complete
agreement with the phenomena, but the exact values of the
optical constants cannot be obtained from electrical measure-
ments. These constants depend in a way which cannot be
foreseen upon the color of the light. In fact not only the
principal velocities a, b, c, but also, in the case of monoclinic
and triclinic crystals, the positions of the axes of optic sym-
metry depend upon the color.
12. Total Reflection at the Surface of Crystalline Plates.
The construction given on page 336 for the refracted wave
front becomes impossible when the straight line & which passes
through A 2 and is perpendicular to the plane of incidence inter-
*Boltzmann, Wien. Ber. 70 (2), p. 342, 1874. Pogg. Ann. 153, p. 531, 1874.
f Schrauf, Wien. Ber. 41, p. 805, 1860.
340 THEORY OF OPTICS
sects one or both of the curves cut from the wave surface 2 by
the bounding surface G. In this case there is no refracted
wave front, but total reflection takes place. The limiting case,
in which partial reflection becomes total, is reached for either
one of the two refracted waves when the line is tangent to
that sheet of the ray surface 2 which corresponds to the wave
in question, i.e. is tangent to the section of the wave surface
by the bounding plane G. In this case, since the point of
tangency T of ($ with 2 lies in the bounding plane G, the
refracted ray is parallel to the boundary (cf. Fig. 89). This
\
"2 Plane of incidence
FIG. 89.
wave then can transfer no energy into the crystal, since the
ray of light represents the path of energy flow (cf. page 313),
and hence no energy passes through a plane parallel to the
ray. Thus it appears from this consideration also that in this
limiting case the reflected wave must contain the entire energy
of the incident wave, i.e. total reflection must occur.
Hence if a plate of crystal be immersed in a more strongly
refracting medium, and illuminated with diffuse homogeneous
light, two curves which separate the regions of less intensity
from those of greater appear in the field of the reflected light.
If the observation is made, not upon the reflected light, but upon
light which, entering the crystal at one side and then falling
at grazing incidence upon the surface, passes out into a more
strongly refractive medium, these limiting curves are much
sharper since they separate brightness from complete darkness.
From these curves the critical angles X and 2 may be
PROPERTIES OF TRANSPARENT CRYSTALS 341
determined. These curves are not in general perpendicular to
the plane of reflection. Special instruments have been devised
for their observation. Fig. 90 represents Abbe's crystal
refractometer. The plate of crystal which is to be investigated
is laid upon the flint-glass hemisphere K of index 1.89.
FIG. 90.
Between the crystal and the sphere a liquid of greater index
than the latter is introduced. K can be rotated along with the
azimuth circle H about a vertical axis. The movable mirror
S makes it possible to illuminate the crystal plate either from
below through K or from the side. The limiting curves of
342 THEORY OF OPTICS
total reflection are observed through the telescope OGGO
which turns with the vertical circle V. For convenience of
observation, the telescope is so shaped that the rays, after three
total reflections within it, always emerge horizontally. The
objective of the telescope is so arranged that it compensates
the refraction due to the spherical surface K of the rays reflected
from the crystalline plate. It forms, therefore, sharp images
of the curves.
The method of total reflection is the simplest for the
determination of the principal indices of refraction of a crys-
talline plate. These indices are obtained at once from the
maximum or minimum values of the angles of incidence which
correspond to the two limiting curves.
Thus if denotes the angle of incidence corresponding to
a limiting curve for any azimuth $ of the plane of incidence
(cf. Figs. 88 and 89), then the line A t A 2 = V : sin 0; for
BA 2 = V (the velocity in the surrounding medium), and
A^A 2 is the distance of the point A l from a line which is tan-
gent to the curve of intersection of the wave surface constructed
about A l with the bounding surface G. Maximum and mini-
mum values of the limiting angles 0, i.e. of the line A 1 A 2J
coincide necessarily with maximum or minimum values of the
length of the ray A^T (cf. Fig. 89), as can be easily shown by
construction. In fact in this case A^A 2 coincides with the ray
A^T) since the tangents must be perpendicular to the radius
vector A^T when this has a maximum or minimum value.
The length A^T of the ray has now in every plane section of
the wave surface the absolute maximum a and the absolute
minimum c. For it appears from the equation of the wave
surface (cf. page 327) that 35 must always lie between a and r,
since otherwise the three terms of equation (53) would have
the same sign and their sum could not be zero. On the other
hand it is also evident that in every plane section G of the
wave surface 35 reaches the limiting values a and c, for, from
Fig. 85, 35 attains the value a at least in the line of intersection
of G with the j/^-plane ; since in the j-s'-plane one velocity has
PROPERTIES OF TRANSPARENT CRYSTALS 343
the constant value $$ = a, while in the line of intersection of
G with the ;rj/-plane $ must attain the value c. In the inter-
section of G with the j^-plane $ = b\ but it is uncertain, as
can be shown from the last of Figs. 85, whether b belongs to
the minimum of the outer or the maximum of the inner limiting
curve. This can be decided by investigating the maxima or
minima of the angle of incidence corresponding to the limiting
curves for two plates of different orientations.* Four such
measurements can be made upon each plate, and three of these
must be common to the two plates. These three correspond
to the three principal velocities a, b, c. Their respective
values may be determined from
A t A 2 = V : sin = a, b, c, . . . (77)
where c) (79) gives
* If the polarization effects be also taken into account, one section of the
crystal is enough. Cf. C. Viola, Wied. Beibl. 1899, p. 641.
344 THEORY OF OPTICS
the maximum value of 25, i.e. it determines the minimum value
of along the limiting curve which arises from a total reflec-
tion of the extraordinary ray. The maximum value of along
this limiting curve determines, therefore, the value of c\ from
the minimum value of it is possible to calculate y, i.e. the
inclination of the face of the crystal to the optic axis. In the
case of negative uniaxial crystals (a < c) the minimum value of
determines the principal velocity c.
Likewise in the case of biaxial crystals the angle between
the face and the axes of optic symmetry can be determined
from observation of the limiting curves of total reflection.
Nevertheless for the sake of greater accuracy it is advantageous
to couple with this other methods, for example, the method
which makes use of the interference phenomena in convergent
polarized light (cf. below).
Conical refraction gives rise to peculiar phenomena in the
limiting curves of total reflection. These may be observed if
the bounding surface G coincides with the plane of the optic
axes. For more complete discussion cf. Kohlrausch, Wied.
Ann., 6, p. 86, 1879; Liebisch, Physik. Kryst. , p. 423; Mas-
cart, Traite d'Optique, vol. 2, p. 102, Paris, 1891.
13, Partial Reflection at the Surface of a Crystalline
Plate. In order to calculate the changes in amplitude which
take place in partial reflection from a plate of crystal it is only
necessary to apply equation (6') and (7) on page 311 together
with the boundary conditions there mentioned.
But since the calculation is complicated (cf. Winkelmann's
Handbuch, Optik, p. 745) only the result will be here
mentioned that there is an angle of complete polarization,
i.e. an angle of incidence at which incident natural light is
plane-polarized after reflection. But the plane of polarization
does not in general coincide with the plane of incidence, as it
does in the case of isotropic media.
14. Interference Phenomena Produced by Crystalline
Plates in Polarized Light when the Incidence is Formal.
Let plane-polarized monochromatic light pass normally through
PROPERTIES OF TRANSPARENT CRYSTALS 345
a plate of crystal and then through a second polarizing
arrangement. This case is realized when the crystalline plate
is placed upon the stage of the Norrenberg polarizing apparatus
described on page 246. The upper
mirror can be conveniently replaced
by a Nicol prism, the analyzer. Let
the plane of vibration of the electric
force within the analyzer be A (Fig.
91), and that within the polarizer P.
The incident polarized light, the
amplitude of which will be denoted
by E y is resolved after entrance into FlG * 9I>
the doubly refracting crystal into two waves of amplitude
E cos and E sin respectively, being the angle which
P makes with the directions H^ and H 2 of the vibrations of
the two waves W l and W 2 within the crystal. The decrease
in amplitude by reflection is neglected. It is very nearly the
same for both waves. These two waves after passing through
the crystal are brought into the same plane of polarization, and
hence after passing through the analyzer have the amplitudes
E cos cos (0 x) t E sin sin (0 j). Now a difference
in phase 6 has been introduced between the two waves by their
passage through the plate. This difference is
dlV V '~ (80)
in which d denotes the thickness of the crystalline plate, V l , V 2
the respective velocities of the two waves within it, V the
velocity of light in air, and 1 the wave length in air of the light
used. Hence, according to page 1 3 1 , the intensity of the light
emerging from the analyzer is
J = 2 {cos 2 cos 2 (0 x) + sin 2 sin 2 (0 X)
+ 2 sin cos sin (0 X] cos (0 X) cos 6} .
If cos d be replaced by I 2 sin 2 $d, the equation becomes
J = E 2 {cos*X sin 20 sin 2(0 x) sin 2 Jtf}. (81)
346 THEORY OF OPTICS
The first term E 2 cos 2 x represents the intensity of the light
which would have emerged from the analyzer in case the
crystal had not been introduced. This intensity J Q will be
called the original intensity; thus
y o = E* cos 2 x ....... (82)
Two cases will be considered in greater detail:
I . Parallel Nicols : x - Then
J t = y o (i - sin 2 20 sin 2 itf). . . . (83)
If the crystal be rotated, the original intensity will be
attained in the four positions = o, r= - , = TT, = y
i.e. whenever one of the planes of vibration within the crystal
coincides with that of the Nicols. In the positions midway
7T
between the above, i.e. = , etc.,
4
Ji = 7o(l - sin2 i<*) = Jo cos 2 R . . . (84)
i.e. with the proper values of tf, i.e. of the thickness of the
crystal, complete darkness may result.
2. Crossed Nicols: X = -. Here J = o and
J x = ^ 2 sin2 2 0sin 2 id .... (85)
Thus, whatever its thickness, the plate appears dark when
its planes of vibration coincide with those of the Nicols. If
this is not the case, it is dark only when 3 = 2hn. In the
7T
positions = , etc.,
y x - E* sin 2 \d ....... (86)
Hence, unless it happens that d zhtt, it is possible to find
the direction of polarization or of vibration within the crystal
by rotating it until the light is cut off*.
Hence a crystalline wedge between crossed Nicols is
traversed by dark bands which run parallel to the edge of the
PROPERTIES OF TRANSPARENT CRYSTALS 347
wedge, unless it is in the position in which the light is wholly
cut off. These bands lie at those places at which the thickness
of the wedge corresponds to the equation $ = 2hn. If the
incident light is white, the bands must appear colored since $
varies with the color.
A plane-parallel plate of crystal between crossed Nicols
must in general appear colored when the incident light is
white. Not only does the amplitude E and the difference of
phase S depend upon the color, but also the angle 0, i.e. the
position of the planes of vibration. However, this latter varia-
tion can in general be neglected on account of the small
amount of the difference in the retardations for different colors.
When the Nicols are crossed it appears from (86) that in white
light for
4
L =
in which 2 is to be extended over the values corresponding
to the different colors. Thus
= white light ...... (87)
Now from (80) its evident that the dependence of <5 upon A
is principally due to the appearance of A in the denominator.
Hence if the approximately correct assumption be made that
V V
-y -- is independent of the color, then
J x = 2P sin* *- T ...... (87')
in which
is approximately independent of A. It appears from a com-
parison of (87') with (78) on page 306 that the plate of crys-
tal shows approximately the same colors as those produced by the
interference of the two waves reflected at the surfaces of a thin
7 f
film of air of thickness . (Newton's ring colors.) But the
348 THEORY OF OPTICS
Newton interference colors of thin plates differ widely from
those produced by the crystal when the difference in the dis-
persion of the two waves within the crystal is large. Then d f
is no longer independent of A. This is, for example, the case
with the hyposulphate of strontium, apophyllite (from the Faroe
Islands), brucite, and vesuvian.
For a given angle
l8 9 J - In Rochelle
salt the angle between the optic axes is for red 76, for violet 56.
354 THEORY OF OPTICS
in accordance with (85), traversed by a black curve, the
so-called principal isogyre, for which sin 20 = o. If the
plane of the optic axes coincides with the plane of polarization
of the analyzer, or the polarizer (the so-called principal posi-
tion^, the principal isogyre is a black cross one of whose arms
passes through the optic axes, while the other, perpendicular
to it, passes through the middle of the field. For, according
to the construction given upon page 322, the directions of
polarization H l and H 2 corresponding to points on this cross
are parallel and perpendicular to the line A^A^ joining the optic
axes. Hence the interference figure is that shown in Fig. 96.
FIG. 96. FIG. 97.
In the second principal position of the crystal, i.e. when the
plane of the optic axes A l and A 2 makes an angle of 45 with
the plane of the analyzer, the principal isogyres are hyperbolae
which pass through the optic axes. Hence the interference
pattern is that shown in Fig. 97. The equation of the prin-
cipal isogyre can be approximately obtained by taking the line
PB, which bisects the angle A 1 PA 2 , as a direction of polariza-
tion //"within the crystal,* P being any point upon the plate
(cf. Fig. 98). Let the directions of the coordinates x and y
* From the rule given on page 322 it is evident that this is only approximately
correct. The problem is more thoroughly discussed in Winkelmann's Handbtich
der Physik, Optik, p. 726 sq.
PROPERTIES OF TRANSPARENT CRYSTALS 355
lie in the planes of polarization of the analyzer and the polarizer
respectively. Also, let PA l = l lt PA 2 / 2 , AjA 2 = /. Then
: BA =
BA =
i.e.
Also, from the triangle A^BP,
sin a : sin
(92)
(93)
But now for the principal isogyre <^^f 1 ^ J P=45, since the
line A^ 2 connecting the optic axes is to make an angle of 45
FIG. 98.
with the coordinate axes, and since, for the principal isogyre,
the line PB is to be parallel to the j-axis. Hence, from (92)
and (93),
I /
sin a = =-,
(94)
Further, from the triangle A t PA 2 ,
I* = I? + I* - 2// 2 cos = & - / 2 ) 2
a
356 THEORY OF OPTICS
i.e., from (94),
or
If the coordinates of the points A l and A 2 of the optic axes are
called /, then
and (95) becomes
*y = / 2 (96)
But this is the equation of an equilateral hyperbola which
passes through the optic axes A t and A 2 and is asymptotic to
the coordinate axes.
These black principal isogyres which cross the interference
pattern are especially convenient for measuring the apparent
angle between the optic axes, i.e. the angle which two wave
normals, which within the plate are parallel to the optic axes,
make with each other upon emergence from the plate. With
the aid of the law of refraction the angle between the optic
axes themselves may be calculated from this, if the mean
principal velocity b within the crystal be known. The apparent
angle between the optic axes is measured by rotating the
crystal about an axis perpendicular to the plane of the optic
axes, and thus bringing the traces of the optic axes succes-
sively into the middle of the field of view, i.e. under the cross-
hairs. The angle through which the crystal is rotated is read
off on a graduated circle. The apparatus constructed for
measuring this angle is called a stauroscope.
In uniaxial crystals a surface of equal difference of path
( = const.) has the form shown in Fig. 99. When the plate
is cut perpendicular to the optic axis, the isochromatic curves
are concentric circles about the optic axis. With crossed
PROPERTIES OF TRANSPARENT CRYSTALS 357
Nicols the isogyre is a black right-angled cross. Hence the
interference pattern is that shown in Fig. 100. From a
measurement of the diameters of the rings the difference in the
FIG. 99. FIG. 100.
two principal indices of refraction of the crystal can be
obtained.
For a discussion of methods of distinguishing the character
of double refraction by means of a plate of selenite for which
d = -, as well as for other special cases, cf. Liebisch, Physik.
Krystallogr. , or Winkelmann's Handbuch der Physik, Optik.
CHAPTER IV
ABSORBING MEDIA
i. Electromagnetic Theory. Absorbing media will be
defined as media in which the intensity of light diminishes as
the length of the path of the light within the medium increases.
The metals are characterized by specially strong absorbing
powers. According to the electromagnetic theory absorption
is to be expected in all media which are not perfect dielectrics.
For the electric currents arising from conduction produce heat
the energy of which must come from the radiant energy of the
light.
The electromagnetic theory given above on page 268 sq.
will now be extended to include the case of imperfect insu-
lators, i.e. to include media which possess both a dielectric
constant e and an electric conductivity R s = RS*', in which R, t R g , d^, --e iA ~ cos cos x '
If in this equation x be replaced by and e' in accordance
with (12), then
i + p-e iA sin tan
i~^- P-*'* = Ve^^sln^' *
Hence when = o, p-/^ i, or z/ = o and p = i.
7T
When 0= -, p/^ + i, i.e. J = o, p = i. Hence the
relative difference of phase A of the reflected light, i.e. its
ellipticity, vanishes at perpendicular and grazing incidence.
That angle of incidence for which the difference of phase A
amounts to is called the principal angle of incidence 0. At
this angle e 1 ^ = i; hence, from (15),
\-\-i-~p sin tan
; = r=== '' O")
I i p Ve sin 2
If this equation be multiplied by the conjugate complex
equation
I __ i . -p sin tan
I + i ~p ~ Ve" sin 2 *
in which e ' denotes the complex quantity which is conjugate
ABSORBING MEDIA 363
to e', the left-hand side reduces to I. Hence the principal
angle of incidence is determined by
sin 4 0- tan 4 = n\i + K^ 2# 2 (i /c 2 ) sin 2 0-f- sin 4 0. (17)
For the numerical calculation it is generally sufficient to
take account of the first term only on the right-hand side of
this equation, since, for all the metals, n\i -\- /c 2 ) has a value
much greater than I, somewhere between 8 and 30. With
this approximation (17) becomes simply
sin tan = n V i + x 2 (18)
This approximation may be obtained directly from (15) by
neglecting in the denominator of the right-hand side sin 2 in
comparison with e' . For, from (n),
4/7 = (I - iK) (I 9 )
so that (15) becomes
i -f- p*f sin tan
I p-eid ~ n(i - iK) * ^ '
Writing
p = tan if} (21)
it appears [cf. (13)] that ?/; represents the azimuth of the
plane of polarization of the reflected light with respect to the
plane of incidence, after it has been made plane-polarized by
any means such as the Babinet compensator (cf. page 257),
Hence $ is called the azimuth of restored polarization.
Now it is easy to c-educe the relation
i -- pe lA cos 2if} /sin A sin 2ip
i - r i^e iA ~ i -f cos A sin 20
so that the following may be obtained from (20):
K = sin A tan 2^\
COS 2 h
n = sin tan ; : = :, /__N
v I + cos A sin 2tf> Y - (22)
i cos A sin 2?/'
n *(i J_ K v\ sin 2 tan 2 0- : -. (
^ i -j- cos A sin 2(/} J
364 THEORY OF OPTICS
From these equations the optical constants n and K of a
metal can be determined with sufficient accuracy from obser-
vations of ^ and A*
The value of ^ which corresponds to the principal angle of
incidence = is called the principal azimuth ip. From the
first of equations (22) it follows that
K = tan 2$ (23)
Inversely, in order to obtain A and i/> from the optical con-
stants, set
tan P = ^ ^-i _ tan Q = K. . . (24)
sin tan
Then from (20), since the right-hand side has the value
cotP./e
tan A sin Q tan 2-P,
cos 2ip = cos Q sin 2P ..... (25)
The reflecting power of a metal is defined as the ratio of
the intensity of the reflected light to that of the incident light
when the angle of incidence is zero. In this case, from
equation (26) on page 284, since n is here to be replaced by
n(i IK) [cf. equation (19)],
Rp __ R,.**, _ n(i - JK) - i
~
If this equation is multiplied by its conjugate complex
equation, the value of the reflecting power R is found to be
) + i -2*
* '
n\i + * + i + 2
Since for all metals 2n is small in comparison with
tf(-\ _]_ /c 2 ), ^ is almost equal to unity, i.e. the reflecting power
is very large. A substance which shows this strong reflecting
power characteristic of the metals (in the case of silver it
* More rigorous equations, in which sin 2 (f> has not been neglected in comparison
with e', are given in Winkelmann's Handbuch, Optik, p. 822 sq,
ABSORBING MEDIA 365
amounts to 95 per cent) is said to have metallic lustre.* This
is more marked the greater the absorption coefficient of the
substance. Since K is different for different colors, some
metals, like gold and copper, have a very pronounced color.
Thus a metal appears red if red light is reflected more strongly
than the other colors. Hence the light reflected from the
surface of a metal is approximately complementary to the color
of the light transmitted by it. In order to observe this it is
necessary to use sheets of the metal which are only a few
thousandths of a millimetre thick. Gold-foil of such thickness
actually appears green by transmitted light.
The more often light is reflected between two mirrors of
the same substance the more saturated does its color become,
for the colors which are most strongly absorbed by the sub-
stance are much less weakened by repeated reflection than the
others. In this way Rubens and Nichols, t and Aschkinass \
have succeeded in isolating heat-waves much longer than any
previously observed. An Auer burner without a chimney was
used as the source of the radiations. After five reflections upon
sylvine an approximately homogeneous beam of wave length
X = 0.06 1 mm. was obtained, this being the longest heat-
wave yet observed. The reflecting power of sylvine for this
radiation is R = 0.80, i.e. 80 per cent. Long heat-waves can
also be isolated by multiple reflections upon rock salt, fluor-
spar, and quartz.
It is important to distinguish between the surface colors
produced by metallic reflection and those which are shown by
weakly absorbing substances with rough surfaces ; for example,
by colored paper, colored glass, etc. These substances appear
colored in diffusely reflected light because the light is reflected
in part from the interior particles of the substance, and hence
* That this effect is actually due to a high reflecting power is proved by the
fact that a bubble of air under water from which the light is totally reflected
looks like a drop of mercury.
f Rubens and Nichols, Wied. Ann. 60, p. 418, 1897.
j Rubens and Aschkinass, Wied. Ann. 65, p. 241, 1898.
3 66
THEORY OF OPTICS
selective absorption is the cause of the color. In such cases
the colors in transmitted and reflected light are the same, not
complementary as in the case of the metals.
3. The Optical Constants of the Metals. Equation (22)
shows how the optical constants n and K of a metal can be
conveniently determined, namely, by observing the vibration
form of the elliptically polarized reflected light when the
incident light is plane-polarized, i.e. by measuring A and ^
by means of a Babinet compensator and analyzing Nicol in
accordance with the method described on page 255 sq. But
care must be taken that the surface of the metal be as clean as
possible, since surface impurities tend to reduce the value of
the principal angle of incidence.* The following table contains
some of the values which Drude has obtained by the reflection
of yellow light from surfaces which were as clean as possible:
Metals.
nx
n
^
R
3.67
0.18
75 4 2 '
4iw f
95-3#
Gold
2.82
0.37
72 18
41 39
85.1
4.26
2.06
78 30
i>2 ?<:
70.1
Copper .
2.62
0.64
71 7C
38 1:7
Tl.2
Steel
7.40
2.41
77 3
27 49
58.5
2.61
0.005
71 19
44 58
99-7
4.96
1.77
79 34
3 "5 41
78.4
The reflecting power R was not measured directly, but cal-
culated from (27).
The optical constants can also be determined by observa-
tions upon the transmitted light. By measuring the absorption
in a thin film of thickness d a value for K : A may be obtained,
as is seen from (10), A denoting the wave length in the metal.
Since now A = A Q : n, and since A , the wave length in air, is
known, ^/crnay also be obtained. But reflection at the bounding
surfaces of thin sheets of metal is accompanied by a great loss
*Cf. Drude, Wied. Ann. 36, p. 885, 1889; 39, p. 481, 1890.
ABSORBING MEDIA 367
in intensity. In order to eliminate this difficulty it is necessary
to compare the absorptions in films of different thickness. The
losses due to reflection are then in both cases nearly the same,
so that a conclusion may be drawn as to the value of UK from
the difference in the absorptions. The difficulty in making
these observations lies in obtaining metal films but a few
thousandths of a millimetre in thickness, which are yet uniform
and free from holes. For this reason the value of UK as deter-
mined by this transmission method usually comes out smaller
than by the reflection method.* But in some cases, t for
example, silver which can be easily deposited upon glass from
a solution the values of HK determined by the two methods
are in good agreement.
As in the case of transparent media, the index of refraction
can be determined from the deviation produced by a prism, {
but in the case of the metals the angle of the prism must be
very small (a fraction of a minute of arc) in order that the
intensity of the light transmitted may be appreciable. Since
Kundt succeeded in producing metal prisms suitable for this
purpose (generally by electrolytic deposition upon platinized
glass), the indices of refraction of the metals have been deter-
mined many times by this method.il Not only is the produc-
tion of these prisms troublesome, but also the. observations are
very difficult, since the result is obtained as the quotient of two
very small quantities. In general the results agree well with
those obtained from observations of reflection; for example,
the remarkable conclusion that for certain metals n < I has
been confirmed.
