,w = A sin <>.
Conversely, any plane polarized wave may be regarded as being
compounded of a right- handed and a left-handed circularly polarized
wave, whose amplitudes, phases and velocities of propagation are
equal.
If S = UTT, or e = e' + ^n\, (7) reduces to v/A = w/A',
which represents a plane polarized wave. From this result, we see
that two plane polarized waves cannot compound into another
plane polarized wave, unless their phases differ by a multiple of
half a wave-length.
The methods by which plane, circularly, and elliptically polarized
light can be produced, will be described in a subsequent chapter.
14. We must now consider a proposition, known as the Prin-
ciple of Huygens.
Let PQ be the front, at time t, of a wave of any form which is
travelling outwards ; and let P'Q' be the front of the wave at time
t'. To fix our ideas we may suppose that the wave is spherical,
and the medium isotropic, but the argument will apply to waves of
any form, which are propagated in an aeolotropic medium.
At time t, the ether in the neighbourhood of P will be in a
state of vibration ; hence P may be regarded as a centre of disturb-
1 According to the definition adopted, the directions of propagation and vibration
are the same as those of translation and rotation of a right-handed screw ; they are
also related in the same manner, as the magnetic force produced by an electric current
circulating round the ray.
14 INTRODUCTION.
ance, which propagates a spherical wave. Draw OPP' to meet |
the front of the wave at time t' ; then at the end of an interval \
t' t after the wave has passed over P, the secondary wave
Q Q'
produced by the element P will have reached P', and the ether in
the neighbourhood of P' will be thrown into vibration. Similarly
if Q be any other point on the wave front at time t, and OQQ' be
drawn to meet the wave front at time t', it follows that at time t' <
the ether at Q f will be set in motion, owing to the secondary wave \\
propagated by Q. We therefore see, that the wave front at time t'
is the envelop of the secondary waves, which may be conceived to ,
diverge from the different points of the wave front at time t.
We are thus led to the following proposition, which was first (
enunciated by Huygens, and which may be stated as follows :
The effect of a primary wave upon any given point in the region
beyond the wave, may be obtained by dividing the primary wave into
an indefinite number of small elements, which are to be regarded as
centres of disturbance, and finding by integration over the front of
the primary wave, the sum of the disturbances produced at the given
point by each of the secondary waves.
15. It is a well-known law of Geometrical Optics, that when
light is reflected at a given surface, the angle of incidence is equal
to the angle of reflection. We shall now prove this law by means
of the undulatory theory.
j
Let AB be any portion of the front of a plane wave, which
is reflected at AC, M any point upon it; and draw MP, BG
perpendicular to the wave front. Draw AD, CD such that the
REFLECTION AND REFRACTION. 15
angles DAG and DC A are respectively equal to the angles BCA
if and BAG] and draw PN perpendicular to CD. Let t t T be the
times which the wave occupies in travelling to P and C.
Then PM=Vt, BC=VT,
also since the triangles ABC. and CD A are equal in every respect,
and the angle D is therefore a right angle,
PN_PC GB-PM
AD~AC CB
hence since AD = CB,
PN=CB-PM = V(T-t\
Now when the original wave reaches P, this point will become
a centre of disturbance, and spherical waves will be propagated ; at
the end* of an interval T t the wave has reached (7, and the
secondary wave which diverges from P has reached N, since we
have shown that PN = V(T t). Hence all the secondary waves
which diverge from points between A and C will touch A C, which
is accordingly the front of the reflected wave at the instant the
incident wave has reached C.
Since we have shown that the triangles ABC and CD A are equal
in every respect, the angle BCA = DAG ; whence the angle of
incidence is equal to the angle of reflection.
16. In order to prove the law of refraction, let V be the
velocity of light in the second medium ; and with A as centre
describe a sphere of radius V'T, and let CE be the tangent drawn
from C to this sphere ; draw PN' perpendicular to CE. Then
PN' _PC _CB-PM _T-t
AE ~ AC~ CB T '
whence PN' = V (T- t);
which shows when the incident wave has reached (7, the secondary
wave which diverges from P in the second medium, will have
reached N'. Hence CE is the front of the refracted wave.
If i, r be the angles of incidence and refraction,
ACsmi=CB=VT
and ACsmr = AE=V'T,
sin i V
whence - = -=> ,
sin r V
which is the law of sines.
16
INTRODUCTION.
When the second medium is more highly refracting than the
first, i > r, whence V > V ; accordingly the velocity of light
is greater in a less refracting medium such as air, than in a more
highly refracting medium such as glass. It also follows that the
index of refraction is equal to the ratio of the velocities of light
in the two media.
The direction of the refracted ray may be found by the following
geometrical construction, which as we shall hereafter show may be
generalized in the case of crystalline media, in which the wave
surface is not spherical.
Let AB be the surface of separation of the two media, A the
point of incidence. With A as a centre describe two spheres
whose radii AB, AC are proportional to the velocities F, V of
light in the two media. Let the incident ray AP be produced to
meet the first sphere in P, and let the tangent at P meet the
plane AB in T. Then if TQ be the tangent from T to the second
sphere, AQ is the refracted ray.
Since the angle
PTA = PAX, and QTA = QAX,
F _ 4? = sin PAX
V'~~ AQ~ s'mQAX'
But PAX = i, whence QAX = r.
,
CHAPTER II.
INTERFERENCE.
0^'
17. WE stated in the preceding Chapter, that we shall assume
that the sensation of light is produced by the vibrations of the
ether, without enquiring for the present into the physical constitu-
tion of the latter. We simply suppose that a medium exists
which is capable of propagating waves, and that when the waves
are plane, the direction of vibration is parallel to the wave front.
We shall now proceed to examine how far this hypothesis is
capable of explaining the interference of light.
We have shown in 12, that the superposition of two waves of
light whose directions, wave-lengths, velocities of propagation and
planes of polarization are the same, but whose amplitudes and
phases are different, may either intensify or diminish the resultant
light. Let us now suppose, that natural light is proceeding from
two sources very close to one another. At a point whose distance
from the two sources is large in comparison with the distance
between them, the waves may be regarded as approximately plane
and parallel to one another, and the displacements may be resolved
into two components at right angles to one another in the front of
the wave. Hence if v, v' be any two components whose directions
are parallel, we may take
v = A cos (x Vt), v' = A' cos (x Vt e) ,
A X
the origin being suitably chosen ; accordingly the resultant of these
two vibrations is
B. O. 2
1 8 INTERFERENCE.
cos(x-Vt-S) ....... (1),
A,
where &* = A* + A'* + 2AA'coa 27re/\ ............... (2),
_ ...
and tan27rS/X = T - rr - K ........................... 3).
A + A cos 2?re/X
If the amplitudes of the two waves are equal, then (2) and (3)
become
gl = 2A cos ?n?/X, S = Je .................. (4),
accordingly the intensity of the resultant light will be zero when
e = (n + J) X, and a maximum when e = n\. It therefore follows
that when the phases of the two waves differ by an odd multiple
of half a wave-length, the superposition of the two waves produces
darkness, and the waves are said to interfere.
In order to produce interference, it is essential that the two
sources of light should arise from a common origin, otherwise it
would be impossible to insure that the amplitudes of the two
waves should be equal, or that the difference of phase should
remain invariable ; accordingly if the light from two different
candles were made to pass through two pin-holes in a card, which
are very close together, interference would not take place 1 ; but if
light from a single candle were passed through a pin-hole, and the
resulting light were then passed through the two pin-holes,
interference would take place if the two pin-holes were sufficiently
close together.
The phenomenon of interference was regarded as a crucial test
of the truth of the Undulatory Theory, before that theory was so
firmly established as it is at present ; inasmuch as it is impossible
to satisfactorily explain on the Corpuscular Theory, how two lights
can produce darkness.
18. We shall now explain several methods, due to Fresnel, by
means of which interference fringes can be produced.
Let AB, AC be two mirrors 2 inclined at an angle TT a, where
a is very small ; let be a luminous point 3 , and let S, H be the
images of formed by the mirrors AB, AC\ also let P be any
point on a screen PN, which is parallel to the line of intersection
of the mirrors, and to SH.
1 The effect of diffraction is not considered.
2 (Euvres Completes, vol. i. pp. 159, 186, 268.
3 In practice the source usually consists of light admitted through a narrow slit.
FRESNEL'S MIRRORS. 19
Since AO = AS = AH, the difference of phase of the two
HAS = OAE - OAB + a = 2a ;
SH = 2a sin a.
SP 2 <=d* + (x+a sin a) 2 ,
streams of light which come from S and H is SP - HP. Let
AO = a, NP x, d the distance of the screen from SH. Then
whence
accord ingly
Now
and since the distance of the screen is large compared with x and
! a sin a, we obtain
SP-HP = 2axsma./d = SH.x/d.
Since the two images are produced by the same source, their^.^ _ _.
amplitudes will be equal, whence the intensity is equal to Jt- I -^
44 2 cos 5
'Z'irax sin a
1.4 /+ c+
Since a is a very small angle, we may write a for sin a, and
from (5) we see that the intensity is a maximum when
x =
and a minimum when w = J (2rc + 1) \d/aa, where n is an integer.
If homogeneous light is employed, a bright band will be
observed at the centre N of the screen, and on either side of this
bright band and at a distance %\d/aa, there will be two dark
'bands ; accordingly the screen will be covered by a series of bright
and dark bands succeeding one another in regular order ; the
distance between two bright bands or two dark bands being equal
22
20
INTERFERENCE.
to ^\d/aa. From this result we see the necessity of a being very
small, for otherwise on account of the smallness of X, the bands
would be so close together as to be incapable of being observed.
If perfectly homogeneous light could be obtained, the number
of bands would be theoretically unlimited, and with light from a
sodium flame, which possesses a high degree of homogeneity,
Michelson has observed as many as 200000 bands, but in practice
it is not possible to obtain absolutely homogeneous light, conse-
quently the number of fringes is necessarily limited.
Since the distance between two bands is equal to %\d/aa, it
follows that the breadths of the bands depend upon the wave-
length, and therefore upon the colour ; hence if sunlight be
employed, a certain number of brilliantly coloured bands will be
observed. At the centre of the system, where x = 0, the difference
of phase of waves of all lengths is zero, and the central band is
therefore white, but its edges are red. The inner edge of the next
bright band will be violet, and its edge of a reddish colour ; but as
we proceed from the centre, the maximum intensity of one colour
will coincide with the minimum of another, and the dark bands
will altogether disappear, and will be replaced by coloured bands
At a still further distance, the colours will become mixed to such
an extent, that no bands will be distinguishable.
19. In Fresnel's second experiment 1 , the light was refracted
by means of a prism having a very obtuse angle, which is callec
a biprism.
Let be the luminous point, and let L be the focus of th
rays refracted at the first face ; let S and $' be the foci of the ray I
refracted at the second faces, and draw LG, LH perpendicular t
1 (Euvres Completes, vol. i. p. 330.
BI-PLATES. 21
these faces. Let OA = a, and let p be the index of refraction.
Then
LA=pa, LG = fj,SG.
Hence if N be the point where SS f cuts A 0,
SN = LS sin a = LG (1 - fjr 1 ) sin a
= (T + fjbo) (1 fjr 1 ) sin a cos a,
where 3T is the thickness of the prism at A.
The light may now be conceived to diverge from the two points
S and $', and therefore if d be the distance of the screen from SS',
the retardation will be equal to
OL* c
-1)0; sin 2a,
which is of the same form as in the last experiment.
The breadths of the bands are accordingly equal to
fi\d
which involves /z, as well as X. Now the index of refraction
depends not only upon the material of which the prism is made,
but also on the colour of the light; hence the breadths of the
bands are affected by the dispersive power of the prism, but are
otherwise the same as in the last experiment.
20. A third mode of producing interference fringes is by
means of bi-plates.
Let CQ, Cq be two thin plates of thickness T inclined at a
very obtuse angle TT - 2a. If be a luminous origin situated on
the axis CA of the bi-plates, a small pencil of light after passing
through the plates CQ, Cq will appear to diverge from two foci
S, H, such that SH is perpendicular to OC.
22
INTERFERENCE.
Let OQRT be any ray, draw QM parallel to SH ; then if i, r
be the angles of incidence and refraction
whence
therefore
OS sin (i r) S - r
-- = - )-. - r;
Tsec r cos (i - a)
(sin i - tan r cos i).
x
cos (* - )
Since i = a very nearly, and a is small, this may be approxi
mately written
Having obtained the value of SH, the calculation proceeds as
before.
21. A fourth method, which was also employed by Fresnel 1 , J
consists of three mirrors, L, Jlf, JV, placed so that L and N intersect j
at a point on Jlf .
The light proceeds from a source S, and is reflected at the
first mirror L, and is then reflected from the third mirror N.
After reflection from N, the light will appear to diverge from a
focus A such that
where 6 is the inclination of SO to L ; and o>, a/ are the angles
which L and JV make with M.
The light reflected at M appears to diverge from a focus B,
such that
whence
AB = SA -SB = 20 + S'A -SB
= 2(a>'-0).
1 (Euvres Completes, vol. i. p. 703.
DISPLACEMENT OF FRINGES.
23
Hence if &>' 6 is small, the distance AB will be small, and
the two pencils proceeding from A, B will be in a condition to
interfere.
22. When interference fringes are viewed through a prism, or
through a plate of glass held obliquely to the screen, the fringes
will be displaced, and we shall now calculate the displacement.
Let T be the thickness of the plate, //, its index of refraction,
its inclination to the screen; also let SQRP be any ray, and
} draw EM parallel to CP. Let R^ be the retardation, and let
| OP = x y OS = c ; then
E, = SQ + pQE + EP.
In calculating R 1} we shall consider /3, and i the angle of
j incidence, to be small quantities, and we shall neglect cubes and
i higher powers.
RM sm(i-r) R M - '
Now
whence
Also
whence
Now d 2 -f (a) c +
whence
SQ + RP=ld* + L
/"ID / * /O\ '
(JJtl COS (I p)
T
P **
QM cos (/3 r)
~'cos(i-)'
cos r cos (i /3)'
= S0> = (SQ + QM + RPy,
Tcos(ff-r)
cos r cos (i /3
la
* i^r -o
'* 24 - ' 'P
We must now find in terms of #. We have
a; + .R^f - c = d tan (i - /3),
or /A (x - c) + TO* -l)i = fid(i - IB),
. /iiBd + u, (x c)
whence ^-/fr-ir*
approximately, accordingly
The value of J2 2 , the retardation of a ray proceeding from H to
P, is obtained by changing c into c ; whence
8 s
The original central band was x = 0, and the central band
which is determined by S = is now given by
which shows that it is shifted through a distance
23. When interference fringes are examined through a prism,
the displacement of the central band is different from the theore-
tical result given by (6). This difficulty was first explained by
Airy 1 , who drew attention to the fact that when no prism is used,
the central band is the locus of the points for which all colours of
the light composing the two pencils have travelled over equal paths.
Now from (6) it appears, that the displacement of the points
which formerly constituted the central band, depends upon p the
index of refraction of the prism ; and this quantity is different for
different colours, being greatest for violet and least for red light.
Since the original central band consists of a mixture of light of
every colour, it follows from (6) that the displacement of the red
portion of the band will be less than that of the violet, and
consequently the portion of the central band which is nearest G
will be red, whilst the farthest portion will be violet. This band
i Phil. Mag. 1833, p, 161.
LLOYD S EXPERIMENT.
25
can therefore no longer be considered the central or achromatic
band.
The actual achromatic band is determined from the con-
sideration, that if the bands of all colours coincide at any particular
part of the spectrum, they will coincide at no other part ; hence if
v be the displacement, measured in the direction CP, of the
original central band, the distance x n of the nth band after ,
displacement will be
x n = v + &'XcZ/2c.
The achromatic band occurs when x n is as nearly as possible
independent of X, that is when dx n fd\ = 0, in which case n must be
the integer nearest to tf?^ ^j^- % ^
<^ <^ &Ls
Since the width h of a band is equal to \d/2c, this may be
written
dv
n
so that the apparent displacement of the achromatic band is 1
, dv
24. The following method of producing interference fringes
was devised by Lloyd 2 .
A luminous point A is reflected from a plane mirror CD at
nearly grazing incidence. The reflected rays accordingly emerge
from a virtual focus B, and the arrangement is therefore equi-
valent to two small sources of light very close together. Let BO,
1 See also Cornu, Jour, de Phys. vol. i. p. 293 (1882). Lord Eayleigh,
achromatic interference bands," Phil. Mag. (5), vol. xxvni. pp. 77 and 189.
2 Trans. Roy. Jr. Acad. vol. xvn.
On
26 INTERFERENCE.
ED meet the screen in Q, g; then since interference is due to the
mixture of the two streams of light, the bands will only exist
between the points Q, q. Moreover since the difference of path is
never zero, there can be no achromatic band.
The achromatic band may however be rendered visible by
placing a thin plate of glass in the path of the direct pencil.
Putting AB=2c the retardation at x is
whence
and x n =
and consequently the position of the achromatic band, which is
determined by dx n jd\ 0, will be given by n, where n is the
integer nearest to
One peculiarity must be, noticed, and that is that the band,
which corresponds to a zero difference of path, is not white but
black. Now when we consider the dynamical theory of reflection
and refraction, it will be found that at grazing incidence, the
amplitude of the reflected light is very nearly equal to that of the
incident light, but is negative. From (2) we see that when
A' A, the intensity of the mixture is proportional to
4J. 2 sin 2 Tre/X, which vanishes when e = 0. The adjoining bright
band is given by e = JX, or
EXAMPLES.
1. A small pencil of light is reflected at three mirrors, so that
the images form a small triangle ABC, of which G is a right angle.
Prove that the intensity at any point (x, y) on a parallel screen at
a distance d, is proportional to
7T , 7TCIX 7TC?/
1 + 8 cos j (ax cy) cos -y- cos j ;
\CU \(t> A-Ct-
where AC=a, BG=c\ and the projection of G on the screen is
the origin, and GA is the axis of x.
EXAMPLES.
27
2. A small source of homogeneous light is reflected in three
mirrors, in such a manner that the images are equally bright and
form an equilateral triangle abc, whose centre of gravity is o.
A, B, C, are the projections of a, b, c, o upon a screen
which is parallel to the plane a, b, c. Show that the intensity at
any point P in the line OA, is to that at 0, in the ratio
1 -f 8in*37r/3d/2X& : 1,
where h is the distance of the screen, and PO = p, oa= d.
CHAPTER III.
COLOURS OF THIN AND THICK PLATES.
25. WHEN light is incident upon a thin film of a transparent
substance, such as a soap-bubble, brilliant colours are observed.
The explanation of this phenomenon is, that the light upon
incidence upon the outer surface of the film, is separated into two
portions, the first of which is reflected by the outer surface, whilst
the second portion is refracted. The refracted portion is reflected
from the second surface of the film, and afterwards refracted by
the outer surface ; and since the thickness of the film is very small,
the difference of the paths of the two portions is comparable with
the wave-length, and the two streams are therefore in a condition
to interfere. Accordingly if sunlight is employed, a series of
brilliantly coloured bands is observed.
In order to obtain a mathematical theory of these bands, we
shall suppose that two plates of glass cut from the same piece, are
placed parallel to one another with a thin stratum of air between
them ; and we shall investigate the intensity of the reflected light.
26. It will be proved in a future Chapter, that when light is
reflected or refracted at the surface of a transparent medium, the
intensities of .the reflected and refracted light are altered in a
manner, which depends upon the angle of incidence and the index
of refraction. The mathematical formulae, which determine the
intensities in these two cases, depend partly upon the particular
dynamical theory which we adopt, and partly upon the state of
polarization of the light. If however the angle of incidence of
the light, which is refracted from the plate into the stratum of
air, is less than the critical angle, we can achieve our object
PRINCIPLE OF REVERSION. 29
without the assistance of any dynamical theories, by the aid of a
principle due to Stokes 1 , called the Principle of Reversion.
Let 8 be the surface of separation of two un crystallized media;
let A be the point of incidence of a ray travelling along I A in the
first medium, and let AR, AF be the reflected and refracted rays ;
also let AR' be the direction of the reflected ray for a ray inci-
dent along FA in the second medium. Then the principle
asserts, that if the two rays AR, AF be reversed, so that RA and
FA are reflected and refracted at A, they will give rise to the
incident ray AI.
Let A be the amplitude of the incident light; and let Ab, Ac
be the amplitudes of the reflected and refracted light, when the
first medium is glass and the second is air ; also let Ae, Af be the
amplitudes of the reflected and refracted light, when light of
amplitude A is refracted from air into glass. Then if AR be
reversed, it will give rise to
Ab z reflected along A I,
Abe refracted along AR'.
Similarly if AF be reversed, it will give rise to
Ace reflected along AR' }
Acf refracted along AI.
Since the two rays superposed along AR' must destroy one
another, whilst the two rays superposed along A I must be
equivalent to the incident ray, we obtain
b + e = 0, 6 2 + c/=l , (1).
27. We are now in a position to calculate the intensity.
Let y = A sin ^Trt/r
be the incident vibration at A ; let i be the angle of incidence, r
that of refraction, D the thickness of the stratum of air.
1 Canil). and Dublin Math. Jonrn. vol. iv. p. 1 ; and Math, and Phys. Papers,
vol. ii. p. 89.
30
COLOURS OF THIN AND THICK PLATES.
The light which is incident at A l is reflected and refracted,
and a portion of the latter is reflected at B lt and the reflected
B l B 2
portion is again reflected and refracted at A, and so on ad infinitum.
It therefore follows, that the light refracted at A which is due to
light incident at A lt is represented by
Acefsin ZTT +
(2).
,T A, /
Also if X' be the wave-length in glass, the vibration at A due
to the light which is reflected at A^ is
'ft AM'
ft \
^.6sm27r -+ ) .
VT X /
t AM\ 2AB AM\
Writing
(2) may be written Acef sin (ft + 8),
where 8 is the retardation of the light which was refracted at A lf
Taking account of (1), and also of the infinite series of reflections
refractions at A z , A 3 ...... etc., we obtain for the resulting
= 46 [sin ^ - (1 - 6 2 ) [sin (^ + S) + 6 2 sin ( + 28) + 6 4 sin ( 2 sin ($ + S) + e 4 sin (< + 28)+ ...... }.
Summing this series, and taking account of (1), we shall find
that the intensity is
A 2 (I b z Y
/* = - - v 1 _ _L^_ /\
22 2 2
Adding (4) and (5^ we see that
#+//=4 ........................ (7)
or the sum of the intensities of the reflected and transmitted lights
is equal to that of the incident light. This result is sometimes
expressed by saying, that the reflected and transmitted lights are
complementary to one another. It must however be borne in mind,
that (7) is not strictly accurate for ordinary transparent media,
inasmuch as a portion of the light is always absorbed in transmis-
sion through the plate ; it only becomes true in the limit for perfectly
transparent substances.
Newton s Rings.
28. The coloured rings produced by thin plates were first
investigated experimentally by Newton, who produced them by
pressing a convex lens down upon a flat piece of glass ; the experi-
ment may also be performed by pressing a prism upon the face
of a convex lens. Since the curvature of the lens is exceedingly
small in comparison with the wave-length of light, the two surfaces
32 COLOURS OF THIN AND THICK PLATES.
may be regarded as approximately parallel, and the preceding
investigation will apply.
Let R be the radius of the lens, the point of contact, and let
OM = p. Then PM = D, and
whence neglecting D 2 , we have D = p*/ZR, accordingly
.cosr ......... ^. '*/... .
and the reflected light vanishes when ^ *"* ' -//? A '
At 0, p = and therefore 8 = 0; whence the central spot is.
black. If homogeneous light is employed, the central spot will be
surrounded by a series of dark rings, whose diameters are pro-
portional to the square roots of the natural numbers.
The intensity will be a maximum, when
8=(2w+l)worp a = (w + i)jRXsecr (10);
accordingly there will be a series of bright rings whose diameters l|
are proportional to Vi> Vf Vf e ^c.
>""
Since the diameters of the rings are also proportional to
(sec r)*, it follows that the rings increase as the angle of incidence
increases.
For light of different colours, the diameters of the rings vary
as X* ; consequently when sunlight is employed, a number of
coloured rings are observed.
The inner edge of the first ring is dark blue, and its outer edge
red ; and the order of succession of the colours of the first seven if
rings was found by Newton to be as follows : (1) black, blue,
white, yellow, red ; (2) violet, blue, green, yellow, red ; (3) purple,
blue, green, yellow, red ; (4) green, red ; (5) greenish-blue, red ; i
(6) greenish-blue, pale red; (7) greenish-blue, reddish-white. This
COLOURS OF THICK PLATES. 33
j list is usually known as Newton's scale of colours ; and the ex-
pression " red or blue of the third order," refers to the colour of
that name seen in the third ring.
In the preceding discussion of Newton's rings, we have sup-
posed that a thin stratum of air constitutes the thin plate ; conse-
quently it is possible to increase the angle of incidence until it
exceeds the critical angle. Under these circumstances, ij; will be
found that *fckoro io dfo^l -a.^ system of coloured rings A a2aTtfeythe
central spot is black ; but the consideration of this question must
be deferred, until we have discussed the dynamical theory of
reflection and refraction.
The system of transmitted rings is complementary to the
reflected system, but is less distinct
Colours of Thick Plates.
29. The phenomenon known as the colours of thick plates was
first observed by Newton 1 , who allowed sunlight, proceeding into a
darkened room through a -hole in the window-shutter, to fall
perpendicularly upon a concave mirror formed of glass quicksilvered
at the back. A white opaque card pierced with a small hole was
placed at the centre of curvature of the mirror, so that the
regularly reflected light returned through the small hole, and a
set of coloured rings was observed on the card surrounding the
hole. The Due de Chaulnes 2 on repeating this experiment,
observed that the brilliancy of the rings was much increased by
spreading over the surface of the mirror a mixture of milk and
water, which was allowed to dry, and thus produced a permanent
tarnish. The colours of thick plates were first explained on the
undulatory theory by Young, who attributed them to the interfer-
ence of two streams of light, one of which is scattered on entering
the glass, and then regularly reflected and refracted, whilst the
other is regularly reflected and refracted, and then scattered on
emerging from the first surface ; but the complete explanation is
due to Stokes 3 , which we shall now consider.
1 Optics, Book ii. part 4.
2 Mem. de V Academic, 1755, p. 136.
3 " On the Colours of Thick Plates," Trans. Camb. Phil. Soc. vol. ix. p. 147.
B. O. 3
34
COLOURS OF THIN AND THICK PLATES.
30. Stokes' investigation is based on the following hypo-
thesis :
In order that two streams of scattered light may be capable of
interfering, it is necessary that they should be scattered, in passing
and repassing, by the same set of particles. Two streams, which are
scattered by different sets of particles, although they may have come
originally from the same source, behave with respect to each other
like two streams coming from different sources.
It will hereafter be proved, that if this law were not true, it
would follow, that if a luminous point were viewed through a plate
of glass, both of whose surfaces were tarnished with milk and water,
coloured rings would be seen ; but on performing the experiment
no rings were observed. Moreover Stokes calculated the retarda-
tion of the stream scattered on emergence relatively to that scattered
at entrance, and found that the dimensions of the rings were
such that they could not possibly have escaped notice had they
been formed. This experiment is decisive, but the truth of the
law is also apparent from- theoretical considerations; for the
dimensions of particles of dust, although small compared with the
standards of ordinary measurement, are not small in comparison
with the wave-length of light, so that the light scattered at
entrance taken' as a whole is most irregular ; and the only reason
why regular interference is possible at all is, that each particle
acts twice in a similar manner, once when the wave enters and
again when it emerges.
31. We shall now work out the problem when the mirror is
plane.
T V
Let L be the luminous point, E the eye of the observer ; let
L Q , E be the feet of^the perpendiculars let fall from L and E on to
the dimmed face of the mirror.
PLANE MIRROR. 35
Let LSTPE be the course of the ray, which is regularly
refracted and reflected at S and T, and scattered on emergence
at P; and let LPVQE be the course of the ray, which is
scattered entering the glass and is then regularly reflected and
refracted.
Let L Q P = s, E P-u, LL = c, EE = h; also let t be the
thickness of the p]ate, JJL its index of refraction, i, r the angles of
incidence and refraction at $; R 19 R 2 the retardations of the rays
LSTPE and LP VQE. Then
= c sec i + 2fit sec r + (Ti 2 + u^ ...... (11),
and c tan i + 22 tan ? = s, smi = fj,sinr ............ (12).
Now experiment shows, that in order to see the rings distinctly
the angle of incidence must be small, whence i, r, s and u are small
quantities. We may therefore, as a sufficient approximation,
neglect powers of small quantities above the second. Expanding
in powers of i, r and u, we obtain from (11)
*<*
E l = c+2frf + A + i(c + 2/rir a -f u?/h) ............ (13).
f . t ' + 4 l>. 1 ~- -3
But (12) gives
whence ^
Again, if i 7 , r' be the angles of incidence and refraction at Q,
= (c 2 + s 2 / + 2//, sec r' + $ sec i, ^
t c *'*'**'(' ^fV * r^-j"' "^ ' ~ c
and ht&ni' + 22 tan r =, sini =a sinr'.
Accordingly we obtain as before *^ ** fr
-( 15 )'
whence R = R 1 -Rt = t . ^ 7-7x ( 16 >
h
The intensity of the light entering the eye is therefore pro-
portional to cos 2 7rR/\.
*K %- c^ u - ^-^- <**> - 32
Xr . ' a- . \ i A ? /?
36 COLOURS OF THIN AND THICK PLATES.
I
Let E Q be the origin, and let E E be the axis of z, and let the
plane xz pass through L. Let x, y be the coordinates of P, and
let E L = a.
,-r
Then
Also let the thickness of the plate be supposed to be so small, that
its square may be neglected. Then substituting in (16), we obtain
^ * . . J 1 _.*!- * -HC *" 4 .*2.*- {
==t_ For a given fringe R is constant; hence the fringes form a"
system of concentric circles, whose common centre lies on the axis
* of x. Hence if a be the abscissa of the centre
ah ah
_*A, Now ah/(h + c) and ah/(h c) are the abscissae of the points, in j
which the plane of the mirror is cut by two lines drawn from the I
eye to the luminous point and its image respectively. We thus j
obtain the following construction for finding the centre of the
system : Join the eye with the luminous point and its image, and
produce the former line to meet the mirror ; then the middle point of
the line joining the two points, in which the mirror, is cut by the two 1
lines joining the eye, will be the centre of the system.
Hence if the luminous point be placed to the right of the
perpendicular let fall from the eye on to the plane of the mirror, j
and between the mirror and the eye, the concavity of the fringes
will be turned to the right. If the luminous point, still lying on
the right, be now moved backwards, so as to come beside the eye \
and ultimately fall behind it, the curvature will decrease until
the fringes become straight, after which it will increase in the
PLANE MIRROR. 37
contrary direction, the convexity being now turned towards the
right.
The circle R = may be called an achromatic line, since at
every point of it the intensity is independent of the wave-length.
\ It evidently passes through the two points mentioned in the last
i paragraph but one.
When the luminous point is situated in the line drawn through
the eye perpendicularly to the mirror, a = 0, and (18) becomes
In this case the achromatic line is reduced to a point ; for the
bright ring of the first order R = X, and therefore the radius of
the ring is equal to
ch
which becomes infinite when c = h. Hence if the luminous point
be at first situated in front of the eye, and be then conceived to
move backwards through the eye till it passes behind it, the rings
will expand indefinitely and then disappear, and will reappear
again when the luminous point has passed the eye.
This result cannot be directly compared with experiment ; but
an analogous experiment was performed by Stokes in the following
manner. Instead of a luminous point, he used the image of the
sun in a small concave mirror, and placed a piece of plate-glass
between the concave mirror and a plane mirror, the surface of
which had been prepared with milk and water. The plate of glass
was situated at a distance of some feet from the plane mirror, and
was inclined at an angle of about 45. The greater part of the
light coming from the image of the sun, was transmitted through
the plate of glass ; and on returning from the large mirror, a
portion of this light was reflected sideways, so that the rings could
be seen by reflection in the plate of glass without obstructing the
incident light. The system of rings thus seen was very beautiful ;
and Stokes found that on moving back the head, the rings
expanded until the bright central patch surrounding the image
filled the whole field of view, and on continuing to move back the
head the rings reappeared again. In the position in which the
central patch filled the whole field of view, the least motion of the
eye sideways was sufficient to bring into the field of view ex-
cessively broad coloured bands.
38
COLOURS OF THIN AND THICK PLATES.
32. We stated in 30, that experiment showed that no rings
could be produced, unless the scattering was caused by the same
set of particles.
To prove this, let LE, the line joining the luminous point and
the eye, be perpendicular to the plate; and let LPpE be a ray
which is regularly refracted at P, and scattered at emergence
and let LQpE be a ray which is scattered at entrance at Q, and
regularly refracted at p. Then
= c sec i + /jit sec r + Ep
~k+lti+Bp +${<# +
where i, r are the angles of incidence and refraction at P. Now
if p = Mp
p = c tan i + t tan r, ^ c I + 7-/ .- c t ** \
whence i = u/r = ^ .
fJLC
Accordingly
0'
Again, if i', r be the angles of incidence and refraction at p,
But LQ*=c
whence jR 2 =
whence
7 r + /u 1 1 / 4-
IS OF MIXED PLATES.
If therefore it were possible for light scattered by different
particles to interfere, it would follow that there would be a series
of rings whose radii are determined by means of (20). No such
rings are however found to exist; and Stokes has shown, by
substituting numerical values in (20), that the dimensions of the
rings were such that they could not possibly have escaped notice
had they been formed; hence the fundamental hypothesis enun-
ciated in | 30 is proved to be true.
Colours of Mixed Plates.
33. The colours of mixed plates were first discovered by
Young 1 , and are produced by interposing between two plates of
glass pressed together, a mixture composed of two different
materials, such as water and air ; and it was found on viewing a
luminous point through the plates, that a system of coloured
rings was produced, which were considerably larger than the
rings produced, when the intervening medium was air. Further
experiments were made by Brewster 2 , who employed various
materials, such as transparent soap and whipped cream; but he
obtained the best results by using the white of an egg beaten up
into froth. To obtain a proper film of this substance, he placed a
small quantity between two glass plates, and after having pressed
it out into a film, he separated the glasses, and held them for a
short time near the fire so as to drive off some of the superfluous
moisture. The two glasses were then placed in contact and
pressed together.
Young attributed the colours of mixed plates to the fact, that
owing to the liquid being divided into an immense number of
separate globules, some of the rays are transmitted through air,
whilst others are transmitted through the liquid ; and since the
velocity of light is less in a liquid than in air, a difference of phase
is produced, and thus the emergent light is in a condition to
interfere.
34. The following theoretical investigation is due to Verdet 3 .
Let SI, S'l' be two parallel rays incident upon the surface of
a mixed plate at an angle i, and let the refracted ray IR be sup-
posed to pass through air, and the ray I'R through the liquid.
1 Phil. Trans. 1802. 2 Ibid. 1838.
3 Lemons d'Optiquc Physique, voh i. p. 155.
40 COLOURS OF THIN AND THICK PLATES.
Also let fi t jju be the indices of refraction of the glass and liquid
referred to air. Then if IK be perpendicular to /'', and ^ be
the retardation R t = iiI'K + plk - IE.
Let r, r be the angles of refraction at /, /'; t the thickness
of the space between the glass plates ; then
TK = t (tan r tan r') sin i,
R 1 = t [fju (tan r tan r') sin i sec r + // sec r'j.
_ sin r p _ sin / ^, / _j
: sin i ' //,' sin i '
whence ^* R^ t (/// cos r' cos r),
which may be written in the form
- i* sin 2 i) J - (1 - // 2 sin 2
;
The intensity will therefore be a maximum or minimum,
according as R 1 is equal to n\ or (n -f J) \, where \ is the wave-
length in air.
At normal incidence, the intensity will be a maximum when
R 1 =n\=t( fJ ,'-l). t-^TT-'
Now if Newton's rings be formed by a prism and a lens, the
radii of the rings will be equal to (2tR) ; whence if p be the radii
of the bright rings seen by transmission when a mixture of air and
liquid is employed p*" 5
If />' be the radii of the bright rings seen by transmission
when the thin plate is composed of air alone, it follows from
(5) and (6), that at normal incidence 2D=n\
EXAMPLES. 41
where D is the thickness ; and since p' 2 = 2DE, this becomes
whence
p*
If the mixed plate consists of froth composed of air and water,
p = f , whence
p = />V6.
The radii of the rings are therefore increased in the ratio *J6
to 1. This result has been found to agree with observation.
EXAMPLES.
1. Two plates of crown and fluid glass, whose refractive
indices are /A, /*', form four parallel plane reflecting and refracting
surfaces. Light of wave-length X in air could pass through each
plate in the same time. A beam of parallel rays, proceeding from
an origin between the two plates, and incident at an angle i on the
front of the crown-glass, is partly reflected once from the front and
partly once from the back, and these two reflected beams are after-
wards reflected once from the back and front of the fluid glass
respectively. Prove that the two beams in their parallel air
courses will differ finally by
wave-lengths, where T is the thickness of the crown-glass, and
sin 4 i is to be neglected.
2. If the eye be placed in the perpendicular from a luminous
point on to a dimmed plane mirror, and the thickness of the glass
be small, prove that the retardation which gives rise to the rings
will be
where e and u are the distances from the eye and luminous point
to the mirror, and x that of the point of scattering from the foot
of the perpendicular.
3. If a plate of glass be pressed down in contact with the
origin, upon a piece of glass the equation to whose bounding
surface is z = o# 2 ?/ 2 , where a is a very small quantity, describe the
appearance presented by the reflected light.
CHAPTER IV.
DIFFRACTION.
35. WHEN light after passing through an aperture, whose
dimensions are comparable with the ordinary standards of measure-
ment, is received upon a screen, the boundary of the luminous
area is well defined ; similarly if an obstacle of sufficient size is
placed in the path of the incident light, a well-defined shadow of
the obstacle is cast upon the screen. It thus appears that, as long
as the apertures or obstacles with which we are dealing are of
moderate dimensions, light travels in straight lines. Now it is
well known that sound does not in all cases travel in straight
lines ; for if a band is playing a piece of music out of doors, a
person seated in a room with an open window can hear the music
distinctly, even though his position may be such as to prevent him
seeing any of the musicians. The objection was therefore raised
against the undulatory theory in its infancy, that inasmuch as
sound is known to be due to aerial waves, and that such waves are
able to bend round corners, a theory which seeks to explain
optical phenomena by means of the vibrations of a medium, ought
to lead to the conclusion that light as well as sound is capable of
bending round corners, which is contrary to ordinary experience.
The reason of this apparent discrepancy between observation, and
what was at first supposed to be the result of the undulatory
theory, arises from the fact that the wave-length of light is
exceedingly small compared with the linear dimensions of such
apertures and obstacles as are ordinarily met with, whilst the
wave-lengths of audible sounds are not small compared with them 1 .
In fact it requires as extreme conditions to produce a shadow in the
case of sound, as it does to avoid producing one in the case of
1 The wave-length of the middle c of a pianoforte is about 4*2 feet,
DIFFRACTION. 43
light. At the same time it is quite possible for a sound-shadow to
be produced. Thus : " Some few years ago a powder-hulk ex-
ploded on the river Mersey. Just opposite the spot, there is an
opening of some size in the high ground which forms the watershed
between the Mersey and the Dee. The noise of the explosion
was heard through this opening for many miles, and great damage
was done. Places quite close to the hulk, but behind the low hills
through which the opening passes, were completely protected, the
noise was hardly heard, and no damage to glass and such like
happened. The opening was large compared with the wave-length
of the sound 1 "
On the other hand it is not difficult to produce a sound-
shadow with an obstacle of small dimensions, by means of a
sensitive flame and a tuning-fork, which yields a note whose wave-
length is so short as to be inaudible ; for although the vibrations
of the air produced by the tuning-fork are incapable of affecting
the ear, yet they are capable of producing a well-marked dis-
turbance of the sensitive flame, by means of which the existence
or non-existence of the sound is made manifest. And if an
obstacle be held between the tuning-fork and the flame, it is
observed that the oscillations of the latter either cease altogether
or appreciably diminish, which shows that a sound-shadow has
been produced 2 .
36. When light passes through an aperture, such as a narrow
slit, whose dimensions are comparable with the wave-length of
light, and is received on a screen, it is found that a well-defined
shadow of the boundary of the aperture is no longer produced.