These small indices of silver, gold, copper, and especially
* W. Rathenau, Die Absorption des Lichtes in Metallen. Dissert. Berlin, 1889.
f W. Wernicke, Pogg. Ann. Ergzgbd. 8, p. 75, 1878. Also the observations of
Wien (Wied. Ann. 35, p. 48, 1888) furnish an approximate verification.
\ For the equations cf. W. Voigt, Wien. Ann. 24, p. 144, 1885. P. Drude,
Wied. Ann. 42, p. 666, 1891.
A. Kundt, Wied. Ann. 34, p. 469, 1888.
| Cf., for instance, Du Bois and Rubens, Wied. Ann. 41, p. 507, 1890.
3 68 THEORY OF OPTICS
of sodium are particularly surprising; they mean that light
travels faster in these metals than in air.
If these optical constants be compared with the demands
of the electromagnetic theory [cf. (u)], a contradiction is at
once apparent. For since e is to equal n 2 (i >c 2 ), the dielec-
tric constant of all the metals would be negative, since
K = tan 2tp, and since 2ip is for all metals larger than 45, i.e.
K > I. But a negative dielectric constant has no meaning.
Also, the second of equations (i i), namely, ri*K = aT, is not
confirmed, since, for example, in the case of mercury, for
yellow light crT = 20, while V = 8.6. For silver crT is
much greater, while ;z 2 /c is much smaller than for mercury.
The same fact is met with here which was encountered
above when the indices of refraction of transparent media were
compared with the dielectric constants. The electromagnetic
theory describes the phenomena well, but the numerical values
of the optical constants cannot be determined from electrical
relations. The extension of the theory, which removes this
difficulty, will be given in the following chapter.
4. Absorbing Crystals. The extension of the equations
for isotropic absorbing media to include the case of absorbing
crystals consists simply in assuming different dielectric con-
stants and different conductivities along the three rectangular
axes of optical symmetry. If the coordinate axes coincide
with these axes of symmetry, equations (12) on page 314 are
obtained, with this difference, that e lt e 2 , e 3 are complex
quantities, if, in accordance with (5) on page 359, the electrical
force is introduced as a complex quantity. To be sure the
equations will not be perfectly general, since the axes of sym-
metry for the dielectric constant do not necessarily coincide
with those for the conductivity. These axes must coincide
only in crystals which possess at least as much symmetry as
the rhombic system. Nevertheless the most general case will
not be here discussed, since the essential elements may be
obtained from the simplification here presented.*
* This is treated more fully in Winkelmann's Handbuch, Optik, p. 8il sq.
ABSORBING MEDIA 369
In order to integrate the differential equations given above,
namely,
let the components u, v, w of the light-vector be represented
by the equations
(a .
u = e= , I
in which m* -f- ^ 2 + / 2 = x an d M, N, 77 may be complex.
These equations correspond to a plane wave whose direction
cosines are m, n, p. V is the velocity of the wave, and K the
absorption coefficient (cf. page 360). Let
(30)
Then Fresnel's law (18) on page 316 may be written
_* = <>. (30
in which, however, # 2 , ^ 2 , C Q * are complex. Plence this equa-
tion splits up into two from which V and K may be calculated
separately as functions of the direction m, n t p of the wave
normal. According to equations (15), (19), and (20) on pages
315 and 317, the following relations hold for the quantities
M, N, II:
Mm + Nn + Up = o, . . . . (32)
Since, by (33), M, TV, II are complex, two elliptical ly
polarized rays correspond to every direction m, n, p. For if it
be assumed that M=M^\ N = N-e i8 *, then ^ # 2 denotes
370 THEORY OF OPTICS
the difference of phase between the components ?/, v of the
light-vector. For plane-polarized light tf, $ 2 o. Equa-
tion (32) expresses the fact that the plane of the vibration is per-
pendicular to the wave normal, (34) the fact that the ellipses
are similar to each other, while their positions are inverted.*
The relation which can be deduced from (31) between the
velocity and the direction m, n, p is very complicated. Hence
Fresnel's law, in spite of its apparent identity with (31), is
considerably modified. But the relations are much simpler in
the case of weakly absorbing crystals such as are always used
when observations are made with transmitted light, t For if /c 2
can be neglected in comparison with I, then co 2 = F 2 (i -[- 2zVc).
Hence setting
then
<2 *F 2 -tf' 2
a 2 -co* ~ a 2 - F 2 -z(2/cF 2 -tf' 2 ) ~~ a* I
Hence (31) splits up into the two equations
f
m
'
f
- FT 2 - v*r - v*f (38)
Equation (37) is Fresnel's law. Hence when the absorp-
tion is small this is not modified. Equation (38) presents K as
a function of m, n, and/. According to (33), when the absorp-
tion is small M, JV, 71 are very nearly real, i.e. the two waves
within the crystal have but a slight elliptic polarization. If
9}, SR, $)3 denote the direction cosines of the principal axis of
*For more complete proof of this, cf. Winkelmann's Handbuch, Optik, p. 813.
f In reflected light the effects of strong absorption are easy to observe, for
example, with magnesium- or barium-platinocyanide. Such crystals show
metallic lustre and produce polariz-a-tion.
ABSORBING MEDIA 37I
the vibration ellipse, then, from (33) and (36), since 9ft is the
real part of M, etc.,
Thus 9ft, Sft, $)3 are determined in the same way as the
direction of vibration in transparent crystals.
In view of (39) and the relation 9ft 2 + 9? + ^ = i, it is
possible to write (38) in the form:
9JL = cos q sin , W 9 cos , 3L = sin q sin . (44)
it * *J * L ^ 7 I<6 * /^ \ ' /
ABSORBING MEDIA 3 73
Hence, from (40), in the neighborhood of the optic axis
2 /c/ 2 = (a 2 cos 2 q + c 2 sin 2 q) cos 2 -f- V 2 sin 2 ,
1 I K45)
2/c 2 fa a C os 2 <7 -j- c'* sin 2 #) sin 2 1- b' 2 cos 2
2 2
These equations show that for any angle $ the value of
A-J is the same as that of /c 2 for an angle #>' = ;r ^. These
equations are indeterminate for the optic axis itself, because
then $ has no meaning. In accordance with the preceding
discussion, the direction of vibration may be taken arbitrarily
(cf. page 319). From (40) it follows that for a wave polarized
in the plane of the optic axes, i.e. vibrating perpendicularly to
these axes, since in this case 9ft = ^ o, 9 = I,
2KJP=b'\ ( 4 6)
but for a wave polarized in a plane perpendicular to the plane
of the optic axes, and therefore vibrating in that plane, since
for this case 9ft = cos q, 9? = o, ^ = sin q,
2/c/ 2 = a' 2 cos 2 q + c'* sin 2 q. . . . (47)
For intermediate positions of the plane of polarization values
of K are obtained which lie between those of K S and Kp.
Hence the absorption of a wave travelling along an optic axis
depends upon its plane of polarization. Upon introduction of
the quantities K S and K p (45) becomes
ib tb ib ib
i = *>.cos^ + *,-sin>-, * 2 - A>-sin a - + /c,.cos 2 -. (48)
For uniaxial crystals (a b, a' b'\ if g represent the
angle between the wave normal and the optic axis, it is easy
to deduce from (40) for the ordinary wave
for the extraordinary wave \- . (49)
374 THEORY OF OPTICS
5. Interference Phenomena in Absorbing Biaxial Crys-
tals. Let a plate of an absorbing crystal be introduced in
convergent light between analyzer and polarizer. Resume the
notation of 14 and 15 on pages 344 and 349, and consider
Fig. 91. A wave W l , vibrating in a direction H v , which
upon entering a crystal has an amplitude E cos 0, upon emer-
_ 27r ^L/
gence from the crystal has the amplitude E cos e ~r v\ , in
which / denotes the length of the path traversed in the crystal.
If d denote the thickness of the plate of crystal, and r l the angle
of refraction of the wave W l , then / d : cos r r Similarly
the amplitude of the wave W 2 is, upon emergence from the
27T _K- 2 .
crystal, E sin e r V t (the length of the path within the
crystal is assumed to be for both waves approximately the
same). After passing through the analyzer the amplitudes 01
the two waves are
cos cos (0 X)-e~ KI o-j
1 cos r* , x
' , \- (50)
E sin sin (0 x)-? K i a< *> cr 2 =
The difference in phase # of the two waves in convergent light
is determined by equation (88) on page 350.
The case of crossed Nicols \x = ) will be more carefully
considered. Assume that the plate of crystal is cut perpendic-
ular to the optic axis A lt and denote by f/> the angle which the
plane A^A 2 of the optic axes makes with the line MA 2 drawn
from a point M, which is near the optic axis in the field of
view,* to the optic axis A^\ then (cf. Fig. 101) the direction
of vibration H^ makes approximately the angle with the
2
direction A^A^ provided A^M is small in comparison with
* The different points of the field of view correspond (cf. p. 351) to the different
inclinations of the rays within the plate.
ABSORBING MEDIA 375
A^A^ If, further, the plane of vibration P of the polarizer
makes the angle a with the plane A^A^ of the optic axes, then
FIG. TGI.
th 7t
in (50) = a , x = ~ The amplitudes of the two
interfering waves are therefore
+ E cos (a - */ 2 ) sin (a - 1%), ~ ^ ) (5 l}
- sin (a - i>/ 2 ) cos (or - ^/ 2 y ~ **
in which
since in the neighborhood of the optic axis V^ = F 2 = /5, and
r is to be small.
Hence the intensity of the light which emerges from the
analyzer is
J= smttea-Me-'W+e- 2 "'"- 2*- ( *' + " i) .cos i}. (52)
4
If the wave normal actually coincides with the optic axis,
the end sought may be obtained from the following considera-
tions: The amplitude E is resolved into components which
are parallel and perpendicular respectively to the plane A^A^
of the optic axes. These components are E cos a and E sin a.
After emergence from the crystal the former has the value
376 THEORY OF OPTICS
E cos a e "'V 7 , the latter E sin at e Ks(T r After passage through
the analyzer the former has the amplitude E cos a.s'm a e ~ 2 "t cr ,
the latter E sin a cos a e ~ **<*. These two waves have no
difference in phase, since the velocity in the direction of the
optic axis is the same for both of them. Hence when the
wave normal is parallel to the optic axis, the light which
emerges from the analyzer has the intensity
y = sin 2* , -'-<- . . (53)
The first factor in (52) placed equal to zero determines the
position of the black principal isogyre /; = 2a. But while the
black isogyre in the unco fared crystals passes through the optic
axis itself, in the pleochroic crystals the point of intersection of
the optic axis with the isogyre is bright, unless a = o or
a = ~, i.e. unless the plate lies in the first principal position.
For, from (53), J 1 differs from zero when sin 2a ^ o, and K p
differs from K S .
The second factor in (52) placed equal to zero shows that
there are dark rings about the optic axis, since the value of
this second factor depends upon cos 6, and cos 6 has periodic
maxima and minima as the distance from the optic axis
increases. Nevertheless even with monochromatic light these
rings are perfectly black only where K I = /c 2 , i.e., according
to (48), when ip = -, for there the second factor actually
vanishes when cos d = I . The whole phenomenon of the
rings is less and less distinct the stronger the absorption, i.e.
the thicker the plate. For the term in (52) which depends
upon the difference in phase d has a factor which can be
written in the form e -(*> 4-^)0". jf the crystal is at all col-
ored, then one at least of the two absorption coefficients /c^and
K S must differ from zero, i.e. for a sufficiently large value of cr
or a sufficiently large thickness d of the plate this term COD-
ABSORBING MEDIA 377
taining cos d vanishes. This second factor in (52) can be
written
F=e- 2 i< J - + e- 2 *- (54)
Although cr is large, these terms may yet have appreciable
values, since K I or /c 2 may be small for certain points M of the
field of view provided either K P or K S is small. It can now be
shown that when ip = o or it, Fis a maximum ; when ^ = ,
a minimum. For, from (48),
3^ *
Therefore maxima or minima occur when ip = o or TT, or when
K I = K 2 , i.e. r/> = . But when ip = o or TT,
= *,;... (55)
n
and when ^ = ,
2
-/!=. /-(*> + * = /? ( 5 6)
Writing ,- 2 "> "= y, then ^ = *-2., \F^ = V^J.
But now, since the arithmetical mean is always greater than
the geometrical (the difference between them increasing as the
difference between x and y, i.e. between K P and K S , increases),
the values ^ = o or n correspond to a maximum, the values
n r T-
ib = to a minimum, 01 r .
2
In addition to the principal isogyre (^ = 2), there is
always a black brush traversing the field of view perpendicular
to the plane of the optic axes \^ -J . This brush coin-
cides with the principal isogyre in the second principal position
of the plate f a J.
37 8 THEORY OF OPTICS
Absorption gives rise to certain peculiar phenomena when
either the analyzer or the polarizer is removed. In the first
case the two amplitudes which emerge from the crystal have
the values E cos ( J#> " K ^ and E sin ( a - t$)e ~ ***. If
these are not brought back to a common plane of vibration,
they do not interfere and the resultant intensity is simply the
sum of the two components, i.e.
J= ?\cos\a - #> " 2Kl" + sin 8 ae~ 2Ks(T \. . . (58)
The following principal cases will be investigated:
I. a o. Then
J 2 |
J' = E 2 e
But since
- sin
therefore
7\ T
= oforip=o or 7t, or for ^ = n
When $ = o or n,
J
when i/J = *2 ,
If, therefore, K P J 2 , i.e.
there is a dark brush perpendicular to the plane of the optic
axes, which is, however, intercepted by a bright spot on the
optic axis. But if K P > K S (type I, andalusite, titanite), then
Jt > Jr ^ n tms case ^ e ^ ar k brush lies in the plane of the
optic axes and is continuous.
ABSORBING MEDIA 379
n
J = 2 {sin2 tye ~ *W + cos* tye ~
J' _ 2. e
When r> = o or TT,
when ^ = */ 2 ,
/ / = 2.^-
If, therefore, ^ < /<-,, / x K S , J l >/ 2 ,
i.e. the dark brush is perpendicular to the plane of the optic
axes and is intercepted by a bright spot on the optic axis.
If both analyzer and polarizer are removed, i.e. if a plate
of biaxial pleochroic crystal cut perpendicular to one of the
optic axes is observed in transmitted natural light, the resultant
intensity is
J = &(e-'W + t- '*.); .... (59)
while along the optic axis itself it is
J' = &(e ~ 2K ^ + e ~ 2K * a ) ..... (60)
For natural light may be conceived as composed of two in-
coherent components of equal amplitudes which vibrate in any
two directions which are at right angles to each other. Hence
in (60) 2E* denotes the intensity of the incident light. Since
now it was shown above [equation (54), page 377] that (59)
has a minimum value when ^ = , it is evident that a dark
brush perpendicular to the plane of the optic axes and intercepted
by a bright spot upon the axis will be seen. These figures
produced in natural light were observed by Brewster as long
ago as 1819. They may be easily seen in andalusite and
epidote.*
* For further discussion of these idiocyclophonous figures cf. Winkelmann's
I!a:i(l! u:b. Optik, p. 817, note I.
380 THEORY OF OPTICS
6. Interference Phenomena in Absorbing Uniaxial Crys-
tals. Let the plate of crystal be cut perpendicular to the
optic axis.
i. Crossed Nicols. Let the plane of vibration of the polar-
izer make an angle with the line AM which connects the
optic axis A with a point M in the field of view of a polarizing
arrangement which furnishes convergent light. Then AM
is the direction of vibration H of the extraordinary ray,
which, after emergence from the crystal, has the amplitude
E cos e ~ Kg(T ', and, after emergence from the analyzer, the
amplitude E cos sin e ~ Ke K e ) there is a dark brush when 0= , i.e. parallel to
the plane of polarization of the polarizer. The dark brush
passes through the axis itself.
j. Transmitted natural light. The intensity of the ordinary
ray is E*e ~ 2K ', that of the extraordinary ray is E*e ~ 2Ke i
or when
r.&,
- J -, b,
47T '
it follows that
.,
. = - J -, b,= L - L v .... (12)
'
The similar expression for e 2 Z 2 is obtained by replacing the
subscript i by 2. Hence, from (7),
A comparison of this equation with (17) on page 269,
O TLT
namely, j x = '- -- , shows that in place of the dielectric
constant e there appears the complex quantity e' which depends
upon the period T( = r 2 TT) ; thus
DISPERSION 387
in which the following abbreviation has been introduced:
The 2 is to be extended over all the ions which are capable
of vibrating. It is possible to assume more than two different
kinds of ions. But in the case of the high periods of light
vibrations and of dielectrics, these kinds are not to be assumed
to be identical with those found in electrolysis.
The meaning of the constants which appear in (15) can be
brought out as follows: If the period is very long, i.e. if
r = oo , a condition which is practically realized in static
experiments or in those upon slow electrical oscillations, it
follows from (15) that
= ,= i+2$' k ..... (16)
In such experiments e is the dielectric constant of the
medium. From (2) and (13) it is evident that $' h can be called
the dielectric constant of the ions of kind h. The resultant
dielectric constant is then the sum of the dielectric constants of
the ether and of all the kinds of ions.
Further, b h is a constant which is associated with the
natural period T h which the ions of kind h would have if their
coefficient of friction a k could be neglected. For in this case
(X = o, a h = r h = o) it follows from (i) that
t> h = *k, T h = T k '.27t ..... (I/)
It has been shown above on page 361 that a complex
dielectric constant indicates absorption of light. If ;/ represent
the index of refraction and K the coefficient of absorption, then
from the discussion there given [equation (n)], and the equa-
tion (15) here deduced,
' l ~r ~ " ~r*
By separating the real and the imaginary parts of this equation,
two relations may be obtained from which n and K may be
calculated.
3 88 THEORY OF OPTICS
2. Normal Dispersion. In the case of transparent sub-
stances there is no appreciable absorption. The assumption
must then be made that for these substances the coefficient of
friction a h is so small that the quantity can be neglected in
comparison with I f ) . This is evidently possible only
when the period T of the light does not lie close to the natural
period T h of the ions; for if these periods are nearly the same,
- = i and absorption would occur even though a h were small.
Transparent substances are to be looked upon as those in which
the natural periods of the ions do not coincide with the periods
of visible light, and in which the coefficients of friction of the
ions are small. If then for this case a h be neglected, the right-
hand side of (18) is real, so that K = o, and the index of
refraction is determined by
T - I
If the difference between the natural and the impressed
periods is great, n 2 can be developed in a rapidly converging
series. The natural periods in the ultra-violet t v must be
separated from the natural periods in the ultra-red ? r . For
the former is a small fraction, hence
,-,/
-? +etc. . . (20)
For the latter is a small fraction, hence
DISPERSION 389
Using these series and introducing in place of r the period
T itself, in accordance with (10) and (17), (19) becomes
- T'2.~ T'3- . . . (22)
Now in fact a dispersion formula with four constants,
namely,
*=-A'T + A+-f t + l , . . . (23)
in which A', A, B, and C are positive, has been found to
satisfy observations upon the relation between n and T for
transparent substances. (23) is easily recognized as the
incompleted series (22), and it is easy to see from (22) why the
coefficients A', A, B, and C must be positive. It also appears
that the term A of the dispersion equation, which does not
contain T, has the following physical significance:
A = i + 2$', ....... (24)
Since by (16) the dielectric constant e has the meaning
e = I + 2Q' h = I + 2& + 2$' r ,
it appears that
e-A = S9' r , ...... (25)
i.e. the difference between the dielectric constant and the term of
the dispersion equation which does not contain T is always posi-
tive and is equal to the sum of the dielectric constants of the ions
whose natural periods lie in the ultra-red. In this way the
discrepancies mentioned above between Maxwell's original
theory and experiment are explained.
Such a difference between e and A must always exist when
the dispersion cannot be represented by the three-constant
equation
^ = A+^ + r t ..... (26)
39 o THEORY OF OPTICS
for the coefficient A' of equation (23) depends upon the ions
which have natural periods in the ultra-red. The behavior of
water is a striking verification of this conclusion. For the
coefficient A' of the four-constant dispersion equation has a
larger value for water than for any other transparent substance ;
and this agrees well with the fact that water absorbs heat-rays
more powerfully than any other substance, and also with the
fact that for water the difference betwen e and A is greater
than for any other substance. If the assumption be made that
there be but one region of absorption in the ultra-red, the posi-
tion of this region can be calculated from A' and e A. For
in this case, from (22), (23), and (25),
e-A=$ r , i.e.T* = -. (27)
Now, according to Ketteler, for water A '0.0128 io 8 -^sec~ 2 ,
in which c = 3 io 10 . Further, e A = 77. From these data
the wave length measured in air which corresponds to the
region of absorption in the ultra-red is calculated as
~ 8 =60.10-,
.e.
X r = 7.75- io- 3 cm. = 0.08 mm. . . (28)
This wave length lies in fact far out in the ultra-red.
Experiment has shown that water has more than one region of
absorption in the ultra-red,* but the order of magnitude of the
wave length which is most strongly absorbed is in fact in
agreement with (28).t
Experiments upon flint glass, fluor-spar, quartz, rock salt,
and sylvine have given further quantitative verifications of the
dispersion equation (19) when rays of long wave length have
been investigated.^ If (19) be written in the form
* F. Paschen, Wied. Ann. 53, p. 334, 1894.
f Rubens and Aschkinass, Wied. Ann. 65, p. 252, 1898.
| Rubens and Nichols, Wied. Ann. 60, p. 418, 1897 J Paschen, Wied. Ann.
54, p. 672, 1895.
DISPERSION 39
i.e. in the form
M h
it is evident that b* must be identified with the dielectric con-
stant e. In the case of the substances just mentioned n* can
be well represented by equation (29) ; for example, for quartz,
for the ordinary ray, the values of the constants are :
J/ x = 0.0106, X* = 0.0106,
M 2 = 44-224, \?= 78.22,
-^=713-55. A 3 2 = 430.56, # = 4-58-
In this \ k = T h - K, and the unit in which \ h is measured is a
thousandth part of a millimetre (^). According to (29) these
seven constants M l , M 2 , J/ 3 , ^ , A 2 , A 3 , 2 must satisfy the
equation
,-, = ,- + $+. . . (30)
A l A 2 A 3
The numerical value of the right-hand side is 3.2, that of the
left 3.6. The difference is due to molecules whose natural
periods of vibration lie so far out in the ultra-violet that -c h = G
for them. If the sum of the dielectric constants of these mole-.
cules be denoted by r , then, from (29),
b* = i + ^ + 2$' h , M h = >;.A A 2.
Hence the following takes the place of (30):
Now the value of the dielectric constant of quartz lies between
4.55 and 4.73, which agrees very well with the value of &.
For fluor-spar
M l = 0.00612, V = 0.00888,
^,= 5099, A 2 2 = 1258,
& = 6.09, = 6.7 tO 6.9.
[Here again (30) is not exactly satisfied.]
392 THEORY OF OPTICS
For rock salt
-^ = 0.018, Aj 2 = 0.0162,
^2= 8 977 A 2 2 =3H9,
&* = 5.18, e = 5.81 to 6.29.
[(30) is approximately satisfied. ' = o. 18.]
For sylvine
M l = 0.0150, A x 2 = 0.0234,
M 2 = 10747, A 2 2
[(30) is not satisfied. According to (30') ' = 0.53.]
The conclusion that the difference between e and A of
equation (25) indicates natural periods of vibration and absorp-
tion in the ultra-red cannot be inverted, i.e. even if the dielec-
tric constant e has the same value as the constant A, which is
independent of the period in the dispersion equation, natural
periods and absorption in the ultra-red are not necessarily
excluded. According to (25) it is only necessary that the
dielectric constants $' r of the kinds of ions which lie in the
ultra-red be very small. Nevertheless appreciable absorption
can occur when r r r > For then in (18) the term $' r : i-~
appears in the expression for e '. By (12) this term has the
value 12 T r yi r : r r , in which r r denotes the frictional resist-
ance defined in (i). The value of this term remains finite even
when r is very small. Thus many substances actually exist,
such as the hydro-carbons, for which the difference between e
and A is small and which yet absorb heat-rays to a certain
extent.
From equations (22) or (23) it follows that ;z 2 continually
decreases as ^increases. This can be observed in all trans-
parent substances: it is the normal form of the dispersion
curve, and hence this kind of dispersion is said to be normal.
3. Anomalous Dispersion. The dispersion is always
normal so long- as the investigation is confined to a region of
DISPERSION 393
impressed periods which does not include a natural period of
the ions. But whenever an impressed period coincides with a
natural period, the normal course of the dispersion is disturbed.
For it follows from (19) that for periods T which are smaller
than a natural period T h , i.e. for which i f J has a nega-
tive value, say , n* contains the large negative term
$'h ' C J while for those values of T which are larger than
T k , i \i assumes the negative value ', so that ri* contains
the positive term -f- $ A ' : '. Hence as T increases contin-
uously ril in general decreases; but in passing through a region
of absorption it increases. Within the region of absorption
(19) cannot be used, but n l and K must be calculated from
(18), a h being now retained in the calculation. In any case
ri* must be a continuous function of T. Hence the general
form of the ri l and K curves is that shown in Fig. 102. The
value of K differs from zero only in the immediate neighborhood
of T h , and there it is larger the smaller the value of a h . For,
from (18), when T = T k ,
27t a h r h
Hence if a h , i.e. r kJ is small, the absorption bands of the
substance are sharp and narrow; but if a h is large, the absorp-
tion extends over a large region of wave lengths but has a
small intensity.
The form of the anomalous dispersion curve shown in Fig.
1 02 represents well the observations upon substances which
exhibit strong selective absorption, for example, fuchsine.*
The gases and the vapors of metals are distinguished by very
narrow and intense absorption bands, and anomalous dispersion
occurs in the neighborhood of these bands.
* Cf. Ketteler, Theoret. Optik, Braunschweig, 1885, p. 548 sq. A good
verification for the case of cyanine is given by PflUger, Wied. Ann. 65, p. 173,
1898.
394
THEORY OF OPTICS
The existence of anomalous dispersion is most simply
proved by the fact that a prism of some substances produces
from a line source a spectrum in which the order of the colors
is not normal. The phenomenon is, however, complicated by
the fact that in the spectrum two colors may overlap. Hence
it is preferable to use Kundt's method in which a narrow hori-
zontal spectrum formed by a glass prism with a vertical edge
is observed through a prism of the substance to be investigated,
the refracting edge of the latter being horizontal. If the dis-
FIG. 102.
persion produced by the second prism is anomalous, the
resultant spectrum is divided into parts which are at different
heights and are separated from one another by dark spaces
which correspond to the regions of absorption.
An objection to this prism method is this, that when the
absorption of the substance under observation is large, only
prisms of small refracting angle can be used. Hence the
method of Mach and Arbes,* in which total reflection is made
use of to determine the anomalous dispersion, is preferable.
A solution of fuchsine is placed in the glass trough G and a
flint-glass prism P placed upon it. The rays from a line
source Z, which lies in a vertical plane, are concentrated by
means of the lens s^ upon the bounding surface between the
glass and the fuchsine solution. The lens s 2 collects the
* Mach and Arbes, Wied. Ann. 27, p. 436, 1896.
DISPERSION 39 5
reflected rays and forms a real image of L upon the screen 5.
This image is spread out into a spectrum by means of a suitably
placed glass prism. This spectrum then shows the distribution
of light indicated in the figure : the curve mnpq represents the
limiting curve of total reflection. The break in the curve
between n and / shows at a glance the effect of anomalous
dispersion. Between n and/ there is a dark band, since, for
the colors which should appear at this place, the index of
refraction of the flint glass is the same as that of the fuchsine
solution, so that no reflection whatever takes place. The
index of refraction within the region of maximum absorption
cannot always be determined by this method, since, on account
of the high absorption, the partial reflection in this region is so
S
FIG. 103.
large (cf. metallic reflection) that it passes continuously into
total reflection, so that no sharp limiting curve appears, n and
K can then be determined from the partial reflection as in the
case of the metals.
A striking confirmation * of the theory here presented has
recently been brought out by the discovery of the fact that for
very long waves (\ 56yw) quartz has a much larger index
(n = 2.18) than for the shorter visible rays. Equation (29)
gives, with the assumption of the values of the constants given
for quartz on page 391, n = 2.20. Hence if the radiation from
an Auer burner be decomposed into a spectrum by means of
a prism of quartz, these long waves are found beyond the violet
* Rubens and Aschkinass, Wied. Ann. 67, p. 459, 1899.
3 g6 THEORY OF OPTICS
end of the spectrum and may therefore be easily isolated by
cutting off the other rays with a screen.
The case inverse to that of narrow absorption bands is that
in which not a h but b k or t h are to be neglected in (18) or
(15), i.e. the case in which the region of absorption is one in
which no natural periods of the ions occur (the impressed
periods are larger than the natural periods could possibly be).
In this case, from (18),
(32)
The last 2, that connected with the index v, refers to the
natural periods which lie in the ultra-violet. If these periods
are assumed to be small in comparison with T, then from (32),
if, as on page 391, 2-S, be called ^,
If only ions of kind h are present, it appears that as T
decreases from T = oo , n decreases continuously, and the
absorption, which covers a broad region, reaches a maximum
for a certain period T. These equations appear to represent
well for many substances the dispersion phenomena as they are
observed by means of long electrical waves ranging between
the limits \ = oo and A, = I cm.*
4. Dispersion of the Metals. In considering conductors
of electricity it is necessary to bear in mind that within these
conductors a constant electrical force produces a continuous
displacement of quantities of electricity, and that these latter
have no definite positions of equilibrium. The idea made use
of in electrolysis, that the displaced electrical quantities are
connected with definite masses (ions), will be applied to the
metals to the extent that the motion of the ions will be
assumed to take place in the metals also as though the ions
*Cf. Drude, Wied. Ann. 64, p. 131, 1898.
DISPERSION 397
possessed inert mass m. But this may be only apparent
Tiass, since the inertia may be accounted for by self-induction
(cf. note, page 383).
The constant $ of these conducting ions must be taken as
infinitely great, since, according to (2), fy is proportional to
the displacement of the ions from their original position be-
cause of the influence of a constant electrical force. The equa-
tion of motion of these ions is therefore obtained from equation
(i) on page 383 by substituting in it l oo . It is, therefore,
3 2 2 a
"&=**-'*'&' (34)
or if the current due to these ions, which according to (5) is
o
j x = e^l^Tf* be introduced,
$<>-* ..... <>
In this equation m is the (apparent or real) mass of an ion, e
its charge, %l the number of ions in unit volume. From (35)
it is evident that if two kinds of conducting ions, one charged
positively and the other negatively, whose resistance factors
are r l and r 2 , respectively, are present, then for a constant
current the following holds :
(36)
in which a is the specific conductivity of the substance measured
in electrostatic units (cf. page 358).
pv rr
For periodic changes, since X = * r ~~> b 7 (35)>
or
~^p ; [ ( 3; )
- r7 + lr
398 THEORY OF OPTICS
Equation (14) on page 386 must then be extended by a
term of this kind so that if, for abbreviation,
m\=m' t ...... (38)
the resultant complex dielectric constant takes the form
<>
If it be assumed that the periods are remote from the
natural periods of the ions of kind /*, so that a h may be neg-
lected, then since e n\\ z/r) 2 , it follows from (39), by
separation of the real and the imaginary parts, that
(41)
From this it is evident that in the case of the metals K may
be greater than I, since the right -hand side of (40) maybe
negative not only on account of the second term, but also on
account of the third term, which is proportional to the mass m)
of the conducting ions. For a given value of m' and r the
right-hand side of (40) becomes negative sooner the smaller r
is, i.e. the larger the specific conductivity. Furthermore, (41)
explains the second difficulty which was mentioned on page
368, namely, that for the metals H I K is smaller than -> H ^~r+ "^r"??
\u^ oy oz i ox* oy 4 oz
i.e. an accumulation of free charge might take place, since in
general for example, in the case of light vibrations the
right-hand side does not vanish.
An unsymmetrical isotropic medium would result if all
the molecules were irregular tetrahedra of the same kind,
the tetrahedra of the opposite kind (that which is the image of
the first) being altogether wanting. The same would be true
if one kind existed in smaller numbers than the other. A
graphical representation of equation (i) may be obtained by
conceiving that because of the molecular structure the paths
of the ions are not short straight lines, but short helixes twisted
in the same direction and whose axes are directed at random
in space. Consider, for example, a right-handed helical path
whose axis is parallel to the .r-axis. The component X drives
the charged ion always toward the left; but a positive Y drives
the ion on the upper side of the helix toward the left, on the
x lower side toward the right. The result
is therefore a force toward the right which
is proportional to , since it depends
v. xL_
upon the difference between the value of
Y above and its value below. Likewise
c a positive Z drives the ion on the front
side of the helix toward the left, on the
back side toward the right. The resultant effect toward the
right is therefore proportional to -f - . These conditions are
represented in equation (i), in which/ 7 would be negative if
the paths of the ions were right-handed helices and if the
coordinate system were chosen as in Fig. 104.
OPTICALLY ACTIVE SUBSTANCES
403
In consideration of equation (i), equation (i) on page 383
would become
If, as on page 385, be assumed to be a periodic function of
the time, then there results, upon introduction of the current
U.\ =
?w ~~ o-, / (3)
in which
7
tf = , O
47T
= T,
(4)
In what follows will be. neglected, which is permissible
if the periods of the light vibrations are not close to the natural
periods of any of the ions. The whole current due to all of
the ions and the ether is then
-?)}>
3r/ )
(5)
in which
e=i+2
(6)
The fundamental equations (7) and (i i) on pages 265
and 267 become therefore, if the permeability >w = I, so that
, etc.,
404
THEORY OF OPTICS
a_a^i\
a* ~ a* JJ "
^ r ^n\__
r "aJJy
a*
(7)
ar
From the same considerations which were given on page
271, it is evident that the boundary conditions to be fulfilled in
the passage of light through the surface separating two differ-
ent media are continuity of the components parallel to the sur-
face of both the electric and magnetic forces.
In this way a complete theory of light phenomena in
optically active substances is obtained.
From equations (7) it follows that
(9)
Hence from equations (7) and (8) there results, by the elimina-
tion of a, /3, y, as on page 275,
(10)
a, /?, y satisfy equations of the same form.
3. Rotation of the Plane of Polarization. If a plane wave
is travelling along the 3-a.x.is, it is possible to set
Z=o
OPTICALLY ACTIVE SUBSTANCES 405
p represents the reciprocal of the velocity of the wave. If the
values in (n) be substituted in (10), there results
These equations are satisfied if
^-, M=tN, . . . . (12)
or if
e-/V 3 =-^-, M=-iN. . . . (13)
Hence in this case the peculiar result is obtained that two
waves exist which have different values of /, i.e. different
velocities. Further, the waves have imaginary j-amplitudes if
they have real ^--amplitudes.
In order to obtain the physical significance of this it is to
be remembered that the physical meaning of X and Y is found
by taking the real part of the right-hand side of (11). Hence
when iN M,
i i
X M cos -(t pz], Y = M sin -(t pz) : . (14)
T r v
when iN = M,
X Mcos-(t pz}, Y= Msin -(t - px). (15)
These equations represent circularly polarized light; and
since, in accordance with the conventions on page 264, the
^r-axis is directed toward the right, the j/-axis upward to an
observer looking in the negative direction of the ^-axis, the
first is a left-handed circularly polarized wave, since its rotation
is counter-clockwise; the second is a right-handed circularly
polarized wave (cf. page 249).
406 THEORY OF OPTICS
Now these two waves have different velocities V, and in
fact, from (12), for the first
and, from (13), for the second
+ (17)
Hence the indices of refraction for right-handed and left-
handed circularly polarized light in optically active substances
must be somewhat different; and a ray of natural light is
decomposed into two circularly polarized rays one of which
is right-handed, the other left-handed. When the incidence
is oblique these two rays should be separated. These deduc-
tions from theory have been actually experimentally verified
by v. Fleischl * for the case of sugar solutions and other
liquids.
The effect of the superposition of two circularly polarized
waves whose velocities are V and V" respectively, one of
which is right-handed, the other left-handed, is
=^X'+X" =
"_ ' . (18)
Y Y'+Y" = 2Mcos-(t-"
Hence in one particular position, i.e. for a certain value of #,
the light disturbance is plane-polarized, since, according to
(18), Jfand Fhave the same phase. The position of the plane
of polarization with respect to the .r-axis is determined from
* E. v. Fleischl, Wied. Ann. 24, p. 127, 1885. It is easier to prove the cir-
cular double refraction of quartz along the direction of the optic axis. In quartz
the constant/" is greater than in liquids.
OPTICALLY ACTIVE SUBSTANCES 407
i.e. this position varies with #. Thus the plane of polarization
rotates uniformly about the direction of propagation of the
light, the angle of rotation corresponding to a distance z being
z p" p' f f
=**. 09)
provided A = Tc denote the wave length in vacuum of the
light considered. Since pc represents the index n of the sub-
stance with respect to a vacuum,
'=
n" and n f denoting the respective indices of refraction of the
substance for a right-handed and a left-handed circularly
polarized wave. Hence, from (19) and (19'),
; 2x~=n" n' ..... (19")
o
If, then, plane-polarized light fall perpendicularly upon a
plate of an optically active substance of thickness 2, the plane
of polarization will be rotated an angle 6 by the passage of the
light through the crystal. The rotation $ may take place in
one direction or the other according to the sign off. n" n'
may be calculated from 6 by (19').
Special arrangements have been devised for measuring this
angle of rotation easily and accurately.* In the half-shadow
polarimeter the field of view is divided into two parts in which
the planes of polarization are slightly inclined to each other.
But even with the use of two simple Nicols, a polarizer and an
analyzer, when the light is homogeneous and sufficiently
intense the position of the plane of polarization can be deter-
mined from the mean of a number of observations to within
* For a description of such instruments cf. Landolt, Das optische Drehungs-
vermogen der organischer Substanzen. Braunschweig, 2d Edition, 1897 ; Mliller-
Pouillet, Optik, p. 1166 sq. Rotation of the plane of polarization has been
practically made use of in sugar analysis.
4 o8 THEORY OF OPTICS
three seconds of arc, provided the setting is made with the aid
of the so-called Landolt band. For when Nicol prisms are
used the field of view is never polarized uniformly throughout,
so that, when the Nicols are crossed, the whole field is not
completely dark, but is crossed by a dark curved line which
was first observed by Landolt. The position of this band
changes very rapidly as the plane of polarization of the light
which falls upon the analyzer changes.*
4. Crystals. In order to obtain a law for crystals, it must
be borne in mind that the constants ^ , r l , which appear in
equations (i) of the dispersion theory on page 383, depend
upon the direction of the coordinates. Also that the terms
which have been added in this chapter and which correspond to
the optical activity can have a much more general form within a
crystal than that given in (i) on page 401. Nevertheless the
assumption will be made that, so far as these added terms are
concerned, a crystal is to be treated like an unsymmetrically
isotropic substance. No objection can be made to this assump-
tion, since the coefficients /of these added terms are so small,
in the case of all the actually existing substances, that the
change of /"with the direction which is due to the crystalline
structure can be neglected.
If the coordinate axes be taken in those directions which
would be the axes of optical symmetry of the crystal if it were
not optically active, the extension of equations (7) and (8)
would be t
(20)
/" o j \ F
i B/?
Of
in which
a/
=-,= ! + 2
-"
-
C5
I
(22)
(23)
In this $' h 9l h , &i9l k ,$t'9l k denote the three different dielec-
tric constants of the ions of kind h along the three coordinate
directions, and r' k , T'^ t" K are proportional to the three periods
of vibration corresponding to the three axes. In (23) $ A , r A
are mean values of ^, ^, \^", and r' h , T^, r'^', respectively.
For the sake of integration set, as on page 369,
v e Y
w e =
(24)
in which //, v, w may be interpreted as the components of the
light-vectors. Then it follows from (20) and (21),* using the
abbreviations
C 2 : e , = a\ C*:e 2 = b\ C*:e, = C \ . (25)
(in which e denotes a mean value of e l , e 2 , e 3 ) that the expres-
* This is more fully developed in Winkelmann's Handbuch, Optik, p. 791.
The normal surface and the ray surface are more fully discussed by O. Weder in
Die Lichtbewegung in zweiaxigen Crystallen. Diss. Leipzig, 1896, Zeitschr. f.
Krystallogr. 1806.
4 io THEORY OF OPTICS
sion for the velocity V in terms of the direction m, n, / of the
wave normal takes the form :
* - r>) + n\ F 2 _ <*)( F 2 - a 2 )
The introduction of the angles g^ and g 2 which the wave
normal makes with the optic axes gives, as on page 320,
2 V* = a 2 + c 2 + (a 2 - <*) cos g^ cos g^
-*r*t **+*. i (28)
2 F 2 2 = a 2 + r + (# 2 2 ) cos ^ cos g z
\ (a* - c*y sin* g l sin" ^ 2 -f- 4^;~.
It appears from this that the two velocities V l and F 2 are
never identical, not even in the direction of the optic axes.
Thus upon entering an active crystal a wave always divides
into two waves which have different velocities. These two
waves are elliptically polarized, and the vibration form of both
is the same, but the ellipses lie oppositely and the direction of
rotation in them is opposite. The ratio h of the axes of the
ellipse is given by
h . _ = - 2
h rj ( 9)
Hence in the direction of an optic axis (g l or g 2 = o)
k = i, i.e. the polarization is circular. But when the wave
normal makes but a small angle with the direction of an optic
axis, the vibration form is a very flat ellipse, since 2rj, even in
the case of powerfully active crystals, is always small in com-
parison with the difference a 2 c 2 of the two velocities.
Biaxial active crystals have not thus far been found in
nature; but several uniaxial active crystals exist. Quartz is
one of these. It exists in two crystallographic forms, one of
which is the image of the other: hence one produces right-
handed, the other left-handed, rotation. The rotation of the
plane of polarization which is produced by a plate of quartz cut
OPTICALLY ACTIVE SUBSTANCES 411
perpendicular to the optic axis is given, as in the case of
isotropic media, by the equation
6 = 2n*- 3 *=j*(n"-n"). . . . (30)
A A
When z = I mm. and yellow light (A 0.000589 mm.) is
used, $ 21.7 = o. I27T radians. Hence in this case
27T = n" n' = o. 12- = 0.000071. . (31)
A -s-
In this #' and n" denote the two indices of refraction which
quartz must have in the direction of its optic axis in conse-
quence of its optical activity. Now a double refraction n" n'
of the magnitude given in (31) has actually been observed in
quartz in the direction of its axis by V. v. Lang. This double
refraction can be conveniently demonstrated by the method
due to Fresnel, in which the light is successively passed through
right- and left-handed quartz prisms whose refracting angles
are turned in opposite directions.
If a quartz plate of a few millimetres' thickness, which is
cut perpendicular to the axis, be observed between crossed
Nicols in white light, it appears colored. For the plane of
polarization of the incident light has been rotated a different
amount for each of the different colors, and all of those colors
must be cut off from the field of view whose planes of polariza-
tion are perpendicular to that of the
analyzer. Hence the color of the
quartz plate changes upon rotation
of the analyzer. In convergent
white light the interference figure
described on page 356 for uniaxial
crystals when placed between
crossed Nicols are observable only
at considerable distance from the
centre of the field. Near the centre
the circular polarization has the
effect of nearly destroying the black FIG. 105.
cross of the principal isogyre. Hence a quartz plate cut per-
412 THEORY OF OPTICS
pendicular to the axis shows, between crossed Nicols in con-
vergent light, the interference figure represented in Fig. 105.
Spiral interference patterns appear when the incident light
is circularly polarized. The calculation of the form of these
spirals, which are known as Airy's spirals, is given in
Neumann's " Vorlesungen liber theoretische Optik, " Leipzig,
1885, page 244.
5. Rotary Dispersion. The rotation d of the plane of
polarization, which is produced by optically active substances,
varies with the color. The law of dispersion can be obtained
from equations (6) and (19) by setting the thickness of the
plate 2=1 and introducing for A , the wave length in vacuum,
A, the wave length in air,* thus
- ~
in which k is a constant.
If the natural periods of the active ions f are so much
smaller than the period of the light used that (r h : ry is neg-
ligible in comparison with I , there results the simplest form 6\
the dispersion equation, namely,
k'
This equation, due to Biot, agrees approximately with the
facts; yet it is not exact. If all the natural periods of the
active ions lie in the ultra-violet, (32) can be developed in
ascending powers of (r h : rf and put into the form
Now in most cases the first two terms of this equation
(Boltzmann's equation) are sufficient; nevertheless this is not
* In view of the small dispersion of air this is permissible.
f By active ions will be understood those kinds of ions whose equations of
motion are of the form (2) above, while those ions will be called inactive for which
he constant / ' in equation (2) has the value zero.
OPTICALLY ACTIVE SUBSTANCES 413
so for quartz, in which 6 has been measured over a large range
of wave lengths, namely, from A = 2,u to A = o.2/*. The
constants k l , k 2 , 3 can have different signs, since the / A '
corresponding to the different kinds of active ions need not
have the same sign.
If some of the active ions have natural periods r in the
ultra-red, then (32) must be developed in powers of (r : r r ) 2 .
The equation then takes the form
= o, ri'K" - H'K' = ~^ A . . (42)
If r is farther from the natural period ? h , and if a h is sufficiently
small, so that it is only necessary to retain terms of the first
order in K or a h , then, from (39) and (41), the law of dispersion
for the difference of the coefficients of absorption takes the
form
*'V'-V = ^^ . . . (43)
OPTICALLY ACTIVE SUBSTANCES 417
As A varies, a change in sign, and also maxima and minima
of n" K" ri 'K' ', may occur, provided there are present several
kinds of ions which have activity coefficients f h of different
signs.
Moreover the difference in the absorptions of the right- and
the left-handed circularly polarized waves is always small in
comparison with the total absorption.
For if./ 2 be neglected, and if only one absorption band is
present, it is easy to deduce, from (16) and (17),
n" K" H'K' 27tf' k
n"*" +**>'- ~r n> -
in which n denotes the mean of ri and n".
But/^ : A is always a small number.
Moreover it is to be observed that it is not necessary that
every active substance which shows an absorption band should
exhibit the phenomena here described. For, in order that this
be the case, it is necessary that the ions which cause the
absorption should be optically active. It is easily conceivable
that absorption and optical activity may be due to different
kinds of ions.
CHAPTER VH
MAGNETICALLY ACTIVE SUBSTANCES
A. HYPOTHESIS OF MOLECULAR CURRENTS
i. General Considerations. Peculiar optical phenomena
are observed in all substances when they are brought into a
strong magnetic field. Furthermore it is well known that
the purely magnetic properties of different substances are very
different, i.e. the value of the permeability // varies with the
substance (cf. page 269). It is greater than I for para-
magnetic substances, less than I for diamagnetic ones. Hence
a magnetic field is said to produce a greater density of the
lines of force in a paramagnetic substance than in the free
ether, and a less density in a diamagnetic substance than in
the free ether. Ampere and Weber have advanced the theory
that so-called molecular currents exist in paramagnetic sub-
stances. According to the theory of dispersion which has
been here adopted, these currents are due to the ionic charges.
When an external magnetic force is applied, these molecular
currents are partially or wholly turned into a definite direction
so that the magnetic lines due to them are superposed upon
the magnetic lines due to the external field.
According to this theory, diamagnetic substances ordi-
narily have no molecular currents. But as soon as they are
brought into a magnetic field, molecular currents are sup-
posed to be produced by induction. These currents remain
constant so long as the external field does not change. The
ionic charges must be assumed to rotate without friction so that
the maintenance of these currents requires the expenditure of
418
MAGNETICALLY ACTIVE SUBSTANCES 419
no energy. The lines of force due to these induced molecular
currents must oppose the lines of the external field, since,
according to Lenz's law, induced currents always flow in such
a direction that they tend to oppose a change in the external
magnetic field.
If it is desired to determine the optical properties of a sub-
stance when placed in a strong magnetic field, it is always
necessary to bear in mind that both in para- and diamagnetic
substances certain ions are supposed to be in rotation and to
produce molecular currents. If e be the charge of a rotating
ion of kind I , and T its period of rotation, the strength of the
molecular current produced by it is
* = e - T
If now such an ion, rotating about a point $)3, be struck by
the electric force of a light-wave, its path must be changed. If
the period of rotation T is very small in comparison with the
period of the light, the path of the ion remains unchanged in
form and period, but the point about which it rotates is changed
from $P to a point *' distant from 9ft in the direction of the
electrical force. The ion then oscillates back and forth
between ^ and 9fi' in the period of the light-wave. The same
mean effect must be produced if the period of rotation is large,
provided it is not a multiple of the period T of the light vibra-
tion. Any rotation of the plane of the path, which is produced
by the magnetic force of the light-wave, may be neglected,
since this is always much smaller than the external magnetic
force. This displacement of the molecular current also pro-
duces a displacement of the magnetic lines of force which arise
from it, so that a peculiar induction effect takes place, an effect
which must be considered when a wave of light falls upon a
molecular current.