If homogeneous light be employed, a series of bright and dark
bands is observed on those portions of the screen, which are quite
dark when the dimensions of the aperture are large in comparison
with the wave-length of light ; and if white light be employed, a
series of coloured bands is produced. Experiments with small
apertures thus show, that light is capable of bending round
corners under precisely the same conditions as sound; and thus
the objections which were formerly advanced against the undula-
tory theory fall to the ground. These phenomena are usually
known by the name of Diffraction, the object of the present
1 Glazebrook, Physical Optics, p. 149.
2 For further information on the Diffraction of Sound, see Lord Bayleigh's
Theory of Sound,, ch. xiv. and Proc. Roy. Inst. Jan. 20, 1888.
44 DIFFRACTION.
chapter is to show, that they are capable of being accounted for
by means of the undulatory theory.
37. Let us suppose that plane waves of light are passing
through an aperture in a screen, whose plane is parallel to that of
the wave-fronts. Each wave upon its arrival at the aperture may be
conceived to be divided into small elements dS. If be any
point at a distance from the screen, it is clear that every element
dS must contribute something to the disturbance which exists at
0. When we consider the dynamical theory of diffraction 1 , it
will be shown that, if we suppose 'that the disturbance existing in
that portion of the wave which passes through the aperture, is the
same as if the screen in which the aperture exists were not present,
or that the wave passed on undisturbed; the vibration at produced
by an element dS of the primary wave, would be represented
by the expression
(1),
where r is the distance of from dS, 6 and c/> are the angles -
which r makes with the normal to dS drawn outwards and with
the direction of vibration respectively, and c sin 2?rF^/X is the I
displacement of the primary wave at the plane of resolution.
In all cases of diffraction, the illumination is insensible unless 1
the inclination of r to the screen is small, which requires that 9 i
should be small and (/> nearly equal to JTT ; we may therefore as a |
sufficient approximation put cos = sin < = 1, and the formula |
becomes
cdS 2?r , Tr .
cos (Vt-r) ..................... (2),
and the resultant vibration at will be obtained by integrating ;|
this expression over the area of the aperture.
The formula (1), which is due to Stokes, will be hereafter
rigorously deduced by means of a mathematical investigation,
which is based on the assumption that the equations of motion of
the luminiferous ether are of the same form as those of an elastic
solid, whose power of resisting compression is very large in
comparison with its power of resisting distortion. It will not
1 Stokes, " On the Dynamical Theory of Diffraction ; " Trans. Camb. Phil. Soc.
vol. ix. p. 1 ; and Math. and Pliys. Papers, vol. n. p. 243.
HUYGEN S ZONES.
45
however be necessary at present to enter upon any investigations
of this character, since all the leading phenomena of diffraction
may be explained by means of the Principle of Huygens 1 .
38. Let be any point towards which a plane wave is
| advancing; draw OP = r perpendicular to the front of the wave,
and with as a centre describe a series of concentric spheres
whose radii are r + JX, . . . r + ^n\. These spheres will divide the
wave-front into a series of circular annuli, which are called
Huygens' zones. Now PM n * =(r + JnX) 2 r 2 , and therefore, if
X 2 be neglected, PM n * = nr\, and the area of each zone is equal to
7TT\.
Let cos K ( Vt r), where K = 2?r/X, be the displacement at
due to the original wave. Then it might be thought, that the
displacement at due to an element at M n would be
A n cos K ( Vt r JftX) ;
this however is not the case, inasmuch, as we shall presently see,
that it is necessary to suppose the phases of successive elements
to be different from that of the primary wave. Let e be
this difference of phase ; then since the amplitude of the zone
is proportional to its area, the displacement produced by the nth
zone will be
irr\A n cos K ( Vt r %n\ + e) = TTT\ (ty l A n cos ic(Vt r-l-e).
Accordingly the total displacement at is
7rr\{A Q -A l -\-A z - ... + (-) n A n ] cos K (Vt - r + e).
Now the amplitude of the vibration produced at by any
1 The so-called Principle of Huygens is not a very satisfactory or rigorous method
of dealing with the question of the resolution of waves. The reader may therefore,
if he pleases, assume for the present the truth of Stokes' law. See Ch. xin. and
also, Proc. Lond, Math, Soc. vol. xxn. p. 317,
46 DIFFKACTION.
zone, is inversely proportional to its distance from 0; we may
therefore write A n = B n /(r + fyi\), and the series becomes
+ . . . - cos K ( Vt - r + e).
'
n + j) X) . 7^7-^V
also since B 2n , B m+l , B 2n+2 are very nearly equal
r + n\ r + (n + I)\ -* r
j x.. - - r A i,
whence every term ot the series is approximately neutralized by
half the sum of the terms which immediately precede and succeed
it ; accordingly the effect of the wave upon a distant point is
almost entirely confined to half that of the central portion PM,
which remains over uncompensated.
It therefore follows that the displacement at is equal to
sin ( Vt - r + e). * ffc
i ^ ' O a a- \ ^ > v
f - y 4- 1\ t f. -* ^- ' *" '^ v ^r ( y^"^ -{*+*) + <-, ^
Since this expression must be equal to the displacement^
produced at by the primary wave, we must have B \ = 1, e = J\. J
We thus obtain the important theorem that the displacement
produced at by any element dS of the primary wave
V
is equal to - sin (Vt r) (3).
A/9" A/
This result may also be obtained, as was done by Archibald
Smith 1 , by integrating over the whole wave-front, for
provided we suppose that cos oo 0, an assumption which is
justified by the result.
1 Camb. Math. Journ. vol. in. p. 46.
r fi~ ^ a c - V "
r^ CYLINDRICAL^ A VE7
39. We must now investigate the corresponding result in the
/case of a cylindrical wave. /? ^ /v *-
x-=o^z. *a
If in (3) we write R = (r 2 -f 2 2 )^ for r, and integrate with respect ^ = /
to s between the limits x and - x , we shall obtain the effect
produced by an infinite linear source at a point 0, whose distance -&F^.
from the source is r ; whence the effect of the source is equal to
the real part of - Ai. ^K(Vt^) ^Ji'W /' "
Ifu = R-r, the integral becomes
.00
e -"< u u n -*du = 2 e~
Jo ^o
Also
Hence if KT is large, which is always the case at a great
distance from the source, all the terms of the series after the first
may be neglect ed^and the integral will be equal to
accordingly the effect of the source at a distance r, is the real
part of
We can now prove by integration, that this expression
reproduces the original wave. Writing x^ + y* for r 2 , we have
ydy=rdr,
6 ~ t
Putting r x = u, this becomes
> tr"*- (u + a$ du
o v
/,
48
DIFFRACTION.
whence expanding in powers of ufx, and supposing K\ to be large,
the integral is approximately equal to
Accordingly the total effect, which is the real part of
2* i - oVw ,(^1 = cos KX _ V
Diffraction through a Slit.
40. Before we discuss the general problem of diffraction, we
shall consider the case in which plane waves are diffracted by a
narrow slit in a screen, which is parallel to the wave-fronts.
Let A be the middle point of the slit, P any point on it,
any point on a screen on which the phenomena are observed.
Let
A0 = r, PO = R, AP = x, PAO=$ir-6.
If the vibration at due to the element at A be
A cos 2-7TX- 1 (r - Vt),
the vibration due to an element dx at P will be
and therefore if a be the breadth of the aperture, the total
disturbance at will be
f* a
= A'
J -i
DIFFRACTION THROUGH A SLIT. 49
Now R 2 = r 2 + a? 2rx sin 0,
and since # is small compared with r, we have to a sufficient -
approximation
Also the difference between A' and A may be neglected, whence
f = A I cos -(r x sin 6 Vt) dx
J -Ja A
JlX 2-7T , Tr N TTO, sin
= -. cos (r Ff) sin _- ,
and therefore the intensity is proportional to
. Tra sin 6
2 niv%2 Q \ \ /
7T" bill U A,
When = 0, P = A*a?, and consequently the projection of the
central line of the slit is bright. The intensity is zero when
sin 6 = mX/a, where m is any integer, and consequently the central
bright band is surrounded by a series of dark bands. Putting
TraX" 1 sin 6 = u, it follows that the intensity is proportional to
u~* sin 2 u, and the positions of the bright bands will be found by
obtaining the maxima values of this expression. Equating the
value of dI 2 /du to zero, we shall obtain
sin u u cos u sin u _
u U*
The first factor corresponds to the minima, and gives u = m7r,
the second gives
tan u = u (7).
The roots of (7), as Lord Rayleigh has shown 1 , may be calculated
in the following manner.
Assume u = (m-\-^)7r y = U y,
where y is a positive quantity, which is small when u is large.
Substituting in (7), we obtain l!s*>)!^
r
1 / y if- \ ' r "
whence tan y = j=. 1 1 + ;+ ^-- + 1.
Expanding tan y in powers of y t we obtain
3 15 315"
1 Theory of Sound, vol. i. 207.
B. O.
50 Dirte ACTION.
This equation is to be solved by successive approximation,
from which it will be found that
The values of U/TT will thus be found to be 1-4303, 2'4590,
3-4709, 4-4747, 5'4818, 6'4844, &c. They were first obtained by a
different method by Schwerd 1 .
Since the maxima occur when u is nearly equal to (in -f J) TT }
it follows that the ratio of the intensities of successive bands to
. the central band is approximately equal to
-A -1- JL &C
9-7T 2 ' 257T 2 ' 497T
We therefore see that the image formed by a slit does not
consist of a bright band bounded by the edges of the geometrical j
shadow, but of a central bright band, surrounded by a number of
alternately dark and bright bands.
Since the minima are determined by the equation
sin 6 = m\/a,
where 6 is very small, it follows that the angular distance between j
two dark bands is X/a. The bands are therefore broadest for red j
light and narrowest for violet light. Hence when sunlight is j
employed, a series of brilliantly coloured bands will be observed,
which will however be necessarily limited in number, owing to the
overlapping of the spectra of different colours.
V 41. It has already been stated, that one of the objections
brought against the undulatory theory in its infancy was, that
inasmuch as sound is known to be produced by aerial waves, it
ought to follow that light should be able' to bend round corners
as sound is known to do, and that an obstacle ought not to be
able to produce a distinct shadow. The results of the last article \
furnish an explanation of this apparent difficulty ; for if X be
large compared with a, as is usually the case with sound, (6) i
becomes
P = AW,
which is independent of 0, and consequently the intensity will be
approximately constant for a considerable distance beyond the
limits of the geometrical shadow. If on the other hand a is
1 Die Beugungserschcinungen, Mannheim, 1835.
APERTURE OF ANY FORM. 51
comparable with X, the intensity will be insensible unless is
small, and diffraction will take place.
The more general problem of diffraction through a large
number of slits, will be discussed under the Theory of Gratings.
42. We shall now proceed to consider the general problem
when the form of the aperture is given.
When light proceeding from a source, passes through a small
aperture of any form, there are three possible cases to consider ;
according as the waves are (i) converging towards a focus in front
of the aperture, (ii) are plane, (iii) are diverging from a focus
behind the aperture; and as the analytical treatment of these
three cases is different, we shall consider the first two cases in the
present chapter, reserving the discussion of the third case for the
succeeding one.
In the first case, let the screen upon which the phenomena are
observed, pass through 0, the focus towards which the light is
converging ; and let it be parallel to the plane of the aperture.
Let f , T; be the coordinates of any point P on the screen
referred to rectangular axes, through 0, and let x, y, z be any
point Q of the aperture ; also let f be the radius of the spherical
wave at the aperture.
Then QP 2 = (x - % ) 2 -f (y - T?) 2 -f z*
Since , 77 are very small compared with f t we may omit f 8 , rf,
whence
Now if cos/cVt be the vibration at the aperture, the vibration
produced at P by an element dS at Q will be equal to
/,*LLi ^ s<^ fi/r-f)A~~*t^
/( ) * J&&(& m~W* **^ * ^ ^r-fj^^y
Integrating this oveVthe area of the aperture, we shall find that
the intensity at P is proportional to
2-7T,
/2 _ J
1 ( [ [ 2?T "I 2
^"^ 2 )] \f }
42
Exactly the same result may- ber~proved to hold good in the ij
H~plane waves, provided / denotes the distance of the aperture Ij
from the screen ; for in this case # 2 + y z may be neglected, and
We shall now discuss several cases.
-
43. Let the aperture be a rectangle, whose sides are the lines |
# = i a J V i& > tnen ^7 integrating (9), we shall find that
o 2 ^ S in 2 7rag/X/.sin 2 7r67;/X/
* 22 '
Each of these factors is of the form w" 1 sin ?/, whence the minima
values of the intensity are given by u = WTT, where m is any i
positive or negative integer, and accordiDgly the field is crossed!
by a series of dark lines whose equations are
% = m\f/a, rj = m\f/b .................. (11).
The intensity is evidently a maximum when f= 0,77 = 0, in
which case 7 2 = (a6/X/) 2 .
To find the other maxima values, we observe that 7 2 is the
product of two factors of the form u~ 2 sin 2 u ; accordingly the
maxima values are the roots of the equation
tan u u,
which has already been discussed in 40.
The diffraction pattern accordingly consists of a central bright I
spot, surrounded by a series of dark lines whose equations are
f = m\f/a, t] = m\f/b ;
and within the rectangle formed by consecutive dark lines, the I
intensity rises to a maximum; but these secondary maxima are]
far less bright than the centre of the pattern.
|
An Isosceles Triangle.
44. Let the vertex of the triangle be the origin, and let the
axis of x be perpendicular to the base ; also let e be the length of
the perpendicular drawn from the vertex to the base, and let the
APERTURE AN ISOSCELES
53
equations of the sides of the triangle /be y = mas. Then if
C = 27T0/X/,
_ X 2 / 2 (sin (f ra??) c sin (f -f m?;) c|
m??
. 27T , .
sin -, (of 4-
^accordingly the intensity
cos mr)C (cos m?;c cos f c)
~<>:v^;
the axis of ?/, f = 0, and the intensity becomes
sin 4
which at the origin is equal to
The intensity at any point on the axis of x may be found
either by evaluating (12) when ?7 = 0, or directly from (9); we
shall thus obtain
?
The case of an equilateral triangle may be worked out in a
similar manner ; but it will be more convenient to suppose that
the origin is the centre of gravity of the triangle, so that the
diffraction pattern is symmetrical with respect to the angular
points of the triangle. The pattern exhibits a star-shaped ap-
pearance, which has been described by Sir J. Herschel 1 .
Circular Aperture*.
45. Let the aperture be a circle of radius c ; also let
Then by (9)
1 Encyclop. Metrop. Art. Light, 172.
a Airy, Trans. Camb. Phil. Soc. 1834.
54 DIFFRACTION.
where $ = // sin (px 4 qy ) dx dy,
C = //cos (px 4 qy) dxdy,
the integration extending over the area of the circle.
To evaluate these integrals, we shall employ the theorem 1 , that
JJF (px 4 qy) dxdy = 2 f (c 2 - x 2 y F {(p* 4 qrfx] dx, or ... (14),
J -c
according as F is an even or an odd function, where the double
integral is taken over a circle of radius c.
Let r be the distance of any point of the screen from the
projection of the centre of the aperture ; then 27rr/X/= (p 2 4 == ........................ < 15 )>
where ^ is Bessel's function of order unity.
The properties of Bessel's functions have been discussed by
the various writers referred to below 2 . The function J l (x) may
be expressed in either of the forms
x [ n
J l (x) = ~ \ cos (x cos 6) sin 2 d>d ............ (16),
' <->=- *r
It is also known that
/,'=-/,
From (15) and (17) we see that when / = (),
1 The theorem may be proved by turning the axes through an angle tan" 1 ^.^
2 Lommel, BesseVsche Functionen. Lord Eayleigh, Theory of Sound. Tod-
hunter, Functions of Laplace, Lame and Bessel. Heine, Kugelfunctionen.
CIRCULAR APERTURE.
55
the origin is therefore a bright spot, whose intensity is proportional
to the fourth power of the radius of the aperture.
Writing x for 2-Trcr/X/', it follows that the minima are deter-
mined by the equation J 1 (x) = 0. The roots of the equation
J 1 (/?r) = have been calculated by Stokes 1 , and are equal to
1-2197, 2-2330, 3'2383, 4*2411, 5'2428, 6'2439, &c. ; from which it
appears that the first dark ring occurs when
r/f= 1-2197 x X/2c.
Since X is very small, it is necessary that the radius of the
aperture should be small in order that this ring should be seen
distinctly.
The maxima are determined by the equation
= . ...(19);
whence by the last two of (18), the maxima are determined by the
roots of the equation J 2 (x) = 0. The value of the intensity in
this case is
7T 2 C 4 * ,
The following table, which has been calculated by Lommel,
gives the values of x for which J 2 (x) = 0, and the corresponding
values of / 2 (a?).
X
J*(x)
00000
1-00000
5-13563
01750
841724
00416
11-61986
00160
14-79594
00078
From this table, it appears that the maximum intensity of the
first bright ring is only about -g^th of that of the central spot.
The diffraction pattern therefore consists of a central bright spot,
surrounded by a series of dark and bright rings; moreover the
central spot is the brightest, and by far the greater portion of the
whole illumination is concentrated in it.
1 Trans. Camb. Phil. Soc, vol. ix. p. 166.
56 DIFFRACTION.
Elliptic Aperture.
46. The corresponding results for an elliptic aperture can be
obtained in a similar manner.
Let the equation of the ellipse be a?/a* + y*/fc = 1, and let
x' = x/a, y' y/b, p =pa, q f = qb. Then the values of 8 and C will
be
S = oh //sin (p'x' + q'tf} dx'dy'
C = ab Jj cos (p'x' 4- q'y') dx'dy',
where the integration extends over the circle a;' 3 + y' 2 = 1. Whence
S=0and
C = 2a6 j _ x (1 - a; 9 )* cos (jp' + 9 /
where
It therefore follows that the curves of constant intensity are
similar to the reciprocal ellipse a?x*
47. These bands were first observed by Fox Talbot 1 , and
are produced when a tolerably pure spectrum is viewed by a
telescope, half the aperture of which is covered by a thin plate of
glass or mica.
The theory of these bands was first given by Airy 2 , but we
shall follow the investigation of Stokes 3 .
We shall suppose, that the object-glass of a telescope is limited
by a screen, in which there is a rectangular aperture, the lengths
of whose sides are 21 and h + 2g+k. Let k be the width of the
thin plate, h that of the unretarded stream ; we shall also suppose
1 Phil. Mag. vol. x. p. 364, 1837. Brewster, Eep. of 1th Meeting of Brit.
Assoc.
' 2 " On the Theoretical Explanation of an apparent new Polarity of Light," Phil.
Trans. 1840 and 1841.
3 " On the Theory of certain Bands seen in the Spectrum," Phil. Trans. 1848,
p. 227 ; Math, and Phy. Papers, vol. n. p. 14.
TALBOTS BANDS.
57
that there is an opaque interval of width 2g between the two
streams, and that the axis of the telescope passes through the
centre of the opaque interval.
Let G be the centre of the opaque plate, the projection of
C on the focal plane of the object-glass ; let be the origin, and
let the axes of x and y be respectively parallel to the sides h+2g+k
and 21 of the aperture.
We shall first consider the light which emanates from any
point of a spectrum whose plane is parallel to the plane xy.
After passing through the object-glass of the telescope, the
light emanating from this point will consis't of a spherical wave,
whose radius is equal to 0(7, which converges to a point 0' as focus.
Let P be any point on this wave, Q any point on the plane xy ;
let- (x, y, z) be the coordinates of P ; (p, q) those of 0' ; (f , T?)
those of Q; also let OG=f.
The displacement at Q due to an element at P is equal to
cdS 2-7T
Now PQa = (f-aj)
-(x-pf-(y-
if f 2 + rf - p 3 - (f be neglected. Whence if
the displacement at Q becomes approximately
(20).
58 DIFFRACTION.
Since the thin plate of glass or mica occupies the space bounded
by the lines x = g + Jc, x = g, y = I, y = I ; it follows that in order
to obtain the resultant displacement at Q due to the whole wave
produced by the point, we must change dS into dxdy,firitof+Ii,
where R the retardation due to the plate is a small quantity
whose square may be neglected, and integrate over the area of the
thin plate, that is from y = I to - 1, and x = g + k to g ; whence
performing the integration, we shall obtain
2cl \f 2*V . irk? . 27r/ T7 Vg
= -=>. ~-f-, sin - J- sm ? sin I Vt - R - f+ *f +
7r% 27T/77 \f \f \ \ /
......... (21).
To obtain the displacement at Q due to the unretarded stream,
we must put R = 0, and integrate (20) from y I to I, and
x = g to g h', accordingly we obtain
. irk? . 2-7r/ h\
m sm
The total displacement at Q due to the two streams of light
is equal to v + v' ; accordingly if we put
cos -
V
........ (23),
the intensity 7 2 will be given by the equation
A-2/2
This is the expression for the intensity due to a point of light
whose geometrical focus is 0'.
48. To obtain the intensity of a line of homogeneous light
which is parallel to the axis of y, we must write Af~ l dq for c 2 , and
integrate from a large positive to a large negative value of g, the
largeness being estimated in comparison with \f/l. Now the
angle Zirrj'l/Xf changes by TT when q changes by \f/2l, which is
therefore the breadth, in the direction of y, of one of the diffraction
bands which would be seen with a luminous point. Since I is not
supposed to be extremely small, but on the contrary moderately
large, the whole system of diffraction bands would occupy but a
very small portion of the field of view in the direction of y, so that
TALBOT'S BANDS. 59
we may without sensible error suppose the limits of q to be - oo and
oo . Since P does not contain g, it follows from (24) that the
resultant intensity due to a luminous line is
(25)
49. To find the total intensity at Q due to a plane area of
homogeneous light, it would be necessary to change A into
Bf~ l dp, and integrate with respect to p between the limits oo and
oo ; but since the bands which we are investigating are produced
by a spectrum, the colour, and therefore B and p, vary from point
to point. The variations of B and \ may however be neglected in
the integration, except in the term p or %7rR/\ because a small
variation of X produces a comparatively large change of phase.
Since p depends upon the position of 0', we shall have
p=f(p)', whence if p and -tar denote the values of p and dp/dj;
at Q, we shall have
approximately.
Let irk/\f=h',
also let u = p f = - f '. Since du = dp, it follows from (23) and
(25) that the intensity is determined by the equation
f rf
/ 2 = -^ I {si n2 M* + sin 2 &'w + 2 sin A'w sin k'u cos (/c/ - g'u)} -
TTj J -QO W"
............ (27).
,00
Now I u^sitftiudu^Trh' ';
7 05
also if we write
cos (// #'w) = cos p cos ^'^ + sin p sin
according as 5 is positive or negative. If therefore we use F(s) to
denote the discontinuous function which is represented by the
above integral, and which is equal to TT or TT according as s
is positive or negative, we get
dw/dg' = JTT (^ + h' + k') + F(g'- h' - k) -F(g' + h' - k)
This equation gives
dw/dg' = 0, from g' = - oo to g' = - (h' + k') ;
= i-TT, from g' = - (h' + k') to #' = - (A/ - &') ;
= 0, from g' = - (# - #) to g' = + (/*' ~ &') ;
= - JTT, from g' = K ~ k' to / = h' + k' ;
= 0, from g' = h' + k' to $r = oo :
the sign ~ being used to denote the difference between h' and k'
when h' > k' ; if A' < A/, the expression h' ~ A/ denotes k' h'.
Now w vanishes when g' = + oo , on account of the fluctuations
of the factor cos g'u under the integral sign ; whence integrating
the value of dwjdg' given above, and determining the constant of
integration, so that w = when g' = oo , we obtain
w = 0, from g' = - oo to g' = - (h f + k') ;
w = ITT (h' + &' + #')> from ^ = - (h' + k') to g' = - (h' ~ k'} ;
w = irk' or irh' (according as h' > k' or h' < k')
from g' = - (h' ~ k') to g' = + (h' ~ k') ;
w = ITT (tf + tf -/), from g' = h' ~ k' to # x = h f + #;
w = 0, from (/' = 7t x + k' to ' = oo .
TALBOT'S BANDS. 61
Substituting, in (28), and putting g' = -rrg/X/in the last of (26),
so that
Q _ ^-^fi^ _ 40 _ /f _ & (29)
we get the following three expressions for the intensity
(i) When the numerical value of g exceeds h + k
(ii) When the numerical value of g lies between h + k and
h-k
J 2 = 2Blf~ 2 [h + k + (h + k- Vg 2 ) cos p 1 } (31).
(iii) When the numerical value of g is less than h ~ k,
/ 2 = 2Blf~- (h + k + 2/t cos p') or 2^/~ 2 (h + k + 2k cos p). . .(32)
according as h or k is the smaller of the two.
50. In discussing these results, Sir G. Stokes says :
" Let the axis of x be always reckoned positive in the direction
in which the blue end of the spectrum is seen, so that in the
image formed at the focus of the object-glass or on the retina,
according as the retarding plate is placed in front of the object-
glass or in front of the eye, the blue is on the negative side of the
red. Although the plate has been supposed at the positive side,
there will be no loss of generality, for should the plate be at the
negative side it will only be requisite to change the sign of p.
" First, suppose p to decrease algebraically in passing from the
red to the blue. This will be the case when the retarding plate is
held at the side on which the red is seen. In this case TO- is
negative, and therefore g < (h -f k), and therefore (30) is the
expression for the intensity. This expression indicates a uniform
intensity, so that there are no bands at all.
" Secondly, suppose p to increase algebraically in passing from
the red to the blue, This will be the case when the retarding
plate is held at the side on which the blue is seen. In this case
TO is positive ; and since TO- varies as the thickness of the plate,
g may be made to assume any value from (4 or < k. As T in-
creases to 2T jP lf the vividness of the bands remains unchanged;
and as T increases from 2T - T to 2T - T z , the vividness decreases
by the same steps as it increased. When T 2T T*,, the bands
cease to exist, and no bands are formed for a greater value of T.
"The particular thickness T may be conveniently called the
best thickness. This term is to a certain extent conventional,
since when h and k are unequal the thickness may range from T l
to 2T T! without any change being produced in the vividness
of the bands. The best thickness is determined by the equation
RESOLVING POWER OF OPTICAL INSTRUMENTS. 63
Now in passing from one band to its consecutive, p changes by 2-Tr,
and p by e, if e be the linear breadth of a band ; and for this small
change of p we may suppose the changes in p and f proportional,
or put dp/dp = 27T/0. Hence the best aperture for a given
thickness is that for which
4# + h + k = 2X//0.
If g = 0, and k = h, this equation becomes h - \ffe. n
The theory of Talbot's bands with a half covered circular
aperture has been discussed by H. Struve 1 .
Resolving Power of Optical Instruments.
52. When a distant object is viewed through a telescope, an
image of the object is formed at the focus of the object-glass
which is magnified by the eye-glass ; and in order that the object
should appear well denned, it is necessary that each point of it
should form a sharp image. The indefiniteness which is- some-
times observed in images is partly due to aberration ; this however
can in great measure be got rid of by proper optical appliances,
but there is another cause, - viz. diffraction, which also produces
indefiniteness, as we shall proceed to show.
If we suppose that the aperture of the telescope is a rectangle,
it appears from 43, that the intensity at the focal point is equal
to (ab/\fY, and therefore increases as the dimensions of the
aperture increase ; on the other hand the distances between the
dark lines parallel to x and y are respectively equal to \f/a and
X//6, and therefore diminish as a and b increase ; accordingly the
diffraction pattern becomes almost invisible as the aperture in-
creases, and the bright central spot alone remains. The effect of
a large aperture is consequently to diminish the effect of diffrac-
tion, and to increase the definition of an image.
When two very distant objects, such as a double star, are
viewed by the naked eye, the two objects are undistinguishable
from one another, and only one object appears to be visible. If
however the two objects are viewed through a telescope, it
frequently happens that both objects are seen, owing to the fact
that the telescope is able to separate or resolve them; and it
1 St Petersburg Trans, vol. xxxi. No. 1, 1883.
64 DIFFRACTION.
might at first sight appear, that a telescope of sufficient power
would be capable of resolving two objects however distant they
might be. This however is not the case, owing to the fact that
the finiteness of the wave-length of light, coupled with the
impossibility of constructing telescopes of indefinitely large dimen-
sions, impose a limit to the resolving powers of the latter.
According to geometrical optics, an image of each double star
will be formed at two points which very nearly coincide with the
principal focus of the object-glass ; but physical optics shows that
two diffraction patterns will be formed, whose centres are the
geometrical images of each star. If the two diffraction patterns I
overlap to such an extent, that the appearance consists of a patch
of light of variable intensity in which the two central bright spots
are undistinguishable, the double star will not be resolved ; but if
the two patterns do not overlap to such an extent as to make the
central spots undistinguishable, the double star will be resolved j
into its two components.
53. In order to investigate this question mathematically, and ,
at the same time to simplify the analysis as much as possible, we
shall suppose that the light from each star consists of plane waves "
which make an angle with the plane xy\ and we shall investi- j
gate the intensity at points on the axis of x. If
-(Xr)- 1 sin K (Vt-r)dS
denote the displacement produced at P, by the element of the
A C
\ K
i
*vf
V
wave which is situated at the centre C of the aperture, the
displacement produced by any other element will be
dxdy . 2-7T , Tr
~ J sm \ Vt - R - x sin 6} .
AT A,
RESOLUTION OF A DOUBLE STAR.
Nowif00=/, ~- ^- 1*
whence the total displacement at P is
[* a sin \Vt-f
and consequently the intensity will be
> / /i\ft *
7T 2 (f /sin 0) 2 X
\^ /
The greatest maximum value of this expression occurs when
f=/sin0 (34),
which gives the position of the central bright spot ; and the first
minimum, which occurs on the negative side of this point, is
given by
f =/(sin0-X/a) (35).
The intensity due to the other component of the double star
will be obtained by changing f into ; accordingly, the greatest
maximum will occur when f = f ', where
f" = -/sin0 (36).
Let us now suppose that .the first minimum of the diffraction
pattern due to the left-hand component of the double star, coin-
cides with the greatest maximum of the right-hand component ;
then f ' = f ", whence by (35) and (36)
9 01 n A "\ In ( S 3. ]> 7\
or since 6 is very small,
2<9 = X/a (38).
By (33), the value of the intensity at either of the bright
points is a 2 6 2 /X 2 /" 2 ; and the intensity of either component at
= is
by (37) ; whence the ratio of the intensity at the middle point to
that of either of the bright points is equal to 8/7T 2 = '8106.
It thus appears that the brightness midway between the two
geometrical images, is about f ths of the brightness of the images
themselves ; and from experiment it appears that this is about the
limit at which there could be any decided appearance of resolution.
Now 20 is the angle which the components of the double star
subtend at the place of observation; and since by (38) 20 = X/a,
B. O. 5
66 DIFFRACTION.
we see that an object cannot be resolved, unless its components
subtend at the place of observation, an angle which exceeds that
subtended by the wave-length of light, at a distance equal to the
breadth of the aperture.
If the distance of the object be such that = X/a, it appears
from (35) that f ' = ; there is accordingly a dark band at the
middle point of the two images, which is more than sufficient for
resolution.
54. We shall now consider the resolving power of a telescope
having a circular aperture.
Let the axis of be drawn perpendicularly to the line of in-
tersection of the fronts of the waves with the plane of the aperture.
Then at points on the axis of f, the intensity due to that component j
of a double star, which lies on the left-hand side will be
where C = -> 1 1 cos -~^ (f -/sin 0) dx dy,
S = ^jj sin ~ (? -/sin 6) dxdy,
and the integration extends over the area of the aperture. From
these expressions we see that S = 0, and
where p = 2-7r(f /sin 0)/\f, and c is the radius of the aperture.
The greatest maximum occurs when p = 0, or f =/sin 0, which
gives the central spot ; and the intensity at this point is equal to
The first dark band to the left of the central spot occurs when
Jj (pc) = 0, or pcj IT = 1*2197 ; in which case
f -/sin 0-(/X/c)x *6098.
If f" be the distance of the central spot due to the right-ham
component,
accordingly if f ' = ",
x '6098 = - nearly ............ (39).
THEORY OF GRATINGS. 67
and if is given by (39)
1 f)^2 f&
C = ^J, (1-8849) = -- x 1-937 ;
3J A, /A,
whence the ratio 'of the intensity at the middle point, to that of
either bright spot is about 7 '5 -H 7r 2 . The corresponding number
for a rectangular aperture was found to be 8/Tr 2 , and 20 was equal
to \/a; whereas in the present case 2# = \/fc. If therefore the
components of a double star subtend at the place of observation
an angle, which is somewhat greater than the angle subtended
by the wave-length of light, at a distance equal to the diameter 2c
of the circular aperture, the telescope will resolve the star. Hence
the resolving power of a telescope having a circular aperture, is
less than one whose aperture is rectangular.
The resolution of a double line is discussed in Lord Rayleigh's
article on Wave Theory 1 .
\f
Theory of Gratings 12 .
55. A diffraction grating consists of a thin plate of glass,
m which a very large number of fine lines have been ruled
rith a diamond very close together; and gratings have been
mstructed, which contain as many as 40000 lines to the inch.
r hen light is incident upon the grating, the lines of the latter
the part of approximately opaque obstacles, and a diffraction
itrum is produced.
For the purpose of presenting the theory in its simplest form,
re shall suppose that plane waves of light, whose fronts are parallel
the grating, fall upon the latter, and are then refracted by a
mvex lens, which is likewise parallel to the grating; and we
^hall examine the appearance on a screen which passes through
:he principal focus of the lens.
If we consider the system of parallel rays, which falls upon the
is and is perpendicular to any plane A$, which makes an angle
with the grating, these rays will be brought to a focus by the
at a point P, which is near the principal focus; and con-
luently each ray will occupy the same time in travelling from
1 Encyc. Brit.
2 Lord Kayleigh, Art. Wave Theory, 14, Encyc. Brit. p. 437.
52
68
DIFFRACTION.
Afj to P. Let ft, Q 2 ... be points on the transparent parts of the
grating, whose distances from A lt A... are equal to x\ let
//
Then the resultant displacement at P is proportional to
+sin 2,r -* -I- ?
+ sin 27r
T
sin J -K
)J
where n is the number of opaque lines of the grating. Integrating
s
.,j\
sin 6
*' x ~J x r t 2-7T/
U * **^** /1 7 _ cos 2?r f - + sin j cos
t 2a + d . n \ ft
- H sm or cos ZTT - +
T X / VT
. J f* ?i(a + d) .. Jl
m0> cos27H- H ^-r --- I m0\
J IT X )J
...... (42).
If 6 be chosen so that
where m is a positive integer, each of the n + 1 pairs of tern
become equal to one another, and the series is equal to
( 2?r ft ma \] ....
z ]cos cos2?r - + 3)} (44).
2-7T sm ( r VT a + d }
(n + I) X
*' /I OX1J. I/ I \ J W I \.*J / I
It therefore appears, that for the directions which are dete
mined by (43), the disturbances produced at P by the transpare;
THEORY OF GRATINGS." 69
portions of the different elements of the grating reinforce" one
another, and that the intensity is a maximum for these directions-
Accordingly when homogeneous light is employed, the diffraction
spectrum consists of a number of bright and dark lines, and the
bright lines occur when the position of A-fl is such that the pro-
jection of the element a + d upon it is equal to any multiple of a
wave-length. The central band is bright, and its intensity J 2 is
equal to (n + l) 2 a 2 , as can at once be seen from (41) by putting
= 0, and then performing the integration.
If I m 2 be the intensity of the mth bright band, it follows from
(43) and (44) that
m .
whence -y- = - - sin 2 (45).
/ 2 \am7rj a + d
If the whole space occupied by the grating were transparent,
the disturbance at P would be
sin 2-7T f - -f - sin J dx, ^ -
and the intensity would be proportional to
i
sn
accordingly the intensity in the direction = 0, is
_____ V
and therefore *i
'
a + d
jp/J ^T^V^ sin2 .^L (46),
~^/ * ^ Jtf^ m ^ a + a
I .. j*stf ^
if the number^oMmes is so large that ft may be treated as
infinite.
Since the sine of an angle can never be greater than unity, it
follows that the amount of light in the mth spectrum can never be
greater than l/raV 2 of the original light. Hence in a grating
composed of alternately opaque and transparent parts, whose
breadths are equal, so that d = a, the central image is the
brightest, and the first lateral spectrum is brighter than any of
the succeeding ones.
70 DIFFRACTION.
56. In practical applications the angle 6 is so small, that
6 may be written for sin 6 ; whence the angular distance between
the centres of two bright bands is equal to \/(a + d). The
breadths of the bands are therefore inversely proportional to
a + d, and will therefore be broad when a 4- d is very small ;
accordingly a fine grating having a very large number of lines,
produces broader bands than one having a smaller number. Since
the breadths of the bands are directly proportional to X, the bands
will be broader for red light than for violet light; accordingly
when sunlight is employed, the outer edges of the central band
will be red, and the inner edges of the two adjacent lateral bands
will be violet.
If the values of the quantities for red and violet light be
denoted by the suffixes r and v,
O r = m\ r /(a + d), O v = m\ l ,/(a + d) ;
whence 6 r v = m (\ r \ v )/(a + d),
which gives the angular value of the dispersion for the two
extreme colours in the mth spectrum. This result shows the-]
importance of having the lines ruled very close together, so that
the dispersion may be as large as possible.
In order that the spectra may not overlap, it is necessary that
the value of 6 V for the (m + l)th spectrum should be greater than
the values of 6 r for the mth ; which requires that
Since \ r is nearly equal to 2X W , overlapping will take place in
the spectra of higher orders than the first.
k ; (^ *- -
^JjJ^J i ' '^c^J* Resolving Power of Gratings^
57. We must now consider the* resolving power of gratings? .'
and shall first sum the series (42). Let
<, 2-Tra/X . sin 6 = a, 2?r (a + d)/\ . sin = % ;
then the first vertical line of (42) becomes
[cos (< + a -f n%) cos ( + a %)
- cos [$ + a + (n +!)%} + cos ( + a)]/4 sin 3
*>y*
RESOLVING POWER OF GRATINGS. ~ffL&3j- ^^
which reduces to * -if ^
whence the resultant displacement at P becomes
$% (cos (6 + A-rcv) - cos (cf> + c
We have already shown, that the intensity will be a maximum
when ___ i x^xt/c^L^u-c- '-*--- *--*' t
(a + d) sin ^ = m\, or % = 2m-7r.
We shall now show, that the intensity will be zero when
(a + d) sin (9 = ( ra 4- ~ ) X
\ ^i ~T" -L/
-OU^u^ -
where 5 = 1, 2...n. For in this case
whence sin J (n 4- 1) % = 0.
Now n is the number of opaque lines, and n + 1 is the number
of transparent lines on the grating ; consequently between the
directions determined by (43), which may be called the principal
maxima, there will be a series of dark lines equal to the number of
opaque lines of the grating, which will be separated by bright
lines, which may be called secondary maxima. The principal
maxima are however far the most distinct, and the secondary
maxima are so faint that they may be left out of consideration.
Let us now suppose, that the incident light consists of a double
line of light of wave-lengths X and X + SX. On account of the
difference of wave-length, the maxima and minima of the two
superimposed spectra will not coincide ; but the want of coin-
cidence will not be capable of being detected, unless the principal
maximum of the mth spectrum light of wave-length X + SX,
coincides with the first minimum succeeding the mth principal
maximum of light of wave-length X. Whence by (43) and (47)
we must have
(a + d) sin 6 = m (X + 8X) = ( m + -- - ) X,
\ n + I/
which gives = 7 N ........................ (48).
X m (n + 1)
This equation gives the smallest difference of wave-lengths
in a double line which can just be resolved ; consequently the
resolving power of a grating depends solely upon the total number
72 DIFFRACTION.
of lines and the order of the spectrum. In the case of the D
lines in the spectrum of sodium vapour, B\/\ = lOOO" 1 , so that to
resolve this line in the first spectrum requires a grating having
1000 (transparent) lines upon it; and in the second spectrum
500 lines, and so on. It is of course assumed in (48) that n + 1
transparent lines are really utilized.
Reflection Gratings.
58. The gratings hitherto considered act by refraction ; but
it is possible to form a diffraction spectrum by means of a re-
flecting surface, on which a large number of fine lines are ruled.
The fine lines act the part of the opaque obstacles in a refraction
grating.
If YP be the incident and PZ the diffracted ray, and if i, (/>
be the angles which the incident and diffracted rays make with
the normal to the grating, the disturbance at any point Q may be
obtained by writing sini+sin^> for sin in (41). Hence the
position of the rath spectrum is determined by
(a + d) (sin i + sin $) = raX.