This inductive effect can be at once calculated if the
number of lines of force associated with a molecular current is
known.
Now this number can easily be found. Let the paths of
420 THEORY OF OPTICS
the molecular currents all be parallel to a plane which is per-
pendicular to the direction R of the external magnetic field.
Consider first a line of length / parallel to the direction R.
Let %l f denote the number of molecular currents due to ions of
kind i upon unit length ; then / $1' denotes the number upon
the length /. These currents may be looked upon as a
solenoid of cross-section q, q being the area of the ionic orbit.
The number of lines of force in this solenoid is *
M = ^nWiq : c.
If now there are W such solenoids per unit area, then the
number of magnetic lines per unit area due to these molecular
currents is
in which 91 is the number of rotating ions of kind I in unit of
volume.
The components of M l in the direction of the coordinates
are
, = ifSt cos (JCr), ft l = -iqK cos (Ky),
y, = -iqW. cos (Ke).
(2)
2. Deduction of the Differential Equations. The discus-
sion will be based upon equations (7) and (n) (cf. pages 265
and 267) of the Maxwell theory, namely,
But while in the extensions of the Maxwell theory which have
thus far been made only the expression j x for the electric cur-
rent density was modified by the hypothesis of the existence
of ions, the magnetic current density s x retaining always the
* The number of lines of force in a solenoid is \itniq, where n is the number of
turns in unit length and i the strength of the current in electromagnetic units.
Since here / is defined electrostatically, c occurs in the denominator.
MAGNETICALLY ACTIVE SUBSTANCES 421
constant value \n>^ here, because of the introduction of the
concept of rotating ions, s x must also assume another form.
47tj x and 471 s x are defined by (12) on page 268 as the change
in the density of the electric and the magnetic lines of force in
unit time.
Now in order to calculate 4?rs x it is necessary to take
account of the fact that it consists of several parts. The
change which is produced directly by a light-wave in the flow
of lines of force through the rectangle dy dz in the ether is
represented by dy dz'-^rr . But another quantity must be added
to this a quantity which is due to the motion, produced by
the light- wave, of the point ^ about which the ions rotate,
since the lines of force M l move with the point P.
In order to calculate the amount of this portion of s x , con-
sider a rectangular element dy dz perpendicular to the ;r-axis,
and inquire what number of lines of force cut
the four sides abed of the rectangle because
of the motion of ^, the components of the
motion being <*;, 77, .
Consider first only the lines of force a l
which are parallel to the ^r-axis. In unit FlG - Io6>
time the number of lines of force which pass into the rectangle
through the side a is (*Vof) dz\ and the number which pass
out through the side c is ( A Y\ are functions of the coordinates. In homogeneous
substances ar lt P l , y l are constant. The number of lines o^,
which in their motion cut the sides a and c, increase the
number of lines which pass through the rectangle by the
amount dy dz~ (i^?j- Similarly the number of lines a l
which in their motion cut the sides b and d of the rectangle
add to the total flow through the rectangle the amount
Because of the component g of the motion of 9$, the lines
of force /?j , which are parallel to the jj/-axis, can cut only the
sides a and c of the rectangle. Now the number of lines which
pass through the rectangle changes only because of a rotation
of the lines ft l about the ^-axis, this change being positive if
the lines f$ l rotate from the -f- direction of y to the -\- direction
of x. The effect of this rotation can be calculated by subtract-
(p, dr\
fii'^t) dz, which gives the number
u* c
of lines which cut the side c in a second, the expression
(o dr\
ytfj- ) dz, which represents the number which cut a in a
off a
second. Since now
the rotation of p l adds to the flow of lines through the rectangle
the amount + dy dz (fti^j )
Similarly the rotation of the lines y l about the j-axis adds
the amount + ^ ^vT'i' to the flow of lines through the
rectangle.
The total flow through the rectangle, obtained by adding
these amounts, is
a/
MAGNETICALLY ACTIVE SUBSTANCES 423
The change in unit time in the number of lines which pass
through an element of unit area perpendicular to the ;r-axis is
therefore, since for a constant external field a lt fi l , y l are
independent of the time /,
4, = |r \ + (X,5 ~ f) - J^ - /,*)[ (4)
Strictly speaking, the current density is modified in a com-
plicated way by the rotation of the ions. But if the ratio of
the period of rotation of the ion to the period of the light is
not rational, it is only necessary, in order to find the mean
effect, to take account of the motion , ty, C of the centre of
rotation 9fi.
The current density j x may therefore be written as above
[cf. equation (7), page 385] in the form
ro
47T " K -dt'
For the motion of a point ^$, which is the mean position of a
rotating ion of kind I, two equations will be assumed. The
first is the same as that given above on page 383, namely,
and corresponds to the case in which 9fi can oscillate about a
position of equilibrium (ions of a dielectric). The second
is equation (34) on page 397, namely,
and corresponds to the case in which ^ moves continually in
the direction of the constant force X, i.e. the case in which e
is the ion of a conductor, for example a metal, m denotes
the ponderable mass of the ion.
If the changes are periodic, so that every X and every is
. /
proportional to /r", there results from (6)
' ' * ^ >
47TT 4^ r 47T
4 2 4 THEORY OF OPTICS
while from (7)
Hence, setting as above
^iQ -M?iQ m
= b = r * ~,=m r , . . (10)
i x>* \ /
(5) gi yes > m case * i s an i n f a non-conductor,
MS I
"i+,a/r-V"^ > '
But if ^ is the ion of a conductor,
In any case it is possible to set
__ . _ .
* ~ 47T 3/ ' ^ ~" 47T 9/ ' ^ ~ 4 7T B/ ' '
in which e' is in general a complex quantity depending upon r.
Moreover from (i), (2), and (8) there results, for an ion of
a non-conductor,
and from (9), for an ion of a conductor,
*ox, - - (is)
In both cases it is possible to set
y = v cos (Kz) X, .... (16)
in which v is in general a complex quantity depending upon r.
A similar expression may be obtained for a, etc. Setting
further
v cos (Kx) = r x , v cos (Ky) = r y , F cos (A>) ^, (17)
MAGNETICALLY ACTIVE SUBSTANCES 425
then from (13), (4), and (16) the fundamental equations (3)
become
When several kinds of molecules are present the same
equations (18) and (19) still hold, but the constants e f and v
are sums ; thus
e* =i+2- h h b + 47TT2 ^7, . . (20)
X + *T ? "*~~T*
q h
<2h , 4^^ v = v.
Hence the fundamental equations (18) and (19) become
(22)
. - (23)
A differentiation of these equations with respect to / and a
substitution in them of the values of g , KT taken from (22)
gives
. . (24)
e^j^F _ d*Y_
-axis in the time dt, then, according to page 384,
the strength of current along dq is / = eW^-;, in which 9?' is
the number of ions in unit length. Hence from (39), since
dl = dr
This is the force acting upon the whole number of ions along
the length drj. The number of these ions is Wdrj. The force
impelling a single ion along the ^r-axis is therefore
(40)
If in addition there is a magnetization in the direction of the
j-axis, a displacement C would add a force
These two terms, (40) and (41), must be added to the
right-hand side of the equations of motion of the ions, (6) and
* If u is not equal to i then must be replaced by the density of the lines of
force, i.e. by the induction.
MAGNETICALLY ACTIVE SUBSTANCES
435
(7) on page 423. If it be assumed that the ions are dielectric
ions, not conduction ions, an assumption which is permissible
for the case of all substances which have small conductivity,
then
t(c>r, 9C\
?W* 9/*V'
and by a cyclical interchange of letters
-^S+Mi^-HH
,3C , '
^ = ^-^--7r 2 ^
-5*-"-
;// -r 5 ^
a/
(42)
2. Deduction of the Differential Equations. The funda-
mental equations (3) on page 420 remain as always unchanged.
Since it has been assumed that there are no rotating ions, the
ions do not carry with them in their motion magnetic lines of
force, hence the permeability ^ I, and the previous relation
(cf. page 269) holds, namely,
* ~57* ^ ns y "^7 ^ ns * = ^7* ' * (43)
Furthermore, as above (page 384),
&
* nj y = 3i
3f
(44)
Equations (3), (42), (43), and (44) contain the complete
theory.*
* The most general equations can be obtained from the theory of rotating ions
presented above in Section A in connection with equation (42). The system of
equations thus obtained would cover all possible cases in which movable ions are
present in a strong magnetic field. For the sake of simplicity the two theories
are separately presented in Sections A and B.
43 6
THEORY OF OPTICS
When the conditions change periodically and the former
abbreviations are used, namely,
(42) becomes
(45)
.-C&) = r.. (46)
If the .s'-axis be taken in the direction of the magnetic field
so that Q x = $ y = o, JQ g = , then, by use of the abbreviations
= $, . . (47)
there results from (46)
-- erj $ = X,
47T
--5-K
~4* '
(48)
If these equations be solved with respect to , 77, and C,
there results
(49)
Hence, from (44),
B-^ /
47^ [i -[-
d^ \
, .
"
. (50)
These equations will be written in the abbreviated form
MAGNETICALLY ACTIVE SUBSTANCES 437
bt
\-tv
IV
3*'
4*JM = e'
(50
3. Rays Parallel to the Direction of Magnetization. In
this case z and / are the only independent variables, and equa-
tions (3), (43), and (51) give
. 3F\ 3 l Ln_ ly _
(52)
if fiX . 3F\ -d/3 it ,,-dY . ?>X\ da
A e w + lv ~wr ~ a? ^l 6 -ar - fV drJ = j
If a and ft be eliminated, there results
(53)
For the sake of integration set, as above on pages 404 and
426,
i t
Then there results, from (53),
i.e. the two sets of equations
= '*(!- iKj = 6 " + r, M= t'N, ) (
= n"\\ - IK")* = e" - v, M= - iN. \
n', K' correspond to left-handed, ", K" to right-handed cir-
cularly polarized waves. From the meanings given to e" and
v in (50) and (51) it follows that
438 THEORY OF OPTICS
If r does not lie close to a natural period, then the
imaginary term in , namely, z-, can be neglected, so that
K' = K" o, and since @ is always small in comparison with
i, and therefore in comparison with @,
From (19) on page 407 the rotation S of the plane of
polarization is
rf = (*" -*'> = (58)
If the mean of n" and n' be denoted by n y then
(59)
Hence, from (57),
n
Thus the index of refraction n is given, to terms of the first
order in $, by
4. Dispersion in the Magnetic Rotation of the Plane of
Polarization. Upon introduction of the values of (9 and <
from (47) in the last equations they become
(63)
Hence, as in hypothesis A, to a first approximation tf is
inversely proportional to ^ 2 .
MAGNETICALLY ACTIVE SUBSTANCES
439
If # 2 can be represented with sufficient accuracy by the
two-constant dispersion equation (cf. page 431)
* = "+ ..... (64)
(A, the wave length in air, is written for A Q ), then, from (62), it
must be possible to represent 6 by the two-constant dispersion
equation
a' and b' must have different signs if but two different kinds of
ions, one charged positively, the other negatively, are present.
This is the simplest assumption that can be made.
The agreement between (65) and observations upon carbon
bisulphide and creosote is shown in the following tables:
BISULPHIDE OF CARBON.
^ = 0.0450, a' = +O.II67, ' = + 0.2379.
Spectr. Line.
d calc.
d obs.
C
0.592
0.592
D
0.760
0.760
E
0.996
I.OOO
F
1.225
1-234
G
1.704
1.704
CREOSOTE.
0.0340, tf' = 0.070, ^=+0.380.
Spectr. Line.
8 calc.
d obs.
C
0-573
0.573
D
0.744
0.758
E
0.987
I.OOO
F
1.222
1.241
G
1.723
1.723
440 THEORY OF OPTICS
The agreement between theory and observation is almost
as good as that obtained by the hypothesis of molecular cur-
rents (cf. page 431).
5. The Impressed Period Close to a Natural Period.
When the period of the light lies close to a natural period,
the friction term cannot be neglected. Assume that T is
close to the natural period 7^ of the ions of kind i, and
write, therefore, r = Vb^i -{- g) ^(i -f~ g), in which g is
small in comparison with i. Then in equation (56), since
be also neglected in
comparison with g or , that
(68)
(69)
The imaginary part of the right-hand side of (68) reaches its
largest value, i.e. a left-handed circularly polarized wave
experiences maximum absorption, when
2^= + 0, i.e. r8 = T/ = r 1 8 (i+0). . . (70)
MAGNETICALLY ACTIVE SUBSTANCES 441
But the maximum absorption for a right-handed circularly
polarized wave occurs when
*g = - 0, i.e. r 2 = V = Tl 2 (i _ 0). . . (71)
Thus a small absorption band in incident natural light is
doubled by the presence of the magnetic field when the direction
of the field is parallel to that of the light. In one of the bands
the left-handed circularly polarized wave is strongly absorbed
so that the transmitted light is weakened and shows right-
handed circular polarization; in the other band the right-handed
circularly polarized light is wanting.
The same result would be reached from the hypothesis A
of the molecular currents.
If g is not small and if 2g is numerically larger than 0, so
that h is negligible in comparison with 2g 0, then in (68)
and (69) K and K" can be placed equal to zero, provided the
right-hand sides are positive. Hence at some distance from
the absorption band
(In order that the right-hand sides may be positive, the
numerical value of A must be greater than that of - -}.
ig 0/
From equation (59) on page 438, the amount of the rotation
of the plane of polarization is
= _ *(^ + 1 ,_ I _),
n A n \ 4T ! -tf > - /
in which
(72)
From this it appears that the rotation 8 has the same sign
upon both sides of the absorption band, and is nearly sym-
metrical with respect to this band, for, at least approximately,
8 depends only upon g*. The same result follows from equa-
442 THEORY OF OPTICS
tion (62). If d is positive, it appears from page 428, that the
rotation takes place in the direction of paramagnetic Amperian
currents. Since the sign of 6 is not determined by the sign of
the small term A' ', but by the much larger term B o, the direction of 6 is opposite to that of the
molecular currents, and further, r t > r rJ i.e. that wave (/)
whose direction of rotation is in the sense of the molecular
currents reaches its maximum absorption for a slower period
T than the wave (r) whose direction of rotation is opposite to
that of the molecular currents. When e l is negative the plane
of polarization is rotated in the direction of the molecular
currents. Then r t < r rJ i.e. in general that wave whose
direction of rotation is the same as that of the rotation d of
the plane of polarization reaches its maximum absorption for
a shorter period than the wave which rotates in the opposite
direction.
All these results have been verified by experiments upon
sodium vapor. These experiments will be discussed later.
For both absorption lines of this vapor (the two D lines) e is
found to be negative. The two D lines of sodium vapor are
then produced by negatively charged ions.
The absorption at a place where g = o may be small pro-
vided 0is large in comparison with h. Then, by (68) and (69),
*=A+A'-%, n"* = A-A'+2
The right-hand sides of these equations must be positive if
they are to have any meaning, i.e. the numerical value of A
D
must be greater than that of -^ The rotation 8 of the plane
of polarization is then proportional to
S ~ ""* ~ "' 2 = B/4> - A'. . . . (73)
MAGNETICALLY ACTIVE SUBSTANCES
443
6 is therefore large since is small. If e l is positive, the
rotation d is in the same direction as the molecular currents,
i.e. within the absorption band the rotation is opposite to that
just outside of the absorption band. Nevertheless the rotation
d need not pass through zero values, for at places where H'K'
and n" K" have large but different values it is meaningless to
speak of a rotation of the plane of polarization.
6. Rays Perpendicular to the Direction of Magnetization.
Let the -axis be taken in the direction of the magnetization,
the ^r-axis in that of the wave normal. Then x and / are the
independent variables and equations (3), (43), and (51) give
= o,
a
c 'dt 9.
Elimination of ft and y gives
e -
e'"$Y
c 'dt
(74)
(75)
If X be eliminated from the first two equations, there results
c* 2 V
= 4 (76)
Setting, for the sake of integration
444 THEORY OF OPTICS
it follows from (75) and (76) that
e"-=fV, e'=fV, M=-~N. . (77)
The velocities of Z and Fare then different, i.e. the sub-
stance acts like a doubly refracting medium. For Z, i.e. for a
wave polarized at right angles to the direction of magnetiza-
tion, the index of refraction and the coefficient of absorption
are obtained from
= n\l - iiff = e' = i +~ ; . . (78)
for a wave polarized parallel to the direction of magnetization
the following holds:
(79)
The difference between n' and w is in general very small,
since it is of the second order in provided is not small.
Hence this magnetic double refraction can only be observed in
the neighborhood of a natural period, since then & is very
small.
7. The Impressed Period in the Neighborhood of a
Natural Period. Set as above r = r,(i + g) = V^(i + ),
and assume that g is small in comparison with I.
Then in every term under the sign 2, save that which
corresponds to ions of kind I, is to be considered a real
quantity which is not very small. ^ is then negligible in
comparison with O 2 .
Hence, using the abbreviations (67) on page 440,
n
n'\\ *V) 2 A
itlf
,
~
MAGNETICALLY ACTIVE SUBSTANCES 445
or
B2 ih
-
Now for a metallic vapor the index of refraction is always
nearly equal to I, even when g is quite small. Hence it fol-
lows (cf. equation for ^ 2 on page 441) that A is almost equal
to i and B must be very small, so that in the second term of
the right-hand side of (80), which contains the small factor ,
B can be neglected in comparison with A. Therefore
n'\i-i K ? = A+ ( ^ + '*> (8.)
1 (2g + th)* <^
The imaginary part, i.e. the absorption, will therefore be
a maximum, provided h is small, when
4^-2 _ 0* = o, i.e. 2g = 0. . , . (82)
Hence when the plane of polarization of the wave is
parallel to the direction of magnetization, there are two absorp-
tion bands, one on each side of the single band which appears
when the magnetic field is not present.
For a wave whose plane of polarization is perpendicular to
the direction of magnetization (78) gives
W2(I _^_ 4 + _A_ . . . (83)
The absorption is a maximum at a place where g = o. Thus
for a wave whose plane of polarization is perpendicular to the
direction of magnetization the absorption is not altered by the
presence of the field.
If 2g is large in comparison with h and 0, K and K' are
very small, and approximately
B _
-- -
hence
B
446 THEORY OF OPTICS
or, since 4^ is large in comparison with 2 , approximately
ri n ' , ., , ....... (84)
**
i.e. the sign of n' n depends upon the sign of^, but is inde-
pendent both of the direction of magnetization and of the sign
of 0. Voigt and Wiechert have succeeded in verifying this
law of magnetic double refraction in the case of sodium vapor.*
8. The Zeeman Effect. Zeeman discovered that when the
vapor of a metal, like sodium or cadmium, is brought to
incandescence in a magnetic field, a narrow line in its emission
spectrum is resolved into two or three lines (a doublet or a
triplet) of slightly different periods. t The doublet is produced
when the direction of the magnetic lines is the same as the
direction of emission, the triplet when these directions are at
right angles to each other. These observations are explained
by the theoretical considerations given above \ in connection
with the law, which will be presented later, that the emission
lines of a gas correspond to the same periods of vibration as
the absorption lines. According to the preceding discussion
the two separate lines of the doublet ought to show right- and
left-handed circular polarization, while'in the triplet the middle
line ought to be polarized in a plane which is perpendicular to
the direction of the magnetization, and the two outer lines in
a plane which is parallel to it. These conclusions are actually
verified by the experiment. From measurements upon the
two triplets into which the two sodium lines (D l and Z> 2 ) are
* W. Voigt, Wied. Ann. 67, p. 360, 1899.
f P. Zeeman, Phil. Mag. (5) 43, p. 226 ; 44, p. 255, 1897.
\ This method of explaining the Zeeman effect is due to Voigt (Wied. Ann. 67,
p. 345, 1899). The differential equations upon which Voigt bases his theory are
the same as those deduced in 2, but he refrains from giving any physical mean-
ing to the coefficients in the differential equations.
This law results both from experiment and from Kirchhoff's law as to the
proportionality between the emission and absorption of heat-rays. The radiation
from a metallic vapor brought to incandescence in a Bunsen flame does not appear
to be a case of pure temperature radiation (cf. Part III), nevertheless theory shows
that even for luminescent rays the emission and absorption lines must coincide.
MAGNETICALLY ACTIVE SUBSTANCES 447
resolved, Zeeman obtained for the distance 2g between the
two outer lines of the triplet, when the strength of the mag-
netic field was = 22,400, the value 2g = 2 : 17,800.
Now, from (82) and (67),
or since r l = Vb l , and consequently, from (45) on page 436,
^ = 47rr 1 V 1 2 : m l , it follows that
2#= = $r l =.~L. (85)
1 cm l 2 n cm^
If the values of 2g y ^>, and 7^ for sodium light be introduced,
there results
-^-= i. 6- io7.
This number represents the ratio of the charge of the ion,
measured in electromagnetic units, to its apparent mass (cf.
note on page 383). From observations upon a cadmium
line (A. = o.48yw) this ratio is determined as 2.4- io 7 .*
Michelson has shown from more accurate observations,
made both with the interferometer and with the echelon spec-
troscope, that in general the emission lines are not resolved
simply into doublets and triplets but into more complicated
forms. t This is to be expected when, as is the case with
* It is to be noted that Kaufmann obtained from the magnetic deflection of the
kathode rays (Wied. Ann. 65, p. 439, 1898) almost the same number (i. 86.10")
for the ratio of the charge to the mass of the particles projected from the kathode.
For the ions of electrolysis this ratio is much smaller (9.5-IO 3 for hydrogen,
4. i io 2 for sodium). This can be accounted for either by assuming that an
electrolytic ion contains a large number of positively and negatively charged par-
ticles (electrons) which are held firmly together in electrolysis but are free to move
by themselves in a high vacuum, or to vibrate so as to give out light ; or that the
electrolytic ion consists of a combination of an electric charge e l of apparent mass
;w t with a large uncharged mass M. In a slowly changing electric field or in a
constant current the electron clings fast to the mass M. But in a rapidly changing
electric field, such as corresponds to light vibrations, only the electron moves, and
in a high vacuum the electron becomes separated from its mass M.
f Cf. Phil. Mag. (5) 45, p. 348. Astrophys. Journ. 7, p. 131 ; 8, p. 37, 1898.
Wied. Beibl. 1898, p. 797.
44 THEORY OF OPTICS
Michelson's experiments, the method of investigation is carried
to such a degree of refinement that the emission lines are
found, even in the absence of the magnetic field, to have a
structure more complicated than is assumed in the above
theoretical discussion, i.e. when an emission line is shown to
be a close double. Furthermore, a theoretical extension of
equation (46) is possible if the influence of the motion of neigh-
boring ions is taken into account. In this case in that equation
the second differential coefficient of the electric force with
respect to the coordinates would appear, and the magnetic
resolution of the absorption and emission lines would be more
complicated.*
A very powerful grating or prism is necessary for observing
the Zeeman effect directly. Hence it is more convenient to
use a method of investigation described by Konig f in which a
sodium flame in a magnetic field is observed through another
such flame outside the field. If the line of sight is perpendic-
ular to the field, the first flame appears bright and polarized.
From Kirchhoffs law as to the equality of emission and
absorption, only those vibrations of the magnetized sodium
flame whose period in the magnetic field is the same as with-
out the field can be absorbed by the unmagnetized sodium
flame. Perhaps the phenomenon observed by Egoroff and
Georgiewsky, J that a sodium flame in a magnetic field emits
partially polarized light in a direction perpendicular to the
field, can also be explained in this way, i.e. by absorption
in the outer layers of the flame, the field being non-homo-
geneous. But even if the field were perfectly homogeneous,
this phenomenon could be theoretically explained, since the
total absorption n ' K' for the waves polarized in the direction of
magnetization, when calculated from equation (80) for all
*Voigt (Wied. Aim. 68, p. 352) accounts for the anomalous Zeeman effects by
longitudinal magnetic effects. What is the physical significance of such an effect
has not yet been shown.
f Wied. Ann. 63, p. 268, 1897.
JC. R. 127, pp. 748, 949, 1897.
MAGNETICALLY ACTIVE SUBSTANCES 449
possible values of g, is found to be somewhat different from
the total absorption HK of the waves polarized in a plane which
is perpendicular to the magnetization when this is calculated
from (83) for all possible values of g*
9. The Magneto-optical Properties of Iron, Nickel, and
Cobalt. Although it has been shown above that in the case
of metallic vapors the conception of molecular currents does
not lead to a satisfactory explanation of the phenomena, yet
this concept must be retained in order to account for the mag-
neto-optical properties of the strongly magnetic metals. This is
most easily proved by the fact that, in the case of these metals,
the magneto-optical effects are proportional to the magnetiza-
tion, and therefore reach a limiting value when the magneti-
zation is carried to saturation, even though the outer mag-
netic field is continuously increased. t The explanation based
upon the Hall effect would not lead to such a limiting value, J
since the magneto-optical effects would then be proportional
to the magnetic induction of the substance, i.e. proportional
to the total density of the lines of force. It is true that,
strictly speaking, the Hall effect is never entirely absent, even
upon the hypothesis of molecular currents; nevertheless the
experimental results show that, in the case of iron, nickel, and
cobalt, the influence of the molecular currents is very much
greater than that of the Hall effect, so that, for simplicity, the
terms which represent the Hall effect will now be neglected.
* Voigt (Wied. Ann. 69, p. 290, 1899) accounts for the phenomenon observed
by Egoroff and Georgiewsky, as well as for the variations in intensity in the
Zeeman effect, by the assumption that the friction coefficient r in equations (42) on
page 435 depends upon the strength of the magnetic field in different ways for
vibrations of different directions. This assumption cannot be simply and plausibly
obtained from physical conceptions.
f This is proved by observations of Kundt (Wied. Ann. 27, p. 191, 1886) and
DuBois (Wied. Ann. 39, p. 25, 1890).
\ This, together with the difference in form of the deduced laws of dispersion,
is the difference between the two theories. They would be identical if the equa-
tions deduced from the hypothesis of the Hall effect were developed only to the
first order in the added magneto-optical terms. This is allowable because in the
case of the metals no narrow absorption bands occur.
450 THEORY OF OPTICS
a. Transmitted Light. When a plane wave passes normally
through a thin film of iron which is magnetized perpendicularly
to its surface, the equations in 3 on page 426 are applicable.
Denote by n and K the index of refraction and the coefficient
of absorption of the unmagnetized metal, by n' and K' the
corresponding quantities for the left-handed circularly polarized
wave, by n" and K" the same quantities for the right-handed
circularly polarized wave. Then from (28) and (29) on page
427, retaining only terms of the first order in v,
p'c = '(i- ,V) = 4
n(i - iK) = Ve.
If v be supposed to have the form
r=a + M, ...... (87)
in which a and b are real, then
n" - ri = (a + bK), n" K" ri K' = (a/c - b). (88)
The second of these equations asserts that the right- and
left-handed circularly polarized waves are absorbed in different
amounts; while the first one, in connection with (19') on page
407 (provided the difference between n" K" and n' K' is small
so that the emergent light is approximately plane-polarized),
shows that the rotation d* of the plane of polarization is de-
termined by
in which it is assumed that A = cT= 2ncr.
The film of metal must be very thin (a fraction of A Q ) in
order that it may be transparent. Nevertheless appreciable
* Unless n" K" and n' K 1 are nearly equal, so that the emergent light is approx-
imately plane-polarized, d has no meaning.
MAGNETICALLY ACTIVE SUBSTANCES 451
rotation is observable; for example, when z = o.332A the
rotation of red light (A = 0.00064 mm.) in the case of iron
magnetized to saturation is <5 = 4.25. This would give for
the rotation produced by a plate of iron i cm. thick the enor-
mous value d = 200 000. From these observations and (89)
there results, for red light and for iron magnetized to satura-
tion, the centimetre being the unit of length,
n(a + bit} = 0.758- io- 6 ..... (90)
The sign of a + bK is positive since the rotation d takes
place in the direction of the molecular currents in paramag-
netic substances.
The relation between the rotation d and the period r or
the wave length A Q is obtained from equations (20) and (21)
on page 425, taken in connection with (87) and (89). It is a
noteworthy fact that d decreases as A decreases.* This result
is seen from equation (89) to be probable, since ;/ and HK
actually decrease rapidly as A decreases, and since, from (21),
it appears that a and b likewise decrease as A decreases, pro-
vided only one kind of conduction ions is particularly effective
in producing the magneto-optical phenomena.
b. Reflected Light (Kerr Effect}. In order that the proper-
ties of the light reflected from a magnetized mirror may be
calculated, the boundary conditions which hold at the surface
of the mirror must be set up. These conditions can be
obtained from the differential equations (18) and (19) on page
425, and the consideration that the surface of the mirror is in
reality a very thin non-homogeneous transition layer in which
these differential equations also hold (cf. page 426).
If the surface of the mirror is taken as the jtry-plane, the
boundary conditions are found, by a method similar to that
used on page 271, to be
Continuity of
a,
(91)
* Cf. experiments of Lobach, Wied. Ann. 39, p. 347, 1890.
452 THEORY OF OPTICS
From these conditions a theoretical explanation of the effect
discovered by Kerr can be deduced.* This effect t consists in
a slight rotation of the plane of polarization of light reflected
from a magnetized mirror, when the incident light is plane-
polarized either in or perpendicular to the plane of incidence.
This can only be due to some peculiar effect of magnetization,
since without magnetization there is complete symmetry and
no such effect would be possible.
10. The Effects of the Magnetic Field of the Ray of
Light. It has been shown above that a powerful external
magnetic field produces a change in the optical properties of a
substance. Now the question arises whether, with delicate
methods of observation, an effect due to the magnetic field of
the light itself might not be detected in the absence of an
external field.
If, first, only the terms representing the Hall effect
be taken into account, i.e. if it be assumed that there are no
molecular currents (revolving ions), then the equations to be
used are (cf. page 435)
4*7, _ 3r_ etc i^___
c -by a*' ' c vt " -dz "" ay
47T/; = - + 4*2dll , . . . . (93)
_,, . . (94)
if e=I + ,- _.* ...... ( 95)
*This deduction was made by Drude, Wied. Ann. 46, p. 353, 1892. The
constant b which appeared there and was assumed to be real must here be taken as
complex, since from (21) on page 425 v is complex. This change makes the
result of the theory identical with that given by Goldhammer, Wied. Ann. 46, p.
71, 1892. The theory is in agreement with practically all of the facts. For the
effect of the surface layer on the phenomenon cf. Micheli, Diss. Lpz. 1900. Ann.
d. Phys. I, 1900.
fKerr, Phil. Mag. (5) 3, p. 321, 1877 ; 5, p. 161, 1878.
MAGNETICALLY ACTIVE SUBSTANCES 453
(94) is the characteristic equation of this problem. This
shows, since rj and C are approximately proportional to Fand
Z, that the differential equations of the electromagnetic field
are no longer linear in X, F, Z, a, fi, y. This means that the
optical properties must depend upon the. intensity of the light.
Such a dependence has never yet been observed, and it can
easily be shown that the correction terms in (94), which
represent the departures from the equation heretofore used,
namely,
are so small that their effect could not be observed. Since the
magnetic force a, /?, y is equal to, or at least of the same
order of magnitude as, the electric force X, F, Z, it is neces-
sary to find the value of ---, -777* i- e - to find the ratio of the
velocity of the ion to the velocity of light. Now approximately,
from (94),
**. >
i.e., when
ft z\
X A - sin 2 n ( -~ y-J ,
(96)
c7
Now, according to page 436, the natural period T Q of the
ion is determined in the following way:
T = " \27t
or
.Q V 2 a
4: (97)
454 THEORY OF OPTICS
A substitution of this value in (96) shows that the largest value
which -- can have as the time changes is
c ot
-
c W ~ 27tT 'me
J"2
If in this @ be set equal to I -- ^-, a substitution which is
permissible provided T is not close to T , it follows that
i 3 T e T*
~ =''*-*-*
e : me has for sodium vapor the value 1.6 io 7 (cf. page
447). This value will be used in what follows. Further, in
the visible spectrum T=2-io~ 15 approximately. Hence
(98) may be written
- ' ' ' (99)
It is first necessary to find a value for A, i.e. for the
strength of field in an intense ray of light. A square metre
on the surface of the earth receives from the sun about 124
kilogrammetres of energy in a second, i.e. 1.22- io 6 absolute
units (ergs) to the square centimeter. But from equation (25)
on page 273, for a plane wave of natural light of amplitude A t
the energy flow dE in unit time through unit surface (cm. 2 ) in
dE(m I sec per cm. 2 ) = A 2 . . . . (100)
* Without using Poynting's equation, the result contained in (100) may be
deduced as follows : The electromagnetic energy which in unit time passes
through I cm. 2 must be that contained in a volume of Fern. 3 , V being the velocity
of light. In air or vacuum V = c. Further, from page 272 the electromagnetic
energy in unit volume of air for the case of natural light is equal to A 2 : ^TT.
Hence dE cA z : \it.
MAGNETICALLY ACTIVE SUBSTANCES 455
From which, if half of the energy of the sun's radiation is
ascribed to visible rays, the maximum strength of the electric
field in sunlight is *
A =A/ . o.6i.io 3 = I.6-IO- 2 = o.oi6.f. . (101)
Hence for intense sunlight
This expression is always small provided T is not close
to T . But even if, for example, T: T = 60 : 59 (sodium
flame illuminated by light of wave length X = 0.0006 mm.),
77 : T 1 T* 30, and the value of (101) is still very small.
If the velocity of a plane wave be calculated from (94), it
is easy to see that its dependence upon the magnetic correction
terms is of the second order, i.e. the change in the velocity of
light produced by an increase in intensity from zero to that of
sunlight would be of the order io- 20 F. Hence the conclusion
may be drawn that an observable magneto-optical effect due to
tJie magnetic field of tJie ligJit itself does not exist. There
might be some question as to this conclusion in the case in
which the period of the incident light very nearly coincides
with the natural period (sodium vapor illuminated by sodium
light). But the absorption which would then take place would
render impossible a decisive test as to whether or not in this
case the index of refraction varies with the intensity.
If now molecular currents (revolving ions) be assumed,
equations (3), (4), (5) on page 420 sq. become applicable.
If it were necessary to consider only one kind of revolving
ion, then, from (31) on page 429, the density y^ of the lines of
force might be set equal to (yu i)y t /* being the permeability
* As a matter of fact this ratio is only about ^.
1 The maximum strength of the magnetic field has the same value. This would
therefore be about T ^ of the horizontal intensity of the earth's magnetic field in
Germany.
456 THEORY OF OPTICS
of the substance. In this it is assumed that the magnetization
of the substance can follow instantaneously the rapid changes
in y. If this should not be the case, it would be necessary to
give /* a value smaller than that which is obtained with a con-
stant field. Hence equations (3) and (4) take the form
8
(I03)
o r y pi y' V ' dt ^ when *'^' -s- are referred to a fixed
system of coordinates, so that finally the relation holds
Now the terms -3-, etc., must appear in equations (2) because
the entire velocity of the ions is composed of the velocity of
translation v x of the substance, and the velocity of the ion with
respect to the substance. This last is represented by - , not
For the components of the magnetic current density the
equations (13) on page 268 hold, namely,
since it is proposed to neglect the effect of any external
magnetic field, and since, in accordance with page 456, the
permeability yu of all substances is equal to unity for optical
periods.
If the substance has no velocity of translation, i.e. if
v x = v y v z = o, then the equation of motion of an ion is
(cf. page 383)
Now it will be assumed that the influence of the substance
upon the ion is not affected by the motion of the substance.
Nevertheless the differential equation must be modified because
of the fact that the ions share in the motion of the sustance,
and a moving ion is equivalent to an electric current whose
components are proportional to ev x , ev y , ev s . The magnetic
460 THEORY OF OPTICS
force a', /?, y acts upon this current. Hence the equation of
motion of an ion is (cf. similar discussion on page 434)*
-
e
-(v y y-v,p). . (5)
d 3
Here, too, it is to be observed that appears, but not ,
dt 9/
since (5) expresses the relative motion of the ions with respect
to the substance.
When the changes in X or are periodic, it is possible to
write
dt
T' is then equal to the period T f divided by 2n. Nevertheless
it is to be observed that this period T' is the relative period
with respect to the moving substance, and not the absolute
period T referred to a fixed system of coordinates. It is
important to distinguish between T and T 1 '; thus, for example,
T' > T when the substance moves in the direction of the
propagation of the light. In the case of plane waves in which
all the quantities are proportional to
in which x, y, and z refer to a fixed coordinate system,
T = T: 27t is proportional to the absolute period T.
* For the reasons discussed on page 455 the terms - , etc., are omitted from
c at
the right-hand side of (4), for they are too small to be considered. For the motion
of the earth v\c=. 10 4 , i.e. it is of an entirely different order of magnitude from
--< : c. Also in Fizeau's experiment with running water, which will be described
later, in which v : c has a still smaller value, it is only the terms which depend
upon -u which have an appreciable effect upon the optical phenomena. The ionic
velocities -^-, etc., do not have such an effect.
dt
BODIES IN MOTION 461
Now, from (3) and (6),
1 Lfi P\ v *+P# y +P&\
t'" t\ oo )'
i.e., if the velocity v is small in comparison with ca,
z: r f
T GO
i =
in which v n denotes the velocity of the substance in the direc-
tion of the wave normal.
If the abbreviations used on page 386, namely,
r m$
a = , o = - s-, ..... (o)
47r' 47te 2
be introduced into (5), there results
. (9)
In equations (2) e9l means the charge present in unit
volume.
If the value of e%l [cf. page 270, equation (20)] obtained
from (the dielectric constant e of the ether is set equal to i)
<>
be substituted in (2), there results
I
If several kinds of molecules are present, the first factor of
the last term of this equation becomes, provided i t be neg-
462 THEORY OF OPTICS
lected, i.e. provided the substance has no appreciable absorp-
tion,
*
In this equation n is the index of refaction corresponding
to the period T' = 2m' when the substance is at rest. Equa-
tion (12) is derived from the theory of dispersion [cf. equation
(18) on page 387]. If now in equation (n) the differential
coefficient -=- be replaced by its value in terms of taken from
dt ut
(3), and if the resulting value for ^nj x be substituted in (i), a
differential equation is obtained for the substance in motion
referred to a fixed system of coordinates. This equation is
much simplified if only terms in the first order in v be retained.
It is always permissible to neglect the other terms, since, even
when v represents the velocity of the earth in space, it is still
very small in comparison to the velocity of light. It is then
d 3
possible to replace -j by -- in those terms in (11) which are
at ot
multiplied by z/, and also to neglect, in the case of homo-
geneous substances, the second term of (i i) which is multiplied
by v x , since approximately, i.e. for v = o, for a periodic
change of condition in such substances the following relation
holds (cf. page 275):
Thus (n) becomes
?>X
But, from (i) and (4),
BODIES IN MOTION 463
hence ^nj x may be written in the form
- (v x X + v y Y
Hence, in view of (i) and (4), there result for a moving, homo-
geneous, isotropic medium whose points are referred to a fixed
system of coordinates the following differential equations:
L - - - { 2 v V - 4- V -- [
\" y
+ ^- +
___ ___ ___ fl /x
^ a/ ~~ a-sr ^ ' c a/ ~~ a^ a^ ' c a/ ~ ^r a^ "
Differentiation of equations (15) with respect to ,t', y, and
-3: respectively and addition gives, with the use of the abbrevia-
tion
rc 2 - I
az
^ " '
- (v x AX + v y A Y + v,4Z) =0. (i 6)
464 THEORY OF OPTICS
In the terms which are multiplied by v x , etc., the following
approximations may be used :
Hence, from (16),
This equation asserts that in the moving substance the elec-
trical force cannot be propagated as a plane transverse wave,
since F is not equal to zero. But the magnetic force, on the
other hand, can be so propagated, since, from (15'),
09)
The differential equations (15) and (15') may easily be
transformed into equations each of which contains but one of
the quantities X, Y, Z, a, ft, y. For example, if the first of
equations (i 5) be differentiated with respect to t, and if and
~ be replaced by their values taken from (15'), there results
In consideration of (18) this becomes
2
The differential equations in Y, Z, -W '"-& ' (27)
If this equation be substituted in (2), then for any number
of kinds of ions, in consideration of (9), (10), and (12),
dX
468 THEORY OF OPTICS
Hence equations (i), (3), (4), and (28) give, in connection
with (19),
-i d e a/ , v,X-v x Y
dY n* - i d f ?> ( . v.Y- v y Z
- -
n 2 dZ tf-i d 9 / , .v x Z - v z X
l_^_ 9 /^ | P.a - ^r\ 3 (^ , ^,/?~ ^ y g \
f dt " 9*\ C J 9A^ <: j'
! \ c\
differentiation ( ) is again denoted simply by ,
\ox) ox
c df
I dc
c df
a* 1
n* dZ'
c df
'by
c df
i_4r 7
7 S 7
(39)
According to (30) and (38) the boundary conditions,
when the boundary is perpendicular to the .s-axis, are that
X' , Y 1 ', cli p f be continuous at the boundary. . (40)
Now equations (39) and (40) have the same form as the
differential equations and boundary conditions of the electro-
magnetic field for the case of a medium at rest. Hence the
important conclusion :
Jf, for a system at rest, X, F, Z, a, /?, y are certain
known functions of x, y, 2, /, and the period T, then, for the
system in motion, X ', Y' , Z 1 ', a', /?', y' are the same functions
BODIES IN MOTION 473
v x x -\- v y -f- v z z
of x, y, z, t ' ^ - , and T; in which now x, y, z
are the relative coordinates referred to a point of the medium,
and Tis the relative period with respect to a point of the moving
medium. From (7) on page 461, the absolute period is in the
latter case to be assumed as T\i -}.
- 7. The Configuration of the Rays Independent of the
Motion. The last proposition is capable of immediate applica-
tion to the relative configuration of the rays. For, in a system
at rest, let the space which is filled with light be bounded
by a certain surface 5 so that outside of 5 both X, Y, Z, and
a, /?, y vanish. Then when the system is in motion X', V,
Z ', and ', /?', y' vanish for points outside of 5, i.e. in the
moving' system also the surface S is the boundary of the space
which is filled with light. Now suppose that 5 is the surface
of a cylinder (a beam of light), an assumption which can be
made if the cross-section of the cylinder is large in comparison
with the wave length. The generating lines of this cylinder
are called the light-rays. According to the ibove proposition,
the boundary of the beam of light, even though it be frequently
reflected and refracted, is unchanged by the common motion
of the whole, i.e. in the moving system light-waves of the rela-
tive period T are reflected and refracted according to the same
hnvs as rays of the absolute period T in the system at rest.
The laws of lenses and mirrors need therefore no modifica-
tion because of the motion. Likewise the motion has no
influence upon interference phenomena. For these phenomena
differ from the others only in that the form of the surface S
which bounds the light-space is more complicated, and, as
above remarked, this form is not altered by the motion.
For crystals * also the configuration of the rays is inde-
pendent of the motion, for the differential equations and
* Whether this is true for naturally and magnetically active substances will not
here be discussed. To determine this a special investigation is necessary.
474 THEORY OF OPTICS
boundary conditions applicable to these can be put into forms
similar to (39) and (40), so that it is only necessary to refer
to the laws of refraction of the crystal at rest.
8. The Earth as a Moving System. The last considera-
tions are especially fruitful in discussing the motion of the
earth through space. For, according to what has been said,
the motion of the earth* can never have an influence of the first
order in v upon the phenomena which are produced with terres-
trial sources of light; for the periods emitted by such sources
are merely the relative periods of the above discussion, i.e.
they are wholly independent of the motion of the earth, so that
the configuration of the rays cannot be altered by this motion.
Now in fact numerous experiments by Respighi,t Hoeck.J
Ketteler, and Mascart || upon refraction and interference (some
of them upon crystals) have proved that the phenomena are
independent of the orientation of the apparatus with respect to
the direction of the earth's motion. On the other hand, when
celestial sources of light are used the effect of the earth's
motion can be detected, for in this case the relative period
depends upon that motion. As a matter of fact the spectral
lines of some of the fixed stars appear somewhat displaced.
This is to be explained by the relative motion of the earth, or
of the whole solar system, with respect to the fixed stars.
For the laws of refraction and interference are concerned with
relative periods, and from equation (7) these are given by
varies
7T I 1, in which T is the absolute period. Thus T
with the magnitude and sign of v n , and hence also the posi-
tion of the spectral lines formed upon the moving earth by
* Substances which show natural or magnetic optical activity are here
neglected.
f Mem. di Bologna (2) II, p. 279.
\ Astr. Nachr. 73, p. 193.
Astron. Undulat. Theorie, pp. 66, 158, 166, 1873.
| Ann. de 1'ecole norm. (2) i, p. 191, 1872; 3, p. 376, 1874.
BODIES IN MOTION 475
refraction or diffraction. This is known as Doppler's Prin-
ciple*
Since the path of the earth about the sun is nearly a circle,
v n is in this case equal to zero. Hence, as has been also
experimentally shown by Mascart,t the motion of the earth
causes no shifting in the Fraunhofer lines of the solar
spectrum .J
9. Aberration of Light. Although, as was shown in 7,
the configuration of the rays is not influenced by the motion of
the earth, yet the direction of the wave normal which corre-
sponds to a given direction of the ray does depend upon that
motion. This has already been shown on page 470; but it is
worth while to here deduce directly the definition of the ray
without using Huygens' principle as was done above. Con-
sider, for example, the case of a plane wave in a system at rest:
all the quantities involved are functions of / ^ ^
CO
In a system at rest p l , / 2 , / 3 are the direction cosines of
both the wave normal and the ray. The physical criterion
for the direction of the ray will be that the light pass through
* In the above it is assumed that the source A is at rest and the point of obser-
vation B in motion. The considerations also hold in case both A and B move.
v n is then the relative velocity of B with respect to A measured in the direction of
the propagation of the light. In this case the rigorous calculation shows that the
actual period T and the relative period T' observed at B stand to each other in
the ratio T: T' = a) v' : GO v, in which v is the absolute velocity of B, v that
of A in the direction of the ray, and oa that of the light in the medium between A
and B. It is only when v' and v are both small in comparison with GO that this
rigorous equation reduces to that given in the text, i.e. to the customary form of
Doppler's principle. Now we know nothing whatever about the absolute velocities
of the heavenly bodies ; hence in the ultimate analysis the application of the usual
equation representing Doppler's principle to the determination of the relative
motion in the line of sight of the heavenly bodies with respect to the earth might
lead to errors. Attention was first called to this point by Moessard (C. R. 114,
p. 1471, 1892).
f Ann. de Tecole norm. (2) I, pp. 166, 190, 1872.
\ No account is here taken of the displacement, due to the rotation of the sun,
of the lines which are obtained from light which comes from the rim of the sun.
In experiments the light from the entire disk of the sun is generally used.
476 THEORY OF OPTICS
two small openings whose line of connection has the direction
cosines p l , / 2 , p y If now the whole system moves with a
velocity v x , v yt v z , there must always be one ray (called a
relative ray when referred to a moving system) whose direction
cosines are/ x , / 2 , p y But according to page 473 this ray is
produced by waves which are periodic functions of
v.x + Vyy + y.*. Pi*+P*y+P** ,,
~7~ ~c*~ "' ' (4I)
This expression corresponds to plane waves for which the
direction cosines of the wave normal p{, p^, p are propor-
tional to
This relation (42) makes possible the calculation of the direc-
tion of the wave normal in the moving system from the
direction of the ray, and vice versa. This relation is also
identical with that deduced on page 471 [cf. (35')], from
Huygens' principle, for the quantities ^ , )3 2 , 3 there corre-
spond to /! , / 2 , / 3 here, and approximately c : GJ = n.
Hence if upon the moving earth a star appears to lie in the
direction p l , p 2 , / 3 , referred to a coordinate system connected
with the earth, its real direction is somewhat different, for this
latter coincides with the direction of the normal to the wave
from the star to the earth, i.e. the position of the star is
obtained from p^ / 2 / p.
The case in which the line of sight to the star and the
motion of the earth are at right angles to each other will be
considered more in detail. Thus set p^ = p 2 = o, / 3 = i,
v y = v z = o, v x = 7>; then from (42), if the velocity in air GO
be identified with c, as is here permissible, the position of
the star is given by
/,' ' /,' : A' = v : O : e, . . . . (43)
i.e. the real direction of the star differs from its apparent direc-
tion by the angle of aberration C which is determined by
BODIES IN MOTION 477
tan C = v : c. This angle of aberration is not changed when
the star is observed through a telescope whose tube is filled
with water, since it has been shown that the relative configura-
tion in any sort of a refracting system is not changed by the
motion.* This conclusion maybe reached directly as follows:
If oj differs appreciably from c, as is the case when the obser-
vation is made through water, then the wave normal in the
water is no longer given by (43), but, in accordance with
(42), by
C*
Pi ' A' : A' = v : : -^ = v : : cn > - - (44)
from which the angle of aberration ' is determined by
tan C' = v : en. The corresponding wave normal in air or in
vacuo makes, however, another angle with the -sr-axis such
that, since the boundary between air and water is to be
assumed perpendicular to the direction of the ray, according
to Snell's law sin : sin ' = n. Since now, on account of
the smallness of and ', the sin is equal to the tan, it follows
that tan v : c, i.e. the angle of aberration is the same as
though the position of the star had been observed directly in
air.