A similar formula holds good when light is incident obliquely
upon a refraction grating.
Rowland's Concave Gratings 1 .
59. In these gratings lines are ruled upon a concave spherical
mirror made of speculum metal, and are the intersections of
parallel planes one of which passes through the centre of the
sphere.
1 Amer. Jour, of Science; 3rd series, vol. xxvi. p. 87. Glazebrook, Phil. 1/ar/,
June and Nov. 1883,
ROWLANDS CONCAVE GRATINGS.
73
In the figure let be the centre of the mirror, Q a source of
light, and let us for simplicity consider the state of things in the
^0/u.^xUv* i^?, and therefore Q' also lies
on the same circle.
The point Q' gives the position of the central diffraction band
J! formed by the grating, but there will also be a series of lateral
spectra arranged along this circle on either side of Q', which we
shall now consider.
\
Let P be any point on the grating, let PR be the diffracted
i ray ; also let RAO = <', POA = t&.
The retardation is QP + PR\QA - AR, which we must
proceed to Calculate. We have
sin 2 Aco 4tau sin i-co sin
L
(u
u cos
sn
= (u + a sin cj> sin ft)) 2 a 2 sin 2 < sin 5
Now ft) is usually a small quantity, whence if' we neglect
powers of sin co higher than the fourth, we may write
4 sin 3 Jft) = sin 3 w + J sin 4 co,
QP 2 = (w + a sin < sin to) 2 + a cos < sin 2 a/a cos (f> - u)
+ J a (a ^ cos $) sin 4 a> ;
accordingly if Q lie on the circle whose diameter is OA, so that
u a cos , the term involving sin 2 o> vanishes, and we obtain
~^ cos $) s i n4 ^1
- , Ar ^>,
8 (^N- a sin <^> sin &)) 2 j
>* r- ^\ -*<
or QP QA = a sin < sin a> + Ja^sin tan <^)sm 4 w.
If E also lie on the same circle, the value of PR - AR will be \
obtained by writing ' for 0, whence
QP + PR-QA-AR
= a (sin sin ') sin o> + |a (sin tan + sin $ tan <^') sin 4 o>. -jj
The advantage of this arrangement is, that the second term ji
involves sin 4 o>, and is therefore exceedingly small ; the accuracy of \
the instrument is therefore far greater than one in which terms I
involving sin 2 o> and sin 3 o> occurred.
Neglecting the second ' term, it follows, from what has gone j
before, that the bright bands are given by the equation
a- (sin < - sin ')= m\.
where cr is the distance between two lines of the grating.
In order that a large part of the field of view may be in focus, I
the eye-piece is placed at 0, whence ' = and
a- sin (x)dx at e, is
proportional to
The total intensity I 2 is obtained by integrating over the
whole area of the source, whence
(52).
We shall now consider the case in which the source is a
uniformly illuminated rectangle of width a, whose centre is in the
axis of the telescope, and whose sides are parallel to SS'.
Putting $(x) = l, and integrating (52) between the limits Ja
and |a, we obtain
- 2 ~x ~\ ............ ^ )'
If /! 2 , 7 2 2 be the intensities at the centres of the bright and
dark fringes respectively, Michelson assumes that the visibility V
of the fringes is
The values of I lt / 2 are obtained by putting y = n\/y and
(n + J) X/7 respectively, whence
F _sin(7r/9a/\)
~~
We have shown in 40, that the maxima values of this
expression occur, when
0al\ = 1-4303, 2-4590, 3'4709, &c.
and the minima when
Since fta = 6a, it follows that if a = \/6, which is the limit of
the resolving power, the fringes will be invisible; but if b, and
consequently fi, gradually increase, the fringes will become visible,
and will again disappear when /3 = 2X/a. The fringes will therefore
alternately appear and disappear as the distance between the slits
increases.
A similar result takes place, when the width of the source
increases, whilst the distance between the slits remains constant.
These results were experimentally verified by Michelson.
FKINGES PRODUCED BY A DOUBLE STAB. 77
62. We shall now consider the case in which there are two
identical sources of equal breadths, which are equidistant from B.
Let s be the distance of the centre of either source from B,
2r its breadth. Then the intensity due to the source whose centre
is x = s, will be obtained by integrating (52) between the limits
s + r and s r ; and is therefore equal to
X 2?r / v . 2-TT^r
*! cos ~x~ ^ y ^ sm ~\~ '
The intensity due to the other source is obtained by changing
the sign of s, whence adding these two results, the total intensity is
2X Ziryy 2?r/3s .
/ 2 = 4 r _j_ cos __^7 cos __^_ sm
TTp A A
T;r sin (27r/3r/X) cos
whence F=
Let a be the angle subtended at A by the line joining the
centres of the sources, then Zs/d = a, whence 27r/3s/\ = ?ra/a , where
a = \/b. Also let 2r/d = a lf then 27rySr/X = TTOL^OLQ ; hence (56)
becomes
By (54), the visibility due to a single source of breadth 2s, is
F = S jn^/,)_
Tra/oto
From these results we see that if a telescope is focused upon
a double star, the fringes will be different from those produced by
a single star ; and the preceding investigation furnishes a method
by means of which the double stars may be detected, which are
incapable of being resolved by telescopes of the largest aperture in
existence. Further information, together with the application of
the theory to spectroscopic measurements, will be found in the
papers by Michelson referred to below 1 .
1 "Measurement of Light Waves," Amer. Jour, of Science, vol. xxxix. Feb. 1890.
" Visibility of Interference Fringes in the Focus of a Telescope," Phil. Mag. March
1891, p. 256. "Application of Interference Methods to Spectroscopic Measure-
ments," Ibid. April 1891, p. 338.
78 DIFFRACTION.
EXAMPLES.
1. Parallel homogeneous light from a source, is intercepted
at right angles by a screen pierced with an aperture in the shape
of a cross, consisting of two equal rectangles of sides 4a, 2a
superposed with their longest sides at right angles, and their
centres coincident. Investigate the phenomena upon a distant
screen.
2. A small luminous body is placed before a convex iens. A
screen pierced with orifices of any form, stands between the
luminous body and the lens. Show that the intensity of the
image on a screen, placed at right angles to the axis of the lens,
at the position where the image is formed, will be proportional to
the sum of the areas of the orifices.
3. Plane waves of homogeneous light of wave-length X
impinge normally on a diaphragm, and are diffracted through an
aperture in the diaphragm, and received on a parallel screen at a
distance d. If the aperture be an annulus of radii a, A t prove
that the intensity of light at a point on the screen, will be
proportional to the square of
Ar- 1 ^ (27rAr/\d) - ar~ l J^ (2?rar/Xd) ;
where r is the distance of the point from the projection of the
centre of the annulus on the screen.
4. Homogeneous light of wave-length X emanates from a
point, and falls on a screen at a distance a from the point, in
which there is a circular hole of radius r; the line joining the
point to the centre of the hole being perpendicular to the screen.
After passing through the hole, the light falls on a parallel screen
at a distance b from the former. A circular ring of glass of
thickness T, and refractive index JJL and outer and inner radii r, r'
is now placed in the hole. Find the change in the intensity of
the illumination at the point on the screen opposite the centre of
the hole, and show that this point will be black, provided r =
and
(//, - 1) T= JX(2w + 1) - J (a + b)r*/ab.
EXAMPLES. 79
5. A screen is placed perpendicularly to the axis of a convex
lens, and the image of a bright point Q on the axis is formed on
the screen at q. If the light is allowed to pass through a small
aperture, whose form is given by y a cos 7r#/2c from x = c
to so = c, the origin being the centre G of the lens, examine the
position of the dark points on the line through q parallel to Cx.
6. A, C, B, D are the middle points of the bounding edges of
a rectangular diffraction grating, AB being parallel to the ruled
lines, and being the central line of an opaque interval. The
grating is blackened over except within the area ACBD. Find
the system of fringes along the line which is the projection of AB
upon the screen, and show how to find them for the projection
of CD.
7. A plane vertical screen is one boundary of a semi-
cylindrical dark chamber. A series of waves of light of given
colour, the fronts of which are parallel to the screen, pass through
it by a very narrow horizontal and rather short slit, of given
length and equidistant from its vertical edges, and illuminate the
opposite wall. Compare the brightness at different points of the
wall in the horizontal plane through the slit, and prove that there
is in this plane a succession of points of perfect darkness.
Prove that there is on the surface of the concave wall a series
of dark bands, the projections of which on the screen are approxi-
mately a series of hyperbolas.
Supposing the colour of the slit to be changed, what alteration
must be made in the length of the slit, in order that the position
of the dark bands may not vary ?
CHAPTER V.
DIFFRACTION CONTINUED.
rr-
63. WE have hitherto supposed that the waves are plane, or
that they are converging to a focus ; we shall now investigate the j
diffraction of light which is diverging from a focus.
We shall first suppose that the problem is one of two di-
mensions.
Let be the focus, AQ the circle 1 representing the wave-front
at the points of resolution ; B any point at which the intensity is
required. Let Q be any point on the wave-front, and let
OA=a, AB = b, AQ = s.
The vibration at B, produced by an element at Q may be taken
to be proportional to
COS 27T (
where S=QB- AB. Now
QB 2 = a? + (a + 6) 2 - 2a (a + b) cos s/a,
1 For the lowest in the figure, read Q.
FRESNEL'S INTEGRALS. 81
whence QB = b + (a + b) s 2 /2ab,
whence the vibration at P is
f
/
J
t
cos 2-7T
(r 2Xa6 I"
and the intensity I 2 is proportional to
I 2 = (/cos I vrv 2 dv) 2 + (/sin %7rv 2 dv) 2 .'..;. . .(1),
where v 2 = 2 (a + b) s 2 /ab\.
These integrals taken from to v are called Fresnel's integrals,
and we shall now discuss their properties.
Fresnel's Integrals.
64. The two definite integrals,
[v rv
I cos7rv 2 dv and I sm^irv^v,
Jo Jo
cannot be evaluated in finite terms except in the single case in
which v = GO , when they are each equal to %. We shall denote
them by C and S, and shall show how their values may be
expressed in the form of a series.
By integration by parts, it follows that
f"u /7j2w+l . a?v z 2^/ 2 C v
I l)^'f~^'^dv ^ i j
accordingly
Putting a = J(l + *)9r*, so that a 2 = ^t7r, and equating the
real and imaginary parts, Ave obtain
where
= cos 7r
............... ' *
1.3.5 1.3.5.7.9 , /QN
.(6).
N = ' - 4-
1.3 i.sis.rT"'
These two series are easily shown to be convergent, and are
adapted for numerical calculation when v is small ; they are due
to Knockenhauer 1 .
1 Die Undulationstheorie des Lichtes, Berlin 1839, p. 36.
B. O. 6
82 DIFFRACTION CONTINUED.
65. When v is large, the integrals may be expressed by means
of certain semi-convergent series due to Cauchy 1 . We have
r e -<**dv = ^
Jo 2a
r e-^dv _ 6~^ 2 _ 2n + l p e-
J v "v^~ "laV^- 1 2a 2 J v v
whence
r 6 -^_ 6 -.wA_ L_+iii-L3i + \
j v e \Wv 2W + 2 W 2W ' ' ') '
Putting a = J(l-M)?r* as before, and equating the real and
imaginary parts, we obtain
cos J 7rv 2 dv = P sin -| TTV* + Q cos J ?rv 2 ,
r 00
I sin \irtPdv P cos ^?rfl 2 + Q sin
, D 1 1.3 1.3.5.7
where P = ---- - +
m;
1 1.5.5 1^3.5.7.9
2^,3 ^-4^,7 '
The first few terms of the series for P and Q converge rapidly
when v is at all large, but the series ultimately become divergent, j
Such series are called semi-convergent series, and it is a known I
theorem that the sum of any number of terms of such a series I
differs from the true value of the function which the series j
represents, by a quantity less than the value of the last term
included ; we may therefore employ these series when v is large.
Accordingly we find
C = i + P sin \-irtf - Q cos i? 2 )
S = J - P cos iTTfl 2 - Q sin 2 ' '
66. Another method has been employed by Gilbert 2 . Put
u = ^7rv' 2 ) then
cos u ,
C = vi r du.
1 [ u cos ^
sp-k~V*
1 1 r e~ ux dx
|\y mu ~
/ ' / / '
V^ V7TJQ *JX
I r r
whence = /0 da? I e~ wa; cos u du.
TryZ 'o Jo
1 C. E. vol. xv. 534, 573.
2 Mem. conronnes de VAcad, de Bruxelles, vol. xxxi. 1,
STRAIGHT EDGE. 83
The integration with respect to u can be effected, whence
~ Iff 00 x^dx f 00 e~ ux c$dx . f e~ ux dx )
= 79 1 I i ^2 ~" cos u I 1 ~~2~ + smu I -T-T > .
Putting # 2 = y, the first integral by a known formula 1 is
^Tj-cosec f TT =
equal to
we thus obtain
where
= _ r
00 I
I
.(6),
.(7).
By proceeding in a similar way, it can be shown that
$ = i 6rsinw Hcosu (8).
When u = 0, it follows that
Now u, which is equal to ^Try 2 , is necessarily positive, and
consequently the factor e~ ux decreases rapidly as u increases. It
therefore follows that the values of G and H are never greater
than J, and converge towards zero as u increases indefinitely.
This property constitutes the superiority of Gilbert's method.
Straight Edge.
67. The first problem we shall consider will be that of
diffraction by a straight edge.
B'
First let the point P lie within the geometrical shadow. Let
BA G be the front of the wave, and let BA = h ; then the
Jo i + y
sin kir '
where 1 > 7c > 0.
62
84 DIFFRACTION CONTINUED.
intensity at P is obtained by integrating from s = h to s = x ,
whence if
the intensity will be proportional to
P=(l cos ^7rv*dv J + ( I sin
At the edge of the geometrical shadow, h = 0, in which case
C = S=Q, and 7 2 = J ; but when V is large, which will always be
the case unless h is so small as to be comparable with \*, it
follows from (4) that the most important term in / 2 is equal to
1/7T 2 F 2 . Whence the intensity at the edge of the geometrical
shadow is proportional to J, and rapidly diminishes as P proceeds
inwards.
For a point Q outside the shadow, we must integrate from
s = h to s = oo , where h is now equal to AC, whence
*>
The maxima and minima values of this expression are deter-
mined by the equation
or
When F = this expression is equal to J and is therefore
positive, and the corresponding value of the intensity is equal
to . Using the series (2) and (3), we see that when F= 1, the
value of this expression is equal to ^ + 8 r or J + M, which is also j
positive ; but if F 2 = f , the expression is equal to G v + 8 V which
in this case is equal to 2N if we employ (2) and (3), or equal
to 2Q if we employ (4) and (5). The expression in question
is therefore negative when F 2 = f , and therefore vanishes and
changes sign for some value of F 2 lying between 1 and f . This
root corresponds to a maximum value of the intensity. The first
maximum according to Verdet 1 occurs when
and the first minimum when
1F 2 = |
1 Lemons d'Optique Physique, vol. i. 90.
CIRCULAR APERTURE OR DISC.
85
The maxima and minima values have also been calculated by
Fresnel, and are shown in the following table :
V
I 2
1st max.
1-2172
2-7413
min.
1*8726
1-5570
2nd max.
2-3449
2-3990
min.
2-7392
1-6867
3rd max.
3-0820
2-3022
min.
3-3913
1-7440
The effect of the screen is therefore to produce a series of
bright bands outside the geometrical shadow of the source ; whilst
inside the shadow, there are no bands, but the intensity diminishes
rapidly to zero.
Circular Aperture or Disc.
68. We have already considered the case of diffraction through
a circular aperture, when light is converging to a focus ; we shall
now discuss the corresponding problem when light is diverging
from a focus.
In the figure to 63, let B be the projection of the centre
of the aperture, AQ the wave-front at the aperture, Q any point
on it; and let P be any point on the second screen, which is
supposed to be parallel to the first screen in which the aperture
exists.
Let OA = a, AB = b, BP = r, the angle which the plane
OAQ makes with the plane OBP ; also let c be the radius of the
aperture.
Then
PQ2 = (> - a sin cos $) 2 + a? sin 2 sin 2 $ + (a + b - a cos 0) 2
= 6 2 + r 2 - 2ra sin 6 cos $ 4- 4a (a + b) sin 2 ^0.
Putting p = a sin 6, and treating p as small, we obtain
,
'
86 DIFFRACTION CONTINUED.
If therefore the vibration at A be denoted by or 1 cos 27r/r,
the vibration at P will be expressed by
Now dS = pdpd very approximately ; if therefore we put
v(a + b)_
"cffixT -*"'
we shall find that the intensity is
where (7 = // cos (^tcp^ lp cos ) pdpd\ , .
S = //sin (i*/> 2 - fy> cos ' '
The above expression for the intensity is of a perfectly general
character, and the limits of integration must be chosen so as to
include the whole of the aperture. When the aperture is a circle
whose centre is A, the limits are from (f> = to 2?r, and from p =
to c; if on the other hand; we are investigating diffraction pro-
duced by a circular disc, the limits of p will be c and GO .
69. Before discussing the general formulas (10) and (11), we j
shall consider two special cases.
(i) Let the diffraction be produced by an aperture of radius ]
c; then at the projection of its centre upon the screen, r I = 0, and
( i i 2?r
= 2-7T cos $Kp- . pdp = sin
JO K
S = (1 -cos^/ec 2 ),
/C
4
whence / 2 = ^- sin 2 J /cc 2 .
(ii) Let the diffraction be produced by a disc, then
whence if p J IK,
L
%7T .
,f r* >2/-/ //i ^ ^^ c i TI
_ ACO/ UjlLr bill
C &
00 Q
x sin J Kx^dx = cos \
whence / 3 = =?--
LOMMEL'S METHOD. 87
and therefore the intensity is the same as if the wave had passed
on undisturbed.
In the middle of the eighteenth century, Delisle 1 had observed
the existence of a bright spot at the centre of the shadow of a
small circular disc ; but this experiment had been so completely
forgotten, that Poisson 2 brought forward the objection to Fresnel's
theory, that it required the intensity at the centre of the shadow
to be the same as if the wave had passed on undisturbed. The
experiment was accordingly repeated by Arago 3 , with a small disc
whose diameter was one millimetre, and the required phenomenon
was immediately observed.
70. The preceding results, which are of a fairly elementary
character, are applicable only to the centre of the projection of
the disc or aperture. The intensity at excentric points forms the
subject of a very elaborate investigation by Lommel 4 , which we
shall proceed to consider.
In the case of a circular aperture or disc
= 2 I cos (i/ep 2 - Ip cos 0) pdpdd>
JQJ
cos (^ /cp 2 ) cos (Ip cos (/>) pdpdcf)
= 211
J QJ
(12).
Similarly S = 2TrfJ (lp)sin(/cp*)pdp ............... (13).
Except in the special case of I = 0, the integrals (12) and (13)
cannot be evaluated unless the limits are infinity and zero. We
shall therefore first obtain their values in this case.
,00
Let u= sce~ a ^J (bx) dx\
Jo
then by a known formula 5
2 r 00
Jo (bx) = - I sin (bx cosh <) d$.
1 Mem. de Vane. Acad. des Sciences, 1715, p. 166.
2 Verdet, Lecons d'Opt. Phys. vol. i. 66.
3 (Euvres Completes, vol. vu. p. 1.
4 Abh. der II. Gl. der Ron. Bayer. Akad. der Wiss. vol. xv. p. 233. In this
paper, and also in another by the same author in the same volume, a large amount
of interesting information concerning Bessel's functions will be found.
5 A proof will be found in Proc. Gamb. Phil. Soc. vol. v. p. 431.
88
DIFFRACTION CONTINUED.
But
/ Te -a?x-
,
J2
' Ol Tl A^/ 1 /r'> 1 f
4a 3
whence
=*rr
TT./O Jo
#e~ aV! sin (bx cosh <) c?c?^
b .[
f: nnsVi fhd.fh
'
Putting a (1 + 1)/2* for a, equating the real and imaginary
parts, and then writing b = I, a 2 = J/c, we obtain
2-7T . I 2 27T Z 2
71. We shall next show, how series may be obtained by which
the integrals G and S may be calculated, when the limits are
c and 0.
It is known that 2 J n ' = J n ^ J n +i,
from which we deduce
j
(15),
,, fi ^
Using (15) and integrating by parts, we obtain
e ftn+i . re
=- J n+1 (lc) e & - - J w (Ip) ei"P= dp.
Accordingly
2?rc
Writing c 2 =y, Zc=2 ..................... (17),
and equating the real and imaginary parts, we obtain
jj
LOMMELS METHOD.
89
where
(19).
These series for U^ U 2 are convenient for numerical calculation
when yjz or rcc/l is small.
72. Series which are suitable for calculation when y\z is large
may be obtained as follows. By (16) we obtain
r J A
L p n
whence
I
accordingly equating the real and imaginary parts, we obtain
cos A y .
C =
2 . z 2 sin
- sin =- + -
.(20),
where
(21).
By (9) and (17) it follows that when y = z t (a + b)/a = r/c ; this
value of r corresponds to the edge of the geometrical shadow.
Under these circumstances we have 1
From (18) and (20) we also find
_2?rc 2 _27rc 2
y 1 ~ y
U iJ
27TC 2 _ 27TC 2
y y
- F
therefore
--(22),
(23).
See Todhunter's Functions of Laplace, p. 320.
90 DIFFRACTION CONTINUED.
73. The functions U lt U z , V 0} V lf although expressed in the
form of infinite series, are of course finite quantities. In fact since
the greatest values of Jo, cos J/ep 2 , and sin ^Kp 2 are unity, it follows
that G and S cannot be greater than ?rc 2 ; and from (22) we see
at once that U 1} U 2 cannot exceed certain limiting values; hence
in calculating the values of these quantities, we may use whichever
of the pairs of series (19) or (21) is most convenient. If we
employ the series (21) we shall obtain the values of F , V ly and
the values of U l} U z can then be found from (23).
Since y = 2?r (a + b) c*/ab\, z = ^7rrc/\b,
Lt follows that z depends upon the position of the point at which
the intensity is to be examined, but y does not. We must
accordingly assign a definite value or series of values to y, and
then calculate the values of U lt U 2 corresponding to each of these
definite values, for different values of r or z. This has been done
by Lommel for a series of values from y = 7r to y = 10?r, for the
functions 2Ui/y and 2U a /y t and from these tables the values of
the intensity can be calculated in the different cases which arise.
74. We shall now show how the intensity may be calculated
in the case of diffraction through a circular aperture.
Since y\z = (a + b) c/ar, it follows that yjz is > or < 1, ac-
cording as the point lies in the bright part of the screen or in the
geometrical shadow ; at points on the edge of the shadow, y = z.
By (18) the intensity is
The points of maxima and minima intensity are given by the j
equation
From (15) and (19) we see that
z TT 2 T z TT
= -- I/*, -y = I/I + - U 2 ,
j *, I
dz y dz y
whence (24) reduces to
^ = 0.
The values of the roots of the equation J^Z/TT) = have been
calculated by Stokes 1 , and the values of the roots of the equation
1 Trans. Carnb* PliiL Soc. vol. ix. p. 166; see ante, p. 55.
CIRCULAR APERTURE.
91
U = for y TT, 27r,...107r have been calculated by Lommel.
The following table gives the results for y = TT and y = 5?r.
/!
*w
I 2
3-8317
+ 1062
0263 min.
4-7154
0320 max.
7-0156
- -0406
0018 min.
t>6
which are what (27) and (28) become, when n \ is written for
11, it is at once seen that they are satisfied.
78. It also follows from {32) and (35), that
(2 \2 / 2 N ^
-) sin a), J_i=(--) cos x ......... (36).
irx) \7rxJ
If in (30) we write y = # 2 , it can be shown that
dy
,. ! /I dY 1
accordingly ty n =
M-fel 1.8...(8n + l)
/2\i 1 . 3...(2w 1) / 1 d\ n cosx
from which we see that /_ n -j is zero when x oo , and infinite
when # = 0.
79. We are now in a position to explain Lommel's method 1 .
The light is supposed to diverge from a linear source 0, and to
be received on a screen. Let B be the projection of on the
1 Abh. der II. Cl. der Ron. Bayer. Akad. der Wiss. vol. xv. p. 531.
96 DIFFRACTION CONTINUED.
screen, then if in the figure to 63, we put = 0, BP = x, AQ = p,
we may prove in precisely the same manner, that
1 26 6 2a6
whence the intensity is proportional to 7 2 = G 1 4- S' 2 ,
where C = fc<
and the origin of p is the intersection of the line OB with the i
wave-front at the point A. The integration extends over the
effective portion of the wave.
80. The two principal problems, which we shall have to
consider, are diffraction through a slit, and diffraction by a long j
narrow rectangular obstacle.
When the slit or obstacle, whose breadth is supposed to be
equal to 2c, is parallel to tbe screen, and is symmetrically placed,
so that its middle line is the intersection of the plane passing \
through the source and B, the integration will be from c to c in ;
the case of a slit, and from oo to c, and -co to c in the case
of an obstacle.
81. When the integration is from c to c, the odd parts of
the integrals disappear, and we thus obtain
re
(7 = 21 cos ^/cp z cos Ipdp
7. .
S = 2 I sin ^icp 2 cos Ipdp
Jo
(39).
The integrals (39) cannot be evaluated in finite terms unless ;
c = oo ; in this case, it may be shown by writing a a (1 + t)/2* in
the integral
cos
that I cos ^tcp* cos Ipdp == (5-) cos [~ J
Jo \**/ \Z
/
, sn * cos = - c - sn -
where
LOMMEL'S METHOD. 97
82. We shall now show how to express the values of C and S
in series. Since
(2 \^
-- ) COS X.
TTX]
it follows that
G = 27r* l J_ l cos
c
= (27T)* (Zp)* J_i (Zp) sin % K p*dp
J o /
Now by (32) t/" n+ j vanishes when #=0, provided n is zero or
any positive integer, whence integrating by parts, and using (15),
we find
* dp = J n+t _ (lp) ei'
whence
Equating the real and imaginary parts, and using (41), we
obtain
/27r u
= (y /
p
where 0.= 2 (-) J n+2J ,(^) ............ (44).
8 = u sin ~ u cos
83. To obtain a series in descending powers of z/y, we must
recollect that J-n-\ is zero when x oo , and infinite when a? = 0.
We must therefore write
C + 18 = 2 [ e* 1 "" 2 cos Zpd/5 - (27r)^ f (^)* J. (Zp) e^ 2 ^ . . .(45).
^0 Jc
By integration by parts, and by (15), we can show that
B. O.
98 DIFFRACTION CONTINUED,
where n is zero or any positive integer ; whence
whence equating the real and imaginary parts, we obtain
COS
) c
sin
sn
cos
(46).
where
V n =
(47).
If therefore we denote the right-hand sides of (46) by C', $C|
respectively, we have by (40) and (45),
.(48).
, Diffraction through a Slit.
84. We are now prepared to investigate the case of diffraction
through a slit.
Since the intensity is proportional to
it follows from (43) that
~
and the maxima and minima values are given by
U *~dz + U $~dz =0
.(49).
DIFFRACTION THROUGH A SLIT.
From (16) and (44) we obtain
99
* ^ 5- * TJ-
dz y f dz y "f
also from (44)
whence (49) reduces to
-(?
Is
Now sfrj = (2/7r)* sin z, which vanishes when z nir, where
n is zero or any positive negative integer ; accordingly there is
a series of bands parallel to the edges of the slit, whose distances
apart are equal to \b\jc. Another system of bands is given by
the roots of the equation U s = 0.
u
Tables have been constructed by Lommel, which give the
values of these roots, when 2/=3, 6, 9... 30, and the following
table gives the results when y = 3 and y = 15.
z
'|
(-Wi^
I' 2
o
7652
8163 max.
7T
1749
0822 min.
_ 3 J4-0127
...
0949 max.
^ ] 2?r
- -0829
0075 min.
7-6130
0223 max.
V STT
+ -0366
0014 min.
2012
1244 max.
7T
1546
. -081 3 min.
27T
3955
1608 max.
y " 15 8-7546
1132 min.
STT
0*
-1739
1157 max.
47T
-0610
0498 min.
Since yjz = (a + b) cfax, it follows that when y > z, the point x
lies within the luminous area; and since 7r = 3.1416, it follows
that when y = 3, the first minimum, and all subsequent maxima
and minima lie in the geometrical shadow. On the other hand
when y= 15, under which circumstances the slit is broader than
.when y 3, there are a succession of maxima and minima within
the luminous area.
72
100 DIFFRACTION CONTINUED.
Diffraction by a Narrow Obstacle.
85. We shall now suppose, that diffraction is produced by a
long narrow rectangular obstacle of breadth 2c.
In this case the intensity is proportional to
where by (46)
V = - c V (Fj sin Jy + F cos
whence /*=_"." (F^+F^ 2 ),
y
and the maxima and minima are determined by the equation
By (15) and (47) we obtain
and
also by (47) F. t + F f =
whence (50) reduces to
In this case also there are a series of maxima or minima value*
corresponding to ^ = 0, TT, 2-Tr... ; whilst another set are given
the roots of the equation Fj = 0. The results are shown in th*
following table.
DIFFRACTION BY A STRAIGHT EDGE.
101
t
.
w>S
j 2
+ 2910
0891 max.
1-35,50
0118 min.
7T
- -3350
1942 max.
2/ " 3 13-7710
1769 min.
57037
6413 max.
V 27T
-6961
6130 min.
+ 0819
0067 max.
1-5426
...
0001 min.
7T
- -0866
0076 max.
y I*- 1 14-6103
...
0011 min.
2.
+ 1024
0108 max.
17-6163
0045 min.
When y = 3, so that the obstacle is narrow, there is a central
bright band, on either side of which are two bands of minimum
intensity, which lie within the shadow; but when y = 12, so that
the obstacle is broader, there are several.
Diffraction . by a Straight Edge.
86. The last problem, which we shall consider, is that of
diffraction by an indefinitely large straight screen, which extends
from the origin to infinity in the negative direction. In this case
C
.(51).
,00
= I cos (^tf/o 2 Ip) dp
Jo
f*
S = sin (^tcp 2 - Ip) dp
Jo
At the projection of the diffracting edge on the screen, I = 0, and
and / 2 = \TT\K.
If the diffracting edge were absent, the limits would be oo and
X) , and we should have at any point of the screen
(7
=1
cos
cos
Ipdp = 2 r^-J sin ( +- ^TTJ ,
102 DIFFRACTION CONTINUED.
Whence if I' 2 be the intensity,
/' 2 =27T/K,
and therefore I 2 = J/' 2 ; or the intensity at the edge of the geo-
metrical shadow, is one quarter what it would be if the obstacle
were removed.
When I is riot zero, the integrals (51) cannot be evaluated in
finite terms; they may however be reduced to Fresnel's integrals.
We have
r 00
C = I (cos Jtf/r cos lp + sin \xcr sin lp) dp
Jo
sn + + sn
Let u = I e-" 2 * 2 sin Zbxdx,
du r
-77 = 2 x<
db J
then = 2 | xe' ^ cos
o
1 ^2bu
a? a 2 '
whence u =
e - *
e a * dx.
o
Writing a = c (1 + 0/2*, we obtain
f i r 6 6 3
I sin c 2 ^ 2 sin %bxdx = I cos
Jo C" J o
f 00 1 [
I cos c 2 ^' 2 sin %bxdx = -
Jo C 2 .'
CVo
The integrals on the right-hand side depend upon Fresnel's
integrals, and accordingly C and S can be expressed by these
quantities.
II
CHAPTER VI.
DOUBLE REFRACTION.
S7. WE have already drawn attention to the fact, that there
are certain crystalline substances, called doubly refracting crystals,
which possess the property of separating a single ray of light into
two rays. We shall now consider the experimental facts connected
with this class of bodies.
One of the best examples of a doubly refracting crystal is a
crystallized form of carbonate of lime called Iceland spar. Crystals
of Iceland spar can easily be split into rhombohedra, the acute
and obtuse angles of the faces of which, are equal to 14? 55' 35"
and 105 4' 25" respectively. The line joining the two opposite
corners, where the obtuse angles meet, is called the optic axis
of the crystal, and is a line with respect to which the properties
of the crystal are symmetrical. Iceland spar therefore possesses
the same kind of symmetry as an ellipsoid of revolution, and
crystals of this class are called uniaxal crystals.
There are certain other kinds of doubly refracting crystals,
which have two optic axes, and which possess three rectangular
planes of symmetry. Such crystals are called biaxal crystals.
104 DOUBLE REFRACTION.
Uniaxal Crystals.
88. When a small pencil of light is incident upon a plate
of uniaxal crystal, it is found that in general there are two
refracted rays. One of these rays is refracted according to the i
ordinary law of refraction, and is consequently termed the ordinary
ray ; whilst the other is refracted according to a totally different ;
law, and is called the extraordinary ray. There are however two j
cases in which there is only one refracted ray, viz. (i) when the
direction of propagation coincides with the optic axis of the s
crystal, (ii) when the face of the crystal contains the axis, and
the pencil is incident normally upon the surface. In both these
cases the ordinary and extraordinary rays coincide, and only one j
refracted ray is consequently observed.
Let us now suppose, that a ray of light is refracted through \
a rhomb of Iceland spar, and that the plane of incidence contains
the axis ; and let the two refracted rays be transmitted through i
a second rhomb. When the two rhombs are similarly situated,
it will be found that there are only two rays after refraction -
through the second rhomb, and that the ordinary ray in the |
first rhomb, gives rise to an ordinary ray in the second, whilst
the extraordinary ray E in the first rhomb, gives rise to an
extraordinary ray E E in the second. If now the second rhomb be
turned through any angle which is less than 90, it will be found
that there are four refracted rays, and that and E each give I
rise to an ordinary and an extraordinary ray 0) E) and E , E E
respectively. When the angle is small, E and E are very faint,
but become brighter as the angle increases, whilst and E E
dimmish in brightness ; and when the second rhomb has been
turned through an angle of 90, and E E will have disappeared,
leaving E and E in possession of the field.
These experimental results show, that doubly refracting crystals,
in addition to dividing an incident ray into two refracted rays, I
also produce an essential modification in the constitution of the j
refracted light.
89. The index of refraction of the ordinary ray is the ratio i
of the sine of the angle of incidence to the angle of refraction,
which as we have already seen is constant for all angles. The I
extraordinary index of refraction is defined as follows. Let the |
HUYGEN'S CONSTRUCTION. 105
plane of incidence contain the axis, and let a pencil of light be
incident at such an angle, that the extraordinary ray is perpen-
dicular to the axis; then the ratio of the angle of incidence to
I the angle of refraction of the extraordinary ray under these
circumstances, is called the extraordinary index of refraction.
The reason of this definition will appear hereafter; it can be
proved at once by geometry, that this ratio is entirely independent
of the inclination of the optic axis to the face of the crystal.
90. The law which determines the refraction of the extra-
ordinary ray in uniaxal crystals was first discovered experimentally
by Huygens 1 , who gave the following construction.
Let A be the point of incidence, AB the direction of the
optic axis; draw AC perpendicular to AB. Let B and D be
points on AB, such that AD JAB is equal to the ordinary index
of refraction, and let C be a point on AC such that AD/ AC is
equal to the extraordinary index. With A as a centre, describe
two spheres whose radii are AB, AD', and describe also an
ellipsoid of revolution, whose polar axis is equal to and coincident
with AB, and whose equatorial axis is AC. Produce the incident
ray to meet the second sphere in /, and let the tangent plane
at / cut the surface of the crystal in a line T. Through T draw
two tangent planes TO, TE to the first sphere and the ellipsoid
respectively, meeting them in and E\ join AO, AE. Then
AO, AE will be the directions of the ordinary and extraordinary
rays respectively.
This construction was discovered by Huygens by a process of
induction, but was afterwards verified by careful measurements.
91. The preceding construction suggests, that the wave-
surface in a uniaxal crystal consists of two sheets, viz. a sphere
1 Traite de la Lumiere.
106 DOUBLE REFRACTION.
and an ellipsoid of revolution, which touch one another at the
extremities of the optic axis; and we shall hereafter see that this
conclusion is borne out both by theory and experiment. When >
the disturbance producing light is communicated to any point of |
the medium, two waves are generated, one of which is spherical
and travels with the same velocity in all directions, whilst the
other is spheroidal, and its velocity is different in different
directions. When a plane wave is incident upon the surface of
the crystal, each point of the surface may by Huygen's Principle,
14, be regarded as the origin of secondary waves, and the
envelop of these secondary waves will consist of two planes TO,
TE, which are the fronts of the ordinary and extraordinary waves
in the crystal. If the equations of the sphere and the ellipsoid j
of revolution, referred to A as origin, and AB as the axis of ;
z t be
We 2 + (^ + 2/ 2 )/a 2 = 1,
ji
then c will be the velocity of the ordinary wave, whilst the
velocity of the extraordinary wave will be equal to the perpen- j
dicular drawn from A on to the wave-front TE. Since the
extraordinary wave-front touches the ellipsoid, it follows that if 6
be the angle which the direction of the wave makes with the
optic axis, and V be its velocity of propagation,
F 2 = c 2 cos 2 + a 2 sin 2 6.
' i i - v ^(f*ar or s ^*y
We therefore see that the directions of the two waves coincide,
whenever they are parallel to the optic axis, or to an equatorial |
in the former case V= c, and in the latter Va. The
i*^^-^ (quantities a and c are therefore called the principal wave
,^****'-^ velocities, and the ratios F/c, F/a are called the ordinary and '
^ j~k*"~ L extraordinary indices of refraction.
*"~ V ~ \! kMttf ' I
4a0- ^ 1%T 92. We have already pointed out, that when common light is
"^C incident upon a crystal, two refracted rays are always produced ; j
^j^ C^JL on the other hand we have shown, that when the incident light j
- 4* vvv consists of the ordinary or extraordinary ray, which is produced by |
^T^ Aji^refracting common light through another crystal, there are always
-4 s *"two positions of the second crystal, in which one of the two rays
\_0is absent.
L*f^
V^' h- We shall now explain how this phenomenon may be accounted I
'i / j^ r for. K ,
f^w,w?& ^ * <&*fr A Jtw* fr'^-N
,.^er ^^^ I* . . * , . /x
POLARIZATION BY DOUBLE REFRACTION. 107
93. We have stated in Chapter I., that light is said to be
polarized, when the elements of ether composing the wave are
vibrating perpendicularly to a fixed plane, which is called the
plane of polarization. Now when common light is refracted
through a crystalline plate, it is supposed that the two refracted
rays are polarized in perpendicular planes ; and that the vibrations
of the ordinary ray are perpendicular to the plane, which passes
through the optic axis and the normal to the ordinary wave-
front ; whilst the vibrations of the extraordinary ray lie in the
plane passing through the optic axis and the normal to the extra-
ordinary wave-front. The plane which passes through the optic
axis and the normal to the wave-front is called the principal plane
for that wave ; we may therefore say, that the ordinary wave is
polarized in the principal plane, whilst the extraordinary wave is
polarized perpendicularly to the principal plane.
94. We are now able to explain why it is, that in certain
positions of the crystal one of the two rays in certain cases
disappears.
For simplicity, let the surface of the crystal be perpendicular
to the optic axis AB ; let xy be the plane of incidence, TO, TE be
the ordinary and extraordinary wave-fronts, and AO, AE the
ordinary and extraordinary rays.
When the incident light is polarized in the plane xy, the
vibrations are parallel to Az. But since we have assumed, that
the vibrations in the extraordinary wave are executed in the
plane xy, it follows that an incident wave, whose vibrations are
perpendicular to this plane, cannot give rise to an extraordinary
wave, but only to an ordinary wave. When, on the other hand,
the incident light is polarized perpetidicularly to the plane xy,
so that the vibrations are executed in that plane, the incident
light gives rise to an extraordinary wave, but riot to an ordinary
wave. If the incident light were polarized in any other plane,
the incident vibrations could be resolved into two components
108 DOUBLE REFRACTION.
respectively in and perpendicularly to the plane of incidence xy,
the first of which would give rise to an extraordinary wave, whilst
the latter would give rise to an ordinary wave.
Since the wave-surface of the ordinary wave is a sphere, the
directions of the ordinary wave and the ordinary ray coincide ;
but since the wave-surface of the extraordinary wave is an ellipsoid
of revolution, the directions of the extraordinary wave and the
extraordinary ray do not coincide within the crystal, unless the i
direction of propagation is parallel or perpendicular to the axis. |
The question whether the vibrations of the extraordinary wave are |j
perpendicular to the ray or the wave-normal is one, which cannot jj
be discussed without the aid of theoretical considerations, but it ||
may be stated that according to Fresnel's theory, the direction s|
of vibration in the extraordinary wave is parallel to EY, that is, ji
perpendicular to the wave-normal.
95. We have thus far given a description of the principal I;
phenomena connected with uniaxal crystals, and of the theoretical !<
explanation by which it is 'proposed to account for them, and in j;
the next chapter we shall show how these phenomena may be jj
explained by means of a dynamical theory. There are however |
certain other experimental facts which demand attention.
In all uniaxal crystals, the radius of the spherical sheet of the j
wave-surface is equal to the semi-polar axis of the ellipsoidal !
sheet; but in Iceland spar, the extraordinary index of refraction is |j
less than the ordinary index, and therefore the ellipsoidal sheet ||
of the wave-surface is a planetary ellipsoid. Such crystals are j
called negative crystals. There are however certain other crystals ;
in which the ellipsoidal sheet is an ovary ellipsoid ; and crystals 1
of this kind are called positive crystals. It therefore follows, that '
* -* *
for negative crystals the ellipsoidal sheet of the wave-surface lies
outside the spherical sheet, whilst the converse is the case for j
positive crystals.
The following is a list of some of the principal uniaxal crystals.
Positive. Negative.
Ice. Beryl.
Lead hyposulphate. Cinnabar.
Magnesium hydrate. Emerald.
Quartz. Iceland spar.
The red silver ores. Ruby.
Sapphire.
Tourmaline.
DOUBLE REFRACTION BY BIAXIAL CRYSTALS. 109
The principal indices of refraction for Iceland spar and quartz
have been determined by Eudberg, for the principal lines of the
spectrum, and are as follows.
Iceland spar
Quartz
B
Mo
*
*
1-65308
1-43891
1-54090
1-54990
C
1-65452
1-48455
1-54181
1-55085
D
1-65850
48635
1-54418
1-55328
E
1-66360
48868
1-54711
1-55631
F
1-66802
49075
1-54965
1-55894
G
1-67617
49453
1-55425
1-56365
H
1-68330
49780
1.55817
1-56772
Biaxal Crystals.
96. The investigations of Brewster and Biot showed the
existence of a certain class of doubly refracting crystals, in which
neither ray is refracted according to the ordinary law. Such
crystals have two optic axes, and are therefore called biaxal
crystals.
The form of the wave-surface for biaxal crystals was discovered
by Fresnel, and is known by his name ; its equation is
where ?- 2 = # 2 + y 2 - + z*. The quantities a, b, c are called the
principal wave velocities in the crystal. This surface will be
discussed in the next chapter, but it is easy to see that if any two
of the three constants a, b, c are equal, the surface splits up into
a sphere and an ellipsoid of revolution.
The following is a list of some of the principal biaxal crystals.
Aragonite. Selenite.
Borax. Sulphur.
Cerceosite (Lead carbonate). Topaz.
Mica.
Nitre.
110
DOUBLE REFRACTION.
97. The next table gives the values found by Rudberg forji
the three principal indices of refraction, of aragonite and topaz,!,
for the principal rays of the spectrum, where p a denotes the ;.
ratio of the velocity of light in air, to that of the principal wave |
velocity a.
Kays
Aragonite
Topaz
*
Mb
Me
M a
Mb
M c
B
1-52749
1-67631
1-68061
1-60840
1-61049
61791
C
1-52820
1-67779
1-68203
160935
1-61144
61880
D
1-53013
1-68157
1-68589
1-61161
1-61375
62109
E
1-53264
1-68634
1-69084
1-61452
1-61668
62408
F
1-53479
1-69053
1-69515
1-61701
1-61914
62652
G
1-53882
1-69836
1-70318
1-62154
1-62365
63123
H
1-54226
1-70509
1-71011
1-62539
1-62745
63506
It will be hereafter shown, that the angle between the optic :
axes depends upon the values of the three principal indices of |
refraction; and since these are slightly different for different |
colours, the positions of the optic axes for different colours will
not coincide. This is called dispersion of the optic axes.
Quartz.
98. When plane polarized light is incident normally on a
plate of Iceland spar, which is cut perpendicularly to the axis,
it is found that the emergent light is polarized in the same plane
as the incident light. There are however certain uniaxal crystals,
of which quartz is the most conspicuous example, which possess
the power of rotating the plane of polarization ; that is to say, the
plane of polarization of the emergent light is inclined at a certain
angle to that of the incident light, which is found by experiment
to be proportional to the thickness of the plate. The construction
of Huygens and the theory of Fresnel do not apply to such
crystals. Crystals of this class require a special theory of their
own, which will be considered in the chapter on rotatory polari-
zation.
DOUBLE REFRACTION BY STRAINED GLASS. Ill
99. Most isotropic transparent media, when subjected to
stress, exhibit double refraction. For example, compressed glass
acts like a negative uniaxal crystal, whose axis is parallel to the
direction of compression ; whilst stretched glass acts like a positive
uniaxal crystal, whose axis is parallel to the axis of extension 1 .
There are also certain crystals, in which the relative position of
the optic axes for different colours varies with the temperature 2 .
1 Brewster, Phil. Trans., 1815, p. 60.
2 Ibid, Phil. Trans., 1815, p. 1 ; Phil. Mag. (3) vol. i. p. 417.
CHAPTER VII.
FRESNEL'S THEORY OF DOUBLE REFRACTION.
100. WHEN the disturbance which produces light is excited at
any point of an isotropic medium, a spherical wave is propagated I
from the centre of disturbance with constant velocity ; but we I
have pointed out in the preceding chapter, that when the dis- I
turbance is excited in a doubly refracting medium, two waves are I
propagated with different velocities, and that when the medium is I
a biaxal crystal, the velocity in any given direction is a function of
the inclination of this direction to the optic axes of the crystal.
The laws regulating the propagation of light in crystals, were I
first investigated mathematically by Fresnel, who showed that the I
wave-surface in biaxal crystals, is a certain quartic surface, which I
reduces to a sphere and an ellipsoid of revolution in the case of I
uniaxal crystals. The theory by means of which Fresnel arrived I
at this result, cannot be considered to be a strict dynamical theory; I
but on account of its historical interest, and also owing to the
fact that experiment has proved that Fresnel's wave-surface is a I
very close approximation to the true form of the wave-surface in
biaxal crystals, we shall proceed to explain its leading features,
and afterwards discuss the geometry of this surface.
101. The theory of Fresnel depends upon the following four
hypotheses, which are thus summarized by Verde t 1 .
(i) The vibrations of polarized light are perpendicular to the
plane of polarization.
(ii) The elastic forces which are produced by the propagation
of a train of plane waves, whose vibrations are transversal and
I
1 Lemons d'Optique Physique. Vol. i. p. 465.
FUNDAMENTAL ASSUMPTIONS. 113
rectilinear, are equal to the product of the elastic forces produced by
| the displacement of a single molecule of that wave, into a constant
factor, which is independent of the direction of the wave.
(iii) When a wave is propagated in a homogeneous medium,
the component of the elastic forces parallel to the wave-front, is
alone operative.
(iv) The velocity of propagation of a plane-wave, which is
propagated in a homogeneolis medium without change of type, is
proportional to the square root of the effective component of the
elastic forces developed by the vibrations of that wave.
102. We have stated in the preceding chapter, that according
to the generally received opinion, the vibrations of the ordinary
wave in a uniaxal crystal are perpendicular to the plane containing
the direction of propagation and the optic axis, whilst the vibra-
tions of the extraordinary wave are executed in the corresponding
plane. Up to the present time no experiments have been described
which prove that this is the case, and consequently for all we
know to the contrary, the vibrations of the ordinary wave might
take place in the principal plane, whilst those of the extraordinary
wave might be perpendicular to that plane. We shall hereafter
show, that there are strong grounds for supposing, that the
vibrations of polarized light are perpendicular to the plane of
polarization ; but for the present FresnePs first hypothesis must be
regarded as an assumption.
103. The second and third hypotheses require careful con-
sideration, and it will be convenient to discuss them together.
Since the motions of the ether which constitute light are of a
vibratory character, it follows that the ether when undisturbed,
must be in stable equilibrium. Hence if F (x, y, z) = V be its
potential energy at any point x, y, z ; and if a particle situated at
this point be displaced to the point x + u, y -f v, z + w, it follows
that dVldx dVjdy = dVjdz = 0; and therefore expanding by
Taylor's theorem,
V = F + (A i? + Btf + Cw* + ZA'vw + 2&wu + ZC'uv\
where F is the constant potential energy when in equilibrium, and
A, B... are positive constants. By properly choosing the axes,
the products may be made to disappear ; whence omitting the
B. O. 8
114 FRESNEL'S THEORY OF DOUBLE REFRACTION.
constant term V , which contributes nothing to the forces, the
value of V may be written
i !
V = ^(a?u < * + 6V + c-w-) (1),
and therefore the forces of restitution are
'f X = o?u y Y=bv, Z = c-w (2).
Hence if we construct the ellipsoid
,, I/" xVT^K < A "
_- < 3 >
"" * ' whose centre is 0, and if we draw a radius OP parallel to the |
3 Z direction of displacement and meeting the ellipsoid in P, and OF I;
L i%r, * lit ^be the perpendicular from on to the tangent plane at P, then!
^ ^_ OF will be the direction of the resultant force.
W ^ The ellipsoid (3) is called the ellipsoid of elasticity ; and it I
from the preceding construction, that the resultant force |
not be in the direction of displacement, unless the displace- a
ment is parallel to one of the principal axes of this ellipsoid.
< C*Z2 /r/
If I, m, n be the direction cosines of the normal to the wave- a
front ; X, //,, v those of the direction of displacement, it follows that |
resultant force of restitution will not be in the plane of the!
^ wave-front. This force may however be resolved into two com- 1
, ponents, one of which is in the plane of the wave, and the other is >
perpendicular to it. The latter component according to the third |
hypothesis will not give rise to vibrations which produce light,!
and therefore need not be considered. The former component)
will give rise to vibrations which produce light ; but it will not
coincide with the direction of displacement, unless the latter
coincides with that of one or other of the principal axes of the
section of the ellipsoid of elasticity by the plane lx -f- my + nz = 0.
For the direction cosines of the force are proportional to a-\, 6 2 //
C 2 v ; and the condition that this line, the direction of the displace
ment, and the normal to the wave-front should lie in the same
plane, is
Z, 77i, n
= 0,
or (fr-tf) + -(c*-a*) + - v (tf-b*) = () (4)
which is the condition, that the line X, //,, v should be a principal
axis of the section of the ellipsoid of elasticity by the wave-front.
VELOCITY OF PROPAGATION. 115
Let us now suppose, that plane waves of polarized light are
incident normally upon a crystalline plate, the direction cosines of
whose face, with respect to the principal axes, are I, m, n. Let
OA, OB be the directions of the axes of the section of the ellipsoid
of elasticity made by the surface of the plate, and OP the direction
of vibration of the incident light. If the second medium were
isotropic instead of crystalline, a single refracted wave would be
propagated, consisting of light polarized in a plane perpendicular
to OP and the surface of the plate ; but if the second medium is a
crystal, a single wave whose vibrations are parallel to OP is
incapable of being propagated, and it is necessary to suppose that
the incident vibrations are resolved into two sets of vibrations,
which are respectively parallel to OA, OB. These two sets of
vibrations are propagated through the crystal with different
velocities (unless the normal to the surface of the plate is parallel
to one of the optic axes), and thus give rise to two waves of
polarized light, whose planes of polarization are at right angles to
one another.
104. If q be the displacement of a particle of ether in either
of the waves, the' equation of motion of that particle will be
and therefore if r be the time of oscillation, I ^p- +*] ~ ^ (d*
2-7T/T = (a 2 X 2 + b 2 /j? +cV)-* = 27rv/X' , fij c ^^^
where v is the velocity of propagation, and X' is the wave length. <*&=-* ^
Hence if we write 2 ( 7ra/X / &c. for a, 6, c, where a, b, c now denoted
the three principal wave velocities, we obtain
From (5) it appears, that the force of restitution a 2 X, fr 2 //,, c 2 v
corresponding to a displacement unity, is equal to a force v 2 along
the direction of displacement, together with some force P along
I, ra, n, the normal ah^ the wave-front ; whence resolving parallel
to the axes, we obtain
IP = (a 2 - v 2 ) X, mP = (b 2 - v 2 ) p, nP = (c 2 -
accordingly since l\ + mjj, + nv = Q, A ~ ''
it follows that *- h
116 FRESNEL'S THEORY OF DOUBLE REFRACTION.
105. Before proceeding to discuss Fresnel's wave-surface, it
will be convenient to consider some preliminary propositions.
We have shown that when polarized light is incident normally
upon a crystal, the incident vibration must be conceived to be ji
resolved into two components, which are parallel to the principal
axes of that section of the ellipsoid of elasticity, which is parallel
to the surface of the crystal ; and that these two sets of vibrations
give rise to two waves within the crystal. Now if the surface
of the crystal is parallel to either of the circular sections of the
ellipsoid of elasticity, every direction will be a principal axis, and
therefore the component force parallel to the wave-front will be in
the direction of displacement ; hence only one wave will be pro-
pagated through the crystal. These two directions are the optic I
axes of the crystal, and therefore the optic axes are perpendicular jl
to the two planes of circular section of the ellipsoid of elasticity.
106. We can now prove the following propositions :
The planes of polarization of the two waves corresponding to the '
same wave-front, bisect the angles between the two planes passing J
through the normal to the wave-front and the optic axes.
Let BAB' be the section of the ellipsoid of elasticity by the
wave-front, ON the wave normal, and OS the intersection of one
of the planes of circular section with the wave-front. The optic
axis corresponding to the circular section through OS is perpeii- I
dicular to OS, and therefore the plane through it and ON cuts the I
plane BAB' in a line OQ, which is perpendicular to OS. Similarly,
if OS' be the intersection of the other plane of circular section
with BAB', and OQ be the projection of the other optic axis, OQ'
VELOCITIES OF EACH WAVE. 117
is perpendicular to OS'. Since 08= OS', the angle SOA = S'OA,
and therefore the angle QOA = Q'OA ; hence the planes of polari-
zation AON and BON bisect the angles between the planes QON
} and Q'ON, which are the planes containing the normal to the
\ wave-front and the optic axes.
107. The difference between the squares of the velocities corre-
sponding to the same wave-front, is proportional to the product of
the sines of the angles, which the normal to the wave-front makes
with the optic axes.
Let OP, OP' be the optic axes ; 6, 6' the angles which they
make with ON] also let X, p, v be the direction cosines of OA.
The equation of the two planes of circular section are a
~
, A1D -
and therefore cos AP = -
(
, . _, X(a a -6 3
and COB4P
Since the optic axis OP lies in the plane QON,
cos A P = cos A Q sin 6,
n X (a 2 - 6 2 )^ + v (6 2 -
whence cos ^ sin (9 =
Similarly, since AQ = AQ'
cos AQ sin 0' =
and therefore
(a 2 - c 2 ) cos 2 AQ sin 6 sin 0' = X 2 (a 2 - 6 2 ) - v 2 (6 2 - c 2 ),
Similarly if v' be the velocity of the other wave,
- (a 2 - c 2 ) sin 2 A Q sin sin 0' = v' 2 - 6 2 ,
whence v 2 - v' 2 = (a 2 - c 2 ) sin sin 0' (7).
108. Another somewhat similar formula may be obtained
as follows.
We have cos = cos PON =
- -
(a? c 2 )*
and therefore
(a 2 - c 2 ) cos 6 cos & = P (a? - 6 2 ) - ri> (6 2 - c 3 ) (8).
*-M
118 FRESNEL'S THEORY OF DOUBLE REFRACTION.
c^ 6'.= , ^v~ ^ jX*A -* .-~*-* .^^V*^^ *^ *
Now v and w' are the two roots of (6), whence
) + m 2 (c 3 + a 2 ) + n~ (a 2 + 6 2 ),
^*c ^, =a 2 +c 2 -(a 2 -c 2 )cos(9cos6l / ............... (9),
by (8), and therefore from (7) and (9) we obtain
cos 2
b > c, is shewn in the figure.
111. Since the wave-surface is symmetrical with respect to
the coordinate planes, it appears that it consists of an outer and
an inner sheet, which intersect at four points in the plane xz.
These four points are singular points, and it will hereafter be
shown, that there is a tangent and normal cone at each of them.
If QR be the common tangent in the plane xz, to the ellipse
and circle, and if xOQ = 6, it follows that
= 6 2 = c 2 cos 2 + a? sin 2 0,
,
Hence OQ is perpendicular to one of the planes of circular
section of the ellipsoid of elasticity, and is therefore one of the
optic axes. The other optic axis lies in the plane of xz, and makes
an angle TT 6 with the axis of x.
+^
DIRECTION OF VIBRATION. ^ f - 121 c ^ V
The line OP is called the ray axis, and its equation isc^r^^ ^ r _
ax (6 2 - c 2 )* = c* (a 2 - 6 2 )* ; ^ 7 '- ^ ^J " *
^'
the ray axes are therefore perpendicular to the circular sections of
c< ** - > M
the reciprocal ellipsoid
112. We shall now prove, that the direction of vibration in
any wave may be determined by the following simple construction.
Draw a tangent plane to the wave-surface parallel to the wave-
front, touching the surface in P ; then if Y be the foot of the
perpendicular from the centre of the wave-surface on to this tangent
plane, PY is the direction of displacement; in other words, the
direction of vibration coincides with the projection of the ray on
the wave- front. *=
*&**-<
We have incidentally proved in 104 that ^^C
(v* - a 2 ) \/l = O 2 - 6 2 ) fju/m = O 2 -c 2 )v/n ......... (23).
Combining these equations with (19) of 109, we see that
(r 2 - a 2 ) \/x = (r 2 - 6 2 ) p/y = (r 2 - c^vjz = k (say). . .(24),
where x, y, z are the coordinates of P.
Since the equation of the tangent plane at P is
Ix + my + nz = v,
it follows that if L, M, N are the direction cosines of PY, then
(x - lv)/L = (y- mv)/M =(z- nv)/N.
But by (19)
r 2 a 2 k ^
whence \/L = fi/M = v/N.
113. The ray and the direction of the resultant force are at
right angles to one another.
For the direction cosines of the ray are proportional to x, y, z ;
and those of the resultant force to a 2 \, b-p, c' 2 v ; and
= k
114. The tangent planes to the wave-surface at the extremities
of the optic axes touch the wave-surface along a circle.
122 FRESNEL'S THEORY OF DOUBLE REFRACTION.
At the extremity of the optic axis OQ, v = b, m = ; and
therefore by (19)
The values of I and n are given by (22), whence by substitution
these equations become
b (r 2 - a 2 ) + x (a 2 - V$ (a? - c 2 )* = 0,
b (r 3 - c 2 ) - z (a 2 - c 2 )* (6 2 - c 2 )* = 0.
These equations are satisfied by the coordinates of the points
of contact of the tangent plane at the extremity of the optic axis
with the wave surface, and since they represent two spheres, it
follows that this tangent plane touches the wave along a circle.
The diameter of this circle is equal to QR (see fig. 110). To
find its value, let OD be that diameter of the elliptic section OCA,
which is conjugate to OR. Then
OJ).OQ = ac,
or OD = ac/b.
Also OjR 2 +OI> 2 = a 2 +c 2 ,
whence Q&=OR>-OQ 2
= a- + c 2 - a 2 c 2 /6 2 - 6 2 ,
115. To find the equations of the tangent and normal cones at
the singular points.
The coordinates of P (see fig. 110) are,
i^V^.%*^. x = c ^ 2 _ ^/(tf - c 2 ) 1 , s = a (6 - - c 2 )*/(a 3 - c 2 )*.
f V yjp-*J Substituting in (19), we obtain
c kVS ^ _ ^ ^2 + ^ ( a a _ 52)1 ( a a _ C 2^ c _ a 2 = 0)
tf nV (^2 _ C 2) ^2 _ C 8)i/ a _ C 2 _ Q
Now ^, m, w are the direction cosines of any normal through P;
whence eliminating v, we obtain
^ ,
whence the equation of the normal cone, referred to P as origin is
c ^/ ac
(25).
TANGENT CONE. 123
Let X, /j,, v be the direction cosines of any generator of the
tangent cone ; then since this generator is parallel to the normal
at some point of the normal cone, it follows that if F \x, y, z) be
the equation of the normal cone,
dFfdat dF/dy dF/dz'
and therefore since I, m, n are proportional to x, y, z in (25), we
obtain
_ X _ _ _ _
21 (6 2 - c 2 ) - n (a 2 + c 2 ) (a 2 - 6 2 )* (6 2 - c^/ac 2m (a 2 - c 2 )
v
~ 2n (a 2 - 6 2 ) - I (a 2 + c 2 ) (oT
and therefore
^ _ 2Xa 2 c 2 (a 2 - 6 2 )* + me (a 2 + c 2 ) (6 2 - c 2 )*
2m ~ 2 (a 2 - c 2 ) (6 2 - c 2 ) (a 2 - &*)*
= 2ra 2 c 2 (6 2 - c 2 )^ + Xac (a 2 + c 2 ) (a 2 - 6 2 )*
7i (a 2 - c 2 ) (a 2 - 6 2 ) (6 2 - c 2 )*
But ZX + m/t + nv 0,
whence
_
6 2 - c 2 4a 2 c 2 a 2 - 6 2 ac (a 2 - & 2 )* (6 2 -
and therefore the equation of the tangent cone is
- 2
a c^ _
a 2 - 6 2 ac (a 2 - 6 2 )* (6 2 - c 2 )^
116. There is a third cone which is also of importance, viz.
the cone whose vertex is the origin, and whose generators pass
through the circle of contact of the tangent plane at the extremity
of the optic axes.
We have shown in 114, that the circle of contact is the
curve of intersection of the two spheres
c 2 ) = ............ (27),
c 2 )^ = ............ (28).
Hence if X, fju, v be any generator of the required cone,
r 2 -a 2 _ __ X(a 2 -fry -
r 2 - c 2 ~ v (b 2 - c 2 )* '
therefore r 2 =
124 FRESNEL'S THEORY OF DOUBLE REFRACTION.
Also
~
whence eliminating r, we obtain
c 2 (a 2 - b 2 ) X 2 + a 2 (b 2 - c 2 ) v 2 + (a 2 + c 2 )(a 2 - 6 2 )* (b 2 - c^\v =(a 2 -
and therefore the equation of the cone is
a 2 (b 2 - c 2 ) # 2 + b 2 (a? - c 2 ) # 2 + c 2 (a 2 - 6 2 ) z 2
- (a 2 + c 2 ) (a 2 - 6 2 )* (6 2 - c 2 )* xz = ......... (29).
Uniaxal Crystals.
117. If in equation (21) we put 6 = c, it becomes
(r 2 - c 2 ) {a 2 ^ 2 + c 2 (i/ 2 -I- z 2 ) - a 2 c 2 } = 0,
which is the form of the wave-surface for a uniaxal crystal. Hence
the wave-surface consists of the sphere
x 2 + f + z 2 = c 2 ,
and the ellipsoid a?a? + c 2 (y 2 -f -s 2 ) = a 2 c 2 ,
the axis of x being the axis of revolution.
Also from (22) we see that when 6 = c, 6 ; whence the two
optic axes coincide with the axis of x, which is therefore the axis
of the crystal.
The ellipsoid is ovary or planetary, according as c > or < a.
In the former case the crystal is positive, and in the latter case
negative.
If a pencil of light be incident upon a uniaxal crystal, the ray
corresponding to the spherical sheet of the wave-surface, will
coincide with the wave normal, and refraction will take place
according to the ordinary law discovered by Snell. Also if X, //,, v
be the direction cosines of the direction of vibration, we obtain
from (5),
c 2 = aTK 1 + c 2 (p 2 -f v 2 )
= a 2 X 2 + c 2 (1 - X 2 ) ;
whence X = 0, which shows that the direction of vibration is
perpendicular to the plane containing the normal to the wave-
front and the optic axis.
The extraordinary ray is in the direction of the radius vector
of the ellipsoidal sheet of the wave-surface, drawn to the point of
contact of the tangent plane, which is perpendicular to the wave
CONICAL REFRACTION.
125
normal; and by 112 the direction of vibration is the projection
ot' the ray on the wave-front. Hence the direction of vibration in
the extraordinary wave, lies in the plane containing the optic axis
and the extraordinary wave-normal, and is perpendicular to the
latter.
We have thus established the laws of the propagation of light
in uniaxal crystals, which were discovered experimentally by
Huygens.
Conical Refraction.
118. The existence of the tangent cone at the extremity of
the ray axis was first demonstrated by Sir W. Hamilton, and this
led to the discovery of two remarkable phenomena, known as
external and internal conical refraction.
119. In order to explain external conical refraction, let us
suppose that a small pencil of light is incident upon a plate of
biaxal crystal, cut perpendicularly to the line bisecting the acute
angle between the optic axes; and let the angle of incidence be
such, that the direction of. the refracted ray within the crystal
coincides with the ray axis.
Let 10 be the ray axis within the crystal, / being the point of
incidence, and the point of exit. At draw the wave-surface
for the crystal, and also the equivalent sphere in air. Produce 10
to meet the crystalline wave-surface in P; then OP will be the
ray axis. To obtain the directions of the refracted rays, draw
tangent planes at P. These tangent planes will meet the face of
the crystal in a series of straight lines T lf T 2 ... ; through each of
these straight lines T lt T z ... draw a tangent plane to the sphere,
126 FRESNEL'S THEORY OF DOUBLE REFRACTION.
and draw OP 1} OP 2 ... joining the points of contact with 0. The
points of contact of the infinite number of tangent planes to the
sphere will lie on a certain spherical curve, and therefore the
refracted rays on emerging from the crystal, will form a conical
pencil whose vertex is 0, and whose generators are the lines
OP,, OP,...,
120. In order to explain internal conical refraction, we must
suppose that the angle of incidence is such, that the direction of
the refracted wave coincides with the optic axis. Since the
tangent plane at the extremity of the optic axis touches the wave-
surface along a circle, the refracted pencil within the crystal, will
consist of a cone of rays, whose vertex is the point of incidence,
and all of whose generators pass through the above-mentioned
circle. The equation of this cone is given by (29). On emerging
from the crystal, each emergent ray will be parallel to the incident
ray, and will form an emergent cylinder of rays.
121. The phenomena of external and internal conical refrac-
tion had never been observed nor even suspected, previously to
the theoretical investigations of Sir W. Hamilton on the geometry
of the singular points of the wave-surface ; and at his suggestion,
Dr Humphrey Lloyd 1 examined the subject experimentally, and
found that both kinds of conical refraction actually existed.
122. The investigations of Sir W. Hamilton, coupled with the
experiments of Dr Lloyd, are undoubtedly a striking confirmation |
of the accuracy of Fresnel's wave-surface ; but it has been subse- '
quently pointed out by Sir G-. Stokes 2 , that almost any theory
which could be constructed, would lead to a wave surface having
conical points, and would therefore account for the phenomenon of
conical refraction. Also a series of very elaborate experiments by
Glazebrook 3 upon uniaxal and biaxal crystals, have shown that
Fresnel's wave-surface does not quite accurately represent the true i
form of the wave-surface in such crystals, but is only a very close i
approximation.
123. The dynamical objections to Fresnel's theory may be
classed under three heads.
1 Trans. Roy. Ir. Acad. vol. xvii. p. 145.
2 Brit. Assoc. Rep. 1862.
3 Phil. Trans. 1879, p. 287 ; 1880, p. 421.
CRITICISMS ON FRESNEL'S THEORY. 127
(i) It is assumed that the potential energy of an elastic
medium, which is displaced from its position of equilibrium, is a
quadratic function of the component displacements ; whereas it
will be shown in a subsequent chapter, that the potential energy
of an elastic medium, which is symmetrical with respect to three
rectangular planes, is a certain quadratic function which involves
the space variations of the displacements, and not the displace-
ments themselves.
(ii) The component of the force of restitution, perpendicular
to the direction of propagation of the wave, is altogether neglected.
And although attempts may be made to justify this by arguing,
that the effect of this force, whatever it may be, cannot give rise
to vibrations which affect the eye, yet the argument is fallacious ;
inasmuch as if such forces existed, they would produce waves of
longitudinal vibrations, which would give rise to transversal vibra-
tions, when light passes from a crystalline medium into another
medium, and thus the sensation of light would be produced by
something which is not light.
(iii) It is not a legitimate way of dealing with the motion of
an elastic medium, to treat a wave as if it were composed of a
number of distinct particles, each of which is acted upon by a
force depending on its displacement. The rigorous theory of
seolotropic elastic media is due to Green, and will be considered in
a subsequent chapter ; but although this theory is rigorous as far
as its dynamics are concerned, it does not offer a satisfactory
explanation of double refraction.
On the Methods of producing Polarized Light.
124. When light falls upon a plate of Iceland spar, it is
divided into two rays within the crystal, which are polarized in
perpendicular planes, and on emerging from the plate, two streams
of plane polarized light are obtained, which are parallel to the
incident rays ; but unless the thickness of the plate is considerable,
these two streams overlap. Since the velocities of the two streams
within the crystal are unequal, their phases on emergence are
different, and consequently the emergent beam is elliptically
polarized, unless the difference of phase amounts to a quarter
of a wave-length, in which case it is circularly polarized.
128 FRESNEL'S THEORY OF DOUBLE REFRACTION.
125. A very convenient method of producing plane polarized
light consists in passing common light through a Nicol's prism, so
called after the name of its inventor, the construction of which we
shall proceed to explain.
There is a transparent substance called Canada balsam, whose
index of refraction is intermediate between the ordinary and
extraordinary indices of refraction of Iceland spar. If therefore
two rhombs of Iceland spar are cemented together with this
substance, it is possible for the ordinary ray to be totally reflected
at the surface of the balsam, so that the extraordinary ray is alone
transmitted.
5
Let AC be the optic axis, ACOF, AC EH the spherical and
spheroidal sheets of the wave-surface; and let the plane of the<;
paper be the plane of incidence, which is supposed to contain the is
optic axis. Let AO, AE be the ordinary and extraordinary rays,
corresponding to a ray incident at A.
Let ABG be the wave-surface of the balsam; then since theji
index of refraction of the latter is intermediate between the I
ordinary and extraordinary indices of the spar, ABG will be a
sphere, whose radius is intermediate between the polar and !
equatorial axes of the spheroid.
In order to obtain the directions within the balsam of the ray< '
corresponding to the ordinary ray, draw a tangent at meeting
the face of the spar in T, and from T draw a tangent to ABG,
and join the point of contact with A ; if however T lies between
F and G, it will be impossible to draw this tangent, and thej
ordinary ray will be totally reflected.
NICOLS PRISM.
129
To obtain the directions within the balsam of the extraordinary
ray, draw a tangent at E meeting the face of the crystal in S, from
$ draw a tangent to ABG meeting it in P, and join A P. If S lie
beyond Q y it will be possible to draw this tangent, and AP will be
the ray corresponding to the extraordinary ray within the balsam.
If this ray is not totally reflected at the second rhomb, it will be
transmitted, and the emergent beam will be plane polarized.
126. In order to construct a Nicol's prism, a rhomb of Iceland
spar is taken, whose length is about double its thickness, and is
cut in two by a plane PE, and the two parts are then cemented
together with Canada balsam. The plane A BCD contains the
optic axis, and is therefore a principal plane; and the plane of
section is inclined to EG at such an angle, that when a ray is
incident at / parallel to BC, the ordinary ray is totally reflected
by the balsam. The extraordinary ray IE is therefore alone
transmitted, and emerges at M parallel to its original direction.
The vibrations of the emergent light accordingly lie in the
plane A BCD, which is called the principal section of the Nicol.
127. A second method of producing plane polarized light is
by means of a plate of tourmaline. Tourmaline is a negative
uniaxal crystal, which possesses the property of absorbing the
ordinary ray, even when the thickness of the crystal is small. If
therefore we take a plate of tourmaline cut parallel to the axis,
B. O. 9
130 FRESNEL'S THEORY OF DOUBLE REFRACTION.
and pass common light through it, the emergent light will be
completely polarized perpendicularly to the principal section of
the plate.
128. A third method consists in using a pile of plates. When
common light is incident upon a plate of glass at an angle equal to
tan" 1 /*, where yu, is the index of refraction, it appears both from
theory and experiment, that the reflected light is nearly, but not
entirely, polarized in the plane of incidence ; and by employing a
pile of plates so as to cause the light to undergo successive
reflections, the component vibrations in the plane of incidence may
be entirely got rid of, and the resulting light becomes plane
polarized.
EXAMPLES.
1. In a biaxal crystal, prove that the cosine of the angle
between the ray axis and the optic axis is
ac + 6 2
b(a+c)'
2. In a biaxal crystal, prove that if v be the velocity of wave
propagation, a, 6, c the principal wave velocities in descending
order of magnitude, A|T, ijr' the angles which the direction of
vibration makes with the two optic axes,
fl 2 = J 2 (a 2 C 2 ) COS ijr COS ^Jr'.
X
3. Prove that the velocity of propagation of the wave in a
biaxal crystal may be expressed in the form
e (a 2 6 2 c 2 -
V/i
(0 - a 2 ) (0 - 6 2 ) (0 - c 2 ) + a 2 6 2 c 2 - <9 '
where a, b, c are the principal wave velocities, and
= #2 _j_ yl _j_ ^ ^ _ a 2,p2 _j. 2^2 ^. C 2^2^
a;, 2/, z being the components of the ray velocity parallel to the
axes of the crystal.
4. Light falls normally through a very small hole on a plate of >
biaxal crystal, of which the parallel faces are perpendicular to one
of the circular sections of the surface of elasticity ; show that if t be ; :j
the thickness of the plate, and the semi-axes of the surface of ;
EXAMPLES. 131
elasticity are proportional to \, 1, X' respectively, the area of
the transverse section of the emergent cylinder of rays will be
5. If v lt v 2 be the velocities of -propagation through a biaxal ; ^
crystal, of the two waves corresponding to a plane wave-front, whose
direction cosines are l,m,n\ prove that
_
~
(a 2 - 6 2 ) (a 2 - c 2 ) ' (6 2 - c 2 ) (6 2 - a 2 ) ' ~ (c 2 - a 2 )(c 2 - 6 2 ) '
6. If one of the directions of vibration in a plane wave inside
a biaxal crystal, make angles a, /?, with the two optic axes, and the
other make angles 7, & ; prove that
cos a cos S + cos @ cos 7 = 0.
7. If a ray be incident on the face of a biaxal crystal in
a plane passing through one of the optic axes, prove that the
directions of vibration within the crystal will be either perpen-
dicular to this axis, or will lie on the surface of a cone of the
second degree.
8. A prism of angle i is cut from a biaxal crystal, whose
principal wave velocities a, -b, c are known. Prove that the
position of either face of the prism relatively to the principal
axes of elasticity of the crystal, may be ascertained thus ; let a
pencil of rays be incident normally on the face, and measure
the deviations 1? 2 of the two rays emergent from the prism,
then will
/
1* .,
' *
where X, p, v are the direction cosines of the face referred to the
iprincipal axes.
9. If a biaxal crystal be cut in the form of a right-angled
jprism, two of whose faces are principal planes, find how a ray must
DC incident at one face, so that the extraordinary ray may emerge
it right angles to the other face. Show also that the minimum
leviation of the extraordinary ray is
Adhere a, 6 are the principal wave velocities in the plane of incidence,
jand u is the wave velocity in the medium surrounding the crystal.
92
132 FRESNEL'S THEORY or DOUBLE REFRACTION.
k 10. A prism is formed of biaxal crystal, the edge being parallel
to a principal axis of the crystal. Show that no extraordinary ray
refracted in a plane perpendicular to the edge of the prism will get
through, if the angle of the prism exceed
2 sin" 1 a/u,
where u is the velocity of light in air, and a the greatest wave-
velocity for rays refracted perpendicularly to the edge of the
prism.
11. A system of extraordinary wave normals in a uniaxal
crystal, lies on the surface of a cone of semi-vertical angle yS, whose
axis makes an angle a with the optic axis. Show th?.t if the
optic axis be the axis of a, and the axis of the cone lie in the plane
xy, the planes of polarization will be normal to the cone
(a 2 + y* + z z ) (y* + z*) cos 2 /3 = {(y* + z*) cos a - xy sin a) 2 .
12. A plate of biaxal crystal is cut parallel to the line bisect- \
ing the acute angle between the optic axes, and a ray of light is j
incident on the plate perpendicularly to its surface. Find the. I
inclination of the face of the crystal to the plane containing the
optic axes, in order that the angle between the two rays within the j
crystal may be a maximum, and prove that the cosine of the angle
is then equal to
13. Show that the locus of the feet of the perpendiculars, let I
fall from the centre of an ellipsoid upon all chords which subtend I
a right angle at the centre, is the solid space between the two I
sheets of a wave surface.
14. The measure of curvature at the extremities of the optic
axes is
15. Prove that the planes through a radius vector of the
wave surface, and the corresponding directions of vibration, bisect
the angles contained by the planes through the same radius vector,
and the two axes of external conical refraction.
16. A prism whose refracting angle is JTT, is cut from a biaxal
crystal with its edge parallel to the axis a. The two rays corre-
EXAMPLES. 133
spending to a ray incident perpendicularly on one of the faces of i
the prism, emerge with deviations S 1} S 2 ; and those perpendicular
to the other face, with deviations B 1} S 3 . Prove that
a (sin B 1 + cos 8j) = 1
- 2(sinS 2 +cos 8 2 )~ 2 } 2 + {6 2 + c 2 -2(sin S 3 + cosS 3 )- 2 } 2
17. If the wave surface be cut by the plane Ix + my + nz = 0,
prove that the radius vector whose equations are
a = y_
- a 2 ) + n?b 2 (c 2 - a 2 )) m {^ 2 a 2 ( c 2 - 6 2 ) + Z 2 c 2 (a 2 - 6 2 )}
z
~ n } W (a 2 - c 2 ) + ra 2 a 2 (6 2 - c 2 )}
will be a maximum or minimum ; and that its length is equal to
18. A thin lens is cut from a uniaxal crystal, the axis of the
lens coinciding with that of the crystal. A pencil diverging from
a luminous point on the axis is refracted directly through it.
Show that after the first refraction, the ordinary and extraordinary
rays converge to a focus, and that the position of this focus for the
extraordinary rays can be found in the same way as for the ordi-
nary rays, by supposing that the curvature of the surface and the
index of refraction, altered by quantities depending solely on the
crystal.
Show that after emerging from the lens, the foci of both sets of
rays coincide.
19. Show that when a line of light is placed before a plate of
biaxal crystal, parallel to the plane containing the optic axes, and
the emergent pencil is observed through a small hole, the luminous
line will be seen in the form of a conchoid.
20. A small pencil of light is incident on a plate of biaxal
crystal, so as to be internally conically refracted. If the ring be a
circle, when the screen is placed parallel to the surface of the plate,
prove that the crystal has been cut by a plane, whose inclination to
the axis of least elasticity is either
r tan
134 FRESNEL'S THEORY or DOUBLE REFRACTION.
21. If 0, & be the angles between the optic axes and either
of the ray axes, prove that
1 ni ab b c
tan i
where ' are the angles between the surface of the crystal, j
and the wave fronts outside and inside the crystal, and
If there be only one position of the screen in which the ring is
a circle, prove that the diameter of the ring will be
'
where T is the thickness of the plate.
\ 23. In a prism of uniaxal crystal of angle JTT, the axis is
perpendicular to the edge, and bisects the angle between the faces.
Show that such a prism may be used, like a Nicol's prism, to
extinguish one ray, and obtain formula for the range of incidence,
within which one and only one ray will emerge.
If the reciprocals of the squares of the (oblate) spheroid of the
wave surface be 1*8 and 1-4, show that the range is approximately
sin- 1 (-894) -sin- 1 (-644).
24. In Fresnel's theory, the planes of polarization of all rays
proceeding in the direction of the circular ridge of the wave
surface, but belonging to the inner sheet, pass through a straight
line.
EXAMPLES. 135
25. A plate of biaxal crystal is cut, so that the normal to the
surface makes angles whose cosines are X, p, v with the principal
axes of the crystal. Show that in order to produce the pheno-
menon of internal conical refraction, the sine of the angle of
incidence must be proportional to
F [{X (6 2 - c 2 )* - v (a 2 - 6 2 )*} 2 + p? (a 2 - c 2 )]*
6(a 2 -c 2 )*
where V is the velocity of light in air.