10. Fizeau's Experiment with Polarized Light. Although
in accordance with the theory the motion of the earth should
have no influence upon optical phenomena save those of aber-
ration and the change in the period of vibration in accordance
with Doppler's principle, and although experiments designed to
detect the existence of such an effect have in general given nega-
tive results, nevertheless Fizeaut thought that he discovered
in one case such an effect. When a beam of plane-polarized
light passes obliquely through a plate of glass, the azimuth of
polarization is altered (cf. p. 286). The apparatus used con-
sisted of a polarizing prism, a bundle of glass plates, and an
analyzer. At the time of the solstice, generally about noon,
* Cf. p. 116 above.
f Ann. de chim. et de phys. (3)58, p. 129, 1860; Pogg. Ann. 114, p. 554,
1861.
478 THEORY OF OPTICS
a beam of sunlight was sent, by means of suitably placed
mirrors, through the apparatus from east to west, and then from
west to east. It was thought that a slight difference in the
positions of the analyzer in the two cases was detected.
According to the theory here given no such difference can
exist. For if in any position of the apparatus the analyzer is
set for extinction, then the light disturbance is limited to a
space which does not extend behind the analyzer. According
to the discussion on page 473, the boundary of this space does
not change because of the motion of the earth, provided the
configuration of the rays with respect to the apparatus remains
fixed ; and this is true even when crystalline media are used
for producing the bounding surface 5 of the light-space.
Hence the position of extinction of the analyzer must be inde-
pendent of the orientation of the apparatus with respect to the
earth's motion. In any case it is to be hoped that this experi-
ment of Fizeau's will be repeated. Until this is done it is at
least doubtful whether there is in reality a contradiction in this
matter between experiment and the theory here presented.
ii. Michelson's Interference Experiment. The time
which light requires to pass between two stationary points A
and B whose distance apart is / is t^ = , where c represents
the velocity of light. It will be assumed that the medium in
which the light is travelling is the ether, or, what amounts to
the same thing, air. If the two points A and B have a common
velocity v in the direction of the ray, then the time of passage
t^ of the light from A to B is somewhat different. For the
light must travel in the time // not only the distance /, but
also the distance over which the point B has moved in the time
//, i.e. the total distance travelled by the light is / -j- vt^, so
that
*/' =/+"'!' (45)
If the light is reflected at B, in order to return to A it
requires a time t such that
t c = I - < (46)
BODIES IN MOTION 479
For this case differs from the preceding only in this, that now
A moves in a direction opposite to that of the reflected light.
Hence the time t' required for the light to pass from A to B
and back again to A is, from (45) and (46),
or
provided the development be carried only to terms of the
v
second order in --. Now although the influence of the com-
mon motion of the points A and B upon the time /' is of the
second order, it should be possible to detect it by a sensitive
interference method.
The experiment was performed by Michelsen in tho year
1 88 1.* The instrument used was a sort of an interferential
refractor furnished with two horizontal arms P and Q set at
right angles to each other and of equal length (cf. Fig. 57,
page 149). Two beams of light were brought to interference,
one of which had travelled back and forth along P, the other
along Q. The entire apparatus could be rotated about a
vertical axis so that it could be brought into two positions such
that first P, then Q coincided with the direction of the earth's
motion. Upon rotating the apparatus from one position to the
other a displacement of the interference bands is to be
expected.
The amount of this displacement will now be more
accurately calculated. Let the arm P coincide with the direc-
tion v of the earth's motion, the arm Q be perpendicular to it.
Let A be the point in which P and Q intersect. The time t'
required for the light to pass the length of P and back is given
by (47). But the time t" required for the light to travel the
* Am. Jo. Sci. (3) 22, p. 120, 1881.
480 THEORY OF OPTICS
length of Q and back is not simply t' r 2/ : c\ for it is neces-
sary to remember that the point of intersection A of the twc
arms P and Q, from which the light starts and to which it
returns after an interval of time t' ', has in this
time changed its position in space. Thus the
distance through which this point A has
moved is vt' (Fig. 107). The first position
of the point A will be denoted by A iy the last
by A 2 . In order that the light from A l may
return to A 2 after reflection at the end of the
3 arm Q, it is necessary that the reflecting
FIG. 107. mirror at Q be somewhat inclined to the wave
normal. The distance travelled by the light is 2s and the
relation holds,
Also, t" = 2s : c denotes the time which the light requires to
travel the length of Q and back. Now, from (47), if terms of
higher order than the second in v be neglected,
hence
'_"-.
t ~ t ~ c ' ~#
If this difference in time were one whole period 7 1 , the
interference fringes would be displaced just one fringe from the
position which they would occupy if the earth were at rest, i.e.
if v = o. Hence if the displacement d be expressed as a
fractional part of a fringe, there results from (49)
in which is the angle of aberration. According to page
116, C = 20. 5" = 20.5. TT : i8o-6o 3 = 0.995. io- 4 radians.
BODIES IN MOTION 481
The displacement produced by turning the instrument
from the position in which P coincides with the direction of
the earth's motion to that in which Q coincides with this
direction should be 2#.
But no displacement of the interference fringes was
observed. The sensitiveness of the method was afterwards
increased by Michelson and Morley* by reflecting each beam
of light several times back and forth by means of mirrors.
The effect of this is to multiply several times the length of the
arms P and Q. Each beam of light was in this way compelled
to travel a distance of 22 metres, i.e. /was 11 metres. The
apparatus was mounted upon a heavy plate of stone which
floated upon mercury and could therefore be easily rotated
about a vertical axis. According to (50) this rotation ought
to have produced a displacement of 2d = 0.4 of a fringe, but
the observed displacement was certainly not more than 0.02
of a fringe, a difference which might easily arise from errors
of observation.
This difficulty t may be explained by giving up the theory
that the ether is in absolute rest and assuming that it shares in
the earth's motion. The explanation of aberration becomes
then involved in insuperable difficulties. Another way of
explaining the negative results of Michelson 's experiment has
been proposed by Lorentz and Fitzgerald. These men assume
that the length of a solid body depends upon its absolute motion
in space.
As a matter of fact, if the arm which lies in the direction
of the earth's motion were shorter than the other by an amount
9
/JL the difference in time t' t" , as calculated in (49), would
2^
* Am. Jo. Sci. (3) 34, p. 333, i g8 7 ; Phil. Mag. (5) 24, p. 449, 1887.
f Sutherland (Phil. Mag. (5) 45, P- 2 3, 1898) explains Michelson's negative
result by a lack of accuracy in the adjustment of the apparatus. But, according
to a communication which I have recently received from H. A. Lorentz, this
objection is not tenable if, as is always the case, the observation is made with a
telescope which is focussed upon the position of maximum sharpness of the fringes.
482 THEORY OF OPTICS
be just compensated, i.e. no displacement of the fringes would
be produced.
However unlikely the hypothesis that the dimensions of a
substance depend upon its absolute motion may at first sight
seem to be, it is not so improbable if the assumption be
made that the so-called molecular forces, which act between
the molecules of a substance, are transmitted by the ether
like the electric and magnetic forces, and that therefore a
motion of translation in the ether must have an effect upon
them, just as the attraction or repulsion between electrically
charged bodies is modified by a motion of translation of the
v 2
particles in the ether. Since -^ has the value io~ 8 , the
diameter of the earth which lies in the direction of its motion
would be shortened only 6.5 cm.
PART III
RADIATION
CHAPTER I
ENERGY OF RADIATION
I. Emissive Power. The fundamental laws of photom-
etry were deduced above (page 77) from certain definitions
whose justification lay in the fact that intensities and bright-
nesses calculated with the aid of these definitions agreed with
observations made by the eye. But it is easy to replace this
physiological, subjective method by a physical, objective
means of measuring the effect of a source of light. Thus it is
possible to measure the amount of heat developed in any sub-
stance which absorbs the light-rays. To be sure this intro-
duces into the photometric definition a new idea which was
unnecessary so long as the physiological unit was used, name-
ly, the idea of time, since the heat which is developed in an
absorbing substance is proportional to the time. According
to the principle of energy, the heat developed must be due to
a cert tin quantity of energy which the source of light has
transmitted to the absorbing substance. Therefore the emis-
sion E of a source Q is defined as the amount of energy which
is radiated from Q into the surrounding medium in unit time.
Now radiant energy consists of vibrations of widely differ-
ing wave lengths. It must be possible to express the amount
483
484 THEORY OF OPTICS
of energy transmitted in unit time by waves whose lengths lie
between A and A + dh in the form E K d\. The factor E^ will
be called the emission for the wave length A.
The emission between the wave lengths A X and A 2 is there-
fore
I
*A,
\, ...... (i)
*,
and the total emission is
/OO
E^d\ ....... (2)
The emission of a body depends not only upon its nature,
but also upon the size and form of its surface. In order to be
independent of these secondary considerations, the term emis-
sive power will be introduced and defined as the emission
(outward) of unit surface.
2. The Intensity of Radiation of a Surface. The funda-
mental law stated on page 77 that the quantity of light is the
same at every section of a tube of light, i.e. of a tube whose
surface is formed by rays of light, appears necessary from the
energy standpoint, since the quantity of light is interpreted as
the energy flow in unit time. For, as was shown on page 273,
the rays of light are the paths of the energy flow, i.e. energy
passes neither in nor out of a tube of light. Hence the flow
of energy must be the same through every section of a tube,
since the same amount of energy must flow out of every
element of volume as flows into it, provided this element
neither contains a source of light nor absorbs radiant energy.
Hence the energy flow which a surface element ds sends
by radiation into an elementary cone of angular aperture dl
may be written in the form [cf. equation (69), page 83]
dL = ids cos dflj ..... (3)
in which denotes the angle included between the element of
surface ds and the axis of the elementary cone, i.e. the dircc-
ENERGY OF RADIATION 485
tion of the rays under consideration, i will be called the
intensity of radiation of the surface ds.
If all parts of a curved radiating surface appear to the eye
equally bright, then, as was shown on page 82, / must be
constant, i.e. independent of the inclination 0. The discus-
sion as to whether or not i is constant when considered from
the energy standpoint will be reserved till later. If, for the
present, / be assumed to be constant, then from (3) the energy
flow which passes from ds into a finite circular cone whose
generating lines make an angle U with the normal to ds is
found to be [cf. (73) on page 83]
L = nids sin 2 U. ..... (4)
Setting U = and dividing by ds, the emissive power e of
ds is obtained in the form
' = ' ........ (5)
Here again /, the total intensity of radiation, must be dis-
tinguished from 4 the intensity of radiation for wave
length A. If e K denote the emissive power for the wave length
A, then
'A = m\ ........ (6)
3. The Mechanical Equivalent of the Unit of Light __ On
page 8 1 the flame of a Hefner lamp was assumed as the unit
of light. Tumlirz * has found the emission within a horizontal
cone of unit solid angle from such a flame to be o. 1483 gram-
calories a second; Angstrom's t value for the same is 0.22
gram-calories a second. If such a lamp be assumed to radiate
uniformly in all directions, then its total emission, i.e. the
energy which it emits in all directions (into the solid angle
47r), is calculated from the value of Tumlirz as
r cal - gr cal
-= 1.86 *
* Wied. Ann. 38, p. 650, 1889.
f Wied. Ann. 67, p. 648, 1899.
4 86 THEORY OF OPTICS
or, since one gram-calorie is equal to 419- io 5 ergs, the value
of E in the C.G.S. system is
6
sec
(7)
Only 2.4 per cent of this energy corresponds to visible
rays.* Hence the light emission amounts to
(8)
sec
Hence if the unit of light is understood to mean the energy of
the light-rays emitted by a Hefner lamp in a second in a hori-
zontal direction within a cone of unit solid angle, i.e. upon
I cm. 2 at a distance of I cm., then
ergf
i unit of light = 1.51 -io 5 . ... (9)
This is then the mechanical equivalent of the unit of light.
The candle-metre is taken as the unit of intensity of illumi-
nation (cf. page 81). It is defined as the quantity of light
which a Hefner lamp radiates upon I cm. 2 at a distance of
i m. The solid angle amounts in this case to i: 100-100.
Hence, from (9),
erg
i candle-metre = 15 - .... (io)
Hence when the intensity of illumination is I candle-metre,
i.e. when an eye is at a distance of I m. from a standard
candle, it receives, assuming that the diameter of the pupil
is 3 mm., about I erg of energy in a second. This rate of
energy flow would require I year and 89 days to heat I gm.
of water i C. This calculation gives some idea of the
enormous sensitiveness of the eye. When the eye perceives a
star of the 6th magnitude it responds to an intensity of illumi-
nation of about i-io~ 8 candle-metres, since a star of the 6th
* In the experimental determination of this number the heat-rays were
absorbed by a layer of water.
ENERGY GF RADIATION 487
magnitude has about the same brightness as a Hefner lamp at
a distance of 1 1 km. In this case the eye receives about
i . io~ 8 ergs per second.
The so-called normal candle (a paraffine candle of 2 cm.
diameter and 50 mm. flame) has an emission about 1.24 times
that of the Hefner lamp.
4. The Radiation from the Sun. According to Langley
about one third of the energy of the sun's radiation is absorbed
by the earth's atmosphere when the sun is in the zenith-
According to his measurements, if there were no atmospheric
absorption, the sun would radiate upon I cm. 2 of the earth's
surface at perpendicular incidence about 3 gr. cal. (more
accurately 2.84) per minute (solar constant). Angstrom
obtained a value of 4 gr. cal. a minute. Hence, making-
allowance for the absorption of the earth's atmosphere, the
flow of energy to the earth's surface is, according to Langley,
about 2 gr. cal. a minute 1 .3 io 6 erg/sec. Pouillet's value,
which was given on page 454, is somewhat smaller. The
energy of the visible light between the Fraunhofer lines A
and H 2 amounts to about 35$ of the total radiation, i.e. the
so-called intensity of illumination B of the sunj without allow-
ing for the absorption in the air, is, from Langley 's measure-
ments,
ersr
B 6.Q. io 5 = 46300 candle-metres. . (11)
sec
If the mean distance of the sun from the earth be taken as
149. io 9 m., the candle-power of the sun is found to be
I.02.I0 27 .
5. The Efficiency of a Source of Light. The efficiency g
of a source of light is defined as the ratio of the energy of the
light radiated per second to the energy required to maintain
the source for the same time.
Thus a Carcel lamp of 9.4 candle-power consumes 42 gm.
of oil in an hour or I.i6-io~ 2 gm. in a second. The heat
of combustion of the oil is 9500 calories per gram, i.e.
4 88 THEORY OF OPTICS
39.7. 1 o 10 ergs. Now equation (8) gives the emission of the
standard unit, hence the efficiency of the lamp is
9.4.I.9-I0 6
Thus the efficiency is very small; only 0.4$ of the energy
contained in the oil is used for illumination.
The electric light is much more efficient. With the arc
light I candle-power can be obtained with an expenditure of
J watt, i.e. 5.IO 6 erg/sec. Hence for the arc light
I.Q. io 6
Z = ? - = 0.38 38$.
5 * io 6
For the incandescent lamp g has about the value 5.5^.
These figures show that it is more economical to use the
heat of combustion of oil to drive a motor which runs a dynamo
which in turn feeds an arc light, than to use the oil directly
for lighting purposes. A Diesel motor transforms about 70$
of the energy of the oil into mechanical energy, and 90$ of
this can be transformed into electrical energy by the dynamo
which feeds the arc light; hence the efficiency of the electric
light, upon the basis of the energy of the oil used, may be in-
creased to
g= 0.38-0.7.0.9 =
In this calculation no account has been taken of the fact
that the carbons in the lamp are also consumed. For an
incandescent lamp of the ordinary construction, which requires
about 3J watts per candle-power, g would be equal to 3.4$
calculated upon the basis of the fuel consumption of the motor.
For a Nernst incandescent lamp which requires I watt per
candle-power,* - would be as high as 12$.
6. The Pressure of Radiation. Consider the case of a
plane wave from a constant source of light falling perpendicu-
* The consumption of energy varies from .5 to 1.8 watts according to con-
ditions.
ENERGY OF RADIATION 489
larly upon a perfectly black body. Such a body is defined as
one which does not reflect at all, but completely absorbs all
the rays which fall upon it, transmitting none.* According
to the theory of reflection given above, an ideally black body
must have the same index of refraction as the surrounding
medium, otherwise reflection would take place, f Moreover it
must have a coefficient of absorption, which must, however,
be infinitely small, since otherwise reflection would take place
(cf. chapter on Metallic Reflection), even though the index of
refraction were equal to that of the surrounding medium.
Hence, in order that no light may be transmitted by the body,
it must be infinitely thick. An approximately black body can
be realized by applying a coat of lamp-black or, since lamp-
black is transparent to heat-rays, of platinum-black; likewise
pitch or obsidian immersed in water, not in air, are nearly black
bodies. The most perfect black body is a small hole in a
hollow body. The rays which enter the hole are repeatedly
reflected from the walls of the hollow body even though these
walls are not perfectly black. Only a very small part of the
rays are again reflected out of the hole. This part is smaller
the smaller the hole in comparison with the surface of the
body.
Let plane waves, travelling along the positive ^-axis, fall
upon a black body . Conceive a cylindrical tube of light
parallel to the -s'-axis and of cross -section q. Let energy flow
in at z = o. This energy will be completely absorbed, i.e.
transformed into heat within the black body, which is supposed
to extend from z a to z = oo. The amount of energy thus
absorbed in any time t is E-q-V-t, if denote the radiant
energy which is present in unit of volume of the medium in
front of , and V the velocity of the waves in this medium.
* A perfectly black body can emit light if its temperature is sufficiently high.
Hence it would be preferable to use the term "perfectly absorbing" instead of
"perfectly black."
f This shows that the definition of a black body depends upon the nature of the
medium surrounding it.
49 o THEORY OF OPTICS
If now the black body be displaced a distance dz in the
direction of light, then the energy which falls upon the body
in the time t is less than before by the amount of the energy
contained in the volume q>dz of the medium, i.e. by the
amount q-dz-E. Hence the amount of heat developed in the
body is smaller than before by the same amount (measured in
mechanical units). But the same amount of radiant energy
always enters the tube in the time t no matter whether the
body ^ is displaced or not. Further, the electromagnetic
energy contained in the volume q-dz, which has been vacated
by the motion of the body, is always the same, i.e. it is inde-
pendent of whether this volume is occupied by $ or not, since
the index of refraction, and therefore also the dielectric con-
stant, of $ is to be identical with that of the surrounding
medium, so that reflection does not occur, i.e. the electric and
magnetic forces at the surface of the body are the same in the
medium and in $. If, therefore, because of the displacement
of $ a distance dz, the same energy which has entered the
light-tube in the time t develops less heat than when $ is not
displaced, then, according to the principle of the conservation
of energy, this loss in heat must be represented by work
gained in the displacement of St If this work be expressed in
the form p-q-dz, p represents the pressure which is exerted
upon St by the radiation. Hence
p-q-dz = q-dz-^Ly
i.e.
/= E (12)
Thus the pressure of radiation which is exerted by plane
waves falling perpendicularly upon a perfectly black body is
equal to the amount of energy of the incident waves contained in
unit of volume of the medium outside.
Since, according to 4, the energy flow from the sun to the
earth amounts to 1.3. io 6 erg/sec, per cm. 2 , this is the amount
ENERGY OF RADIATION 49*
of energy contained in 3-IO 10 cm. 3 of air. Hence the energy
in i cm. 3 is
Therefore the sun's rays exert this pressure upon I cm. 2 of a
black body. This pressure is about equal to a weight of
4-io~ 5 mgr., i.e. it is so small that it cannot be detected
experimentally. Nevertheless this pressure is of great theoret-
ical importance, as will be seen in the next chapter.
7. Prevost's Theory of Exchanges. Every body, even
when it is not self-luminous, radiates an amount of energy
which is greater and contains more waves of short period the
higher the temperature of the body. If, therefore, two bodies
A and B of different temperatures are placed opposite to each
other, then each of them both radiates and receives energy.
The temperatures of the two bodies become equal because
the hotter one radiates more energy than it receives and
absorbs from the colder, while the colder receives more than
it radiates. This conception of the nature of the process of
radiation was first brought forward by Prevost.
If, therefore, the emission of a body A be determined by
measuring the rise in temperature produced in a black body
which absorbs the rays from A, the result obtained depends
upon the difference in temperature between the bodies A
and B. The rise in the temperature of B would be so much
more correct a measure of the entire emission of A the smaller
the amount of energy which B itself radiates. Hence if it is
desired to measure the energy of the light-rays from a source
A, whose ultra-red rays are all absorbed in a vessel of water,
it can be done by measuring the absorption in a black body B
which has the same temperature as the water. For at the
temperature of a room the body B emits only long heat-rays,
and it receives from the water as many of these rays as it
emits. On the other hand the total emission of a source of
light is somewhat greater than that which is represented by
492 THEORY OF OPTICS
the absorption of the body B at the temperature of the room ;
nevertheless, in considerat.on of the greater temperature of
the source (the sun or a flame), the result of the measurements
is practically independent of the variations in temperature of
the body B. But the temperature of B must be taken into
account in measuring the emission of a body A which is not
much hotter than B. This subject will be resumed in the next
chapter.
CHAPTER II
APPLICATION OF THE SECOND LAW OF THERMO-
DYNAMICS TO PURE TEMPERATURE RADIATION
i. The Two Laws of Thermodynamics. The first law of
thermodynamics is the principle of energy, according to which
mechanical work is obtained only by the expenditure of a
certain quantity of energy, i.e. by a change in the condition of
the substance which feeds the machine. Although this law
asserts that it is impossible to produce perpetual motion, i.e.
to make a machine which accomplishes work without produc-
ing a permanent change in the substance which feeds it, yet a
machine which works without expense is conceivable. For
there is energy in abundance all about us ; for example, con-
sider the enormous quantity of it which is contained as heat in
the water of the ocean. Now, so far as the first law is con-
cerned, a machine is conceivable which continually does work
at the expense of heat withdrawn from the water of the ocean.
Now mankind has gained the conviction that such a machine,
which would practically be a sort of perpetual motion, is
impossible. In all motors which, like the steam-engine,
transform heat into work, at least two reservoirs of heat of
different temperatures must be at our disposal. These two
reservoirs are the boiler and the condenser. This latter may
be the air. In general heat can be transformed into work
only when a certain quantity of heat Q is taken from the
reservoir of higher temperature and a smaller quantity Q' is
given up to a reservoir of lower temperature.
Hence the following law is asserted as the result of
universal experience : Mechanical work can never be continually
493
494 THEORY OF OPTICS
obtained at the expense of heat if only one reservoir of heat of
uniform temperature is at disposal. This idea is the essence
of the second law of thermodynamics.
Only one consequence of this law will be here made use of.
If a system of bodies, so protected that no exchanges of heat or
work can take place between it and the external medium, has at
any time the same temperature in all its parts, then, if no
changes take place in the nature of any of the bodies, no
difference of temperature can ever arise in the system. For
such a difference of temperature might be utilized for driving
a machine. If, then, this difference of temperature should be
equalized by the action of the machine, it would again arise of
itself in such a system, and could again be used for the pro-
duction of work, and so on indefinitely, although originally but
one source of heat at uniform temperature was at disposal.
This would be in contradiction to the second law. It is
important to observe that heat originally of one temperature
could be used in this way for the continual production of work
only if the nature of the bodies of the system remained un-
changed. For if this nature changes, if, for example, chemi-
cal changes take place, then the capacity of the system for
work ultimately comes to an end. A condition of equality
can indeed be disturbed by chemical changes; this is not,
however, in contradiction with the second law. This phe-
nomenon can be observed in any case of combustion.
2. Temperature Radiation and Luminescence. Every
body radiates energy, at least in the form of long heat-rays.
Now two cases are to be distinguished: either (i) the nature
of the body is not changed by this radiation, in which case it
would radiate continuously in the same way if its temperature
were kept constant by the addition of heat. This process
will be called pure temperature radiation. Or (2) the body
changes because of the radiation, in which case, in general,
the same radiation would not continue indefinitely even though
the temperature were kept constant. This process is called
luminescence. The cause of the radiation does not in this case
THE SECOND LAW OF THERMODYNAMICS 495
lie in the temperature of the system, but in some other source
of energy. Thus the radiation due to chemical changes is
called chemical luminescence. This occurs in the slow oxida-
tion of phosphorus or of decaying wood. The phenomenon of
phosphorescence which is shown by other substances, i.e. the
radiation of light after exposure to a source of light, is called
photo-luminescence. Here the source of energy of the radia-
tion is the light to which the substance has been exposed,
which has perhaps produced some change in the nature, for
instance in the molecular structure, of the substance, which
change then takes place in the opposite sense in producing
phosphorescence. The radiation produced in Geissler tubes
by high-tension currents is called electro-luminescence.