26. If the boundary planes of a plate of biaxal crystal are
perpendicular to the axis c, show that in the case of internal
conical refraction, the area of a cross section normal to the
generators of the emergent cylinder is
where d is the radius of the equivalent sphere in air, and T the
thickness of the plate.
27. If 0, & are the angles between the direction of a ray and
the two ray axes, and v, v' are the two ray velocities corresponding
to this direction, show that
1 __ j. _ a 2 -c 2 . . &
v* v' z ~ a?c 2 l
where a, c are the greatest and least optical constants.
28. If the two faces of a prism formed of a biaxal crystal be
perpendicular to one another, and one contain the two axes of
elasticity a, c and the other 6, c; and if /j, a , ^ be the two refractive
indices for the ordinary ray, when the planes of refraction are
perpendicular to the axes a and b respectively; show that 8, the
minimum deviation of the extraordinary ray is given by
29. If A, JJL, v be the direction cosines of one of the two lines
of vibration of the plane front of a wave in a biaxal crystal, and
X', fi', v those of either of the two lines of vibration of a plane
front, intersecting the former plane front at right angles, and
passing through the line X, fi, v, prove that
^ (ft* - c 2 ) + ^ (c 2 - a 2 ) + V - (a 2 - b 2 ) = 0,
(c- - a 2 )-
136 FRESNEL'S THEORY OF DOUBLE REFRACTION.
30. A plate of uniaxal crystal is bounded by planes inclined
at a small angle a to the axis, and a pencil of rays is incident in
the direction of the axis ; prove that the difference of retardation is
where n and /// are the principal refractive indices for the ordinary
and extraordinary rays, and T the thickness of the plate.
CHAPTER VIII.
COLOURS OF CRYSTALLINE PLATES.
129. IN the present chapter, we shall discuss one of the most
striking and beautiful phenomena in the whole science of Optics,
viz. the production of coloured rings by thin crystalline plates.
These rings were discovered by Arago 1 in 1811, and we shall first
give a general explanation of their formation.
When plane polarized light is incident upon a crystalline plate,
the incident vibrations, upon entering the plate, are resolved into
two components, which are polarized in perpendicular planes, and
travel through the plate with different velocities; hence the
phases of the two components upon emergence are different. If
the angle of incidence is small and the crystal is thin, the two
emergent rays are sensibly superposed ; but since they are polar-
ized in different planes, they are not in a condition to interfere.
If however the emergent rays are passed through a Nicol's prism,
each ray on entering the prism is again resolved into two
components, which are respectively parallel and perpendicular to
the principal section of the Nicol; the two latter components
cannot get through the Nicol, whilst the two former components
being brought into the same plane of polarization by the Nicol, and
being already through the action of the crystalline plate in different
phases, are in a condition to interfere. We thus perceive, how it
is that coloured rings are produced by the action of a thin crystal-
line plate.
The apparatus (frequently a Nicol's prism), which is used
to polarize the light which falls upon the crystal, is called the
1 (Euvres Completes, vol. x. p. 36.
138 COLOURS OF CRYSTALLINE PLATES.
polarizer, and the second Nicol is called the analyser. The planes of
polarization of the light, which emerges from the polarizer and
the analyser, are respectively called the planes of polarization and
analysation.
130. We are now prepared to consider the mathematical
theory of these .rings.
Let OA, OB be the principal planes of the crystal at any point
0, the former of which corresponds to the ordinary ray, and the
latter to the extraordinary ray. Let OP be the direction of
vibration of the incident light, so that if the light is polarized
by a Nicol, OP is its principal section ; and let OS be the principal
section of the analyser. Let PO A = a, POS /3 ; also let the
vibration which is incident upon the crystal be represented by
sin 27r/T.
B
On entering the crystal, the incident vibration is resolved
into
cos a sin 2?r^/T
along OA, which constitutes the extraordinary ray, and
sin a sin %7rt/r
along OB, which constitutes the ordinary ray. The waves corre-
sponding to these rays travel through the crystal with different
velocities, and therefore on emergence, the two vibrations may be
written ^oA ~->$3 W^.- fi.6 ***,,,
cos a sin 2?r (t/r Ej\) t and sin a sin 2?r (t/r 0/\)
where X is the wave length in air, and and E are the thicknesses
of two laminae of air, such that light would occupy the same times
in traversing them, as are occupied by the ordinary and extra-
ordinary waves in traversing the crystalline plate.
On entering the analyser, only those vibrations can pass through
which are parallel to OS] whence resolving the two vibrations in
INTENSITY OF EMERGENT LIGHT.
this direction and putting = t/r 0/\, the resultant vibration on
emerging from the analyser is represented by
cos a cos (a /3) sin 2?r { + (0 E)/\] + sin a sin (a j3) sin 27r<.
The intensity of the emergent light is therefore represented by
I 2 = {cos a cos (a - /3) cos 2?r (0 - E)/\ + sin a sin (a - $} 2
+ cos 2 a cos 2 (a - ft) sin 2 2?r (0 - E)/\ * ^
= cos 2 ft- sin 2asin2(a-/3)sin 2 7r(0-jE r )/X
If in this expression we write ^TT + ft for /3, which amoul
turning the analyser through an angle of 90, we obtain
/ /a = sin 2 ft + sin 2a sin 2 (a -ft) sin 2 TT (0 - #)/\. . . (2),
whence / 3 + / /2 =l.
We therefore see, that the effect of turning the analyser
through an angle of 90, is to transform each colour into ii
complementary one.
131. Let us now suppose, that the incident light consists
small pencil of rays converging to a focus, every fay of whjfch*
makes a small angle with the normal to the crystalline plate.
Let P be the point of incidence, i the angle of incidence; PQ 5 5
the front of the incident wave. At the end of time t, let Til TO, ^ '
v
\ s ^
TE be the fronts of the incident, ordinary and extraordinary waves \
and let v, u, 114 be their velocities of propagation. Draw PO, PE\ ^
perpendicular to TO, TE ; and let r, r' be the angles which these/ ^ '
perpendiculars make with the normal to the plate. Then ) ' I v
(3),
'hence
sim
v
sin T
u
sin r'
u
.(4).
140 COLOURS OF CRYSTALLINE PLATES.
If the thickness of the crystal and the angle of incidence are
small, the ordinary and extraordinary rays will be superposed on
emergence. Hence if OM be the wave-front on emergence corre-
spending to the incident wave PQ, the difference between the
times which the ordinary and extraordinary waves occupy in
travelling from P to OM, is equal to ' __
*>T *..Wy PO/u-PE/u'-EM/v, T2I1 ^J'_
ich is equal to (0 E)/v ; whence = JJL^^u
- E = (v/u) PO - (v/u) PE - EM. * "j~^_
- Accordingly if T be the thickness of the plate, ~ V^^
A Vv V "^ c-o'T/i, c&tm
0s/u=aU^ v^ -V^T" ^TTT^
"*"^ / ^Jl TV? / ^il Tl 7
-s . T ', - jT(tan r-tan r) sin i
sm r cos r sin r cos r
^^-^^
(5).
We thus see that the mathematical solution of the problem is
reduced to the determination of the angles r, r', in terms of the
angle of incidence and the optical constants of the crystal. Their
values depend upon the particular kind of crystal under considera-
tion, and the inclination of the face of the plate to the directions
of the principal wave velocities in the crystal.
Uniaocal Crystals.
132. We shall now consider the coloured rings produced, when
the crystalline plate consists of any uniaxal crystal, except crystals
of the class to which quartz belongs.
PLATE PERPENDICULAR TO THE AXIS.
141
Plate cut perpendicularly to the Axis.
133. Let the axis of x be perpendicular to the faces of the
plate ; then in the present case, the wave surface in the crystal
consists of the sphere
x 2 + f + z* = b 2
and the ellipsoid
aW + b 2 (f + z 2 ) = a 2 b 2 .
In negative crystals such as Iceland spar, the velocity of the
ordinary wave is less than that of the extraordinary, and therefore
a > b ; hence the ellipsoidal sheet of the wave-surface lies outside
the spherical sheet, and is planetary. The reverse is the case with
positive crystals, for which the ellipsoidal sheet is ovary.
In the figure, P is the point of incidence, PB the axis of the
crystal ; let PQ be the front of the incident wave, and let TO, TE
be the fronts of the ordinary and extraordinary waves at the end
of unit of time. Then if p be the length of the perpendicular
drawn from P on to TE,
a 2 sin 2 r' + b 2 cos 2 r' =p 2 = PT 2 sin 2 / = v 2 sin 2 r'/sin 2 i
by (3) and (4) ; therefore
cot r 1 = (v 2 a 2 sin 2 i)$/b SIIML
Similarly b = PO = PT sin r = v sin r/sin i,
whence cot r = (v 2 b 2 sin 2 )*/& sin i ; /^ ^7
accordingly we obtain from (5)
0-E = Tb~ l [(v 2 - b 2 sin 2 *)* - (> 2 - a 2 sin 2 t)*} (6)7
e of inciteiCeTFa small quantity ; if therefore
expand the right-hand side of (6), and neglect sin 4 1, we obtain
- E = i (T/fo;) (a 2 - 6 2 ) sin 2 i
r/i) j.
9 XI
142 COLOURS OF CRYSTALLINE PLATES.
134. Let us suppose, that the polarized light previously to
being refracted by the crystal, consists of a small conical pencil
proceeding from a focus, whose distance from the plate is d, and
let p be the distance of the points of incidence of any ray from the
projection of the focus on the plate ; then
approximately. Whence
^IXl/VV
The intensity of the light on emerging from the analyser
is given by (1) ; hence the equation of the isochromatic curves,
(or curves of equal intensity if monochromatic light be used), is
sin 2a sin 2 (a - ft) sin 2 ^
Pxi
u' 2 = a 2 cos 2 r' + b 2 cos 2 EPB + a 2 cos 2
but cos EPB = sin r' cos 4- a 2 sin 2 a
^ ^X,*^ ' , o*^*/
, Also by the second of (4)
*< u' = v sin r' cosec i,
V
i}*/a sin i \
whence cot r' = {?j 2 (a 2 sin 2 to + 6 2 cos 2 ) sin 2
an( ^ T/?i^ /
0E=T [b~* (v 2 - b 2 sin 2 1)* - or 1 fv 2 - (a 2 sin 2 w 4- b 2 cos 2 /to) sin 2 i}*].
Expanding and omitting sinVi, we obtain ^
- E = T (a b) {v 2 + 4& sin 2 i (a sin 2 o> - 6 cos 2 a>)}/abv.. .(9).
-, v ,-n the present case, the angles a and /3 are the same for all rays
incident upon the crystal ; whence if a = \nir, so that the axis is
either parallel or perpendicular to the plane of polarization of the
incident light, the intensity of the emergent light is constant and
equal to cos 2 /3; consequently if the planes of polarization and
analysation are parallel, so that ft = 0, the intensity of the field is
the same as that of the incident light ; but if these planes are
perpendicular, so that /3 = ^TT, the field is perfectly dark. A
precisely similar effect is produced when a-j3 = ^n7r, in which
case the axis is parallel or perpendicular to the plane of analysa-
tiou. It therefore follows that brushes, which are produced by the
dependance of the factor sin 2a sin 2 (a - /3) upon the position of
the incident rays, do not exist in this case.
The conditions most favourable for the production of rings are
when a = JTT, and fi has either of the values or JTT.
When a = JTT, /3 = 0, the intensity is equal to cos 2 TT (0 E)j\ ;
and therefore the dark rings are given by the equation
and the bright rings by
Now if #, y be a point on one of the rings referred to the
direction of the optic axis as axis of x, we have
tan to = yjx, tan i (a?
TWO PLATES SUPERPOSED. 145
whence to a sufficient approximation,
sin a) sin i = y/d, cos &> sin i = as/d t
and therefore the equation of the isochromatic curves is
and are therefore hyperbolas, whose asymptotes are the straight
lines
y (b fa/fa
Since the right-hand side of (10) does not in general vanish,
the asymptotes do not usually form part of the system of iso-
chromatic curves.
If a = JTT, /3 = JTT, the intensity is equal to sin 2 TT (0 E)j\ ;
whence the dark rings are given by the equation E = n\, and
are therefore complementary to the bright rings in the preceding
case.
136. When the axis is neither parallel nor perpendicular to
the surface of the plate, the calculation becomes more complicated.
The isochromatic curves are of the fourth degree, which ap-
proximate to circles when the axis is nearly perpendicular to the
plate, and to hyperbolas when the axis is nearly parallel to the
plate. For the mathematical investigation, the reader is referred
to Verdet's Lemons d'Optique Physique, vol. II. p. 161.
Two Plates Superposed.
137. We shall now suppose, that light passes through two
)lates cut parallel to the axis, which are of the same thickness and
are cut from the same piece of crystal, and that their principal
planes are at right angles.
In the figure to 130, let OA, OB be the principal planes of
the first and second plates respectively; then we have shown in
130, that on emergence from the first plate, the vibrations may
be represented by
cos a sin 2?r (t/r E/X)
along OA, and
sin a sin 2-rr (t/r - 0/\)
along OB.
B. O. 10
146 COLOURS OF CRYSTALLINE PLATES.
The first of these waves, which is the extraordinary wave in the
first plate, becomes the ordinary wave in the second plate ; whilst
the second becomes the extraordinary wave. Hence if 0', E'
denote the retardations produced by the second plate measured by
their equivalent paths in air, the vibrations on emergence will be
represented by
cos a sin STT [t/r - (0' + E)j\]
along OA, and
sin a sin 2?r {t/r - (0 + E')/\}
along OB.
Since the crystals are of the same thickness and of the same
material, we must have = 0'; whence if < = t/r (0 4- E)/\,
these become
cos a sin 27r, and sin a sin 2-jr }< + (E E')/\],
and therefore on emerging from the analyser, the resultant vibra-
tion is
cos a cos (a - /3) sin 27r + sin a sin (a - ft) sin 2?r {<]> + (E - E')l\] ;
accordingly the intensity is equal to
I 2 = cos 2 /3 - sin 2a sin 2 (a - /3) sin 2 TT (E - E')/\.
The value of the quantity E' is obtained from (9) by putting
\TT a for o>, whence
E - E' = JT(a 2 - 6 2 ) sin 2 * (cos 2 a> - sin 2 a>)/av.
The appearance presented on a screen can be discussed in this
case in the same manner as in the preceding. The most favourable
cases for the production of the rings are when a = JTT, and ft = or
JTT. In the first case the intensity is equal to cos 2 TT (E E')l\
whence the bright rings are given by the equation
E - E' = n\
or a? - f = 2n\avd 2 / (a 2 - 6 2 ) T
where n is zero or any positive or negative integer. The iso-
chromatic curves therefore consist of the two systems of rectangular
hyperbolas, which are included in the equation a? 2 y- = A; 2 ,
together with their asymptotes y=%.
-BIAXAL CRYSTAL^/* ^
y ft*--7-^*- i-^^V^ay
Biaxal Crysfats.r
138. We shall first consider the rings and brushes produced^
by a plate of biaxal crystal, such as nitre or aragonite, whose optic
axes make a small angle with one another, and which is cut
perpendicularly to the axis of least elasticity.
We shall first find the form of the brushes.
4
n r
+* f^'-t-t
l\
^
/
In the figure, let be the point of incidence for any ray ; let
OQ be either of the wave normals within the crystal corresponding ^
to this ray. Let the plane of the paper be any plane parallel to
the face of the crystal, and let A, B and C be the points where the - u'~ l (v 2 - it* sin 2 i)}
which is equal to
u u
Since i, and the difference between u and u' are small, we may
neglect the terms in sin 2 i t whence (5) becomes
---) (13).
u u J
1 Lemons d'Optique Physique, vol. u. p. 170; see also Berlin, Ann.de Chim.et de j
Phtjs. vol. LXIII. p. 57 (1861).
PLATE OF NITRE OR ARAGONITE. 149
Since the angle between the optic axes, and also the angle of
incidence are very small, the two rays, and also the two wave
normals corresponding to any incident ray, may be approximately
supposed to coincide with OQ ; hence in the formulae (10), of 108,
2u* = cf + c 2 - (a 2 - c 2 ) cos (d + 0'\
2u' 2 = a? + c 2 - (a 2 - c 2 ) cos (6 - 6'),
we may suppose that 6, 9' denote the angles QOA and QOB
respectively.
Since 0, & are small, we have
u'* = c 2 + J (a 2 - c 2 ) (6 - 0')\
11 (a 2 - c 2 ) 66'
whence --- / = ~s-V
u u 2c 3
(a*-*)QA.QB
2c 3 OA*
The isochromatic curves are determined by the condition that
0-E = n\\
and consequently by (13) and (14), they consist of a family of
lemniscates, whose foci are the extremities of the optic axes.
The form of the rings and brushes when a = ^7r, f3 = ^7r are
shown in figure 3 ; and when a = JTT, $ = JTT in figure 4, of the
plate at the end of Chapter IX.
It may be worth while to point out, that the rings and brushes
are both included in the equation
in which the curves f = const, are lemniscates, and y = const, are
rectangular hyperbolas.
140. When the optic axes form an angle which is very nearly
equal to 180, the crystal approximates to a uniaxal crystal, which
is cut parallel to the axis, and it may be anticipated that the rings
are hyperbolas. This we shall show to be the case.
The velocity of propagation corresponding to the wave-front
I, m, n is determined by the equation
72 nv)2 2
F^ + wb + Wb= (15) -
150
COLOURS OF CRYSTALLINE PLATES.
In the figure, let G> be the angle which the plane of incidence
corresponding to any ray makes with the plane containing the
optic axes ; i, r the angle of incidence and refraction. Then
I = cos o> sin r, ra = sin &> sin r, n = cos r ;
also V=v sin r cosec i.
Writing (15) in the form ,
V 4 - F 2 {P (6 2 + c 2 ) + m 2 (c 2 + a 2 ) + ri 2 (a? + 6 2 )}
+ We 2 + ra 2 c 2 a 2 4- n 2 a 2 6 2 = 0,
and then substituting the above values of I, m, n, V, we shall
finally obtain after reduction
a 2 Z> 2 cot 4 r sin 4 i + [{a 2 6 2 + c 2 (6 2 cos 2 a> + a 2 sin 2 a>)} sin 2 1
v 2 (a 2 + 6 2 )] cot 2 r sin 2 1 + c 2 sin 4 i (6 2 cos 2 a> + a 2 sin 2 w)
- v 2 sin 2 1 (6 2 cos 2 w + a 2 sin 2 w + c 2 ) + v 4 = (16).
This is a quadratic equation for determining the two values of
cot 2 r sin 2 i, corresponding to a given angle of incidence i.
Let h? = 6 2 cos 2 a> + a 2 sin 2 a>.
Then
2a 2 6 2 cot 2 r sin 2 i = tf (a 2 + 6 2 ) - (a 2 6 2 + c 2 ^ 2 ) sin 2 \ H . . . (1 7),
where
H* = {v 2 (a 2 + 6 2 ) - (a 2 6 2 + c 2 /t 2 ) sin 2 i} 2 - 4a 2 6 2 (c 2 /i 2 sin 4 *-^ 2 sin 2 i
= (a 2 6 2 - c 2 ^ 22 sin 4 - 2 a 2 sin 2 ia 2
= }6 2 (a 2 - c 2 ) cos 2 &> + a 2 (b 2 - c 2 ) sin 2 w) 2 sin 4 i
- 2(a 2 -6 2 ) { (a 2 -c 2 ) cos 2 w+a 2 (b 2 - c 2 ) sin 2 ft)} v 2 sin 2 i + ?; 4 (a 2 - 6 2 ) 2
+ 4aV (a 2 - 6 2 ) (6 2 - c 2 ) sin 2 co sin 2 i
= [> 2 (a 2 - 6 2 ) - {6 2 (a 2 - c 2 ) cos 2 &> -f a 2 (6 2 - c 2 ) sin 2 w} sin 2 i] 2
(a 2 - b 2 ) (b 2 - c 2 ) sin 2 w sin 2 i (18).
INCLINATION OF THE OPTIC AXES LARGE. 151
We have shown in 111^ that if 2 A be the angle between the
optic axes, cos A = (6 2 - c 2 )V( 2 c 2 )* ;
hence if A be nearly equal to 90, b c will be very small, and we
may therefore neglect the term (6 2 c 2 ) sin 2 i ; accordingly (18)
may be written H = v 2 (a 2 6 2 ) 6 2 (a 2 c 2 ) cos 2 co sin 2 i.
Now
2 6 2 + c 2 & 2 = 6 2 (a 2 + c 2 ) cos 2 &> + 2a 2 c 2 sin 2 co + a 2 (6 2 - c 2 ) sin 2 ;
the last term may be omitted when multiplied by sin 2 i t whence by
(17) we obtain
b 2 cot 2 n sin 2 i = v" - (6 2 cos 2 co + c 2 sin 2 co) sin 2 z,
2 6 2 cot 2 r 2 sin 2 * = v*~b" c 2 (6 2 cos 2 co + a 2 sin 2 a>) sin 2 1 ;
whence
(cot i\ cot r 2 ) sin % = 7 (a 6) v 2 \ a (b 2 cos 2 a> + c 2 sin 2 a>)
T- (6 2 cos 2 w + a 2 sin 2 a>) I sin 2 i .
To find the isochromatic curves, we must write ocfd = cos co sin i,
y/d = sin o> sin i, and we shall find that these curves are determined
by the equation
6 2 (c 2 ab) a? + ac 2 (a b)y z = const.
In the preceding investigation, we have tacitly supposed that
the axis of least elasticity is perpendicular to the plate, in which
case a > b > c ; accordingly ab > c 2 , and the curves are hyperbolas.
Also since the constant on the right-hand side may be either
positive or negative, we see that there are two systems of hyper-
bolas.
The investigation would however equally apply to a crystal
such as nitre or aragonite, whose optic axes make a small angle
with one another, and which is cut perpendicularly to the greatest
axis of elasticity. In this case, we must suppose that c is the
greatest and a is the least principal wave velocity ; whence c 2 > ab,
but a < b, so that the curves are still hyperbolas.
141. When a biaxal crystal is cut perpendicularly to either of
the optic axes, the isochromatic curves are, as might be expected,
approximately ellipses, which are symmetrical with respect to this
axis. The equation to these curves can be shown to be
(6 2 (a 2 - c 2 ) - (a 2 - 6 2 )} # 2 + a 2 (6 2 - c 2 ) f = const.,
the velocity of light in air being taken as unity (see Verdet, vol,
II. p. 179).
152 COLOURS OF CRYSTALLINE PLATES.
Circularly polarized Light.
142. We have hitherto supposed that the incident light is
plane polarized, and that it is analysed by an instrument which
could plane polarize common light. If however the light which
has passed through the polarizer, or the light which emerges from
the crystal, is passed through an apparatus which could circularly
polarize plane polarized light, the rings and brushes undergo
certain modifications, which we shall proceed to consider.
143. Circularly polarized light may be either produced by
passing plane polarized light through a Fresnel's rhomb, which is
an instrument which will be explained in a subsequent chapter;
or by passing it through a quarter undulation plate, which consists
of a thin plate of uniaxal crystal cut parallel to the axis, of such a
thickness, that it produces a difference between the retardations of
the ordinary and extraordinary waves, which is equal to a quarter
of a wave-length.
Let 7 be the angle which the principal section of the quarter
undulation plate makes with the plane of polarization of the incident
light. Then since E = V, the vibrations on emergence parallel
and perpendicular to the principal section of the quarter undula-
tion plate, are
sin 7 cos 2-7T (tjr 0/X), and cos 7 sin 2?r (t/r 0/X).
We therefore see that the effect of the plate is to convert plane
polarized light into elliptically polarized light ; if however 7 = JTT,
the emergent light is circularly polarized.
144. We shall first suppose, that the quarter undulation plate
(or the Fresnel's rhomb) is placed between the polarizer and the
crystal, so that the light incident upon the latter is circularly
polarized. The vibrations incident upon the crystal may be taken
to be cos 27r/T in the principal plane, and sin ^Trt/r perpendicularly
to the principal plane ; hence if = 2-rr (t/r 0/X), the vibration
on emerging from the analyser is
cos {(/> + 2?r (0 E)/\] cos a + sin sin ,
where a is the angle which the principal section of the analyser
makes with the principal plane of the incident ray. Whence the
intensity of the emergent light, is proportional to
7 2 =l-sin2sin27r(0-#)/X (19).
CIRCULARLY POLARIZED LIGHT. 153
From this expression we see, that P can never vanish unless
dn 2a sin 2?r (0 E)j\ = 1 ; hence there are no brushes.
When the crystal is a plate of Iceland spar, cut perpendicularly
the axis, E varies as r 2 , whence (19) may be written
7 2 =l-sin2asin&r 2 (20),
diere & is a constant.
K we assign any constant value to r, say r 2 = (2n -f J)TT/&, the
intensity along this circle is zero at the points a = JTT, or JTT, and a
maximum when a = f TT, or JTT. When r 2 = mr/k, P is constant,
and equal to 1; and when r 2 = (2n 4- f ) TT/&, the intensity is a
maximum when a = ITT or JTT, and a minimum when a = |7r or
|TT. Hence the general appearance of the pattern is, that the
brushes are absent, whilst the rings in the first and third quadrants
are pulled out, whilst those in the second and fourth are pushed
in.
Any other case can be discussed in a similar manner ; and the
appearance is of an analogous character, when the incident light is
plane polarized and circularly analysed.
145. We shall lastly consider the case in which the light
is circularly polarized and circularly analysed. In order to accom-
plish this, we must place another quarter undulation plate between
the crystal and the analyser, with its principal plane inclined at an
angle of 45 to the principal plane of the latter.
Let 7 be the angle between the principal section of the quarter
undulation plate, and that of the crystal. Then on emerging from
the plate, the vibrations parallel and perpendicular to the principal
section are of the form
cos x sin 7 sin (^ + 2?r (0 E)/\] cos 7,
and sin ^ cos 7 cos {^ + 2?r (0 E)j\\ sin 7 ;
whence on emerging from the analyser, the vibration is
sin (% + 7) -sin { % -f 7 + 2-rr (0 - E)/\] ;
accordingly J 2 = 4 sin 2 TT (0 - E)/\.
It therefore follows that the rings are of the same form as when
the light is plane polarized and analysed, but that there are no
brushes.
154 COLOURS OF CRYSTALLINE PLATES.
EXAMPLES.
1. If a horizontal ray is first polarized in a vertical plane,
then passed through a plate of crystal with its axis inclined at an
angle JTT to the vertical, then through a film which retards by a
quarter undulation, light polarized in a vertical plane ; show that
the emergent light is polarized in a plane inclined to the vertical,
at an angle equal to half the retardation of phase due to the plate
of crystal.
2. Plane polarized light is incident normally on a plate of
uniaxal crystal cut parallel to its axis, and is then passed through
a parallel plate of crystal, which could circularly polarize plane
polarized light. Prove that the emergent light will be plane
polarized if
tan a = tan /9 sin 2-7T&/X tan 7 cos %7rk/\ ;
where a is the angle between the principal plane of the first plate
and the plane of polarization of the incident light, /3 is the angle
between the principal plane of the second plate and the plane of
polarization of the emergent light, 7 is the angle between the
principal planes of the first and second plates, and k is the
equivalent in air to the relative retardation of the ordinary and
extraordinary rays caused by the first plate.
3. A small beam of circularly polarized light is incident on
one of the parallel faces of a plate of uniaxal crystal, which is cut
parallel to its axis, the angle of incidence being small ; and the
crystal is then made to revolve round a common normal to its
plane faces, whilst the direction of the incident pencil remains
unchanged. It is found, that when the axis of the crystal lies in
the plane of incidence, the emergent light is circularly polarized
in the opposite direction to the incident light ; and when the axis
of the crystal is at right angles to the plane of incidence, the
emergent light is circularly polarized in the same direction as the
incident light. Prove that if the axis of the crystal were inclined
at an angle JTT to the plane of incidence, the emergent light would
be polarized either in or perpendicularly to that plane.
4. If n equal and similar plates of a crystal be laid upon each
other, with their principal directions arranged like steps of a uni-
EXAMPLES. 155
form spiral staircase, and a polarized ray pass normally through
them ; prove that the component vibrations of the emergent
ordinary and extraordinary rays are of the form
X cos 27r/T + Fsin 27rZ/r,
where X and F are of the form A cos ny + B sin ny, where
cos 7 = cos B cos a ; a being the angle between the principal
directions of two consecutive plates, and 2S the difference of phase
between the ordinary and extraordinary rays in passing through
one plate.
Determine also the condition, that a ray originally plane
polarized may emerge plane polarized.
5. The extraordinary wave normal OQ in a uniaxal crystal,
whose optic axis is OA, makes a constant direction with a given
direction OP. Show that the mean of the displacements irre-
spective of sign, which are parallel to the plane POA as the
position of OQ varies, will be a minimum, when OA and OP are
at right angles to one another.
6. If a biaxal crystal is bounded by two parallel planes per-
pendicular to the axis of greatest elasticity, and if 0, be the
angles of inclination to this axis of the two emergent rays, situated
in the plane containing the optic axes at the point of emergence,
prove that
6 2 c 2 cosec 2 6 = a? (a 2 cot 2 + c 2 ).
7. A pencil passing through a feebly doubly refracting plate,
is denned by two small holes through which it has to pass, the
holes being situated in a line perpendicular to the plate, and on
opposite sides of it ; show that whatever be the law of double
refraction, when the thickness of the plate and the distances of the
holes vary, the angle in air between the two pencils which can
pass, varies as
h + fj, (k + A?')
where h is the thickness of the plate, and k, k' the distances of the
holes from the surfaces respectively next them.
8. The surfaces of a plate of uniaxal crystal are nearly per-
dicular to the axis of the crystal ; show that if polarized light be
incident nearly perpendicularly to the faces, and afterwards analysed
and received on a screen, the rings will be sensibly the same as
156 COLOURS OF CRYSTALLINE PLATES.
would have been formed if the surfaces had been perpendicular to
the axis, but shifted in the direction of the projection of the axis
through a distance proportional to fia, where fj, is the refractive
index for the ordinary ray, and a the angle between the axis of the
crystal and a normal to the surfaces of the plate.
9. Plane polarized light of amplitude c passes in succession
through two plates of crystal cut parallel to the axis, and is then
analysed by a Nicol's prism. The inclinations of the principal
planes of the two crystals, and of the Nicol's prism to the plane of
polarization of the incident light, are a, a -f /3, and a + -f 7 ; p and
q are the retardations of phase due to the two crystals respectively.
Prove that if AO be drawn equal to c cos a cos $ cos 7,
AB = c sin a sin ft cos 7, BG = c cos a sin /9 sin 7,
and CD = c sin a cos ft sin 7 ;
and if AB, BG, CD make with AO angles respectively equal to p,
q and p + q ; then OD will be the amplitude, and the supplement
of CDO will be the retardation of phase of the emergent ray.
10. The end of a Nicol's prism, in which air is substituted for
balsam, is a rhombic face inclined at an angle JTT to the axis of
the crystal, and the prism is sawn so that the layer of air contains
that axis. If the axes of the ellipsoid in the wave-surface cor-
responding to a sphere of unit radius in air be (2'3)~^, (2'6)~^, the
cosines of the angles of incidence for the extinction of the ordinary
and extraordinary rays are respectively equal to
A * A 7 46 *
V5 V2' 8 V 65~V2'
CHAPTER IX.
ROTATORY POLARIZATION.
146. WHEN plane polarized light is transmitted at normal
incidence through a plate of Iceland spar, which is cut perpen-
dicularly to the axis, the plane of polarization of the emergent
light coincides with that of the incident light. It was however
discovered by Arago 1 in 1811, that there are certain uniaxal
crystals, of which quartz is the most notable example, which
possess the power of rotating the plane of polarization. It thus
appears, that crystals of the. class to which quartz belongs possess
certain peculiarities, which distinguish them from ordinary uniaxal
crystals, such as Iceland spar.
The subject of the rotation of the plane of polarization by
crystals was afterwards studied experimentally by Biot 2 , who
established the following laws.
I. The rotation of the plane of polarization produced by a
plate of quartz cut perpendicularly to the axis, is directly propor-
tional to the thickness of the plate, and inversely proportional to the
square of the wave-length of the particular light employed.
II. If an observer looks along the direction in which the light
is travelling, there are certain varieties of quartz which rotate the
plane of polarization towards his right hand, whilst there are others
which rotate it towards his left hand.
The former class of crystals are called right-handed, arid the
latter left-handed.
1 Mem. de la prem. classe de VInst. vol. xn. p. 93 ; see also (Euvres Completes x.,
p. 36.
2 Mem. de VAcad. des Sciences, vol. n. p. 41.
158
ROTATORY POLARIZATION.
From this definition it follows, that if an observer, who is
looking through a Nicol's prism at a ray of plane polarized light,
places the Nicol in the position of extinction, and then inserts a
plate of right-handed quartz, he must turn the Nicol towards his
left-hand in order to bring it into the position of extinction ;
whilst if the plate of quartz is left-handed, he must turn the
Nicol towards his right.
Many continental writers adopt a definition, according to
which, a plate of quartz is considered right-handed, when the
observer has to turn his Nicol towards the right, in order to bring
it into the position of extinction ; but I have decided after much
consideration, to adopt a definition, in which the directions of
propagation and rotation are related in the same manner, as the
magnetic force produced by an electric current circulating round
the ray.
147. Since the rotation varies inversely as the square of the
wave-length, the position of the plane of polarization will be
different for different colours. If sunlight be employed, the
different colours will be superposed on emergence, and the emergent
light will be white ; but if the emergent light be examined by a
Nicol's prism, placed so that its principal section is parallel to the
plane of polarization of any particular colour, that colour will be
extinguished, and the light on emergence from the Nicol will
appear coloured. The following table gives the values found by
Broch 1 , for the rotations of the principal lines of the spectrum,
produced by a plate of quartz one millimetre in thickness.
Kays
Eotations
Product of rotation
multiplied by X 2
B
15 18'
7238
C
17 15'
7429
D
21 40'
7511
E
27 28'
7596
F
32 30'
7622
G
42 12'
7842
From this table it appears that the law, that the rotation is
inversely proportional to the square of the wave-length, is only
approximate.
1 Ann. de Chim. de la Pliys. (3), xxxiv. p. 119.
QUARTZ AND MAGNETIZED MEDIA. 159
148. On account of the smallness of the wave-length of all
visible parts of the spectrum, it follows from the first law, that the
rotation will amount to a large number of right angles, if the
thickness of the plate is considerable. Unless therefore the plate
is sufficiently thin for the rotation to amount to less than 180,
the observer is liable to be mistaken as to whether the direction
of rotation is to the right or to the left. In fact it appears from
the table, that for the line G of the spectrum, a plate of quartz
only one millimetre in thickness produces a rotation of 42 12'.
By employing plates of different thicknesses all cut from the same
specimen of crystal, or by employing light of different colours, all
chances of error can be eliminated.
149. The photogyric properties of quartz depend upon the
angle which the ray traversing the crystal makes with the axis ;
they are most marked when the ray is parallel, and disappear
when the ray is perpendicular to the axis ; for the latter direction,
quartz behaves in the same manner as Iceland spar. There
are also certain liquids, such as oil of turpentine, essence of lemon,
common syrup &c., which possess the power of rotating the plane
of polarization of light ; and this property is independent of the
direction of the transmitted light.
150. Faraday * discovered, that when plane polarized light is
transmitted through a transparent diamagnetic medium, which is
placed in a field of magnetic force, whose direction is parallel to
that of the ray, a rotation of the plane of polarization takes place ;
and the direction of rotation is the same as that in which a
positive electric current must circulate round the ray, in order to
produce a magnetic force in the same direction as that which
actually exists in the medium.
It was afterwards discovered by Verdet 2 , that certain ferro-
magnetic media, such as a strong solution of perchloride of iron in
wood spirit, produce a rotation in the opposite direction to that of
the current which would give rise to the magnetic force.
151. There is an important distinction between the rotation
produced by quartz, turpentine &c., and that produced by a
magnetic field. If, when a ray is transmitted through quartz in a
given direction, the rotation is from left to right, it is found that
1 Experimental Researches, xixth series, 21462242.
2 C. E. vol. LVI. p. 630 ; vol. LVII. p. G70 ; Ann. de Chim. et de Phys. (3), vol.
LXIX. 415 ; Mem. de VInst. vol. xxxi. pp. 106, 341.
160 ROTATORY POLARIZATION.
when the ray is transmitted in the opposite direction, the rotation
is from right to left. If therefore the ray after passing through
a plate of quartz, be reflected at perpendicular incidence by a
mirror, and thus be made to return through the plate in the same
direction, the rotation will be reversed, so that on emerging a
second time from the plate, the plane of polarization will be
restored to its original position. But when the ray is transmitted
through a magnetic field, the direction of rotation in space is |
always the same, whether the ray is propagated along the positive |
or negative direction of the magnetic force ; if therefore the ray |
be reflected, and be made to return through the magnetic field,
the rotation will be doubled.
The photogyric properties of a magnetic field will be more
fully considered, when we discuss the electromagnetic theory of
light.
152. The dynamical theories which have been proposed to
account for rotatory polarization will be considered later on. At
present we shall show how these phenomena can be explained by
geometrical considerations.
Fresnel assumed, that the only kind of waves, which media of
this class are capable of propagating without change of type, are
circularly polarized waves. If light polarized in any other manner
is incident upon the medium, Fresnel supposed that the wave
is immediately split up, on entering the medium, into two waves
which are circularly polarized in opposite directions, and are
transmitted with different velocities.
Let us therefore suppose, that a plane polarized wave is
incident normally at a point upon a plate of quartz cut per-
pendicularly to the axis. Let the incident displacement be
parallel to y and equal to a sin , where = 27rt/r ; and let the
axis of z be the axis of the quartz.
The incident wave may be conceived to be made up of the
four displacements
u = ^a cos <, u = ^ a cos $,
v = J a sin , v' = ^ a sin c/>.
By 13, the displacements (u, v) represent a right-handed
circularly polarized wave, while the displacements (uf, v') repre-
sent a left-handed wave. Their combination is equivalent to a
single plane polarized wave.
FRESNEL'S INVESTIGATION. 161
Let d be the thickness of the plate; V l} Fa the velocities of
propagation of the two waves. Then on emergence, the two waves
are represented by
d\ . 27r/ d
%= fa cos
2-7T/ d\ . 2-7T
T
accordingly the displacement on emergence becomes
2-7T f 7 / 1 IV) -rrd / 1 1 \
u = HI + u 2 = a sin -j \d ( -p- + ^ } \ sm I -=- ^.- 1 ,
T { \r i ^2/J T V "i K 2 /
2?r
whence - = tan
^ 77(^/1
- = n u
v
7(^/1 1 \
u = .
T \V t VJ
We therefore see that the emergent light is plane polarized,
and that the plane of polarization is rotated through an angle
m
Hence according as V z > or < "Pi, the rotation will be towards
the right-hand or the left-hand of a person who is looking along
the positive direction of the axis of #, which is the direction in
which the wave is supposed to be travelling. From this result,
coupled with the definition in 146, it follows that in right-handed
quartz, the velocity of the left-handed circularly polarized wave is
greater than that of the right-handed wave ; whilst the converse
is the case with left-handed quartz.
Equation (1) is in accordance with the experimental fact, that
the rotation is proportional to the thickness, but it does not give
any information respecting the dependance of the rotation upon
the wave-length, inasmuch as the relation between F l5 F 2 and T is
unknown. Experiment however shows that "Pi F 2 must be a
function of the period.
153. From the table on page 158, it follows that a plate of
quartz one millimetre in thickness rotates the plane of polarization
of the mean yellow rays (that is the rays midway between the
lines D and E), through an angle of about 24. Hence a plate
B. O. 11
162 ROTATORY POLARIZATION.
whose thickness is 15 mm. produces a rotation equal to 360 ; and
therefore for a plate of quartz of this thickness
where V is the velocity and X is the wave-length in air.
If the plate is right-handed, F 2 > V 1 ; also since the difference
between the velocities is small, we may put
where h is a small quantity, and therefore
F_ V__Vh
F 7 ~ ~V "F* '
v 2 V i r.\
approximately. Accordingly (2) becomes
The ratio V/V 1 is sensibly equal to the ordinary index of
refraction of quartz, which is 1*555 or about f ; whence
For the mean yellow rays, X = ^ x 10~ 3 mm. ; whence
from which it appears, that the difference between the two
velocities is less than the twenty thousandth part of one of the
velocities of the two circularly polarized waves; and that the
difference between the two indices of refraction is less than 1*00005,
which is less than the index of refraction of the least refrangible
Theory of Coloured Rings.
154. Fresnel confined his attention to the case of plane]
polarized light transmitted through a plate of quartz in a direction
parallel to the axis. The refraction of polarized light, transmitted
in an oblique direction, was first studied by Airy 1 , whose investi-
gations we shall proceed to consider.
When a plate of quartz is cut parallel to the axis, and the
plane of incidence is perpendicular to the latter, Airy found thatf
no appreciable difference existed between the action of quartz, and|
y
1 Trans. Camb. Phil. Soc., Vol. n. pp. 79, 198; see also Verdet, Lecons d'Optiqvm
Physique, Vol. u. pp. 237265.
AIRY'S THEORY OF COLOURED RINGS. 163
that of any other uniaxal crystal ; but when the plate is cut
perpendicularly to the axis, and the incident rays are inclined at a
small angle to the axis, he found that the two refracted rays were
elliptically polarized in opposite directions. Quartz is a positive
crystal ; and if the plate is right-handed, it follows from 13, and
also from the definition in 146, that the left-handed elliptically
polarized ray corresponds to the ordinary ray, and the right-handed
one to the extraordinary ray. The converse is the case when the
plate is left-handed.
The elliptic polarization is only sensible, provided the angle
of incidence does not much exceed 10 ; if it is considerably
greater, the refracted light appears to be plane polarized, and the
position of the planes of polarization is the same as in the case of
ordinary uniaxal crystals. It therefore follows, that the ratio of
the major and minor axes of the two ellipses, which is equal to
unity at perpendicular incidence, increases rapidly with the angle
of incidence ; so that when the latter ceases to be small, the two
ellipses do not sensibly differ from two straight lines at right angles
to one another. It also follows, that the major axis of the elliptic
vibrations of the ordinary ray, is perpendicular to the plane
containing the optic axis and the ordinary wave normal, whilst
in the case of the extraordinary ray, the major axis lies in the
plane containing the optic axis and the extraordinary wave
normal.
We have already shown, that the difference between the
ordinary and extraordinary indices of refraction is very small ; it
was therefore assumed by Airy, that the two ellipses corresponding
to the same incident ray are always similar.
When light is transmitted parallel to the axis through an
ordinary uniaxal crystal, the velocity of propagation of the two
waves is the same ; this arises from the fact that the wave surface
consists of a sphere and a spheroid, the former of which touches
the latter at the extremities of its polar axis. But this result does
not hold good in the case of quartz. Airy therefore assumed, that
, the wave surface in quartz consists of a sphere and a spheroid,
[which do not touch one another; and since quartz is a positive
crystal, the spheroid is prolate, and consequently the spherical
heet lies wholly outside the spheroidal sheet. From this hypo-
thesis, it follows that the difference of path of the two elliptically
ijoolarized waves is equal to the corresponding expression for
112
164 ROTATORY POLARIZATION.
ordinary uniaxal crystals, together with a certain quantity, which
is independent of the angle of incidence, but which is a function
of the period.
155. By the aid of the foregoing hypotheses, Airy succeeded
in giving a mathematical explanation of the rings and brushes
which are produced, when polarized light is passed through a
plate of quartz ; but before we consider his investigations, it will be
desirable to give a description of the peculiarities of these rings.
I. When the planes of polarization and analysation are per-
pendicular, and a thin plate of quartz is placed between them, a
set of circular coloured rings is observed. The centre of the
pattern consists of a coloured circular area, and the colour depends
upon the thickness of the plate. If the thickness is '48 of an inch,
the central tint is pale pink ; with thicknesses of '38, "26 and '17 of
an inch, the colours of the central spot are a bright yellowish green,
a rich red plum colour, and a rich yellow. The colours of the
successive rings, beginning from the centre, appear to be nearly the
same as in Newton's -scale, commencing with the colour of the i
central spot. At a considerable distance from the centre, four,
faint brushes commence, which intersect the rings in the same
directions as the black cross in Iceland spar.
II. When the polarizer and analyser are initially crossed, and
the latter is made to rotate, a blueish short-armed cross appears j
in the centre, which on continuing to turn the analyser becomes
yellow, and the rings become enlarged. When the inclination of
the planes of polarization and analysation is 45, the rings are
nearly square, and the diagonals of the square bisect the angles
between these planes.
III. When incident light is circularly polarized, the rings
consist of two spirals mutually intertwining one another.
IV. When two plates of quartz of equal thickness, one of
which is right-handed and the other left-handed, are attached
together, and placed between the polarizer and analyser, four
spirals proceeding from the black cross to the centre make their
appearance. These curves are usually known as Ahy's spirals.
156. We shall now proceed to give the mathematical investi-
gation of these phenomena.
We shall suppose, that a small pencil of convergent light i&
incident upon the crystal at a small angle. Let be the point oi
VALUE OF THE INTENSITY. 165
incidence of any one of the rays, OA the intersection of the plane
passing through the ray and the optic axis, at the point where the
central ray of the pencil meets the crystal ; also let OP, OS be the
principal sections of the polarizing and analysing Nicols.
B
Let .
If we suppose the crystal right-handed, the incident wave will
be resolved into a right-handed elliptically polarized wave, which
corresponds to the extraordinary wave, and a left-handed ellip-
tically polarized wave, which corresponds to the ordinary wave;
also since the two ellipses of vibration are similar, and their major
axes are at right angles, it follows that if (u, v) and (u', v f ) respec-
tively denote the displacements parallel to OA, OB in the two
waves, and & is a quantity lying between zero and unity, we may
put
u = mk cos (< +f*>), v = m sin (< + //,))
u' = nk~ l cos ( 4- v), v' = n sin (< + 1>) } '
The resultant of these four displacements must be equal to
sin <, and must be parallel to OP, whence
(u + u') cos a + (v + v') sin a = sin
(u + u') sin a (v + v') cos a = 0.
These equations must hold good for all values of t and therefore of
$ ; whence equating the coefficients of sin < and cos (/>, we obtain
(m cos fju + n cos v ) sin a (mk sin p nk~ l sin v) cos a = 1,
(m sin //, + n sin v) sin a + (mk cos fju nk~ l cos v) cos a = 0,
(m cos fji + n cos v) cos a + (mk sin //, nk~ l sin v) sin a = 0,
(m sin fi + n sin v) cos a + (ra& cos /* 71&" 1 cos ^) sin a = 0.
From these equations we find, that
m cos p + n cos v = sin a, mk cos /A wAr 1 cos v 0,
m sin fjb + n sin i> = 0, w& sin //, W&" 1 sin i/ = cos a ;
166 ROTATORY POLARIZATION.
accordingly
sin a . &cosa
m cos ^ =
sin a . & cos a
Substituting in (3), we obtain
k
u = ^ - r 2 (sin a cos + k cos a sin ),
I "T~ A/
v = - - ^ (sin a sin k cos a cos ),
u' = ^ - j- (k sin a cos cos a sin <),
1 + K
k
v' = n , (A; sin a sin + cos a cos <).
JL -\- K
The first pair of these equations represents the extraordinary
wave, and the second pair the ordinary wave.
If D and G are the equivalent paths in air for the ordinary and
the extraordinary waves, and if we put
B = 27rD/X, 7 = 2wG7X ;
it follows, that on emerging from the crystal, we must replace by
8 for the ordinary wave, and by 7 for the extraordinary.
Hence on emerging from the analyser, the displacements become
(u + u') cos (a 0) + (v + v) sin (a - ft),
or
(1 + k' 2 )- 1 [{k sin a cos ($ 7) + k? cos a sin ((f> 7) k sin a cos ( 8)
+ cos a sin ( B)} cos (a ft)
4- {sin a sin ( 8)} sin (a )].
Writing (/> 7 + 7 5 for $ S, picking out the coefficients of
cos ($ 7) and sin (c/> 7), and then squaring and adding, and
writing for brevity ty and ^ for a ft and 7 S, the intensity / 2 is
determined by the equation
(1 + A; 2 ) 2 / 2 = {(k sin a k sin a cos % + cos a sin ^) cos ^
+ ( k cos a + k 2 sin a sin % + k cos a cos ^) sin i/r} 2
+ {(& 2 cos a + & sin a sin ^ + cos a cos ^) cos ^
+ (sin a + k 2 sin a cos ^ k cos a sin ^) sin T/r} 2
= { sin ^ (cos a cos ^r + & 2 sin a sin ty) & cos % sin /3 + & sin /3)' J
VALUE OF THE INTENSITY. 167
+ [cos % (cos a cos i/r 4- k 2 sin a sin i/r) + & sin ^ sin /3
+ sin a sin ijr + Jo 2 cos a cos -^r) 2
= (cos a cos -fy + & 2 sin a sin -^) 2 + 2& 2 sin 2 ft
+ (sin a sin ->|r + & 2 cos a cos -^) 2
+ 2 (cos a cos i/r + k* sin a sin ty) x
{cos % (sin a sini/r + & cos a cos ty) +k sin ^ sin /3}
+ 2k sin % sin ft (sin a sin -\|r + & 2 cos a cos -\Jr) - 2& 2 cos % sin 2 fi.
Replacing cos ^ by 1 2 sin 2 ^ ^, this becomes
= (1 + & 2 ) 2 cos 2 j3 + A? (1 4- & 2 ) sin 2/3 sin % + 4A; 2 sin 2 J x sin 2 y8
4 sin 2 J% (cos a cos i/r + k 2 sin a sin -^r) (sin a sin -^r + & cos a cos ifr)
- (1 + & 2 ) 2 cos 2 + k (1 + & 2 ) sin 2/3 sin % - 4& 2 cos 2/3 sin 2 fa
- (1 - & 2 ) 2 sin 2a sin 2-^ sin 2 J%,
whence dividing by (1 + & 2 ) 2 , the value of P may be put into the
form
(2k
cos /3 cos fa + j-j-^ sin ^ si
(J _ y.2\ 2
^ - r , ) (
I + & /
sn
2
cos 2 3 sin 2a sin 2ir sin 2
Restoring the values of ^ and %, this finally becomes
f 2k ) 2
7 2 = jcos j3 cos i(S 7) ^- ,- ^ 2 sin j3 sin J (8 -
cos2 2a - sin2 - ......
157. Returning to 155, we see that the first case which has
to be examined, arises when the Nicols are crossed, so that ft = ^TT ;
in this case (4) becomes
In the neighbourhood of the centre of the field of view, k
is nearly equal to unity, and the preceding expression shows, that
7 2 is very nearly independent of a ; hence the centre of the field
consists of a bright patch. As we proceed from the centre, k
rapidly diminishes to zero, and therefore at some distance from
the centre, the intensity is approximately equal to
sin 2 2a sin 2 1(7 -8),
and therefore vanishes when a = \nnr. It therefore follows, that at
a certain distance from the centre, four dark brushes make their
168 ROTATORY POLARIZATION.
appearance, which divide the field into quadrants, but which do
not extend right up to the centre ; also since the intensity is not
absolutely, but only approximately, zero when a. = ^UTT, the brushes
are much fainter than in the case of Iceland spar.
We shall now ascertain the form of the rings, which are deter-
mined from the condition that
S 7 = 2n7r,
or D-G = n\.
If the plate were an ordinary uniaxal crystal, it follows from
133 that we should have
where T is the thickness of the plate ; and therefore the retardation
vanishes at the centre. We know that this is not the case with
quartz, and Airy therefore assumed, that the retardation is equal
to the sum of this expression, together with a quantity, which is
directly proportional to the thickness of the plate and inversely
proportional to the wave-length in air. We therefore put
(6),
where H is a constant.
In the neighbourhood of the centre, the intensity is very nearly
equal to
Bin f ir(D-0)/X,
and since this expression does not vanish when r = 0, the centre can
never be black ; also since TT (D - G)/\ is of the form (A/\ + 5/X 2 ) T,
it follows that the colour of the central spot will vary with the
thickness of the plate.
At some distance from the centre, k will be small ; whence the
intensity of the rings will be a maximum or minimum according
as
D-G = (n + %)\ or n\',
accordingly the isochromatic curves are circles.
The pattern therefore consists of a bright coloured circular spot
in the centre of the field, surrounded by coloured circular rings ;
and the rings are interrupted by four faint brushes at right angles
to one another, which commence at the circumference of the circular
spot.
PLATE PERPENDICULAR TO THE AXIS. 169
158. If /3 = 0, so that the planes of polarization and analysa-
tion are parallel, the expression for the intensity becomes
from which it appears that the rings are interrupted by two white
brushes, and that the colours are complementary to those in the
former case.
The forms of the curves are shown in figures 5 and 6 of the
plate at the end of this Chapter.
159. When the planes of polarization and analysation are
neither parallel nor perpendicular to one another, the isochromatic
curves are of a more complicated character. In order to get an
approximate idea of their form, let
2/Han/3
TT^ :=tan *'
Substituting the values of sin /3, cos/3 deduced from this equa-
tion in the first term of (4), it becomes
* (S - ?> -
cos 2 /3 + sin 2 [ cos 2 J (3 - 7 +
If we suppose that a small variation of 8 7 does not produce
any sensible alteration in the value of k, the maximum or minimum
value of the intensity for given values of a and /3 will be found by
differentiating / 2 with respect to 8 7, on the supposition that k is
constant ; we thus obtain
tan (8 7 + i|r)
(1 + 2 ) 2 cos 2 /3 + 4 2 sin 2 /3 + (1 - 2 ) 2 cos 2 (2a - ft) ,
Q Y (i + py cos 2 ^ + 4A; 2 sin 2 yS - (1 - kj cos 2 (2a - yS) ' '
= tan H (say).
Let Q be any point of the coloured image, QP the direction of
vibration of the incident light, QS the principal section of the
analyser ; then if is the centre of the field, OQ is the principal
170 ROTATORY POLARIZATION.
section corresponding to the ray Q. Now as we proceed along OQ,
the intensity depends upon the value of S 7 ; and we see from (8),
that the points of maximum and minimum brightness occur, when
S 7 has a value which depends upon a and k. Now if OQ = r, it
follows from (6), that
where A and B are constants. Hence as OQ revolves around the
origin, the points of maximum and minimum brightness will not
be equidistant from 0, but will lie on a sort of square curve.
The equation of these curves is
Now fl is a maximum when a = |/3 + Jw7r, and a minimum
when a = J/3 + (^n + J) TT ; and therefore H is a maximum when
OQ bisects the interior and exterior angles between the planes of
polarization and analysation, and is a minimum when OQ makes
an angle \TT with its four preceding positions. The best position
of the analyser for viewing these curves is when /3 = JTT, in which
case the isochromatic curves form a sort of square, whose diagonals
bisect the angles between the planes of polarization and analysa-
tion.
In the central portions of the field, k is very nearly equal to
unity, and from (7) it appears, that the intensity will be least when
8 7 + 2-^r = (2n + 1) TT ; consequently there will be a dark spot in
the central portion. Now for points equidistant from the centre,
8 7 has the same value ; and we see from (7), that for all points
which are near the centre, the intensity is approximately greatest
when a = %/3 + JWTT, and least when a = J/3 + (-J-n + J) TT ; hence in
the centre of the field there is a dark cross, whose arms coincide
with the diagonals of the square curves.
160. We shall in the next place investigate the rings and
brushes which are produced when the incident light is circularly
polarized, and we shall suppose that the plate and the polarization
are right-handed.
Let the incident light be
u = cos , v = sin (f> ;
where < = 27rt/r.
On entering the quartz at oblique incidence, the incident light
is resolved into two elliptically polarized waves, one of which is
CIRCULARLY POLARIZED LIGHT. 171
right-handed and the other left-handed ; and therefore within the
crystal we may put
u = mk cos (< + //,), v 1 = m sin (< -f- //,),
u 2 = nk~ l cos ($ + v), V 2 = n sin (< + v).
Since these two sets of vibrations are equivalent to the incident
vibrations, we must have
u = U! + u 2 , v = v + v 2 ;
whence equating coefficients of cos <, sin <, we get
mk cos fj, nk~* cos v =1,
mk sin //, nAr" 1 sin z> = 0,
m sin /u- + n sin z> = 0,
m cos /ji-\-n cos v = 1,
from which we deduce
fi = 0, * = 0.
_ 1 + &
- '
Hence, on emerging from the crystal, the two rays are repre-
sented by the equations
k(l+k) 1+k .
Ul = cos ^ ~ ^' Vl = sm ( ~ 7) '
If a be the angle which the plane analysation makes with the
principal section of the crystal, the light on emerging from the
crystal is represented by
(M! + u 3 ) cos a + (v + v a ) sin a ;
that is ^| 2 {*(! + k) cos ( - 7 ) + (1 - 4) cos (< -8)),
sma
+ 1 + J 2 K! + *) sm ((/> - 7) - k (1 - A;) sin(< - 8)}.
Replacing 0-8 by < - 7 + 7-8, and putting ^ for 7 - 8, the
intensity is determined by the equation
(1 + #)* / 2 = [{1 + k - & (1 - k) cos % } sin a - (1 - k) sin % cos a] 2 ,
+ [{(1 - k) cos x + & (1 + &)} cos a & (1 - k) sin % sin a] 2
= {(1 + &) 2 - 2k (1 - k 2 ) cos x + ^ (1 - &) 2 } sin 2 a
+ {(1 - kf + 2A? (1 - & 2 ) cos x + # (1 + &) 2 } cos 2 a
(1 &*) sin 2a sin ^,
172 ROTATORY POLARIZATION.
whence
1 =l -
+ #y cos 2a CQS x ~ ITI 2 sin 2a sin * ........ (9)>
161. If in this expression we put
(1 + & 2 ) tan 2a = 2k tan 2^
and restore the value of %, we obtain
2fe(l-fr) 2fr(l-& 2 )cos(8- 7 -2^)
22 r2222 22 "'
Let us now draw a line from the centre making an angle a with
the principal section of the analyser. Then if we consider a series
of points on this line, which are not very distant from one another,
we may suppose that k is approximately constant for such points.
From (10) we see that 7 2 is a maximum or minimum according as
8 _ 7 _ 2^/r = 2ft7r or (2n + 1) TT ;
and since S 7 = -4r 2 -f- B,
the points of maximum intensity are determined by
^lr 2 + = 2w7r + 2^ ..................... (11).
In the neighbourhood of the centre, k does not differ much
from unity, and we may therefore as a first approximation put
ijr = a ; whence writing 6 for a, the equation of the isochromatic
curves becomes
Ar* = 2mr + 20-B.
This equation represents a spiral curve which commences ati
the origin. The form of the curve when n = l is shown in the!
figure ; if we put n = 2, we obtain a second spiral which is derived |
AIRY'S SPIRALS. 173
from the former by turning it through two right angles. For
values of n greater than two, the two spirals will be found to be
reproduced.
162. The fourth case which we shall consider, arises when
plane polarized light is incident upon two plates of quartz of equal
thickness, one of which is right-handed, and the other left-handed ;
and the planes of polarization and analysation are parallel.
The displacements in the two elliptically polarized waves on
emergence from the first plate are given in 156 ; and we must
recollect that on emergence, we must write 4- % for (/> in the
values of (u', v'}.
Since the second plate of quartz is left-handed, the sign of k
must be reversed, and therefore on entering the second plate we
must write
U = mk cos ( + yu,), V = m sin (0 + //,)
for the ordinary wave, and
U' = nk~ l cos (>|r + v), V = n sin (^ + v )
for the extraordinary wave where i|r = < + %.
The four quantities m, n, ft, v must be determined by equating
the coefficients of sin \jr, cos ^r in the equation
u + u'=U+U', v + v'=V + V.
Having obtained the values of m, n, //-, v, we must write ^ + %
for.^r in the expressions for U', V.
Since the planes of polarization and analysation are supposed
to be perpendicular, the displacement on emergence from the
analyser will be
( U+ U') cos a + ( V+ V) sin a ;
we must therefore form this expression, and then write down the
sum of the squares of the coefficients of sin , cos $, which will
give the intensity.
The actual calculations are somewhat tedious, but on performing
the above operations, it will be found that
4fk ) 2
~ -^ cos 2a sin J (8 - 7 ) - 2 sin 2a cos J (8 - 7 )| .
174
ROTATORY POLARIZATION.
163. This expression can vanish in two ways. In the first
case sin l (8 7) = 0, which requires that
8 7 = 2n7r,
which represents a series of circular rings, which are black if
homogeneous light be employed, but coloured if white light be
used.
In the second case the intensity will vanish when
In the neighbourhood of the centre, k does not differ much
from unity, and we may therefore take as a first approximation
& 7 = 4a 4- 2n7r ;
whence writing 6 for a, the equation of the isochromatic curves are
Ar 2 + B = 40 + 2/iTT,
which is the equation of a spiral curve.
Let us first suppose that n = ; then it follows that when
6 = ^B, r = ; so that the spiral commences at the origin, and the
distances of successive points from the origin increase with 6.
When 6 = ITT, r 2 = (2ir - B)/A.
Next let n = 1, then r = 0, when 6 = \B ^TT ; and when
6 = IB, r 2 = (27r - B)/A. We therefore see that the spiral corre-
sponding to n = 1 is equivalent to the spiral corresponding to
n = 0, turned from left to right through a right angle.
B. 0.
[To face page 174.
OFTHE
UNIVERSITY
OF
AIRY'S SPIRALS. 175
Similarly for n = 2, n = 3 there are two other spirals, whose
positions can be obtained by turning the spiral for which n =
backwards through two right angles, and three right angles
respectively. When n = 4, the original spiral is reproduced.
The forms of these spirals, which after their discoverer are
usually known as Airy's Spirals, are shown in the figure on the
last page, and also in figures 7 and 8 of the plate. At a con-
siderable distance from the centre, a faint black cross makes its
appearance, whose four arms are parallel and perpendicular to
the plane of polarization of the incident light ; also the spirals
disappear and are replaced by circular rings. Now at a distance
from the centre, k is nearly equal to zero ; whence the intensity
becomes
7 2 = sin 2 2a sin 2 (8 - 7),
which vanishes when 7 8 = nir
or a = ^mr.
The first equation gives the circular rings, whilst the latter
equation gives the brushes.
CHAPTER X.
FRESNELS THEORY OF REFLECTION AND REFRACTION.
164. WHEN common light is incident upon the surface of a
transparent medium, such as glass, it can be proved experimentally,
that the proportion of the incident light which is reflected or
refracted, depends upon the angle of incidence ; and that the
amount of light reflected is greater when the angle of incidence is
large, than when it is small. It is also known, that when light
proceeding from a denser medium, such as glass, is incident upon
a rarer medium, such as air, at an angle greater than the critical
angle, the intensity of the reflected light is very nearly equal to
that of the incident light, and the reflection is said to be total.
When the incident light is polarized in the plane of incidence, the
effect produced by a reflecting medium is not very different from
that produced upon common light; but when the light is polarized
perpendicularly to the plane of incidence, it is found that the
intensity gradually diminishes from grazing incidence, and very
nearly vanishes, when the angle of incidence is equal to tan" 1 //,,
where yu- is the index of refraction of the reflecting substance ; as
the angle of incidence still further increases, the intensity of the
reflected light increases to normal incidence.
165, That the intensity of light polarized perpendicularly to
the plane of incidence is zero for a certain angle of incidence, was
first discovered by Malus, who while examining with a prism of
Iceland spar the light reflected from one of the windows of the
Luxembourg palace at Paris, observed that for a certain position of
the prism, one of the two images of the sun disappeared. On
turning the prism round the line of sight, this image reappeared ;
and when the prism was turned through 90, the second image
BREWSTER'S LAW. 177
reappeared. More accurate experiments were afterwards made by
Brewster 1 , who discovered that when the reflector is an isotropic
transparent substance, and the incident light is polarized per-
pendicularly to the plane of incidence, the intensity of the reflected
light is zero, or very nearly so, when the angle of incidence is equal
to tan -1 /x. This discovery is known as Brewster's law, and the
angle tan" 1 //, is called the polarizing angle.
166. Brewster's law has been tested by Sir John Conroy 2 for
transparent bodies in contact with media other than air, in the
following manner. A glass prism was placed in contact with
water and with carbon tetrachloride respectively, and the polariz-
ing angles were determined. Their values, as found by experi-
ment, were as follows :
Polarizing angle in air 57 14'
in water 49 41'
in carbon tetrachloride 46 32'.
The polarizing angles were then determined experimentally
for water and carbon tetrachloride in contact with air, and the
values of the polarizing angles for glass in contact with these
substances were then calculated. The results were as follows :
Polarizing angle in air observed 57 14'
calculated from observations in water 57 28'
in carbon tetrachloride 57 | 01'.
These results show, that within the limits of experimental
error, Brewster's law holds good for glass in contact with water
and carbon tetrachloride, as well as air; and that in all probability,
it is true for most transparent bodies.
167. Crystalline substances, such as Iceland spar, also possess
a polarizing angle as well as a critical angle. In isotropic media,
the critical angle is equal to sin" 1 //, ; but in doubly refracting
media, the values of the polarizing and critical angles cannot be so
simply expressed.
168. Metallic substances, such as polished silver, possess a
quasi-polarizing angle, since there is a particular angle of incidence
at which the intensity of light polarized perpendicularly to the
plane of incidence is a minimum.
1 Phil. Trans. 1815, p. 125. See also Lord Eayleigh, "On Reflection from
Liquid Surfaces in the Neighbourhood of the Polarizing Angle," Phil. Mag.
Jan. 1892.
2 Proc. Roy. Soc. vol. xxxi. p. 487.
B. O. 12
178 FRESNEL S THEORY OF REFLECTION AND REFRACTION.
169. In order to explain these experimental facts, it is
necessary to determine the intensities of the reflected and refracted
lights. This was first effected by Fresnel ; and although his theory
is not rigorous, it will be desirable to give an account of it in
the present Chapter. Other theories based upon speculations
respecting the physical constitution of the ether, which are
developed according to strict dynamical principles, will be con-
sidered in subsequent chapters ; and it will be found that most of
them give results, which are substantially in accordance with
those obtained by Fresnel.
170. We shall first calculate the rate at which energy flows
across the reflecting surface.
Let the incident vibration be
w = A cos (x Vt),
A,
and let us consider the energy contained within a small cylinder i
whose cross section is dS, and whose sides coincide with the direc-
tion of propagation. If T denote the amount of kinetic energy per
wave-length
X
Since this amount of kinetic energy flows across dS in time r,
the rate at which kinetic energy flows across dS is T^A^pdS/Xr.
Let dS' be any oblique section of the cylinder, which makes
an angle e with dS, then if we assume that the energy of the wave
is half kinetic and half potential, the rate at which energy flows
across dS' is
27rA*V 2 pcosedS'l\T (1).
The mean energy for unit of volume is
27rU 2 F 2 / >/V (2),
as has already been shown in 10.
171. We are now prepared to consider the problem of reflec-
tion and refraction.
Let the axis of x be normal to the reflecting surface, and let
the axis of z be perpendicular to the plane of incidence.
POLARIZATION IN THE PLANE OF INCIDENCE. 179
Let the incident light be polarized in the plane of incidence,
X
hen the incident, reflected and refracted waves may be taken to
be the real parts of w, w', w lf where
) \
Vt) ( ..................... 3
Since the reflected and refracted waves are forced vibrations
produced and maintained by the incident wave, it follows that the
; periods of the three waves must be the same ; whence
K V=K 1 V 1 , or F/X = FJ/X! (4).
Since the traces of all three waves on the plane x = move
together, it follows that the coefficients of y must be the same in
all three waves, whence
/em = Km! = tfjWi. (5).
If i, i', r be the angles of incidence, reflection and refraction,
I = cos i, I' = cos i', I-L = cos r,
m sin i, m' = sin i' y m l = sin r,
and therefore from (5), i = i' t
sin i KI F
sin r K Fj
m
which is the law of sines.
172. We now require two equations connecting the amplitudes
of the reflected and refracted waves. In order to effect this, Fresnel
i assumed, (i) that the displacements at the surface of separation are
the same in the two media, (ii) that the rate at which energy flows
] across this surface is continuous.
122
180 FRESNEL'S THEORY OF REFLECTION AND REFRACTION.
The first condition gives
-A, (6),
and by (1) the second condition gives
Vt_ ^2 Y 2 p\~ l cos i A /2 V 2 p\~ l cos i = A x 2 F^Xr" 1 cos r
T * N , -rr /H7X
or (^ 2 -^. /2 ) Fpcosi = ^l 1 2 F 1 /3 1 cosr (7).
Fresnel's third assumption was, i/ta Ae product of the velocity !
wo tf&e s^ware roof o/^e density is constant for all media; whicj/l
! - ^ Accordingly (7) becomes fr
.(8).
v
Splying (6) and (7), we obtain
""N ~\ , _ A sin (i r)
/ ^*? " rr^vi? i /v.\
/^
2^1. sin r cos i' I
-^"^ ^BulTif /, /', /i denote the square roots of the intensities, it
^ /iollows from (3) of 10, that
A-^i'A ...do).
^1|02 ^. p* Aipi*
,. T/
Accordingly / =
sin (t + r)
/ V iJL >
sin (i + r)
These formulae give the ratios of the intensities of the reflected
and refracted light to that of the incident light.
173. When the second medium is more highly refracting than
the first, as is the case when light proceeding from air is reflected
at the surface of glass, r is always real ; but in the converse case, r
is imaginary when the angle of incidence exceeds the critical angle.
For if fju be the index of refraction from air to glass, and light is
internally reflected and refracted at the surface of glass in contact
with air,
sin i = fjb~ l sin r,
cos r = (1 frf sin 2 1)*.
Since fj, > 1, it follows that cos r is imaginary when i >
Under these circumstances, the expressions for the amplitudes oi
CHANGE OF PHASE.
181
the reflected and refracted waves become complex, and their inter-
pretation in former times was supposed to be a matter of con-
siderable difficulty. The true explanation is this. The incident,
reflected and refracted waves are the real parts of the right-hand
sides of (3) ; if therefore A' and A l are real, the reflected and
refracted waves are
w' = A' cos K ( Ix + my Vt)
w l = A 1 cos ! (Ijjos + m$ - Vjt) ;
but if A' and A l are complex, we must write A' = a + t/3,
A I = OL I + ifti, and the reflected wave is
w r = a cos (# + m?/ F) /3 sin K ( Ix + my
= (a 2 + /3 2 )* cos {* (- Ix+my- Vt) + tan" 1 ft /a]
which shows that there is a change of phase.
To find a, ft, we have from the first of (9)
, cos i i (/u, 2 sin 2 i 1)*}
~
^ ^
h
whence if
we obtain
where X is the wave-length in glass ;
whence tan~V3/a = 2-Tre/X.
Also (a 2 + ft' 2 )% = A, so that the reflected wave becomes
w = A cos (x cos i + y sin i Vt e)
which shows that the reflection is total, and is accompanied by a
change of phase whose value is given by (12).
174. To find what the refracted wave becomes, we have from
(9)
whence
Also
#.*
COS *
COS
2
i /* cos
o
^! = IK^ cos 7" = (ft 2 sin' 2 i 1)^,
182 FRESNEL'S THEORY OF REFLECTION AND REFRACTION.
whence the refracted wave is
CQS 27T g . n . _ ^ _ ......
(JL* - 1)* A,
Since a; is negative in the second medium, it follows that the
wave penetrates only a very short distance, and becomes insensible j
at a distance of a few wave-lengths 1 .
175. The preceding theory is rigorous from a dynamical point .
of view ; but when we consider the corresponding problem in
which the incident light is polarized perpendicularly to the plane
of incidence, we shall find that a difficulty arises, which will be
considered in 180.
The displacements in the three waves are given by (3), and
they lie in the plane of xy and are perpendicular to the direction
of propagation of the waves.
The condition that the displacements parallel to y should be
continuous gives
(A A') cosi = AI cosr .................. (14).
Combining this with (8) we get
A tan (i r)
. , _
'
2 A cos i sin r <'
sin (i + r) cos (i r\
* -f, J. tan (i V)
whence I = ^
tan ( + r)
sin (i + r) cos (i r)
The first of these formulae shows, that the intensity of the
reflected light vanishes when
7r or tan i = t.
1 Another explanation, differing only in form, is as follows. The hypothesis that
the reflected wave, corresponding to the incident wave
A cos 27TX- 1 ( - x cos i + y sin i - Vt), is A' cos 27TA- 1 (x cosi + y sin i - Vt),
tacitly involves the assumption, that reflection is unaccompanied by a change of
phase. The fact that the amplitude becomes complex when the angle of incidence
exceeds the critical angle shows, that this assumption is erroneous in this particular
case. We ought therefore to assume that a change of phase takes place, both in
the reflected and refracted wave ; and we shall find, that the changes of phase are
zero, when the angle of incidence is less than the critical angle, and have the above
values when it exceeds it.
POLARIZING ANGLE. 183
176. When light of any kind is incident upon a transparent
reflecting surface, the vibrations may be resolved into two com-
ponents respectively in and perpendicular to the plane of incidence;
and the first of (16) shows, that the component of the reflected
vibration in the plane of incidence vanishes, when the angle of
incidence is equal to tan" 1 p. It therefore follows, that if common
light be incident at this angle, the reflected light will be polarized
in the plane of incidence. This is the law which was established
experimentally by Brews ter.
177. Airy observed, that certain highly refracting substances,
such as diamond, never completely polarize common light at any
angle of incidence, but the proportion of polarized light is a
maximum at the polarizing angle. The subject has been further
investigated experimentally by Jamin 1 , who found that for most
transparent substances, Brewster's law is true as a first approxi-
mation only. It is therefore not possible to completely polarize
light by a single reflection, but this may be accomplished by
successive reflections from a pile of plates. Jamin also found,
that when light which is plane polarized in any azimuth, is re-
flected from a transparent substance, the reflected light frequently
exhibits slight traces of elliptic polarization ; this shows, that
reflection produces a difference of phase in one or both of the
components of the reflected light 2 . The reflection and refraction
of light incident perpendicularly upon a glass plate have been
experimentally investigated by Rood 3 , and his results show that
Fresnel's formulae are very approximately correct.
178. When light proceeding from glass, is reflected at the
surface of a rarer medium such as air, at an angle greater than
the critical angle, it will be found that the values of A', A given
by (15) become complex; and it can be shown in the same manner
1 Ann. de Chimie et de Phys. (3), xxix. pp. 31 and 263; Ibid. (3), xxx. p. 257.
2 Owing to the extreme smallness of the wave-length of light, compared with
the ordinary standards of measurement, it is probable that if the surface of a
polished reflector were magnified to such an extent, that the wave-length of light
were represented by one inch, the surface of the reflector would appear to be
exceedingly rough and uneven. It is therefore by no means improbable, that the
secondary effects observed by Jamin, may be due to the fact, that our mathematical
machinery is too coarse-grained to take into account inequalities of the reflecting
surface, which though excessively minute, are not small compared with the wave-
length of light.
3 Amer. Jour, of Science, vol. i. July 1870.
!
184 FRESNEL'S THEORY OF REFLECTION AND REFRACTION.
that the reflection is total, and is accompanied by a change of
phase. In fact the reflected wave is
w f = A cos (cc cos i + y sin i Vt e
A/
where
and the refracted wave is
_
tan ~ ' ~
\
COS I
.(17),
[cos 2 i + (J? O 2 sin 2 i - 1)}*
i-i)ia. 2-7T, . . Tra
cos - (y sin i Vt -,
179. The change of phase which accompanies total reflection,
was experimentally verified by Fresnel in the following manner.
Let ABCD be a rhomb of glass, of which the angles at B
and D are greater than the critical angle ; and let light polarized
in a plane which makes an angle of 45 with the plane of incidence
(that is the plane of the paper), be incident normally upon the
face AB, and after undergoing two reflections emerge at the face
DC. The vibrations in and perpendicular to the plane of incidence
after emergence, will be represented by
\
If therefore e l e = J\,
the emergent light will be circularly polarized. Now if
TT (e l - e)/\ = S,
FRESNEL S RHOMB. '
we obtain from (12) and (17)
tan 8 =
whence
cos 28 =
Hence if 8 = JTT, this becomes
4//, 2 sin 4 i (2
This equation gives a real value of sin i for value&of p
between 1/4 and 1*6.
Fresnel employed a rhomb of St Gobain glass, for
^ = 1-51, which gives * = 48 37' 3" or 54 37' 20". Now the
angles at B and D of the rhomb are each equal to the angle
of incidence ; if therefore a rhomb of glass, whose index of re-
fraction is 1*51, and whose acute angles are equal to 54 37' 20"
be employed, and light polarized as described above is incident
normally on the face AB and is reflected twice, the emergent
light ought to be circularly polarized. This result was found to
agree with experiment.
If the incident light is polarized in any other plane, the
! emergent light will be elliptically polarized.
If a be the azimuth of the plane of polarization, and the
! emergent elliptically polarized light be passed through a second
rhomb, the reflected light will be plane polarized, and the plane
of polarization will be rotated through an angle 2a.
Theories of Neumann and MacCullagh 1 .
180. We must now consider the difficulty alluded to at the
commencement of 175.
The surface conditions assumed by Fresnel are, (i) continuity
of the rate at which energy flows across the reflecting surface,
(ii) continuity of the components of displacement parallel to this
surface. Now when the incident light is polarized in the plane
of incidence, there is no component displacement perpendicular to
this surface; but when the light is polarized perpendicularly to
1 Neumann, Abhand. Berlin AJcad. 1835.
MacCullagh, "On Crystalline ^Reflection and Kefraction." Trans. Boy. Irish
i Acad. vols. xvni. p. 31, and xxi. p. 17.
186 FRESNEL'S' THEORY OF REFLECTION AND REFRACTION.
the plane of incidence, it is impossible to evade the conclusion,
that the components perpendicular to the surface ought also to be
continuous. In fact a discontinuity in the normal displacement,
would involve something analogous to an area source in Hydro-
dynamics, and there are no grounds for supposing that anything
of the kind occurs.
The condition that the normal displacements should be con-
tinuous, is
( A + A') sin i = A l sin r.
Multiplying this by (14) we obtain
(A 2 A'' 2 ) sin i cos i = A-? sin r cos r (18).
From (7) the condition of continuity of energy may be written
( A 2 A'' 2 ) p sin i cos i = A^ p 1 sin r cos r,
and in order that this may be consistent with (18), we must have
p = pi. Accordingly Neumann and MacCullagh assumed this
condition in their theories of reflection and refraction; and we shall
now trace the consequences of this hypothesis.
When the vibrations are perpendicular to the plane of inci-
dence, the equations are
(A 2 - A 2 ) sin 2i = Af sin 2r,
whence A > = 4*Q^> }
'1,2* [ ( 19 >-
sin (i + r) cos (i r))
When the vibrations are in the plane of incidence
.(20).
. A. sin z't '
"i = ~~- 7"^, \
sin (t + r)
It follows from (2), that on this theory, the intensity of light
in all transparent media is proportional to the square of the
amplitude, and accordingly (19) and (20) give the intensities of
the reflected and refracted light. Neumann and MacCullagh
further supposed, that the vibrations of polarized light are in
instead of perpendicular to the plane of polarization, and on this
supposition the formulae (19) and (20) are in complete agreement
with (16) and (11) given by the theory of Fresnel.
EXAMPLES. 187
181. The two hypotheses of Neumann and MacCullagh are
singularly seductive, inasmuch as it will hereafter be shown, that
they enable the laws of the propagation of light in crystals, and
also the reflection and refraction of light from crystalline surfaces,
to be determined in accordance with Green's rigorous theory of
elastic media ; whereas the contrary assumption, that the density
of the ether is different in different media, leads to a variety
of difficulties in the application of this theory. There are however
grave objections to these hypotheses ; for in the first place, there
are strong grounds for supposing, that the vibrations of polarized
light are perpendicular to the plane of polarization ; and in the
second place, Lorenz and Lord Rayleigh have shown, as will be
explained in Chapter XII., that the hypothesis of equal density,
leads to the conclusion that there are two polarizing angles,
which is contrary to experiment.
EXAMPLES.
1. A thin layer of fluid of thickness T } floats on the surface of
a second fluid of infinitesimally greater refractive power. Light is
incident perpendicularly on the layer; show that the intensity of
the reflected light is
yU,
where //, and p 4 Sp are the refractive indices of the layer and of
the fluid which supports it respectively, and a, A are the amplitude
and wave-length of the incident vibration. "
2. If in the separating surface of two media, there be a
straight groove ^of small depth c, inclined at an angle a to the
plane of incidence, prove that there will be a groove in the re-
fracted wave of depth c sin (i r) cosec i, inclined at an angle
tan" 1 (tan a cosec r) to the plane of refraction, where i and r are
the angles of incidence and refraction.