From what was said in I it is clear that the second law of
thermodynamics leads to conclusions with respect to pure tem-
perature radiations only. From the conception of heat
exchanges mentioned on page 491 it follows, for example,
that if an equilibrium of temperature has once been established
in a closed system, of bodies, it can never be disturbed by pure
temperature radiation. But a disturbance of the equilibrium
might be produced by luminescence.
In what follows only pure temperature radiations will be
considered.
3. The Emissive Power of a Perfect Reflector or of a
Perfectly Transparent Body is Zero. Consider a very large
plate of any substance K enclosed between two plates of per-
fectly reflecting substance 55. A perfectly reflecting body is
understood to be one which reflects all of the radiant energy
which falls upon it. Let K and 55 have originally the same
temperature. K and 55 may be thought of as parts of a large
system of uniform temperature which is closed to outside influ-
ences. If now K emits energy, it also receives the same
amount back again by reflection from 55. Assume that the
absorption coefficient of K is not equal to zero. The absorb-
ing power a of a. body * or of a surface may be denned as the
* The absorbing power a must be c a .r*fully distinguished from the coefficient
49 6 THEORY OF OPTICS
ratio of the energy absorbed to the energy radiated upon it
from without. If the incident energy is i, then the quantity
absorbed is a, the quantity reflected I a, provided the body
transmits no energy. Hence this quantity I a is the reflect-
ing power r = I a, provided the body is so thick that no
energy is transmitted ; otherwise r < \ a.
The energy reflected to K from the mirrors 55 is now par-
tially absorbed in K and partially reflected to 55. This
reflected part is again entirely reflected back to K from 55,
and so on. It is easy to see, since 55 absorb no energy, that,
when a stationary condition has been reached, the body K
reabsorbs all the energy which it emits. If, therefore, the
mirrors 55 also emitted energy, the temperature of the body
K would rise, since then K would absorb not only all the
energy which it itself sends out, but also a part of the energy
emitted by 55. On the other hand the temperature of the
mirrors would fall, since they radiate but do not absorb. Now
since, according to the second law, the original equilibrium of
temperature cannot be disturbed by pure temperature radiation,
the conclusion is reached that the emissive power of a perfect
mirror is zero. If, therefore, a system of bodies is surrounded
on all sides by a perfect mirror, it is completely protected from
loss by radiation. In a similar way the conclusion may be
reached that the emissive power of a perfectly transparent body
is zero. For conceive an absorbing body K surrounded by a
transparent body, the whole being enclosed within a perfectly
reflecting shell, then the temperature of the transparent body
must fall if it emits anything, since it does not absorb.
4. Kirchhoffs Law of Emission and Absorption. Con-
sider a small surface element ds of an absorbing body at the
centre of a hollow spherical reflector of radius I, which has at
opposite ends of a diameter two small equal openings dQ
(cf. Fig. 1 08).
of absorption mentioned on page 360. A metal, e.g. silver, has a very large
coefficient of absorption /<, but an extremely small absorbing power a, since silver
reflects almost all of the incident light.
THE SECOND LAW OF THERMODYNAMICS 497
Let ds be small in comparison with dl. The energy
radiated by ds through each of the openings dfl is, according
to (3) on page 484,
dL = ids cos 0dQ, (i)
in which is the angle between the normal to ds and the line
connecting the middle points of ds and dl. i is called the
intensity of radiation from ds in the direction 0. Whether or
FIG. 108.
not i depends upon will not here be discussed. All the
energy which ds emits in other directions it again receives and
completely absorbs because of the repeated reflections which
take place at the surface of the hollow sphere. Suppose now
that the hollow sphere is surrounded by a black body K' ,
whose outer surface is a perfect reflector. K' then radiates
towards the interior only. Part (dE r ) of the energy emitted
from K' passes through the two openings dl to the element
ds and is there partially absorbed. The element ds subtends
at a surface element ds' of the black body a solid angle
dfl r = -3 cos (2)
if r denotes the distance between ds and ds' . The energy
radiated from ds' to ds is then
dL' = i'ds' cos 0W/y, (3)
49 8 THEORY OF OPTICS
in which /' represents the intensity of radiation of the black
surface at an angle 0' from its normal. The sum of all the
surface elements ds' which radiate upon ds is
2ds' = r*d& : cos 0', ..... (4)
in which r and 0' are to be considered constant for the different
elements of surface ds' . Hence the entire energy radiated
from K' through the opening dO, upon the element ds is
dE' = 2dL' = i'-ri-dldG,', .... (5)
or, from (2),
dE' = i'dClds cos ...... (6)
Similarly the energy which comes to ds from the other side is
dE" = i"dlds cos 0, ..... (7)
in which i" and i' must be distinguished if they depend upon
0' and if 0' is different on the two sides of the enveloping
black body.
If there is originally equilibrium of temperature, it cannot
be disturbed by the radiation. The energy 2dL sent out by
ds through the two openings dfl must be compensated by the
energy a(dE' -\- dE"} absorbed, a being the absorbing power
of ds corresponding to the direction 0. According to the
second law and (i), (6), and (7),
2 = (*' +*"). . ..... (8)
This equation must remain unchanged when the enveloping
black body K' changes its form, thus varying 0'. Hence
i'(=. i") must be independent of 0', i.e. the intensity of radia-
tion i' of a black body is independent of the direction of radia-
tion. Hence, from (8),
''=*'' ........ (9)
If different black bodies be taken for the surface ds' , while
the substance ds remains unchanged, then, according to (9), i'
must always remain constant, i.e. the intensity of radiation of
a black body does not depend upon its particular nature, but is
THE SECOND LAW OF THERMODYNAMICS 499
always the same function p of the temperature.* Hence (9)
may be stated as follows :
The ratio between the emission and the absorption of any
body at a given angle of inclination depends upon the tempera-
ture only : this ratio is equal to the emission of a black body at
the same temperature. These laws are due to Kirchhoff.f
They hold not only for the total intensity of emission, but also
for the emission of any particular wave length, thus
(9')
For if a. perfectly transparent dispersing prism be placed
behind the opening dl outside of the hollow sphere (page
497), then one particular wave length from ds can be made to
fall upon the black body, the others being returned by perfect
mirrors through the prism and the opening dD, to ds. Then
within a small region of wave lengths which lie between A and
X __|_ d^ the considerations which lead to equation (9) are
applicable.
Equations (9) and (9') must hold for each particular
azimuth of polarization of the rays. For if a prism of a trans-
parent doubly refracting crystal be introduced behind dl, the
waves of different directions of polarization will be separated
into two groups. One of these groups may now be allowed
to fall upon a black body while the other is returned by a suit-
ably placed perfect mirror. The above considerations then
lead to equation (9'), which therefore also holds for any par-
ticular direction of polarization.
5. Consequences of Kirchhoff's Law. If a black body is
slowly heated, there is a particular temperature, namely, about
525 C., at which it begins to send out light. This is at first
light of long wave length (red); but as the temperature is
raised smaller wave lengths appear in appreciable amount (at
* This function can depend upon the index of refraction of the space through
which the rays pass. This will be considered later. Here this index will be
assumed to be I, i.e. the space will be considered a vacuum.
f Cf. Ostwald's Klassiker, No. 100.
500 THEORY OF OPTICS
about 1000 the body becomes yellow, at 1200 white).*
Now equation (9') asserts that no body can begin to emit light
at a lower temperature than a black body, but that all bodies
begin to emit red rays at the same temperature (about 525 C.)
(Draper's law).\ The intensity of the emitted light depends,
to be sure, upon the absorbing power a K of the body at the
temperature considered. Polished metals, for example, which
keep their high reflecting power even at high temperatures
emit much less light than lamp-black. Hence a streak of
lamp-black upon a metallic surface appears, when heated to
incandescence, as a bright streak upon a dark background.
Likewise a transparent piece of glass emits very little light at
high temperature because its absorbing power is small. If a
hollow shell with a small hole in it be made of any metal, the
hole acts like a nearly ideally black body (cf. page 489). It
must therefore appear, at the temperature of incandescence, as
a bright spot upon the surface of the hollow shell, since the
metal has but a small absorbing power.
In the case of all smooth bodies which are not black, the
reflecting power increases as the angle of incidence increases ;
hence the absorbing power must decrease. Hence, according
to (/), the intensity of emission i of all bodies wJiich are not
black is greater when it takes place perpendicular to the surface
than when it is oblique. Hence the cosine law of emission holds
rigorously only for black surfaces.
At oblique incidence, as was shown on page 282, the
* The first light which can be perceived is not red but a ghostly gray. This
can be explained by the fac that the retina of the human eye consists of two
organs sensitive to light, the rods and the cones. The former are more sensitive
to light, but cannot distinguish color. The yellow spot, i.e. the most sensitive
point of the retina, has many cones but few rods. Hence the first impression of
light is received from the peripheral portions of the retina. But as soon as the eye
is focussed upon the object, i.e. as soon as its image is formed upon the yellow
spot, the impression of light vanishes, hence the ghostliness of the phenomenon.
\ Every exception to Draper's law, as for example phosphorescence at low
temperatures, signifies that the case is not one of pure temperature radiation, but
that, even when the temperature remains constant, some energy transformation is
the cause of the radiation.
THE SECOND LAW OF THERMODYNAMICS 501
reflecting power, and therefore the absorbing power, depends
upon the condition of polarization of the incident rays. Hence
the radiation emitted obliquely by a body is partially polarized.
That component of the radiation which is polarized in a plane
perpendicular to the plane defined by the normal and the ray
must be the stronger, because it is the component which is less
powerfully reflected, and is therefore more strongly absorbed.
In the case of crystals like tourmaline, the absorbing power,
even at perpendicular incidence, depends upon the condition
of polarization of the incident light. If, therefore, tourmaline
retains this property at the temperature of incandescence, a
glowing tourmaline plate must emit partially polarized light
even in a direction normal to its surface. Kirchhoff has ex-
perimentally confirmed this result. To be sure the depend-
ence of the absorption upon the condition of polarization is
much less at the temperature of incandescence than at ordi-
nary temperatures.
Kirchhoff made an important application of his law to the
explanation of such inversion of spectral lines as is shown in the
Fraunhofer lines in the solar spectrum. For if the light from
a white-hot body (an electric arc) be passed through a sodium
flame of lower temperature than the arc, the spectrum shows
a dark ZMine upon a bright ground. For at high tempera-
tures sodium vapor emits strongly only the ZMine, conse-
quently it must absorb strongly only light of this wave length.
Hence the sodium flame absorbs from the arc light the light
which has the same wave length as the ZMine. To be sure it
also emits the same wave length, but if the sodium flame is
cooler than the arc, it emits that light in smaller intensity than
the latter. Hence in the spectrum the intensity in the position
of the ZMine is less than the intensities in the positions cor-
responding to other wave lengths which are transmitted with-
out absorption by the flame.* According to this view the
Fraunhofer lines in the solar spectrum are explained by the
* For further discussion cf. Muller-Pouillet, Optik, p. 333 sq., 1897.
502 THEORY OF OPTICS
absorption of the light which comes from the hot centre of the
sun by the cooler metallic vapors and gases upon its surface.
Nevertheless this- application of Kirchhoff 's law assumes that
the incandescence of gases and vapors is a case of pure tem-
perature radiation. According to experiments by Pringsheim
this does not seem to be in general the case. This point will
be further discussed in I of Chapter III.
6. The Dependence of the Intensity of Radiation upon
the Index of Refraction of the Surrounding Medium. Con-
sider two infinitely large plates PP' of two black substances
placed parallel to one another. Let the outer sides of PP' be
coated with a layer of perfectly reflecting substance 55' so
that radiation can pass neither out of nor into the space PP' %
It has thus far been assumed that the space into which the
radiation is to take place is absolutely empty, or filled with a
homogeneous perfectly transparent medium like air. Instead
of this the assumption will now be made that an empty space
:%^%%^
P'
FIG. 109.
adjoins P, while a perfectly transparent substance, whose index
is n for any given wave length A, adjoins P' '.* Let the
boundary of this medium be the infinitely large plane E
(cf. Fig. 109), which is assumed to be parallel to the plates
PP' in order that P may be everywhere adjacent to a vacuum.
Now, according to page 83, an element of surface ds upon
P radiates into a circular conical shell, whose generating lines
make the angles and -f- d
i.e. from (10), (11), and (12), since, according to page 498,
the intensities of radiation i and i' are independent of the
angles and #,
;/ /"/
A / /
iin cos ^0 (i r^,) z 7 / sin^cos j ^/j (i r x ). (13)
Now it is to be noted that for angles j, for which
sin r > r = T > since in this case total reflection takes place
n
at E. Hence it is only necessary to extend the integral (13)
from x = to X X> where sin ~x = -. It will for the present
5 o 4 THEORY OF OPTICS
be assumed that n is constant for all wave lengths. Hence in
(13) and X can be thought of as a corresponding pair of
angles of incidence and refraction for which the following
holds:
sin : sin x = n, ..... (14)
and the integration can then be carried out with respect to
between the limits = o and
(18)
The 2 is to be extended over all periods between T = o and
T= oo.
Between the two bodies P and P' conceive a layer intro-
duced which is transparent to a certain wave length A, but
reflects other wave lengths. Equation (18) must always hold,
but the functional relation between r^ and T varies according
to the thickness and nature of the layer. Now in order that
(i 8) may hold as r& is indefinitely varied, every term of the 2
in (18) must vanish, i.e. for every value of T*
i T :i T =n* (19)
According to Kirchhoff's law (9'), for a body which is not
black the ratio of the emission 4 to the absorption a K is pro-
portional to the square of the index n of the surrounding
medium. Since the change of a K with n may be calculated
from the reflection equations, the relation between z' A and n is
at once obtained. In any case, then, for bodies that are not
black the intensity of radiation is not strictly proportional to n 2 .
7. The Sine Law in the Formation of Optical Images of
Surface Elements. If ds' is the optical image of a surface
element ds formed by a bundle of rays which are symmetrical
* Equation (17) can also be obtained by the method employed on page 497 if
the space outside of the hollow sphere be conceived as filled with a medium differ-
ent from that inside the sphere, but the calculation is somewhat more complicated.
Since in such an arrangement the waves of different periods T may be separated
from one another by refraction and diffraction, (19) results at once from (17) in
consideration of the conclusions upon page 497.
5o6 THEORY OF OPTICS
to the normal to ds and have an angle of aperture u in the
object space, u' in the image space, then the whole energy
emitted by ds within the bundle under consideration must fall
upon ds' \ and inversely, ds' must radiate upon ds, since the
rays denote the path of the energy flow. Hence if ds and ds'
be considered black surfaces of the same temperature, and
coated on their remote sides by perfectly reflecting layers,
then, since no difference in temperature between ds and ds 1
can arise because of the radiation, the energy dL sent out from
ds must be equal to the energy dL received by it from ds' . If
now ds lies in a medium of refractive index , ds' in one of index
', and if the intensity of emission of a black body in vacuo be
denoted by z' , then, by (17), the intensity of emission of ds is
i =. n 2 i , that of ds' , i' = #'%. Moreover, from (4) on page
485,
dL = Tr-ds't-sm 2 u, dL = Tt-ds' -t' -sin 2 u'.
Hence, since dL dL ',
ndsnH^ sin 2 u = nds'ri 2 / sin 2 u',
i.e.
dsn* sin 2 // = ds'ri* sin 2 u' (20)
This is the sine law deduced on page 61 [cf. equation
(46)]. The deduction there given, which was purely geomet-
rical, is more complicated than the above, which is based upon
considerations of energy.
8. Absolute Temperature. As was noted on page 493,
work can be obtained, with the aid of a suitable machine, by
withdrawing a certain quantity of heat W l from a reservoir i,
and giving up a smaller quantity of heat W 2 to another reser-
voir 2, which is colder than i. In this process the machine
may return to its original condition, i.e. it may perform a
so-called cycle. The principle of the conservation of energy
then demands that the work A performed be equal to the
difference between the quantities of heat W l and W 2 when
these are measured in mechanical units, i.e.
A = W l -W, (21)
THE SECOND LAW OF THERMODYNAMICS 507
Now compare two machines M and M' , both of which
withdraw in one cycle the same quantity of heat W l from reser-
voir I. They may, however, give up different quantities W 2
and W to reservoir 2. In that case the two quantities of
work A and A f done by them are different, for from (21)
A=W l -W 2 , A' = W l - W 2 f .
Now consider J/to be so constructed that it can be made
to work backwards (i.e. let it describe a reversible cycle']. In
so doing it withdraws the quantity of heat W 2 from reservoir
2, gives up the quantity W^io reservoir I, and performs the
work A. If now a cycle of machine M' be combined with
such an inverted cycle of machine M t the resultant work
accomplished is
A' - A = W 2 - W 2 r (22)
This process can be conceived to be repeated indefinitely.
Hence according as W 2 W 2 is positive or negative heat is
continually withdrawn from or added to reservoir 2, while on
the whole heat is neither withdrawn from nor added to reser-
voir i. Hence in this case reservoir I may be assumed to be
finite and may be considered to be part of the machine which
describes the cycle ; while reservoir 2 may be conceived to be
the surrounding medium, for example the water of the ocean,
whose heat capacity may be considered infinite. If now
A' A were greater than o, then a machine would have been
constructed which, with the aid of one infinitely large heat-
reservoir, would do an indefinite amount of work. But by the
second law of thermodynamics this is impossible (cf. page
493), hence*
A' A A', ... (23)
i.e. of all machines which take up a quantity of heat W l at a
definite temperature and give up heat to a colder reservoir, and
* That in general the equality A A' does not hold is evident from a con-
sideration of many irreversible processes, e.g. friction. As soon as useless heat is
developed A' < A.
5o8 THEORY OF OPTICS
which work in a cycle, that machine does the largest amount of
work which describes a reversible cycle. In the case of such
a machine, the work A which is obtained from a given quantity
of heat W l taken from the higher reservoir is therefore per-
fectly definite, since it is a finite maximum, i.e. this work A is
determined by the amount of heat W l taken up and by the tem-
peratures of the two reservoirs, and is wholly independent of
the nature of the machine. Evidently A must be proportional
to W so that the relation holds,
A = WJ^, r,) ...... (24)
in which f denotes a universal function of the reservoir tem-
peratures measured according to any scale whatever. A
combination of (21) and (24) gives
-/IV,, rj),
W l : W t =#T lt rJ, .... (25}
in which is a universal function, i.e. one which is independ-
ent of the nature of the machine.
Now it can be easily shown that this function must be
the product of two functions, one of which depends only upon
TJ , the other only upon r 2 . For if another machine be con-
sidered which works reversibly between the temperatures r 2
and r 3 , taking up the amount of heat W 2 and giving up the
amount W^ then, by (25),
W,: W 3 =(r iy T 3 ) = $ 1 :3 3 (30)
Hence in (2 5) 0(r l , r 2 ) = $ L : $ 2 and there results
W,_ ^
^2 ~ V
$j and $ 2 are functions of the two reservoir temperatures r l
and r 2 measured upon any scale. ^ and $ 2 are called the
absolute temperatures of the reservoirs. The ratio of the abso-
lute temperatures of any two bodies means then the ratio of
the quantities of heat which a machine working in a reversible
cycle withdraws from one and gives up to the other of these
bodies, provided the bodies may be considered infinitely large
so that their temperatures are not appreciably changed by the
gain or loss of the quantities of heat W^ or W 2 .
Since this merely defines the ratio of the absolute tempera-
tures of the two bodies, it is necessary to establish a second
relation in order to establish a scale of temperature. This
relation is fixed by the following convention: The difference
between the absolute temperatures of melting ice and boiling
water, both at atmospheric pressure, shall be called 100. It
* It is desirable to write the second factor -. instead of $ 3 , because then the
^3
parameter r 2 disappears from (29), as can be seen at once by writing
0(r x , r 2 ) = ^ : S 2 and 0(r 2 , r 2 ) = -& 2 : >,.
5 io THEORY OF OPTICS
is shown in the theory of heat that the absolute temperature is
approximately obtained by adding the number 273 to the tem-
perature measured in centigrade degrees upon an air-thermom-
eter.
9. Entropy. Consider again a machine M which, in per-
forming a reversible cycle, takes up the quantity of heat W l at
the absolute temperature ^ and gives up the quantity W 2 at
the absolute temperature $ 2 . If heat be always considered
positive when it is given up by the machine, then, from (31),
W W
If now there be combined with this a similar machine
which works between the temperatures 3 3 and 3- 4 , then, from
(32)-
W W W W
-~-\ ^--\ --| * = o. . . . (33)
In general, then, it may be said that when a reversible
cycle is described, in which the elements of heat dW are given
up at the temperatures 3-,
x MS Cx r/r/
= o, .... (34)
in which the sum or the integral is to be extended over all the
quantities of heat given up, and denotes the corresponding
absolute temperatures of the machine or of the reservoirs.*
Hence if a reversible cycle between two different conditions
I and 2 of a body be considered, it is possible to write, in
accordance with (34),
6W
(35)
(35')
* In a reversible process the temperature of the machine must be the same as
that of the source, otherwise an exchange of heat could not take place equally well
in either direction and the process would not be reversible.
THE SECOND LAW OF THERMODYNAMICS 511
in which 5 represents a single-valued function of the state of
the body, and dS the differential of this function. For then,
according to (34), the right-hand side of (35') always reduces to
zero as soon as a cycle is described in which the final condi-
tion 2 of the substance is identical with the initial condition I .
This function 5 of the state of a body or of a system of bodies
is called the entropy of the body.
The energy E is also a function of the state of the body.
It is defined by means of the assertion of the first law of ther-
modynamics, that in any change of the body the work 6A
done by the body plus the heat $W given up (measured in
mechanical units) is equal to the decrease dE in the energy
of the body, i.e. it is defined by the equation
= -dE (36)
10. General Equations of Thermodynamics. It is con-
venient to choose as the independent variables which determine
the state of a body or of a system, the absolute temperature $
and some other variables x, whose meaning will for the pres-
ent be left undetermined, x will be so chosen that when the
temperature changes in such a way that x remains constant,
no work is done by the body. Then, since A does not change
when -x remains constant, the following relations hold:
dA = Mdx, 6W = Xdx-\- Yd$. . . (37)
6x and d-B represent any changes in x and $; dA and dW, the
corresponding work done and heat given up by the body.
The process will be assumed to be reversible, i.e. the equations
(37) will be assumed to hold for either sign of dx and 6$.
Now from (35), (36), (37),
X Y
dS = -~-3x H K-S Y dS
0- = "~aP "~a^ * ' ' (39)
M+X=-^, Y= - |0-. . . (40)
Differentiation of these equations gives
d(*A) 3( F A) 'd(M-\-X)_ dY
or, after a few transformations,
ii. The Dependence of the Total Radiation of a Black
Body upon its Absolute Temperature. Consider a cylinder
whirh has unit cross-section and length x and whose walls con-
sist of a perfectly black body. Let these walls be covered with
perfect mirrors so as to prevent radiation into the space out-
side. Within the cylinder temperature equilibrium will occur at
a certain temperature $. Let the energy in unit volume at this
temperature be denoted by ^'($) This radiant energy exerts
a definite pressure upon the walls of the cylinder. It was
shown above on page 490 that the pressure exerted upon a
black surface by plane waves at normal incidence is equal to
the energy contained in unit volume. If the radiation is irreg-
ular, taking place in all directions, the normal pressure due to
any set of waves may be resolved into three rectangular com-
ponents in such a way that one is perpendicular to a surface s
of the walls of the cylinder. Only this component exerts a
pressure upon s. Consequently the whole pressure upon s is
not #($), but #($).*
If unit area of the cylinder wall moves a distance 8x out-
ward, the work done is
dA = %i/>($)3x (43)
* For a deduction of this factor cf. Boltzmann, Wied. Ann. 22, p. 291, 1884;
or Galitzine, Wied. Ann. 47, p. 488, 1892.
THE SECOND LAW OF THERMODYNAMICS 5 '3
Again, if the temperature of the entire cylinder is increased an
amount 6$, while x remains constant, the energy increases by
(44)
since the volume of the cylinder is x. No work is done so
long as x remains constant.