What is the corresponding quantity for the reflected wave ?
Explain why the image of a candle from rough glass becomes
red, as the angle of incidence is diminished.
188 FRESNEL'S THEORY or REFLECTION AND REFRACTION.
3. A ray polarized at right angles to the plane of incidence
falls on a refracting surface ; if the intensities of the reflected and
refracted rays are equal, and the tangent of the polarizing angle
lies between 1 and 3, prove that the corresponding angle of
incidence is least, when the refracting medium is such that its
polarizing angle is JTT.
4. Circularly polarized light is incident in the usual manner
upon a Fresnel's rhomb, so cut that after one reflection in the
rhomb, the incident ray emerges from it perpendicularly to the
cut face. A uniaxal crystal cut parallel to the optic axis is placed
in the path of the emergent ray, with its faces normal to it, and
with its principal plane inclined at an angle JTT to the plane of
incidence and reflection in the rhomb. Show that the intensities
of the refracted rays are as */2 1 : \/2 4- 1.
5. If light polarized perpendicularly to the plane of incidence,
falls on a thin plate of air between two plates of different kinds of
glass, prove that there are 'two angles at which the colours will
disappear, and that between the two angles a change takes place
in the order of the colours.
6. A ray of circularly polarized light is incident at the plane
surface of separation of two media. If e and e are the excentricities
of the elliptically polarized light reflected and refracted, and i and
r the angles of incidence and refraction, show that
(1 - e 2 ) (1 - e*) = cos 2 (i + r).
CHAPTEE XL
GREEN'S THEORY OF ISOTROPIC MEDIA.
182. THE various dynamical theories of the ether, which have
been proposed to explain optical phenomena, may be classed under
three heads ; (i) theories which suppose that the ether possesses
the properties of an elastic medium, which is capable of resisting
compression and distortion; (ii) theories based upon the mutual
reaction of ether and matter; (iii) the electromagnetic theory
advanced by the late Prof. Clerk-Maxwell, which assumes that
light is the result of an electromagnetic disturbance. We shall
now proceed to consider the first class of theories.
183. The dynamical theory proposed by Green 1 , assumes that
the ether is an elastic medium, which is capable of resisting
compression and distortion. It therefore follows, that if the ether
is in equilibrium, and any element is displaced from its position
of equilibrium or is set in motion, the ether will be thrown into
a state of strain, and will thereby acquire potential energy. Now
the potential energy of any element of the ether, must necessarily
depend upon the particular kind of displacement to which it is
subjected ; hence the potential energy per unit of volume must
be a function of the displacements, or their differential coefficients,
or of both. If therefore we can determine the form of this function,
the equations of motion can be at once obtained by known dy-
namical methods.
According to the views held by" Cauchy, the ether is to be
regarded as a system of material particles acting upon one an-
other by mutually attractive and repulsive forces, such that the
1 Trans. Camb. Phil. Soc. 1838 ; and Math. Papers, p. 245.
190
GREEN'S THEORY OF ISOTROPIC MEDIA.
mutual action between any two particles is along the line joining
them; but inasmuch as the law of force is entirely a matter of
speculation, Green discarded the hypothesis of mutually attracting
particles, and based his theory upon the assumption that; In
whatever way the elements of any material system may act upon
one another, if all the internal forces be multiplied by the elements
of their respective distances, the total sum for any assigned portion
of the medium will be an exact differential of some function. This
function is what is now known as the potential energy of the
portion of the medium considered ; and Green showed that in its
most general form, it is a homogeneous quadratic function of
what, in the language of the Theory of Elasticity, are called the
six components of strain, and therefore contains twenty-one terms,
whose coefficients are constant quantities. For a medium which
is symmetrical with respect to three rectangular planes, the
expression for the potential energy involves nine independent
constants ; whilst for an isotropic medium it involves only two ;
one of which depends upon the resistance which the medium
offers to compression, or change of volume unaccompanied by
change of shape, whilst the other depends upon the resistance
which the medium offers to distortion or shearing stress, unac-
companied by change of volume.
184. The general theory of media, which are capable of
resisting compression and distortion, is given in treatises on
Elasticity ; and it will therefore be unnecessary to reproduce
investigations, which are to be found in such works. There are
however one or two points, which require consideration ; and we
shall commence by examining the stresses, which act upon an
element of such a medium.
EQUATIONS OF MOTION OF THE ETHER. 191
Let the figure represent a small parallelepiped of the medium.
The stresses which act on the face AD are,
(i) A normal traction X x parallel to Ox ;
(ii) A tangential stress or shear T x parallel to Oy ;
(iii) A tangential stress or shear Z x parallel to Oz.
Similarly the stresses which act upon the faces ED and CD,
are Y y , Z y) X y and Z Zt X z , Y z .
These are the stresses exerted on the faces AD, BD, CD of
the element by the surrounding medium ; the stresses exerted
by the medium on the three opposite faces will be in the opposite
directions.
185. In order to find the equations of motion, let u, v, w be
the displacements parallel to the axes, of any point x, y, z ; p the
density, and X, Y, Z the components of the impressed forces
per unit of mass. Then resolving parallel to the axes, we obtain
the equations
(1).
These equations express the fact, that the rates of change of
the components of the linear momentum of an element of the
medium, are equal to the components of the forces which act upon
the element. It is however also necessary, that the rates of change
of the components of the angular momentum of the element
about the axes, should be equal to the components of the couples
which act upon the element. Whence taking moments about the
axis of x, we obtain
d? " z 3!
{y (IZ X + mZ y + nZ z ) - z (IY X + mY y + nY z )} dS. . .(2),
where dS is an element of the surface of the portion of the
medium considered, and /, m, n are the direction cosines of the
normal at dS.
d z u
nY 0-
dX x
\ ^
, dX z
P dt*
pA T
oY-\-
dx
dy
dz
P dt*
P 1
n7 4-
dx
dZ x
dy
dZy
dz
{ dZ,
p dp
dx
dy
dz
192 GREEN'S THEORY or ISOTROPIC MEDIA.
Transforming the surface integral into a volume integral 1 , and
taking account of (1); (2) reduces to
Jff(Z y -Y t )da;dydz = (3),
which requires that Z y = Y g . It can similarly be shown, that
X Z = Z X> and Y x = X y . These results show, that the component
stresses are completely specified by the six quantities X Xj Y y) Z z ,
Y Z) Z X) X y> which we shall denote by the letters P, Q, R, 8 f T, U.
Equations (1) may now be written
.(4).
propagation is seen from (13) to be equal to (m + w)*/p* ; whilst '
the other involves rotation and distortion, without change of
volume, and whose velocity of propagation is (n/p)%. The first
type of waves depends partly on the rigidity and partly on the
elasticity of volume ; whilst the second type depends solely on
the rigidity, and is therefore incapable of being propagated in a
medium devoid of rigidity, such as a perfect gas. Hence if any
disturbance, which involves change of volume and distortion, be
communicated to a portion of the medium, two distinct trains of
waves will be produced ; one of which consists of a condensation
DILATATIONAL AND DISTOKTIONAL WAVES. 195
and rarefaction, which is propagated with a velocity (m
whilst the other consists of a distortion or change of shape, which
is propagated with a velocity (n/p)%.
189. Let us now suppose that a train of plane waves is
propagated through the medium, the direction cosines of whose
fronts are I, m, n. Since equations of the form (13) and (14) are
satisfied by the function
where F is the velocity of propagation, we may suppose that the
resultant displacement S is
$ = F(lx + my + nz Vt) ;
hence if X, /z, v be the direction cosines of the direction of displace-
ment, we shall have
u = S\, v = S/JL, w = Sv.
Whence S = (l\ + ra/z + nv) S',
If the direction of displacement is in the front of the wave, so
that the vibrations are transversal, 4 /^ ^ t ^^i v <^
l\ 4- m/j, + nv = 0,
whence S = 0.
If on the other hand the displacement is perpendicular to the
front of the wave, so that the vibrations are longitudinal,
l = \ )m = /ji) n = v " <^t-~^
whence f = 0, 77 = 0, ?= 0, S = S'.
It therefore follows that when the vibrations are longitudinal,
dilatational waves unaccompanied by rotation or distortion are
alone propagated; whilst if the vibrations are transversal, distor-
tional waves unaccompanied by condensation or rarefaction are
alone propagated.
Since the phenomenon of polarization compels us to adopt the
hypothesis, that the vibrations which constitute light are trans-
versal and not longitudinal, we must suppose that the portion of
the disturbance, which consists of distortional vibrations, is alone
capable of affecting the eye.
132
196 GREEN'S THEORY or ISOTROPIC MEDIA.
190. Let us now suppose, that a wave of light consisting of
transversal vibrations in the plane of incidence, is refracted
through a prism. Then it is not difficult to show, that the
incident wave will give rise to two refracted waves, which
respectively consist of transversal and longitudinal vibrations;
and since the velocities of propagation of these two waves are
different, their indices of refraction will be different, and thus the
two refracted rays will not coincide. The refracted wave, whose
vibrations are normal to the wave-front, will be divided on
emergence at the second face of the prism into two more refracted
waves, one of which will consist of transversal and the other of
normal vibrations. Thus, even though we assumed that waves jj
consisting of normal vibrations are incapable of affecting the eye ;
it would follow, in the first place, that a wave of normal vibrations
might give rise to a wave of transversal vibrations, and consequently
the sensation of light might be produced by something which is
not light ; and in the second place, that whenever light polarized
perpendicularly to the plane of incidence is refracted through a
prism, there ought to be two refracted rays. This is altogether
contrary to experience, hence our theory of the ether without
some further modification is defective. When we discuss the hi
reflection and refraction of light, it will be proved that the refracted i
wave whose vibrations are normal, will become insensible in the t i
second medium, at a distance from the face of the prism which is
equal to a few wave-lengths, provided the ratio of the velocity of
propagation of the normal wave to that of the transversal wave, \
is either very great or very small ; that is whenever (m + n)[n is
very large or very small. Now (m + n)/n = (k + $ri)/n, hence this
ratio will be large, if k is large compared with n. But if a uniform ]
hydrostatic pressure be applied to every point of the surface of a
spherical portion of the medium, k is the ratio of the pressure to
the compression produced. If therefore k is large compared with
n, the power which the medium possesses of resisting compression,
must be very great in comparison with its power of resisting
distortion. On the other hand, if the ratio (k + %ri)/n were
positive and very small, it would be necessary for A; to be a
negative quantity, whose numerical value is equal to or slightly
less than $n. But inasmuch as in this case, an increase of
pressure would produce an increase of volume, and as no known
substance possesses this property, Green concluded that k is very
large compared with n. The difficulty of satisfactorily accounting
CRITICISMS ON GREENS THEORY. 197
for waves of normal vibrations, whose existence we are forced
to admit, is thus to a great extent, though not entirely, overcome.
For since the velocity of light in vacuo is about 299,860 kilometres
per second, the velocity of propagation of the condensational
waves would be enormously greater than those of the distortional
waves ; and it is therefore not unreasonable to suppose, that they
are incapable of affecting our eyes. At the same time, the
amount of energy existing in the universe, which is due to these
waves, must be very large ; and the assumption, that this large
; amount of energy is incapable of producing any effect of which
our senses are capable of taking cognizance, is not very satis-
factory.
Sir W. Thomson has lately proposed a theory, in which it is
assumed that & is a negative quantity, which is numerically equal
to, or slightly less than f n. This theory will be considered when
we discuss double refraction; for the present we shall confine our
attention to Green's theory, in which k is supposed to be large
compared with n.
191. Green's hypothesis has sometimes been supposed to
, involve the assumption, that the ether is very nearly incompressible.
This however is an altogether erroneous view, for as a matter
i of fact the ether might be more compressible than the most
highly compressible gas; all that is necessary is, that the ratio
of the resistance to compression to the resistance to distortion
should be very large. Moreover since the velocity of light in
vacuo is about 299,860 kilometres per second, p must be very
small in comparison with n. That k, n and p must all be
exceedingly small quantities compared with ordinary standards, is
proved from the fact, that the most delicate astronomical observa-
tions have not succeeded in detecting with any certainty, that any
resistance is offered by the ether to the motions of the planets;
although it has been suggested that the irregularities, which are
observed in the motions of some of the comets, may be referred to
this cause. But whether this be so or not, Green's hypothesis
requires us to suppose, that p, n and k are exceedingly small
j quantities in ascending order of magnitude, such that n is large in
comparison with p, and k is large compared with n.
CHAPTER XII.
APPLICATIONS OF GREEN'S THEORY.
192. IN the present chapter, we shall apply Green's theory to
investigate the reflection and refraction of light at the surface of
two isotropic media, the theory of Newton's rings, and the
reflection of light from a pile of plates.
We have stated in 10, that the intensity of light is measured
by the mean energy per unit of volume. Hence if T and W
denote the kinetic and potential energies of a plane wave per ;
unit of volume, which is being propagated parallel to x, we have
Hence if w = A cos (x Vt),
A/
then
^sin'^-FO;
and since n = V 2 p, it follows that the kinetic and potential
energies are equal. Accordingly the mean energy per unit of
volume is
which measures the intensity.
POLARIZATION IN THE PLANE OF INCIDENCE. 199
Reflection and Refraction 1 .
193. We are now prepared to consider the problem of reflection
id refraction.
Let the axis of x be perpendicular to the reflecting surface, and
let the axis of z be parallel to the intersections of the wave-fronts
with the same surface ; and let us first suppose, that the incident
light is polarized in the plane of incidence.
x
Since u and v are zero, it follows from (12) of 187, that the
equations of motion are
d 2 w TTo /
dtf
df)
in the first medium and
(2)
.(3)
dt* J tf 2 df
in the second, where V 2 = n/p, V-? n^p-^.
The conditions to be satisfied at the surface of separation are,
that the displacements and stresses must be the same in both
media. These conditions will often be referred to, as the con-
ditions of continuity of displacement and stress.
The first condition gives
w = Wj_ (4),
and the second
dw _ dw l
dx dx
.(5).
1 Green, Math. Papers, pp. 245, 283. Hon. J. W. Strutt (Lord Kayleigh), Phil.
Mag. Aug. 1871. Kurz, Pocjg. Ann. vol. cvni. p. 396.
200 APPLICATIONS OF GREEN'S THEORY.
Let A } A', A! be the amplitudes of the incident, reflected and
refracted waves, then we may write
where K = 2-7T/A,, /q =
The vibration in the second medium is a forced vibration
produced and maintained by the incident waves, hence the
coefficients of t must be the same in all three waves. Also the .
coefficients of y must be the same, since the traces of all three
waves on the plane oc move together. Hence
But l = cos i,
K i Vi ) Km/ = A
jiT/ii . .
(8).
V = cos i, I
i = cos r \
(9).
m' = sini, m
a = sinrj
From these equations we see that
V sin i \
.(10).
i snr j
The first of these equations is the well-known law of sines,
whilst the second expresses the condition, that the period of the
refracted wave must be equal to that of the incident.
Substituting from (6) and (7) in (4) and (5), we obtain
A+A =A l}
(A A') ntc cos i = A^n^ cos r,
the last of which may be written
A A , AM tan i
A. A = ;
n tan r
whence ^, = _4 (n. tan '-n tour)
n-L tan i + n tan r
. 2An tan r
H! tan i + n tan r
If /, /'.. 7j be the square roots of the intensities, it follows from
(1) that
Up to the present time, we have not assumed that any
relation exists between n and n v . If however we assume with
Green, that the rigidities are the same in all isotropic media, and
HYPOTHESIS OF NEUMANN AND MAC OULLAGH. 201
that refraction is consequently due to a difference of density, we
must put n = HI and we obtain
Jsinff-r)
'
sm (i + r)
rhich are the same as Fresnel's formulae.
If on the other hand we adopt the hypothesis of Neumann and
MacCullagh, that the density of the ether is the same in all media,
and that refraction is consequently due to a difference of rigidity,
it follows that
n _ V' 2 _ sin 2 i
n 1 ~Vf~ sin 2 ~r '
and we obtain /' = - '" V '/ (15),
tan (i + r)
T _ * SID- *& (~\R\
sin (i + r) cos (i r)'
These expressions are the same as Fresnel's formulas for the
intensity of light polarized perpendicularly to the plane of
incidence.
194. The preceding formulae do not enable us to decide the
question, whether the vibrations of polarized light are in or
perpendicular to the plane of polarization ; or whether the
hypothesis of Green on the one hand, or of Neumann and
MacCullagh on the other, is the best representative of the facts.
For the present we shall adopt Green's view, and shall proceed to
calculate the change of phase which occurs, when the angle of
incidence exceeds the critical angle.
If ju is the index of refraction
hence if p : > p. /JL >1, and r is always real ; but if p 1 < p, p < 1, and
the angle of refraction becomes imaginary, when the angle of
incidence exceeds the critical angle.
When p 1 < p, we shall write //T 1 for //,, so that jt denotes the
index of refraction from the rarer into the denser medium, and
ft sin i sin r.
202 APPLICATIONS OF GREENS THEORY.
Since tbe expressions for the amplitudes of the reflected and
refracted waves become complex, we must write
........................ (18),
in which a, ]3, a 1} &, are real. Now
j = cos r = i (/A 2 sin 2 i 1)*
*i = *//*>
whence ticjh = (2-TT/XyLt) (^ sin 2 i - 1)* = KC^ (say),
the lower sign being taken, because x is negative in the second
medium.
The boundary conditions give
- a - 1 cos * =
, Equating the real and imaginary parts, we obtain
^. cos ^ - o 2 A cos 2 1
' ire ^ (p? sin 2 1 -
Let tan-- ==- . = - : '- ............... (19),
, \ cos i
then a + ^ = ^e"^, a x + i& = 2^e' lJre cos TT^/X ;
whence the reflected wave is
2_
w = ^1 cos (x cos * + y sin i Vt e)
A*
and the refracted wave is
ZAu, cos i rit!n*-n*fl 2?r . T ,, , x
^ " O* 8 -!)* 6V C S "X" (2/ Sm * " ~ ie)< ' <( )v
From these equations we see, that when the angle of incidence
exceeds the critical angle, the reflection is total and is accompanied
by a change of phase, whose value is given by (19).
Since the refracted wave involves an exponential term in the
amplitude, it becomes insensible at a distance from the surface
which is equal to a few wave-lengths.
All the foregoing results are in agreement with Fresnel's
formulas.
VIBRATIONS IN THE PLANE OF INCIDENCE. 203
195. The investigation of the problem, when the light is
polarized perpendicularly to the plane of incidence is more
difficult.
In this case w = Q, and by (12) of 187, the equations of
motion in the upper medium are
6?u ^ d fdu dv^
d?v . . d du dv
ith similar equations for the lower medium. In these equations
k is the resistance to compression, which Green supposes to be
very large compared with n.
The boundary conditions are
U=U L , v = v, (22),
4 \du . /, 2 \dv / 7 .4 \du l . /, 2
du dv\ _
t dx)
of which the first two express the conditions of continuity of
displacement, and the last two the conditions of continuity of
stress.
We have therefore four equations to determine two unknown
quantities. Now we have already shown, that elastic media are
capable of propagating waves of two distinct types ; viz. dilatational
waves, which involve condensation and rarefaction, and distortional
waves, which involve change of shape without change of volume.
When the vibrations of the incident wave are not parallel to the
reflecting surface, there will be a dilatational as well as a
distortional reflected and refracted wave, whose amplitudes must
be determined by (22) and (23); accordingly we have four
unknown quantities and four equations to determine them.
When the resistance to compression is very large, 8 or dujdx + dvjdy
is very small, but we are not at liberty to treat the latter quantity
as zero, because k8 is finite ; we must therefore introduce the
dilatational waves.
T dd> d^lr d(f> d^lr /n4\
Let u = -5 K-TT-J V -J =- v/"*/
dx dy dy dx
jj^ /T, I 4 \/ 1^2 / ^9^
204 APPLICATIONS OF GREEN'S THEORY.
so that U and V are the velocities of the dilatational and distor-
tional waves respectively. Then
^ + ^ = V 2 , - = V 2 >|r,
dec dy dy dx
where, by (21), < and ^ respectively satisfy the equations
(26}
dty_
dt*~
with similar equations for the lower medium.
Let us now assume that in the first medium
= K (lx+my- Vt) '( - Ix+my - Vt)
and in the second medium
In these equations the coefficients of y and must be the same
in all the waves, whence
K m = /CjT/ij, /cV=K l V l .................. (29),
also since the dilatational wave is propagated with velocity U,
it follows that its wave-length is equal to U\/V which is therefore
very large compared with \.
Substituting from the second of (27) in the first of (26), we
obtain
F 2 = (a 2 + m 2 )^ ........................ (30),
and since V/U is very small, we shall have to a sufficient
approximation
ia = ra ;
the lower sign being taken, because x is positive in the upper
medium.
Similarly from the second of (28) and (26), we shall obtain
*i 2 V? = K? (of + m*) U? = (tfa^ + * 2 O U, 2 ...... (31).
Whence approximately
#!&! = /cm ;
because x is negative in the lower medium.
v -
* -
VIBRATIONS "i THE* PLANE OF INCIDENCE. 205
*~\ (3
uc my- t J
We may therefore write
i _ rtf
(32).
From (22) and (24) we obtain
B / m-(A-A')l = B l m-A lKl l 1 / f c\ ( "
We shall now assume with Green, that n = n l ; whence, since
the continuity of u involves the continuity of dujdy, the last of
(23) reduces to
dvjdx = dv l /dx,
or - B'm* -(A + A') iP = Ejtf - A^lf/ii* .(34).
The first of (23) may be written ^ A
which, through the continuity of v and dv/dy, becomes
Substituting in this from the second of (27) and (28), we
obtain R'Q V*#* - /] a Y t *.
B'p U Z K* (a 2 + m 2 ) = B IPI Ufic? (of + mf) ; &* *> = fy ^[ m
by (30) and (31) ; since
F 2 /o = V l 2 p l = n, and K = 2?r/X.
Introducing the index of refraction /t, which is equal to
(35) may be written
-//"* *~ U. 1/J -I t * A
From this equation, (34), and the lSstT6f (33), we obtain * (
B'= L - ^a^ 1 ^i = ^ 1 2^ + i ( 36 )-
Using these in (33), we obtain
(tan
^-^ = H"a^~" Xr: '^~ tanih
and therefore
= A , 1 /A 2 4- an -^ -
' .(37).
tan r
206 APPLICATIONS OF GREENS THEORY.
In order to realize these formulae, let
2A = A.Re^ 2A' = A,R'e-' ;
then A' I A = (RjR) e- <*+>, A, = ZA/Re" ;
whence if M = (fj? - 1)/O 2 + 1),
M (uP 1) tan i tan r
tan e = ^- T
yu, 2 tan r 4- tan i
= Mtan(i-r) ........................ (38),
since //, = sin I'/sin r.
Similarly tan e' = - Jf tan (i + r) ..................... (39),
whence
cos {* (- ^ + y - Vt ) - e ~
cos /ci ^ + m - Fx - e
These expressions show, that when light is polarized per-
pendicularly to the plane of incidence, the reflected and refracted
waves experience a change of phase, which is determined by (38)
and (39).
The amplitudes of the reflected and refracted waves are
determined by the equations
R' 2 _ (^ cot i - cot r) 2 + M 2 (^ - 1) 2
R* ~ (fj? cot i + cot r) 2 + M* (tf - 1) 2
-
and
_4_ _ _ 4 sin 2 r cos 2 i , .
R 2 ~ sin 2 1 sin 2 (i + r) (cos 2 (i - r) +M sin 2 (i - r) cosec 2 r} ""'
196. These equations do not agree with the results furnished
by Fresnel's theory unless M=0.
According to Fresnel's formula, when light polarized per-
pendicularly to the plane of incidence is incident at the polarizing
angle, the intensity of the reflected light is zero; experiment
however shows that this result is not rigorously correct, inasmuch
as the intensity of the reflected light does not absolutely vanish,
but attains a minimum value. On the other hand Green's
formula deviates too much the other way, and shows that too
much light is reflected at the polarizing angle.
CHANGE OF PHASE. 207
197. To calculate the change of phase, when light is re-
flected at the surface of a rarer medium at an angle greater
than the critical angle, we shall denote as before the index of
refraction from the rarer to the denser medium by /u- ; we must
therefore write /JT I for //, in (37).
Now, sin r = //. sin i,
p sin i
tan r = +
sn *
-!)*'
Kill tan i i (u? sin 2 i 1 )*
and ~~r = i - = - -
Ki tan r fi cos i
also since iK-h is a real positive quantity, and I = cos i, the upper
sign must be taken ; accordingly (37) become
yL6 2 //, COS I
/A COS I (JL
These equations may be written in the form
2A = A,Re el \ 2A' = AiRr 1 ^ ;
from which we get A' = ^e~ 2t7reA ,
where tan ^- = //, (p 2 tan 2 1 - sec 2 *)* - ^ ^ tan t . . . . (43).
A< fJf + 1
Accordingly the reflected wave is
2_
i// = A cos ( Ix + my T 7 ^ e),
A,
which shows that the reflection is total, and is accompanied by a
change of phase, whose value is given by (43). The first term of
the change of phase agrees with that which we have already
obtained from Fresnel's theory, whilst the second is small unless
the medium is highly refracting, or the angle of incidence is
large.
When the incident light is plane polarized in any azimuth,
the incident vibrations may be resolved into two components,
which are respectively in and perpendicular to the plane of
incidence; and from (19) and (43), we see that when the light
is totally reflected, total reflection is accompanied by a change of
phase, whose value is not the same for the two components.
208 APPLICATIONS OF GREEN'S THEORY.
Accordingly the reflected light under these circumstances is
elliptically polarized. This remark will be found to be of import-
ance, when we consider the selective reflection, which is produced
by certain of the aniline dyes.
198. We have already pointed out, that in the theories of
Neumann and MacCullagh, it is assumed that p = pi, in which
case reflection and refraction would be due to a difference of
rigidity. We have also seen, that in Fresnel's theory the con-
ditions of continuity of displacement violate the condition of
continuity of energy, unless p = p^ It therefore becomes important
to enquire what results would be furnished by Green's theory,
qn the supposition that the density of the ether is the same
in all media. This point has been examined by Lorenz 1 and
Lord Rayleigh 2 , and the results are decisive against the hypo-
thesis in question.
In this case, the boundary conditions (33), which express
continuity of displacement, remain the same as before; whilst
(23) become
,,
dy) dy \dx dy I dy
From the first equation, we obtain
& DV (a 2 + m 2 ) - 2 FVm {B' ............. .'.(53),
A! L q~
and is equal to or 6', according as the light is polarized in or
perpendicularly to the plane of incidence.
203. The corresponding formulae for the transmitted light
can be obtained in a similar manner. When the superficial wave
arrives at the second surface, its amplitude is equal to cq ; the
amplitude of the refracted portion is cqf, and that of the reflectec
portion is cqe ; whence after one reflection at the first plate, anc
one refraction at the second plate, the portion refracted at the
latter becomes cfePf. Hence the amplitude of the refractec
portion is
Substituting the value of b, this becomes
__ 2tq sin 20
(1 - g 2 ) cos 20 + 4 (1 + f) sin 20
_ 2q sin 2(9 {(1 + (f) sin 20 + * (1 - g 2 ) cos
(1 - 2 ) 2 + 4# 2 sin 2 20
Whence the intensity of the transmitted light is
and the transmitted wave is
Pi c s ^ (x cos i y sin i + Vj
where tan = - cot 20 . . . . (55).
A! 1 + q 2 sin 2 20'
'
(56) -
Now the distinctness of the black spot, which is produced by
reflection, depends upon s being large; and since in the neigh-
bourhood of the critical angle, we have from (56), s. 2 = s l /jt, 4 , it
follows that the spot is much more conspicuous for light polarized
2L4 APPLICATIONS OF GREEN'S THEORY.
perpendicularly to the plane of incidence, than for light polarized
in that plane. As i increases, the spots seen in the two cases
become more and more nearly equal, and become exactly alike
when i = i' t where cosec a i'= J(l +/* 2 ). When i becomes greater
than i', the order of magnitude is reversed, and when i ^TT,
s l = s. 2 fj,*, so that the inequality becomes large. It must however
be recollected, that this statement refers to the relative magnitudes
of the spots, for when the angle of incidence is nearly equal to |TT,
the absolute magnitudes of the spots become very small.
All the conclusions deduced by the above theory have been
verified experimentally by Stokes, and he has also discussed the case
in which the incident light is polarized in a plane, making a given
angle with the plane of incidence.
The Intensity of Light reflected from a Pile of Plates.
POLARIZATION BY A PILE OF PLATES.
215
208. Let p be the fraction of tlie light, which is reflected at
the first surface of a plate ; then 1 p is the fraction of the light
which is transmitted.
Since the light reflected by a plate is made up of that which is
reflected at the first surface, and that which has suffered an odd
number of internal reflections, it follows that if the intensity of the
incident light be taken as unity, the intensity of these various
portions will be
p, (1 - p} 2 pg' 2 , (1 - p)*py, etc.
Hence if R be the intensity of the reflected light,
R = p + (i - pypf- (i + />y + PY +)
Similarly if T be the intensity of the transmitted light,
.(59).
209. The value of p depends upon the particular theory
of light which we adopt, but in any case it may be supposed to be
a known function of i the angle of incidence and fju the index of
refraction ; the value of g depends upon i and p, and also upon q,
which may be supposed to have been determined by experiment.
To complete the solution, we have therefore to solve the following
problem : There are m parallel plates, each of which reflects and
transmits given fractions R and T of the light incident upon it ;
light of intensity unity being incident upon the system, it is required
to find the intensities of the reflected and refracted light.
Let these be denoted by (m), ty (m) ; and consider a system of
m-rn plates, and imagine them grouped into two systems of
m and n plates respectively. Since the incident light is repre-
sented by unity, < (m) will be the intensity of the light reflected
from the first group, whilst -fy (m) will be transmitted. A fraction
(ri) of the latter will be reflected by the second group, whilst a
portion \jr (n) will be transmitted ; and the fraction (f> (m) of the
light reflected by the second group will be reflected by the first
group, whilst the fraction yfr (m) will be transmitted, and so on. It
therefore follows, that the intensity of the light reflected by the
whole system will be
< (m) + (^mY (/> (n
216 APPLICATIONS OF GREEN'S THEORY.
and the intensity of the light transmitted will be
ty (m) n)~ (0ra) J ^ (n) 4- ...
The first of these expressions is equal to (m + ii\ whilst the
second is equal to ty (m + w) ; whence summing the two geometrical
series, we obtain
.......... (60) -
In the special problem under consideration, m and n are
positive integers; but we shall now show how to obtain the
solution of these two functional equations, when m and n have
any values whatever. From (60) we obtain
(m + n) (1 - (m) (n)} = (m) + (n) {(^m)* - (*)-] . - .(63).
In order that (60) and (61) may hold good for a zero value of
one of the variables, say n, we must have (0) = 0, ^ (0) = 1. If
however we put n = in (63), the equation reduces to an identity ;
we must therefore differentiate (63) with respect to n, and then
put n = 0. Accordingly we find
0' (0) (m) {1 - 20 (m) cos a + (0m) 2 { + 0' (m) cos a - (m) $ (m)
= (1 - 20 (m) cos a + (0m) 2 } 0' (0) cos a.
Dividing out by (m) cos a, since the solution (m) = cos a
would lead to ty (m) = (7, we obtain
$ (m) = 7 (0) {1 - 20 (m) cos a + (0m) 2 ] (64).
POLARIZATION BY A PILE OF PLATES. 217
Integrating this equation, and determining the arbitrary
constant from the condition that 0(0) = 0, and writing ft for
'(0)sin a, we obtain
A ( \- ^ m P (t'fC\
v* sin (a + mft)
Substituting in (61) and reducing, we find
sin a , ,
xjr(w)= . , ~ (66).
Equations (65) and (66) may be written in the form
0?0_ _ ty( m ) _ 1 / QtJ\
sin mft sin a sin (a + mft)
When in = 1, $ (?tt) = R, ^ (m) = T, where the values of R and
T are given by (58) and (59) ; and therefore to determine the
arbitrary constants, we have the equations
R T 1
sin ft sin a sin (a + ft)
(68).
210. Equations (67) and (68) give the following quasi-
| geometrical construction for solving the problem : Construct a
triangle, in which the sides represent in magnitude the intensities
\ of the incident, reflected and transmitted light in the case of a
: single plate; then leaving the first side and the angle opposite
to the third unchanged, multiply the angle opposite the second,
by the number of plates ; then the sides of the new triangle will
;. represent the corresponding intensities in tlie case of a system of
plates. This construction cannot however be actually effected,
inasmuch as the first side of the triangle is greater than the
1 1 sum of the two others, and the angles are therefore imaginary.
To adapt the formula to numerical calculation, it will be
\ convenient to get rid of the imaginary quantities. Putting
we have by ordinary Trigonometry
whence putting (1 + R>- T* + &)/2R = a .................. (70),
we have e ta = cos a + i sin a = a Tl .
218 APPLICATIONS OF GREENS THEORY.
Choosing the lower signs, we have
2R sin a = - tA, t ta = a ;
1 + ^-12* flsina *
also cos p = -- -- , sin p = / -
Whence if (1 + T*- & + A)/2r=6 ............... (71),
we shall have e^ = &,
arid (67) becomes
b m - b~ m a - a' 1 ab m - or 1 b~ m
211, From this equation we see, that the intensity of the
light reflected from an infinite number of plates is or 1 ; and I
since a is changed into a" 1 , by changing the sign of a or A,
we have
a- 1 = (!.+ .#+ a** -A)/2jR ............... (73),
which is equal to unity in the case of perfect transparency, -j
Accordingly substances, such as snow and colourless compounds ^
thrown down as chemical precipitates, which are finely divided
so as to present numerous reflecting surfaces, and which are
transparent in mass, are brilliantly white by reflected, light.
212. The following tables, taken from Stokes' paper, give the i
intensity of the light reflected from, or transmitted through, a
pile of m plates for the values 1, 2, 4 and oo of ra for three
degrees of transparency, and for certain selected angles of in-
cidence. The refractive index is taken to be equal to 1'52 ;
8 = 1 e~ qT is the loss by absorption in a single transit through
a plate at perpendicular incidence, so that 8 = corresponds to
perfect transparency ; also the value of p is supposed to be
calculated from Fresnel's formulae, so that
sin 2 (i - r) tan 2 (i r) /t _ . ,
p = ~ ;. ; or ;. ; ............... (74),
sm 2 (i + r) tan 2 (i + r)
according as the light is polarized in or perpendicularly to the
plane of incidence. The angle OT is the polarizing angle tan" 1 /JL ;
(f> and -v/r denote the intensities of the reflected and transmitted
light, the intensity of the incident light being taken as 1000.
For oblique incidences, it is necessary to distinguish between
DISCUSSION OF THE RESULTS.
219
light polarized in and perpendicularly to the plane of incidence,
and the suffixes 1 and 2 refer to these two kinds respectively.
5=0
in
1.1
i=*
< = + *>
*
0i
*
0i
*
02
+*
*^.
1
82
918
271
729
300
700
1
999
701
2
151
849
426
574
459
541
2
998
542
4
262
738
598
402
628
372
4
996
373
8
416
584
749
251
771
229
8
992
231
16
587
413
856
144
870
130
16
984
132
32
740
260
922
78
931
69
32
968
071
00
1000
1000
1000
1000
000
II.
5 =-02
5='l
m
<=0
i = v
i =
torn
*
0i
i
Mfc
4
0i
*,
t'i ' tllt'2
I 80
900
265
711 \ 976
728
74
826 245
639
881
725
2 i 145
815 410 544 ! 953
571
125
686 351
435
777
559
4 244
679 555
355 ! 908
391
185
479 427
215
604
357
8 ' 364
490 656
182 824
221 \ 229 ! 237 451
57
365
156
16 464
276 695
58 679
086 243
59 453
4
133
030
32 509
97 699
7 461
014
244
4
453
18
001
GC
1000
699
000 244
453
000
213. In discussing these tables Sir G. Stokes says : "The
intensity of the light reflected from a pile consisting of an infinite
number of similar plates, falls off rapidly with the transparency
of the material of which the plates are composed, especially at
small incidence. Thus at a perpendicular incidence, we see from
the above table that the reflected light is reduced to little more
than one half, when 2 per cent, is absorbed in a single transit ;
and to less than a quarter, when 10 per cent, is absorbed.
" With imperfectly transparent plates, little is gained by
multiplying the plates beyond a very limited number, if the
object be to obtain light, as bright as may be, polarized by
reflection. Thus the table shows, that 4 plates of the less
220 APPLICATIONS OF GREENS THEORY.
defective kind (for which 8 '02) reflect 79 per cent. ; and 4
plates of the more defective kind (for which = '!) reflect as
much as 94 per cent, of the light, that could be reflected by
a greater number; whereas 4 plates of the perfectly transparent
kind reflect only 60 per cent.
" The table also shows, that while the amount of light
transmitted at the polarizing angle by a pile of a considerable
number of plates is materially reduced by a defect of transparency,
its state of polarization is somewhat improved. This result might
be seen without calculation. For while no part of the transmitted
light which is polarized perpendicularly to the plane of incidence
underwent reflection, a large part of the transmitted light polarized
the other way was reflected an even number of times ; and since
the length of path of the light within the absorbing medium is
necessarily increased by reflection, it follows that a defect of
transparency must operate more powerfully in reducing the
intensity of light polarized in, than of light polarized perpen-
dicularly to the plane of polarization. But the table also shows,
that a far better result can be obtained, as to the perfection of the
polarization of the transmitted light, without any greater loss of
illumination, by employing a larger number of plates of the more
transparent kind."
214. We shall now confine our attention to perfectly trans-
parent plates, and consider the manner in which the degree of
polarization of the transmitted light varies with the angle of
incidence.
The degree of polarization is expressed by the ratio
which we shall denote by %. When % = 1, there is no polarization ;
and when ^ = 0, the polarization is perfect in a plane perpendicular
to the plane of incidence. Now when <7 = 1, it follows from (58)
and (59) that
20 1-0
P *P rn_ *- P
*~~ 1 > """" T 9
whence R + T=l; accordingly from (70) and (71), a and b are
each equal to unity, and (72) becomes indeterminate. Now when
a and 6 are nearly equal to unity, a and ft become indefinitely
small, whence (67) becomes
< (m) _ ^ (m) _ 1
mft a a + mft '
PERFECTLY TRANSPARENT PLATES. 221
Also from (68), @ = Ea/T, whence
__
l- -"
p
Let i r = 0, i-t-r = o-, //, sin r = sin i
then
cfo' dr dO da-
tan i - tan r tan i + tan r
whence d0 = sin 0d ;
and since cos (and there-
fore i) decreases, or as m increases. For m = 1, i = JTT, and
%i = A 6 " 2 5 f r m = > cos ; we may therefore put
dd> dd) dd)
u i= j v i = j > w i = j ( 2 )'
dx dy dz
where < is some function of cc, y, z and t.
Since the displacements u?,, v 2 , W 2 do not involve dilatation, it
follows that
- -\ r-^ -j =-^ = (3).
dx dy dz
224 DYNAMICAL THEORY OF DIFFRACTION.
From (1) we see that
also putting u = d/dx + u 2 in (1), and taking account of (4), we
l
(5),
with two similar equations for v 2 , w. 2 . It therefore follows that
the most general solution of (1) is
d(p d
dx dy dz
where is any function of x, y, z and t which satisfies (4), and
u 2 , v 2 , w 2 are similar functions, which satisfy equations of the form
(5), subject to equation (3).
Propagation of an Arbitrary Disturbance.
216. We shall now apply equations (1) to obtain the solution
of a problem which was first investigated by Sir G. Stokes, viz. -
the propagation of an arbitrary disturbance in an elastic medium 1 .
Let us suppose that the medium is initially at rest, and that a
disturbance is excited throughout a certain volume T of the
medium. The subsequent character of the disturbance is com-
pletely determined, when the initial displacement and the initial
velocity of every element within T is known. Let P be any point
within T, the point at which the disturbance is sought, and let
us first consider that portion of the disturbance which depends
upon the dilatation 8.