A comparison of (43) with (37) and of (44) with (38) shows,
since by (38), when dx = o, dE Yd, that
(45)
It follows, therefore, from (42), since */> depends only upon
and not upon x, that
Integration of this equation with respect to $ gives
3*=*H-* ...... (46)
An integration constant need not be added, because when
# = o the body contains no heat, and hence no radiation can
take place. It follows from (46) that
3^ dB dty
.4* : = *a' Le -4^r= -f\
hence
4/^-0 = Igty + const.,
or
0(fl) = C-& ....... (47)
If now a small hole be made in the wall of this cylinder,
radiation will take place from the hole as though it were a
black body (cf. page 489).* The intensity of radiation i must
* This also occurs if the walls of the cylinder are not perfectly black. Hence
in this case also ^>(f>) is the energy in unit volume for the condition of temperature
equilibrium, and %i/> is the pressure on the wall of the cylinder. Only if the walls
514 THEORY OF OPTICS
evidently be proportional to the energy in unit volume
within the cylinder. Hence the intensity of radiation of a
black body is
i=a-W, ...... (48)
i.e. the total intensity of emission of a black body is proportional
to the fourth power of its absolute temper attire.
This law, which Stefan* first discovered experimentally
and Boltzmann deduced theoretically in a way similar to the
above, has been since frequently verified. The most accurate
work is that of Lummer and Pringsheim.t who found by bolo-
metric measurements that within the temperature interval 100
to 1300 C. the radiation from a hole in a hollow shell followed
the Stefan-Boltzmann law. It is of course necessary in such
experiments to take account of the temperature of the bolome-
ter (cf. page 491). The radiation of the small surface ds upon
the surface ds' at a distance r amounts, when ds and ds' are
perpendicular to r [cf. the definition of intensity of radiation,
equation (3), page 484], to
.dsds'
dL = z -g .
The radiation from ds' upon ds amounts, if /' denote the
intensity of radiation of ds' , to
of the cylinder had been perfect mirrors and no heat had been originally admitted
into the cylinder would the energy in unit volume ip o. The energy in unit
volume would reach the normal value if) if the walls of the cylinder contained a
spot, no matter how small, which was not a perfect mirror. If this spot were per-
fectly black, the pressure upon it would be ^^. But in that case every part of the
cylinder wall, even that formed of perfect mirrors, would experience the same
pressure, since otherwise the cylinder would be set into continuous motion of trans-
lation or rotation.
* Wien. Ber. 79, (2), p. 391, 1879. Stefan thought that this law held for all
bodies. It is only strictly true for black bodies.
fWied. Ann. 63, p. 395, 1897.
THE SECOND LAW OF THERMODYNAMICS 515
Hence if i and i' follow the law (48), the total quantity of heat
transmitted in unit time to the element ds' is
dW = dL - dL' = a ~ (3 4 - S' 4 ), . . (49)
in which $' denotes the absolute temperature of ds' .
The constant a has recently been determined in absolute
units by F. Kurlbaum * by means of bolometric measurements.
In these experiments the temperature to which the bolometer
was raised by the radiation was noted ; the radiation was then
cut off, and the bolometer raised to the same temperature by
a measured electric current. The radiation is thus measured
in absolute units by means of the heat developed by the current.
Kurlbaum found that the difference between the emissive power
of unit surface of a black body between 100 and o, i.e. the
difference between the energy radiated in all directions, was
gr-cal
^100-^0=0.01763^^ ..... (50)
Now [cf. equation (5), page 485] e = ni, in which i is the
intensity of radiation. Further, I gm-cal = 419- io 5 ergs,
hence
0.01763- 419- io 5
'100 - *o = (373 4 - 273 4 ) = - ~- -,
i.e. the radiation constant a for a black body in absolute
C. G. S. units is
*=I.7I.IO-V.ec, ..... (50
or, in gm-cal,
* = 0.408. 10- U^/.ee ..... (5lO
12. The Temperature of the Sun Calculated from its
Total Emission. If the sun were a perfectly absorbing (i.e. a
black) body which emitted only pure heat radiations, its tem-
* Wied. Ann. 65, p. 746, 1898.
5i6 THEORY OF OPTICS
perature could be calculated from the solar constant (page 487)
and the absolute value of the constant a.* If $ denote the
absolute temperature of the sun, $' that of the earth, then from
(49) and (51') the solar constant, i.e. the energy radiated in a
minute upon unit area of the earth, would be
. . (52)
But
ds -f*=7
in which is the apparent diameter of the sun = 32'.
If, therefore, Langley's value of the solar constant be
taken, namely, dW ' 3 gm-cal per minute, t the effective tem-
perature of the sun would be $ 6500, i.e. about 6200 C.
If Angstrom's value be taken, namely, 4 gm-cal per minute,
the effective temperature would be about 6700 C.
13. The Effect of Change in Temperature upon the Spec-
trum of a Black Body. The spectrum of a black body is
understood to mean the distribution of the energy among the
different wave lengths. The investigation will be based upon
the principle of the equilibrium of temperature within a closed
hollow shell. The intensity of radiation of a black surface
(conceived as a small hole in the wall of the hollow shell) is
proportional to the energy in unit volume within the shell.
Following the method used on page 5 1 3 (cf. note I ) it appears
that the temperature at which temperature equilibrium is
attained is not dependent upon the nature of the walls of the
hollow shell, provided they do not consist entirely of perfect
mirrors.
The effect of a change in temperature upon the spectrum
* The temperature obtained by this calculation is called the effective tempera-
ture of the sun. Its actual temperature would be higher if its absorbing power is
less than I, but lower if luminescence is involved in the sun's radiation.
f $' can be neglected, since, according to (52), '* is small in comparison
with^V
THE SECOND LAW OF THERMODYNAMICS 517
of a black body can now be determined by means of the fol-
lowing device, due to W. Wien.*
Conceive a cylinder of unit cross-section within which two
pistons 5 and S', provided with light-tight valves, move.
Let AT and K' be two black bodies of absolute temperatures $
Km 1 I ^A7
H
~- \
' \
s ' ^
*""* "
FIG. no.
and \9 -f- #$. Let the side walls of the cylinder, as well as the
pistons 5 and S', be perfect mirrors. Let also the outer sides
of K and K' be coated with perfect mirrors. Let there be a
vacuum within the cylinder.
At first let S' be closed and 5 be open. Then Eradiates
into the spaces I and 2, K' into 3. The energy in unit volume
is greater in 3 than in 2 because the temperature of K' is greater
by d than that of K. Let now 5 be closed and moved a
distance 8x toward 5', until the energy in unit volume in 2 is
equal to that in 3. The value which dx must have in order
that this condition may be fulfilled will now be calculated. If
( denote the original amount of radiant energy contained in
space 2, then the original energy in unit volume in this space is
-
a x
Hence the change in energy in unit volume corresponding to
a change in x is
d **
Now d& is the work which is done in pushing forward the
piston 5. Hence, from page 512, df& = i'/'&r. Hence
a
a x
a x
*Berl. Ber. 1893. Sitzung vom 9 Febr.
5 i8 THEORY OF OPTICS
But, by (47), ty is proportional to the fourth power of $, hence
If, therefore, the energy in unit volume in space 2 is to be made
equal to that in 3 by a displacement 8x of the piston S, a
comparison of (53) and (54) gives
Now from the second law of thermodynamics the conclusion
may be drawn that, if the total radiant energy in unit volume
is the same in spaces I and 2, the distribution of energy
throughout the spectrum must be the same within the two
spaces.
For if this were not the case there would be waves of some
wave lengths which would have a larger energy in unit volume
in 3 than in 2. For it would be possible to place in front of
the valve in S r a thin layer which would transmit waves of the
length considered, but reflect all others. If then the valve
were opened, a greater quantity of energy would pass from 3
to 2 than in the inverse direction, and the energy in unit
volume would become greater in 2 than in 3. Suppose now
that S f were closed, the layer removed, and the piston S'
pushed back by the excess of pressure in 2 until the energy in
unit volume in the two spaces became again equal. Let the
work which would be thus gained be denoted by A. Then
let S f be again opened and brought into its original position.
This operation would require no work. Let then S f be closed
and 5 pushed back to its original position. In this operation
the same work would be gained which was expended in the
displacement of 5 through the distance dx. If, finally, the
valve in S were again opened, the original condition would be
restored; no heat would have been taken from or added to the
body K, but a certain amount would have been withdrawn from
K' (by radiation through the layer before the valve in S'}.
Further, a certain amount of work A would have been gained.
THE SECOND LAW OF THERMODYNAMICS 519
But, according to the second law, work A can never be
gained by means of a cycle in which heat is withdrawn from
only one source K', the heat being thus entirely transformed
into work. Hence the conclusion that when the two spaces 2
and 3 contain the same quantity of energy in unit volume, the
distribution of energy in their spectra is always the same.
But, according to Doppler's principle, the distribution of
energy in the spectrum is changed by the motion of the
piston 5. Let the total energy in unit volume in space 2 be
given by
(56)
then the expression 0(A, 5)- T= -
a x
and also the change tfjA in the wave length A. due to the
motion of 5 is
A= - Jl-^ . (60)
a x
When dx is positive d^ is negative, i.e. the wave length is
shortened.
Moreover, it must be remembered that only one third of
that part of the energy which is represented by (56) and which
corresponds to the wave length A can be looked upon as due
to waves which travel at right angles to 5 (cf. page 512).
The waves which travel parallel to 5 undergo no change in
wave length because of the motion of 5. If, therefore, that
part of the energy which is originally present in space 2 and
which corresponds to waves whose lengths lie between A and
\ + d\ is
dL = 0(A, S)(\ - dX y $)dl ..... (62)
The energy which corresponds to the wave length A at the
temperature $ -f~ <^> i- e - after the motion of the piston, is the
same as the energy corresponding to the wave length A dA
at the temperature $. But now, from (60) and (55),
A &r # 6x
: $ 4 instead of in ^.
THE SECOND LAW OF THERMODYNAMICS 523
The above law then asserts that for a black body one and
the same curve expresses the functional relationship between
*/> : 4 and A at any temperature. Now, from (56),
ff) r
& ' )
I/O
(65)
Hence 0(A., -0) : $ 5 must be a function of A$, thus
(66)
If, therefore, for any temperature $ the curve of the dis-
tribution of energy be plotted using A$ as abscissae and
0(A., $) : $ 5 as ordinates, then this curve holds for all tempera-
tures, and it is easy to construct from this curve the actual
distribution of energy for other temperatures, when the A's are
taken as abscissae and the 0's as ordinates. Hence the follow-
ing theorem:
If at a temperature $ the maximum radiation of a black
body corresponds to the wave length A w , then at the temperature
$' it must correspond to a wave length h' m stick that
** = *:' (6 7 )
Further, it follows from (66) and (67), if the function
which corresponds to the wave length A w be denoted by W ,
that
W :0: = ^:^ 5 ; (68)
i.e. if two black bodies have different temperatures, the intensity
of radiation of those wave lengths which correspond to the
maxima of the intensity curves for the two bodies are propor-
tional to the fifth power of the absolute temperatures of the
bodies.
14. The Temperature of the Sun Determined from the
Distribution of Energy in the Solar Spectrum. Equation
(67) has been frequently verified by experiment.* The mean
*C Paschen and Wanner, Berl. Ber. 1899, Jan., Apr.; Lummer and Prings-
heim, Verb. d. deutsch phys. Ges. 1899, p. 23. For low temperatures, cf.
Langley, Ann. de chim. et de phys. (6) 9, p. 443, 1886. With the use of a bolom-
5 2 4 THEORY OF OPTICS
value of A W 5 as determined from a number of experiments in
good agreement is X m $ = 2887, the unit of wave length being
o.ooi mm. Since now, according to Langley, the maximum
energy of the sun's radiation corresponds to the wave length
\' m = 0.0005, it would follow that the temperature of the sun
is
5' =5774 = 5501 C.
This result is of the same order of magnitude as that calculated
on page 5 16. It is, however, questionable whether the sun is
a perfectly absorbing (black) body which emits only pure tem-
perature radiation. If chemical luminescence exists in the
sun, its temperature may be wholly different.
15. The Distribution of the Energy in the Spectrum of a
Black Body. The preceding discussion relates to the change
in the distribution of the energy in the spectrum of a black
body with the temperature ; but nothing has been said about
the distribution of the energy for a given temperature. In
order to determine the law of this distribution W. Wien pro-
ceeds as follows : *
If the radiating black body be assumed to be a gas, then,
upon the assumption of the kinetic theory of gases, Maxwell's
law of the distribution of velocity of the molecules would hold.
According to this law the number of molecules whose veloci-
ties lie between v and v + dv is proportional to the quantity
&. f */Pdv> (69)
in which ft is a constant which can be expressed in terms of
the mean velocity v as follows :
a* = l/?' (70)
eter cooled to 20 C. he found that the maximum radiation of a blackened
copper plate at a temperature 2 C. corresponded to X m = 0.0122 mm. From
AwjO = 2887 it would follow that at 2 C. A^ = 0.0107. To be sure the copper
plate was not an ideal black body and it was only its maximum relative to a
bolometer at 20 that was measured. This relative maximum corresponds to
a smaller A than the absolute maximum, as can be seen by drawing the intensity
curves.
* Wied. Ann. 58, p. 662, 1896.
THE SECOND LAW OF THERMODYNAMICS 525
According to the kinetic theory the absolute temperature is
proportional to the mean kinetic energy of the molecules, i.e.
fl~p~/?2 ....... (71)
Now Wien makes the hypotheses :
1. That the length A of the waves which every molecule
emits depends only upon the velocity v of the molecule.
Hence v must also be a function of A.
2. The intensity of the radiations whose wave lengths lie
between A and A + d\ is proportional to the number of
molecules which emit vibrations of this period, i.e. propor-
tional to the expression (69). If this intensity of radiation be
written in the form
0(A, $)a\,
then from (69), (70), and (/i), since v is a function of A,
/(A)
0(A, 3) = F(X)-e * .* . . . . (72)
Since now, from (66), : O 5 must be a function of the argu-
ment A$, it follows that F(\) = ^ : A 5 and /(A) = c 2 : A, so
that the following law of radiation results :
)= J -jp -- , ..... (73)
and the total radiation is
/*
= ^y -
j5 ^ A ..... (74)
#/* radiation must hold for all black bodies whether
they be gases or not, since, as was shown on page 498, the law
of radiation of a black body does not depend upon the nature
of the body.
This law has been frequently verified by experiment, t
* Planck deduces the same radiation law from electromagnetic theory (BerL
Ber. 1899; Ann. dePhys. I, 1900).
j- Cf. note on page 523. Recently certain deviations from Wien's law have
been found (cf. Lummer and Pringsheim, Verh. deutsch. phys. Ges. I, p. 23, 215,
1899 ; Beckmann, Diss. Tubingen, 1898 ; Rubens, Wied. Ann. 69, p. 582).
526 THEORY OF OPTICS
That wave length, A m , at which the intensity of radiation is a
maximum
from (73),
maximum is determined from the equation = o. Now,
hence
-
3A "~ A 2 S A*
Hence the relation obtains,
*-S = ' a : 5 ....... (75)
Since A w # has the value 2887 (cf. page 524),
* a = J 4435 ...... (76)
when the unit of wave length is o.ooi mm.* In cm.,
' 2 = 1-4435 ......
Writing - = j, ^ = *', (74) becomes
i = c l \
J
But
Hence
and
^ j \" " 4 ** (77)
* According to Beckmann (Diss. Tubingen, 1898) and Rubens (Wied. Ann. 69,
p. 576, 1899) the constant c v when calculated from the emission of waves of great
length, is considerably larger. According to this Wien's law is not rigorously
correct.
THE SECOND LAW OF THERMODYNAMICS 527
If this equation be compared with (48) on page 514, it
appears that
* = 6c l :ef, (78)
in which a is the constant of the Boltzmann-Stefan law of
radiation. Now from equation (51), page 515,
Hence in consideration of (76') the constant ^ has the value
in C.G.S. units
c v = \ ac ^ i.e. *! = I.24-IO- 5 . . . (79)
The law of radiation (73), which is universal, furnishes a
means of establishing * a truly absolute system of units of
length, mass, time, and temperature a system which is based
upon universal properties of the ether and does not depend
upon any particular properties of any body. Thus universal
gravitation and the velocity of light represent two universal
laws. The absolute system is then obtained from the assump-
tion that the constant of gravitation, the velocity of light, and
the two constants c l and c 2 in the law of radiation all have the
value i.
* Planck, Berl. Ber. 1899, p. 479.
CHAPTER III
INCANDESCENT VAPORS AND GASES
i. Distinction between Temperature Radiation and
Luminescence. The essential distinction between tempera-
ture radiation and luminescence has already been mentioned
on page 494. What is now the criterion by which it is possi-
ble to decide whether a luminous body shines by virtue of
luminescence or by pure temperature radiation ?
In the case of luminescence Kirchhoff's law as to the pro-
portionality between emission and absorption is not applicable;
nevertheless even in this case the emission of sharp spectral
lines is accompanied by selective absorption of these same
lines, since both are closely connected with the existence of
but slightly damped natural periods of the ions.
A criterion for the detection of luminescence can be
obtained from measurements of the absolute value of the
emissive power or of the intensity of radiation. For if the
intensity of radiation of a body within any region of wave
lengths is greater than that of a black body at the same
temperature, and within the same region of wave lengths, then
luminescence must be present. By means of this criterion
E. Wiedemann,* F. Paschen,t and E. Pringsheim $ have shown
that the yellow light which is radiated when common salt is
burned in the flame of a Bunsen burner is due at least par-
tially to chemical luminescence (according to Pringsheim the
*Wied. Ann. 37, p. 215, 1889.
f Wied. Ann. 51, p. 42, 1894.
JWied. Ann. 45, p. 428, 1892 ; 49, p. 347, 1893.
528
INCANDESCENT J/APORS AND GASES 529
reduction of the sodium from the salt). The latter concludes,
after many experiments, that in general, in all methods which
are used for the production of the spectra of gases, the in-
candescence is a result of electrical * or chemical t processes.
Nevertheless at sufficiently high temperatures all gases and
vapors must emit temperature radiations which correspond to
KirchhofT's law,J since otherwise the second law of thermo-
dynamics would be violated. It is, to be sure, possible that
the absorption, and hence also the temperature radiation,
when chemical processes are excluded, is small, and gives
possibly no sharp spectral lines because the absorbing power
reaches an appreciable value only because of chemical pro-
cesses. For example, it would be conceivable that the natural
vibration of the ions, which occasion strong selective absorp-
tion, become possible only upon a change in the molecular
structure of the molecule.
2. The Ion-hypothesis. According to the electromag-
netic theory, the vibrations of the ions produce electromagnetic
Vaves of their own period, i.e. light-waves of a given color.
The attempt will be made to find out whether this hypothesis
can be carried to its conclusions without contradicting other
results deduced from the kinetic theory of gases.
Consider a stationary condition, in which the vibrations of
the ionic charges have a constant amplitude. Since this
amplitude would necessarily diminish because of radiation and
* E. Wiedemann has shown that a low temperature exists in Geissler tubes
(Wied. Ann. 6, p. 298, 1879).
f Pringsheim (Wied. Ann. 45, p. 440) obtained photographic effects from CS 2
flame at a temperature of 150 C. Pure temperature radiation could in this case
have produced no photographic effect. According to E. St. John (Wied. Ann. 56,
p. 433, 1895) the effectiveness of the Auer burner does not depend upon lumi-
nescence, but is due to the use in the flame of a substance of little mass, small con-
ducting power, large surface, and large emissive power. But according to Rubens
(Wied. Ann. 69, p. 588, 1899) the Auer burner is probably chemically active for
long waves.
| According to Paschen (Wied. Ann. 50, p. 409. 1893) CO 2 and water vapor
show pure temperature radiation. Their absorbing power for certain regions of
wave lengths is also very great.
530 THEORY OF OPTICS
friction, it is necessary to suppose that it is kept constant by a
continuous supply of energy. In the case of temperature
radiation this supply of energy comes from the impacts of the
molecules ; in the case of luminescence, from chemical or elec-
trical energy.
If the distance between two equal electric charges (meas-
ured in electrostatic units) of opposite sign (they may be at
rest or in motion) undergoes a periodic change of amplitude /
and period T, then, according to Hertz,* the electromagnetic
energy emitted in a half-period is
<'>
in which A denotes the wave length in vacuo.
Hence the amount of energy radiated in unit time from two
oppositely charged ions is
1 6 2 / 2 1 6
Now, according to measurements of E. Wiedemann,t the
energy emitted in a second, in the two ZMines, by I gm. of
sodium is
L^ = 3210 gr-cal = 13.45- io 10 ergs. . . (3)
The atomic weight of sodium is 23. It is next necessary
to calculate the absolute weight of an atom of sodium.
According to Avogadro's law, in every gas or vapor, at a
given temperature and pressure, there exists the same number
of molecules in unit volume. This number, at a pressure of
I atmosphere and at o C., is calculated from the kinetic
theory:): as N = io 20 in a cm. 3 . According to Regnault I cm. 3
of air at o C. and atmospheric pressure weighs 0.001293 gm.
*Wied. Ann. 36, p. 12, 1889. A different numerical factor is here given be-
cause T is defined differently.
f Wied. Ann. 37, p. 395, 1889.
JCf. Richarz, Wied. Ann. 52, p. 395, 1894.
INCANDESCENT SAPORS AND GASES 531
Hydrogen is 14.4 times lighter than air; hence the weight ^
of one molecule of hydrogen is given by
T A A 7
14.4
g = 9-io- 23 gr.
Since a molecule of hydrogen (H 2 ) consists of two atoms, the
weight of an atom of hydrogen is 4. 5'io~ 25 gm. An atom
of sodium is 23 times heavier; hence it weighs 1.03 io~ 23 gm.
Sodium is a univalent atom. Each atom is connected with
one ion whose charge will be denoted by e. If, therefore, two
atoms with charges e are required to produce one vibrat-
ing system, then in one gram of sodium there are present
: 1.03- io~ 23 = 4.85. I0 2 * such systems. Hence, from (2)
and (3),
16 e*l*
rfcjr- 4.85- io 22 = I3.45-I0 10 . . . . (4)
Now e is a universal constant, since it represents the electrical
charge which is connected with a univalent atom (it is the
charge corresponding to a valence i); for since, according
to Faraday's law of electrolysis, a given electrical current
always decomposes the same number of valences in unit time,
the charge corresponding to a valence I must be a universal
constant which does not depend upon the special nature of the
atom. Now an electric current of I ampere decomposes in
one second o. 1160 cm. 3 of hydrogen at o C. and atmospheric
pressure. Now the quantity of electricity carried in a second
through any cross-section of a conductor conveying I ampere
of current is -fa electromagnetic units or 3 I o 9 electrostatic
units. Half of this flows as positive electricity in one direction,
half as negative in the other. Hence in o. 116 cm. 3 of
hydrogen at o C. and atmospheric pressure, the total positive
charge is 1.5. io 9 electrostatic units, the negative charge being
the same. In I cm. 8 there would therefore be 1.29- io 10 units.
Since, according to page 530, the number of molecules in a
cm. 3 is N = io 20 , and since each molecule contains a positive
53 2 THEORY OF OPTICS
and a negative charge, the charge of a univalent ion (the
element of electric quantity) is
e = I.29-IO- 10 .* ...... (5)
The introduction of this value into (4) gives, since c = 3. io 10
and A 0.000589, for the value of /,
/ 1.13- io- 11 cm ...... (6)
The diameter of a molecule as calculated from the kinetic
theory is about . ..... (,8)
Now the decrease in energy d& must at least be equal to
the decrease d& which is due to radiation. Hence, from (17)
and (18), there results, if dl is set equal to yl,
e> 16 n*
-' --
Introducing the value of / from (6),
INCANDESCENT SAPORS AND GASES 535
i.e., from (15),
r ^ 1.6- io~ 9 ,
It will be shown below that r must be considerably above
the lower limit thus determined, and that, for the value of /
used, the damping of the ionic vibrations, because of their own
radiation, would be altogether negligible.
Even if / were assumed to be of the order of magnitude
of the diameter of a molecule, i.e. if /= 2-io~ 8 , then
y 2- io~ 8 , while it is probable that y is considerably larger.
4. The Radiation of the Ions under the Influence of
External Radiations. Under the influence of an external
force of period T 2nr and of amplitude A the ions take up
a motion of the same period whose amplitude may be written
[cf. (8) and the abbreviations (12)]
I A. := (20)
/i f j v o 19 O \ /
The energy emitted in unit time by a layer of thickness dz
and of area I is, according to (2) on page 530,
. (21)
On the other hand the energy A 2 enters the layer in unit
time (cf. page 454; the electric energy is equal to the mag-
netic), while the energy A' 2 passes out, provided A' repre-
4?r
sents the amplitude of the impressed electric force after it has
passed through the layer dz. Hence
dz
A i A ^TtnK
A'=A-e A.
536 THEORY OF OPTICS
The energy absorbed in unit time within the layer amounts
then to
,
THE UNIVERSITY OF CALIFORNIA LIBRARY