By equations (3) and (6), it follows that 8 = V 2 , where <
satisfies (4); and it will be more convenient to consider the
function than 8, for when the former function is known, the
portions of the displacements which depend upon the dilatation
can be immediately obtained by differentiation.
If .r, y, z be the coordinates of 0, the initial values of c/> and <,
will depend entirely upon the position of 0, and we must therefore
have
1 Trans. Camb. PMl. Soc. Vol. ix. p. 1, and Math, and Pnys. Papers, Vol. n.
p. 243.
POISSONS SOLUTION. 225
where / and F are given functions. We therefore require the
solution of (4) subject to (7).
217. The solution of (4), which was first obtained by Poisson,
may be effected as follows. The symbolic solution is
(f> = cosh (atV) x + sinh (atV) ty,
where ^ and ty are functions of x, y, z\ we therefore obtain
from (7)
and J T = =
Accordingly the solution becomes
* = co,h (*)/+ *-. .............. (8).
We must now show how the operations denoted by the
symbolic operators may be performed.
With the point as a centre, describe a sphere of radius r,
and let a, ft, 7 be the coordinates of any point P on this sphere
relatively to ; also let us temporarily denote d/dsc, djdy, d/dz by
\, [Jb, V.
Consider the integral
where the integration extends over the surface of the sphere. If
through the point a, /3, 7, a plane be drawn, whose direction
cosines are proportional to X, //,, v, and if p be the perpendicular
from on to this plane, it is known that
\cL + fJt,l3 + vy = (X 2 + i# + v^ p = Vp.
Also if be the angle, which the radius drawn from to the
point P makes with p, then p = r cos 6, dS = 27rr 2 sin Odd', whence
f the integral
r cos
.
r\
tl
But if dl be the elementary solid angle subtended by dS at 0,
dS = r~dl ; whence putting r = at, and restoring the values of
X, ft, v, we obtain
B. O. 15
226 DYNAMICAL THEORY OF DIFFRACTION.
Now the operation denoted by the exponential factor on the
right-hand side of the last equation, can be performed by means of
the symbolic form of Taylor's theorem ; we thus obtain
I rr tL+pA+y *_
(x, y, z) = -r 1 1 e dx d v dz F(x, y :
If I, m, n be the direction cosines of OP, we shall have a = lat,
ft = mat, 7 = nat ; and therefore the portion of which depends
upon the initial velocities is
lat, y + mat, z -f nat) dfl.
From the form of (8) it is at once seen, that the portion of <
depending on the initial displacements may be obtained by
changing Pinto/, and differentiating with respect to t\ we thus
obtain
1 1 F (x '+ lat, y + mat, z + nat) dl,
4?rJJ
(9).
This equation determines the value of at time t, at any point
of the medium whose coordinates are x, y, z> in terms of the
initial values of and <. The portions of the displacements
which depend upon the dilatation are obtained by differentiating
(9) with respect to x, y, z.
218. If the initial disturbance is confined to a portion T of the
medium, the double integrals in (9) will vanish, unless the sphere
whose centre is and whose radius is at cuts a portion of the
space T. Hence if be outside T, and if r lt r 2 be respectively the
least and greatest values of the radius vector of any element of
that space, there will be no dilatation at until at = r 1 . The
dilatation at will then commence, and will last during an
interval (r 2 r^/a, and will then cease for ever.
219. If /i, /a, ft denote the initial values of u 2 , v 2 , w 2 , which
are the portions of the displacements which depend upon the
distortion; and if F lt P 2 , F 3 denote the initial values of u 2 , v s , iu 2 ,
then since u 2 , v 2 , w 2 each satisfy equations of the same form as (4)
with b written for a, it follows that the values of these quantities
PROPAGATION OF AN ARBITRARY DISTURBANCE. 227
at time t are determined by equations of the same form as (9). It
must also be recollected that/!,/ a ,/ 8 and also F 1} F 2 , F 3 satisfy (3).
If therefore we write for brevity
F(at) for F(x + lat, y + mat, z + nat),
the complete value of u will be
with similar expressions for v and w.
220. The initial velocities are determined by the equations
dF
dF
'
where F 19 F 2 , F 3 satisfy (3) ; and since our object is to find the
values of u, v, w at any subsequent time in terms of the values of
the initial displacements and velocities, we must proceed to
eliminate the F's and /'s from (10). It will however be sufficient
to perform this operation for those parts of u, v, w which depend
upon the initial velocities, for when this is done, the portions
depending upon the initial displacements can be obtained by
differentiating with respect to t and changing u Qt v , w into
( o, Vo, W .
221. Let a, /3, 7 denote the coordinates of any point P
relatively to ; let OP = r, and let I, m, n be the direction cosines
of OP ; then at points on the surface of the sphere r = at, we have
a = lat, &c. ; also if
I the first term of (10) becomes
152
228 DYNAMICAL THEORY OF DIFFRACTION.
By Green's Theorem,
where V 2 = d' 2 jda? + d?\df& 4- d*/dy-, dv is an element of the normal
to the sphere r = at, and the upper or lower sign is to be taken,
according as the volume integrals extend throughout the space
external or internal to the sphere.
Putting = #, ^ = r" 1 , and applying the theorem to the space
outside the sphere r = at, we obtain
Putting = %, "^ = 1, and applying the theorem to the space
e the sphere r - at, we obtain
ing d%/dr between (13) and (14), we obtain
(r < at) - -~
11) and (12).
f U Q , v , W Q be the initial velocities,
V V (16)
x '"
Now the function ^, and consequently the functions u , v , w , ]
when they occur in a triple integral, are functions of the position |
of the point whose coordinates are x+ a, y + ft, ^ + 7; whence
d/dx = d/doL, and accordingly we may write d/da, &c. for d/dx, &c. j
Hence substituting the value of ^ from (16) in (15), integrating ;
by parts, and observing that the two surface integrals which appear
ik the integration cancel one another, we obtain
r>at.}
Integrating the right-hand side again by parts, it follows that
PROPAGATION OF AN ARBITRARY DISTURBANCE. 229
if if*' be the portion of u which depends upon the initial velocity
^- ' v I "7 * # V
of dilatation, ^^ ? X^^/J ^ ^ "* ^ ^^ *'
/Y/y. d . d . d y\ 7 1Q ,
+ 7- K> i H v T- ^ + ^o -y- -^ I dadSdy.
4<7rJJJ\ dar 8 da r 3 da. r 3 ]
\ x^^ ^ ~ ^/
Let qo be the initial velocity along OP, so that
V + ftWo,
and let (qo) at denote the value of q at a distance a^ from 0, then ^~
the surface integral
also the triple integral can easily be shown to be equal to ^ **, ~~A< ^
, - 3lq) r-' dad/Sdy ;
whence we finally obtain for the portion of u depending upon tfre^/ ^ ^
initial rate of dilatation
t rrr
+ -T (w 3%) r~ 3 dadftdy, (r>at) (17). - - ^^i
wJJJ __ '
xu^
222. We must now find the portion of u due to the initial, tr *
velocities of rotation. . r d fl
^ i-^:
Applying Green's theorem to the space outside the sphere **.
t, by writing ^ for ^ in (13), we obtain
- ..* /
4
^-!
, (r>60 ...(18).
o- \ . z 3
Since ^- ' + -T-5 + -r^ = 0,
rfa d/3 7
by (3), we have
1 . 2
=+
DYNAMICAL THEORY OF DIFFRACTION.
L J a '
'
Adding this to (18), we obtain
, (r>bt) ...... (19).
~? ^ Now if ^ OJ ^o> So be the initial velocities of rotation,
From the last article it follows, that the first triple integral
= -jjl fe)w dO, - jjj (u - 3%) r~ 3 dadfa, (r > bt).
Since dadfldy = c?rc?bt) ............ (20).
To obtain the portion of u due to the initial velocities, we
must add the right-hand sides of (17) and (20), and must recollect,
that in (17) the limits of r are oo and at, and in (20) the limits
are oo and bt ; we thus finally obtain
w/ = ' (? )a! dn +
, (bt t it) *
ai i~
It therefore follows that
5
The triple integral is taken throughout the space bounded by
the two spheres whose common centre is 0, and whose radii are
respectively equal to at and bt. If therefore we write r*drdl for
an element of volume, the triple integral may be written
/Y
JJJbt
~fi-jj
and therefore its differential coefficient with respect to t, is + r ^
/Y * /Y
j ^ ^ -f r 1 JJ (3^ - Uo ) at da - -' JJ (3^ -
Hence the portion of u which depends upon the initial
placements is ^//_T ((( / \
<** ir)j (* * -*?o). . J-CL ^',
+
^iJfo,
>^ *
-1 H( 2 + btp- p, - ibt ^) da \_ %<
^TT JJ \ ar ar/bt
The complete value of the displacement u, due to the initial
displacements and velocities, is obtained by adding the values of
u, u" given by (21) and (22). The values of v and w can be
written down from symmetry.
232 DYNAMICAL THEORY OF DIFFRACTION.
224. Sir G. Stokes has applied these results to the solution
of two important problems, viz. (i) the determination of the
disturbance produced by a given variable force acting in a given
direction at a given point of the medium ; (ii) the determination
of the law of disturbance in a secondary wave of light.
We shall now proceed to consider the first problem.
, i Disturbance produced by a given Force.
225. Let P be the point at which the force acts ; and let T
be a small space described about P, which will ultimately be
supposed to vanish, and let be a point outside T at which the
value of the disturbance is sought ; also let D be the density of
the medium.
Let t be the time of observation, measured from some previous
epoch ; and let t r be the time, which the dilatational wave occupies
in travelling from P to 0.
Let f(t) be the given force, and F(t) the velocity at P pro-
duced by the force during a very small interval of time dt', then
the usual equation of motion gives
,dv
Now if we consider the state of things which was going on at
P at a time t 1 ago, we must in this equation write t t' for t, and
dt f for dt ; also $v = F(t- 1'), whence
This equation gives the value of the velocity communicated
during the interval Stf in terms of the force.
Let be the origin, OP = r ; also let I, m, n be the Direction
cosines of OP, and l' t m, n' those of the force ; and let & be the
angle between OP and the direction of the force, so that
k = IV + mm' + nri.
Since the disturbance may by virtue of (23) be regarded as
one which is produced by a given initial velocity, the resulting
disturbance at is determined by (21) ; also since
?o = kF,
DISTURBANCE PRODUCED BY A GIVEN FORCE. 233
; it follows that the first term of (21) becomes da ^ <*
snce r =
' -, &
Since the force is supposed to have commenced to act an
infinitely long time ago, we must integrate this expression with
respect to t' between the limits r/a and oo ; but since the force
is confined to the indefinitely small volume T, f(t t') will be
insensible except for values of tf comprised between the narrow
limits r-Ja and r 2 /a, where r 1} r 2 are the least and greatest values
of the radius vector drawn from to T. We may therefore omit
the integral signs, and replace SrdS by T, and we thus obtain for
the value of the first term of (21),
**/({-*} ...(24).
4t7rDa 2 r J \ aJ
If we denote by t", the time which a distortional wave occupies
in travelling from P to 0, and treat the second term of (21) in a
similar manner, we shall obtain
In order to find what the triple integral in (21) becomes, we
see from (17) and (20) that it may be written
T- (i - %o) r~ 3 dad/3dy (r > at')
The first term of this accordingly becomes
t'
(r > bt"\
I' - Slk)f(t - t') dt'r~ s dad/3dy.
Since f(t t') is insensible except throughout the space T,
we may write T for dctdfidy, and omit the integral signs ; we thus
obtain
and this has to be integrated with respect to t' between the limits
r/a and oo . This term thus becomes
/:
-** =
234
DYNAMICAL THEORY OF DIFFRACTION.
Treating the second term in the same manner, and remembering
that the limits of t" are r/b and - oo , and adding we obtain
Hence collecting all the terms we obtain
U == ~~, i^ ~ / I t ~~~ ) ~P ~ TTv7 / I V T
(26) -
The values of v and w are obtained by putting m, m' ; n, n'
respectively for I, l f . If therefore we take OP for the axis of x, and
the plane passing through OP and the direction of the force as
the plane nz, and put for the inclination of the force to PO, we
shall have
, ra = 0, ft = 0; l' = k =
ra' = 0, n' = sin c/>.
Whence
cos
..(27).
226. In discussing this result Sir G. Stokes says :
"The first term in u represents a disturbance which is pro-
pagated from P with a velocity a. Since there is no corresponding
term in v or w, the displacement, as far as relates to this dis-
turbance, is strictly^ normal to the front of the wave. The first
term in w represents a disturbance which is propagated from P
with a velocity b, and as far as relates to this disturbance, the
displacement takes place strictly in the front of the wave. The
remaining terms in u and w represent a disturbance of the same
kind as that which takes place in an incompressible fluid, in
consequence of the motion of solid bodies in it. If / (t) represent
a force which acts for a short time, and then ceases, / (t t') will
DISTURBANCE PRODUCED BY A GIVEN FORCE. 235
differ from zero only between certain narrow limits of t, and the
integral contained in the last terms of u and w will be of the order
r, and therefore the terms themselves will be of the order r~ 2 ,
whereas the leading terms are of the order r~ l . Hence in this
case the former terms will not be sensible beyond the immediate
neighbourhood of P. The same will be true if / (t) represent a
periodic force, the mean value of which is zero. But if f (t)
represent a force always acting one way, as for example a constant
force, the last terms in u and w will be of the same order, when r
is large, as the first terms.
"It has been remarked, that there is strong reason for believing
that in the case of the luminiferous ether, the ratio of a/b is
extremely large if not infinite. Consequently the first term of u\
which relates to normal vibrations, will be insensible, if not
absolutely evanescent. In fact, if the ratio a/b were no greater
than 100, the denominator in this term would be 10000 times
as great as the denominator of the first term of w. Now the
molecules of a solid or gas in the act of combustion are probably
thrown into a state of violent vibration, and may be regarded,
at least very approximately, as centres of disturbing forces. We
may thus see why transversal vibrations should be alone produced,
unaccompanied by normal vibrations, or at least by any which are
of sufficient magnitude to be sensible. If we could be sure that
the ether was strictly incompressible, we should of course be
justified in asserting that normal vibrations are impossible.
"If we suppose that a = oo , and /()== F sin 27r&/\, we shall
obtain from (27)
F sin 6 . 2-7T /L . F\ sin 2?r
W= -
sn
^(28),
and we see that the most important term of u is of the order \/irr
compared with the leading term of w, which represents transversal
vibrations properly so called. Hence u aod the second and third
terms of w, will be insensible, except at a distance from P
comparable with X, and may be neglected ; but the existence of
236 DYNAMICAL THEORY OF DIFFRACTION.
terms of this nature, in the case of a spherical wave whose radius
is not regarded as infinite, must be borne in mind, in order to
understand in what manner transversal vibrations are compatible
with the absence of dilatation or condensation."
Determination of the Law of Disturbance in a Secondary Wave
of Light.
227. Let us suppose, that plane waves of light are travelling
through an elastic medium. Let the axis of x be parallel to the
direction of propagation of the waves, whilst the axis of z is
parallel to the direction of vibration; then the displacement at
any point of the medium may be denoted by
w=f(bt-x).
Let P be a fixed point, which we shall choose as the origin ;
a point whose coordinates are x, y, z ; dS a small element of
the plane yz, which contains P. We require to find that portion
of the total disturbance at 0, which is due to the element dS at P.
The disturbance at dS consists of a displacement f(bt) and
a velocity bf (bt). In order to find the disturbance at due to the
velocity, let t' be the time which the disturbance occupies in
travelling from P to ; then if PO = r, we shall have r = bt';
also let I, m, n be the direction cosines of OP measured from 0,
so that I, m y n are the direction cosines of OP measured
from P.
We shall thus have
also since the dilatational terms are to be omitted on account of
the largeness of a, the displacement corresponding to that part of
the disturbance which is due to the velocity, which existed at P
at time t' ago, is given by (20). Since the volume integral varies
as r~ 3 , it must be omitted ; whence recollecting that the signs of
l y m, n in (22) must be reversed, we obtain for the portion de-
pending on dS,
In order to find dl in terms of dS, let us consider a thin film
comprised between dS and a parallel surface, whose thickness is
bdt'. Then the volume of this slice is bdt'dS; but this volume is
SECONDARY WAVES OF LIGHT. 237
also equal to r^dQdr ; and since r = bt', it follows that r*dl = dS ;
whence
Treating v and w in a similar manner, we obtain
mndS
Equations (29) and (30) show that lu + mv + nw = 0, from
which we see that the displacement takes place in a plane through
0, perpendicular to PO ; also since u/v = l/m, it takes place in a
plane through PO and the axis of z t which is the direction of
vibration of the primary wave. Putting n = cos <, so that < is
the angle between PO and the axis of z, the magnitude of this
displacement is
'
since the signs of I, m, n in (22) have to be changed.
In order to determine this differential coefficient, let x r , y f , z'
be the coordinates of P referred to any origin, then
p = nw Q = nf (bt bt' #'),
and ^ = -l^, = -lnf(U-U f -x')',
d/T QjOC
where the accent in /' denotes differentiation with respect to bt.
Transferring the origin to P, and recollecting that btf = r, we
obtain
, PndS /./,,. x
whence u = ~~i / (^ r )
238 DYNAMICAL THEORY OF DIFFRACTION.
Treating v and w in the same manner, we obtain
ImndS jf
.(33).
This is the portion of the displacement at 0, which depends
upon the displacement at P. If we denote it by f 2 > and put
Z = cos 0, we see that its direction is the same as that of , and
its magnitude is
(34).
229. By combining the results of (31) and (34) we obtain the
important theorem, which was enunciated in 37.
Let u = Q, v = Q, w f(bt sc) be the displacements correspond-
ing to the primary wave ; let P be any point in the plane yz, dS an
element of that plane adjacent to P ; and consider the disturbance
due to that portion only of the incident disturbance, which passes
continually across dS. Let be any point of the medium
situated at a distance from P, which is large in comparison with
the wave-length of light ; let PO = r, and let this line respectively
make angles 6 and. with the direction of propagation of the
incident light, and with the direction of vibration. Then the
displacement at will take place in a direction perpendicular
to PO and lying in the plane zPO, and if f be the displacement at
reckoned positive in the direction nearest to that in which the
incident vibrations are reckoned positive,
r) (35).
In particular if
f(bt x) = c sin - (bt x)
A.
we shall have
/v7., we obtain
* * QjOLii CO COS \J
, c. *v toT _H =
t 4 .. dt 1 sin 2 sin 2 o) '
^ ' ^rom this equation we observe, that as t increases from to
^~ 7r/2o), dot-ci/dt increases /n value. It therefore follows that as a t -
increases, the planes 01 vibration of the diffracted rays will not be
2** - distributed uniformly, but will be crowded towards the plane
'.perpendicular to Jthe plane of diffraction, according to the law
^expressed by th/ above equation.
angles which the planes of polarization of the
diffracted rays, (if the diffracted rays prove to be really
polarized), make with planes perpendicular to the plane of
diffraction, can be measured by means of a pair of graduated
instruments furnished with Nicol's prisms; and the readings of
^Ifj'*. U
VIBRATIONS OF POLARIZED LIGHT. 241
the instrument, which is used as the analyser, will show whether
the planes of polarization of the diffracted rays are crowded towards
the plane of diffraction, or towards the plane perpendicular to the
plane of diffraction.
Let tzr, a be the azimuths of the planes of polarization of the
incident and diffracted rays, both measured from planes perpen-
dicular to the plane of diffraction. Then if the vibrations of
polarized light are in the plane of polarization, the planes ZOA
and ZOD will respectively be the planes of polarization of the
incident and diffracted rays; accordingly on this hypothesis we
should have r = -, a = a d and therefore by (37)
tan a = cos 6 tan r ;
and we have already shown that in this case, the planes of
polarization will be crowded towards the plane perpendicular to the
plane of diffraction. But if the vibrations of polarized light
are perpendicular to the plane of polarization, we shall have
r = JTT + t -, a = \TT + a d , in which case ^ r *C - ^
/i A = cos-(a*-*)
is the real part of
n P n + (w + l)f n+1 P B+1 }...(38),
where n is zero or any positive integer, P n is a zonal harmonic,
*T= 27T/X, and
.(39),
Since the function tynPn is the velocity potential of a multiple
source of sound of the ?ith order, it follows that the effect of the
element may be represented by three multiple sources of orders
n 1, n, n + 1. If in (38) we put n = 0, and realize, the result is
dS ,. , 2-7T , dScosO 2?r /
~ (1 +C S ^ Sm ~\ ( at ~ r) +
where r is the distance of the element from a point P, and 6 is
the angle which the direction of r makes with that of propagation.
If X is small compared with r, as is always the case in optical
problems, the first term is the most important.
1 Encycl. Brit. Art. "Wave Theory," pp. 452454.
2 Proc. Lond. Math. Soc. vol. xxn. p. 317.
RESOLUTION OF PLANE WAVES. 243
232. From the preceding result, it might be anticipated, that
Stokes' formula is equivalent to the combination of a simple and
a double source of light ; and we shall now show that this is the
case.
We have shown in 215, equation (6), that the most general
solution of the equations of motion of an elastic solid is given by
the equations
u = d(f)/dx + tt'e l(cW , v = d(f>/dy + v'e LKbt , w = d^jdz + w'e lltbt ,
where < is the function which determines the longitudinal wave ;
and u' t v', w each satisfy an equation of the form
(V 2 + 2 K = (41),
subject to the condition
;r - + ^r +^= =0 (42).
dx dy dz
It is well known *, that (41) are satisfied by
where X n , Y n , Z n are three solid harmonics of positive degree n,
and ty n is the function defined by (39) and (40). The function ^
can also be shown to satisfy the following equations, viz.
/vt * /n A f** ,/.. f**
...(44),
(2 71 + 1) ^r w = t*T (^ n .
which are frequently useful.
Let (f> n , % n , be any positive solid harmonics of degree n ; then
by means of a process similar to that employed by Lamb 2 , it can
be shown that
d
= r- \ dx ' y dz " dy) ' n + 2 dx
(45)
with symmetrical expressions for v', w'.
The simplest way of verifying this result is to recollect, that
a solid harmonic is a homogeneous function of x, y, z of degree n ;
1 Lord Rayleigh, Theory of Sound, Ch. xvii. ; Stokes, On the communication of
* vibrations from a vibrating body to the atmosphere, Phil. Trans. 1868.
2 Proc. Lond. Math. Soc. vol. xm. p. 51 ; see also, Basset, Hydrodynamics,
\ vol. n. pp. 316318.
162
244
DYNAMICAL THEORY OF DIFFRACTION.
we thus see that the first term of u' is of the form r
n , whilst
the second is of the form r~ n ~ 2 i|r 7l+ JT n+2 . The above expression
therefore satisfies (41). Also if we differentiate the expressions for
u' t v', w' with respect to cc, y, z, and take account of (44), it will be
found that (42) is satisfied.
233. The simplest solution is obtained by putting n = 0, in
which case
whence
= const.
Ae~ iier
U = '~r~ + V j 1 + ^ + (wrr-
.(46).
This expression may be regarded as giving the value of u for a
simple source of light, and it corresponds to a source in hydro-
dynamics, or to an electrified point. The expression is, however,
more analogous to a doublet or a magnet, inasmuch as a simple
source of light has direction as well as magnitude. The direction
cosines of the axis of the source are proportional to A, B, C ; anu r
if we suppose that its axis is parallel to z, we shall have A=B=().
Also, in optical problems, \ is usually so small in comparison with
r, that at a considerable distance from the source, powers of (ar)" 1
may be neglected ; whence, writing F=&C, we shall obtain
Fxz
In the figure, let P be the point x, y, z :
also let 6 = POx, <>
INTERPRETATION OF STOKES' THEOREM. 245
draw PT perpendicular to OP in the plane POz. Then the
preceding equations show, that the direction of vibration is along
PT, and its magnitude is equal to
rr
sn <>.
r
Restoring the time factor e iKbt and realizing, this becomes
F %TT
- sin (f> cos - (U - r) .................. (48).
V A/
This is the expression for the disturbance produced by a simple
source of light at a point r, whose distance from the source is large
compared with the wave-length.
The motion is, as might be expected, symmetrical with respect
to the axis of z, and vanishes on that axis where <> = or TT ; and
it is a maximum on the plane xy where = %TT.
234. In order to obtain the most general expression for a
singular point of the second order, we must put n = 1 ; whence
& = (B- G) x- + (C - A) if + (A-B)& + ZA'yz + VBzx +
arid
The expression for a singular point of the second order accord-
ingly contains eight constants, and is therefore a function of
considerable generality. Let us now suppose, as a particular case,
that
then, if we confine our attention to points at a considerable dis-
tance from the origin, we may put
'hence
.(50).
246 DYNAMICAL THEORY OF DIFFRACTION.
It therefore follows, that the magnitude of the displacement
represented by (50) is
- e~ lKr sin cos (51),
and that its direction is along PT. Restoring the time factor,
adding (48) and (51), and writing cdS/2\ for F, we obtain
CClO ,- /i\ i -"T /i i \
^ (1 -f cos 6) sin cos - (bt r),
&W1P A*
which is Stokes' result.
We therefore see that Stokes' expression for the disturbance
produced by an element of a plane wave of light is equivalent to
the combination of a simple and a double source.
At the same time, if we were to carry out the investigation on
the same lines as I have done in the case of sound in the paper
referred to, there can, I think, be little doubt, that we should
find that there is an infinite- number of combinations of multiple
sources which would produce the required effect, and consequently
Stokes' law although the simplest, is only one out of an infinite
number. The question is not, however, of very much importance
in the case of light, inasmuch as, in problems relating to diffraction,
we may with sufficient accuracy take sin = cos 0=1, in which
case the disturbance due to the element will be
c^f cos 27r
\r \ '*
corresponding to the wave
w = c sin (bt x).
A,
' '
Scattering of Light by Small Particles.
235. The physical explanation of the intensely blue colour
of the sky, which cannot fail to have attracted the attention of
those who have resided in warm countries, has formed the subject
of various speculations. It has also been found by experiment,
that a beam of light which is emitted by a bright cloud, exhibits
decided traces of polarization, and that the direction of maximum
polarization is perpendicular to that of the beam. The experi-
SCATTERING OF LIGHT. 247
merits of Tyndall 1 on precipitated clouds, point to the conclusion,
that both these phenomena are due to the existence of small
particles of solid matter suspended in the atmosphere, which
modify the waves of light in their course; and we shall now
proceed to give an account of a theory due to Lord Rayleigh 2 , by
means of which these phenomena may be explained.
236. The theory of Lord Rayleigh in its original form, was an
elastic solid theory; but it is equally applicable to the electro-
magnetic theory of light 3 , since we shall hereafter see, that
the equations, which are satisfied by the electric displacement, are
of the same form as those which are satisfied by those portions
of the displacements of an elastic solid, upon which distortion
unaccompanied by dilatation depends.
If we suppose that the particles are spherical, it follows that
when a plane wave of light impinges upon a particle, the latter
will be thrown into a state of vibration ; and the only possible
motion which the particle can have, will consist of a motion
of translation in the plane containing the directions of propagation
and vibration of the impinging wave, and a motion of rotation
about an axis perpendicular to this plane. If the ether be
regarded as a medium, which possesses the properties of an
elastic solid, the motion of the particles will give rise to two
scattered waves, one of which will be a longitudinal wave, and
therefore produces no optical effects, whilst the other will be
a distortional wave, which will give rise to the sensation of light.
If on the other hand, the ether be regarded as an electromagnetic
medium, only one wave, viz. an optical wave, will be propagated.
In order to obtain a complete mathematical solution, it would be
necessary to introduce the boundary conditions 4 , and to proceed on
1 Phil. Mag. May 1869, p. 384.
2 Ibid. Feb., April and June 1871 ; Aug. 1881.
3 See Phil. Mag. Aug. 1881.
4 If the ether be regarded as a medium which possesses the properties of an
elastic solid, three suppositions may be made respecting the boundary conditions.
(i) We may suppose that no slipping takes place, which requires that the
velocity of the ether in contact with the sphere should be equal to that of the sphere
itself ; but, inasmuch as there are reasons for thinking that the amplitudes of the
vibrations of the matter are very much smaller than those of the ether in contact
with it, except in the extreme case in which one of the free periods of the matter is
equal to the period of the ethereal wave, this hypothesis is improbable.
(ii) We may suppose that partial slipping takes place. This hypothesis is
248 DYNAMICAL THEORY OF DIFFRACTION.
the same principles as in the corresponding acoustical problem 1 .
It will not however be necessary to enter into any considerations of
this kind, if we assume that the principal effect of the incident
wave is to cause the particle to perform vibrations parallel to the
direction of vibration of this wave.
237. To fix our ideas, let us suppose that the direction of
propagation of the primary wave is vertical, and that the plane of
vibration is the meridian. The particle will accordingly vibrate
north and south, and its effect will be the same as that of a simple
source of light, whose axis is in this direction. Accordingly if <
be the angle which any scattered ray makes with the line
running north and south, it follows from (48), that the displace-
ment will be of the form
F . ' 27T/ 7 , \
- sin (f> cos ( bt r J,
and is therefore a maximum for rays, which lie in the vertical
plane running east and west, .for which (/> = JTT; whilst there is no
scattered ray along the north and south line for which = 0. If
the primary wave is unpolarized, the light scattered north and
south is entirely due to that component which vibrates east and
west. Similarly any other ray scattered horizontally is perfectly
polarized, and the vibration is performed in a horizontal plane.
In other directions, the polarization becomes less and less com-
plete as we approach the vertical, and in the vertical direction
altogether disappears.
238. The preceding argument also shows, that the vibrations
of polarized light must be perpendicular to the plane of polariza-
tion. For if the light scattered in a direction perpendicular to
that of a primary wave be viewed through a Nicol's prism, it will
be found that no light is transmitted, when the principal section is
open to the objection that the law of slipping is unknown, and would therefore
involve an additional assumption; and also that it would introduce frictional
resistance.
(iii) We may suppose that perfect slipping takes place. In this case the
boundary conditions are continuity of normal motion, and zero tangential stress.
This hypothesis has much to commend it on the ground of simplicity, since the
action of the ether on the matter consists of a hydrostatic pressure, and in the
case of a sphere is consequently reducible to a force ; whereas, if no slipping
or partial slipping took place, the action would (except in special cases) consist of a
couple as well as a force.
1 Lord Kayleigh, Theory of Sound, vol. n. 334.
SCATTERING OF LIGHT. 249
parallel to the direction of the primary wave. Hence the
vibrations of the extraordinary wave in a uniaxal crystal, lie in
the principal plane.
239. We must now consider the colour of the scattered light.
The experiments of Tyndall showed, that when the particles of
foreign matter were sufficiently fine, the colour of the scattered
light is blue. The simplest way of obtaining a theoretical
explanation of this phenomenon, is by means of the method of
dimensions. The ratio I of the amplitudes of the scattered and
the primary light, is a simple number, and is therefore of no
dimensions. This ratio must however be a function of T the
volume of the disturbing particle, p its density, r the distance
of the point under consideration from it, b the velocity of propaga-
tion of light, and p the density of the ether. Since / is of no
dimensions in mass, it follows that p and p' can only occur under
the form p[p', which is a number and may be omitted ; we have
therefore to find out how / varies with T, r, \ and b.
Of these quantities b is the only one depending on the time ;
and therefore since / is of no dimensions in time, b cannot occur.
We are therefore left with T, r and X.
Now it is quite clear from dynamical considerations, that /
varies directly as T and inversely as r, and must therefore be
proportional to T/X 2 r, T being of three dimensions in space. In
passing from one part of the spectrum to another, X is the only
quantity which varies, and we thus obtain the important law :
When light is scattered by particles, whose dimensions are
small compared ivith the wave-length of light, the ratio of the
amplitudes of ike vibrations of the scattered and incident light,
varies inversely as the square of the wave-length, and the ratio
of the intensities, as the inverse fourth power.
From this law we see, that the intensity of the blue light is the
greatest. Hence the blue colour of the sky may be accounted for
on the supposition, that it is due to the action of minute particles
of vapour, and also probably to the molecules of air, which
scatter the waves proceeding from the sun.
250 DYNAMICAL THEORY OF DIFFRACTION.
Common Light.
240. The distinguishing feature of common light is, that
it exhibits no trace of polarization ; and the theory of sources
of light given in 232 furnishes an explanation of the reason
why it is, that the light emitted from an incandescent substance
is unpolarized.
The molecules of an incandescent body are in a violent state of
vibration; each molecule may therefore be regarded as a centre of
disturbance, which produces ethereal waves. The most general
form of the waves produced by any molecule is given by (45), but
for simplicity, we shall confine our attention to the first term
of this series for which n = 0. It therefore follows from (46), that
at a distance from the molecule, which is large compared with the
wave-length of light, the displacements would be represented by
the equation
- + ^Ur 2 -3 (Ax + B-U + Cz} x\ cos =f (bt-r)
\ cos "V ^ ~
with symmetrical expressions for v and w, where A, B, G are
proportional to the direction cosines of the direction of vibration
of the molecule.
This expression represents a spherical wave of light, whose
direction of vibration lies in the plane passing through the line
of vibration of the molecule, and the line joining the latter with
the eye of the observer.
But owing to a variety of causes, amongst which may be
mentioned collisions, which are continually taking place between
the molecules, the line of vibration of any particular molecule is
perpetually changing, so that the angular motion of this line is
most irregular. These changes take place in all probability with
a rapidity, which is comparable with the period of waves of light,
so that it is impossible for the eye to take cognizance of any
particular direction. Moreover the light which is received from
an incandescent body, is due to the superposition of the waves
produced by an enormous number of vibrating molecules, the lines
of vibration of each of which are different, and are continually
changing. Hence the actual path which any particle of ether
describes during a complete period is an irregular curve, whose
form changes many million times in a second. We thus see why
it is that common light is unpolarized.
COMMON LIGHT. 251
241. We can now understand why interference fringes cannot
be produced by means of light coming from two different sources.
For the production of these fringes requires, that there should be
a fixed relation between the phases of the two streams; but
inasmuch as the two streams are affected by two distinct sets of
irregularities, no fixed phase relation between them is possible.
If however the two streams come from the same source, the
irregularities by which the two streams are affected are identical,
and consequently a fixed phase relation will exist between them.
EXAMPLES.
1. A luminous point is surrounded by an atmosphere containing
a number of small equal particles of dust, the density of whose
distribution varies inversely as the nth power of the distance from
the point, and which scatter the light incident upon them. Show
that except in the immediate vicinity of the luminous point, the
(n + I)/(n + 3)th part of the whole light scattered by the dust will
be polarized.
2. Establish the truth of Stokes' expression for the effect of a.n
element of an infinite plane wave at a point Q, by integration over
the whole wave-front.
If the wave be finite, and all points of its boundary be at the
same distance a from Q, prove that the displacement at Q will be
jsin ~ (vt #) T~i (a + x) (a? + x 2 ) sin
where x is the distance of Q from the wave at the plane of
resolution.
3. In a biaxal crystal the ratios of the axes of the ellipsoid of
elasticity are slightly different for different colours, so that the
angles between the optic axes for yellow and violet are a, a + .
The normal to a wave-front of white light in such a crystal makes
angles lt 6 2 with the mean optic axis, and the planes through the
normal and the optic axes make an angle o> with one another.
Show that the directions of polarization lie within a small angle
/sin #! sin 2 \ sin
\sin 0. 2 sin O sin a
CHAPTER XIV.
GREENS THEORY OF DOUBLE REFRACTION.
242. THE theory of double refraction proposed by Green 1 , is
the theory of the propagation of waves in an sBolotropic elastic
medium.
We have stated in Chapter XI., that the potential energy of
such a medium is a homogeneous quadratic function of the six
components of strain ; and we shall now proceed to examine this
statement.
Let be any point of the medium, and let OA, OB, OG be
the sides of an elementary parallelepiped of the medium when
unstrained. Then any strain which acts upon the medium, will
produce the following effects upon the element.
(i) Every point of the element will experience a bodily
displacement.
(ii) The three sides OA, OB, OC will be elongated or
contracted.
(iii) The element will be distorted into an oblique parallele-
piped.
Let u, v, 10 be the component displacements at 0; e, f, g the
extensions of OA, OB, 0(7; a, b, c the angles which the faces
OCA, OAB, OBG make with their original positions. Since a
bodily displacement of the medium as a whole, cannot produce
any strain, it follows that the potential energy due to strain
cannot be a function of u, v, w ; but since any displacement, which
1 Tram. Camb. Phil. Soc. 1839 ; Math. Papers, p. 291.
POTENTIAL ENERGY. 253
produces an alteration of the forms (ii) or (iii) must necessarily
endow the medium with potential energy, it follows that the
potential energy due to strain, must be a function of the six strains
e,f, g, a, b, c.
243. The most general form of the potential energy W, is
given by the equation
W=W 1 +W Z +W, + ...,
where W n is a homogeneous n-tic function of the strains. It is
evident that W cannot contain a constant term of the form W Q)
for when the medium is unstrained, the potential energy is zero,
The most general expression for Wi is
where E, F ... are constants. Now Green supposed, that if the
medium were subjected to external pressure, the first three terms
of W l might come in ; but it appears to me that this hypothesis is
untenable. For if P be the stress of type e, then
de de
accordingly if W contained a term W l} stresses would exist, when
the medium is free from strain. If the medium were absolutely
incompressible, the stresses might undoubtedly contain terms
independent of the strains. For if a portion of such a medium
were enclosed in a rectangular box, and stresses E, F, G, A, B, C
were applied to the sides of the box, of such magnitude as to
preserve its rectangular form, no displacement, and consequently
no strain would be produced, on account of the incompressibility
of the medium ; but the internal stresses would contain terms
depending on the values of the surface stresses. These surface
stresses could not however give rise to any terms in the potential
energy, inasmuch as they do no work. If on the other hand, the
medium were compressible, the effect of the surface stresses would
be to produce displacements, and consequently strains depending
upon them, in the interior of the medium ; hence the internal
stresses P, Q, ... could not contain any terms independent of the
strains, and the term W 1 could not exist. We have already
pointed out, that in order to get rid of the pressural or dilatational
wave, it is unnecessary to make the extravagant assumption, that
the medium is incompressible ; all that it is necessary to assume
is, that the constants upon which compressibility depends, are very
254 GREEN'S THEORY or DOUBLE REFRACTION,
large in comparison with those upon which distortion depends.
Under these circumstances, we conclude that W l is zero, and that
the internal stresses do not contain any terms independent of the
strains. Also since the terms W 3) TT 4 ... would introduce
quadratic and cubic terms into the equations of motion, they will
be neglected.
244. The potential energy is therefore a homogeneous quad-
ratic function of the six strains, and accordingly contains twenty-
one terms. Biaxal crystals, however, have three rectangular
planes of symmetry; and as Green's object was to construct a
theory which would explain double refraction, he assumed that
the medium possessed this property. Whence the expression for
W reduces to the following nine terms, and may be written
2 W = E& + Ff 2 + Of + 2E'fg + 2Fge + Wef
+ Aar+ Bb~ + Cc* (1),
where e, f, g, a, 6, c are the six strains.
The coefficients in the expression for W are all constants,
depending on the physical properties of the medium. The first
three, E, F, G are called by Rankine 1 coefficients of longitudinal
elasticity ; the second three, E', F', G' are called coefficients of
lateral elasticity ; whilst the last three, A, B, C are the three
principal rigidities.
245. The waves which are capable of being propagated in an
isotropic medium, have already been shown to consist of two
distinct types, which are propagated with different velocities ; viz.
longitudinal waves, which involve dilatation unaccompanied by
distortion ; and transversal waves, which involve distortion un-
accompanied by dilatation. Waves of the first type depend upon
the dilatation S, and do not involve rotation ; hence the rotations
97, f are zero, and the displacements are the differential
coefficients of a single function <. Waves of the second type
depend upon the rotations f, 77, f, and do not involve dilatation ;
hence 8 is zero, and the displacements must therefore satisfy the
equation
du dv dw
~^ I 7 r ~T~ U)
dx dy dz
which is the condition, that the displacement should be perpen-
dicular to the direction of propagation.
1 Miscellaneous Scientific Papers, p. 107.
EQUATIONS OF MOTION. 255
246. Let us now consider a portion of a crystalline medium,
which is bounded by a plane ; and let plane waves whose vibrations
are transversal, be incident normally upon the medium. The
incident wave will produce a train of waves within the medium,
which, as will presently be shown, will involve dilatation and
distortion, unless certain relations exist between the coefficients.
But since the disturbance which constitutes light, consists of a
vector quantity, whose direction is perpendicular to the direction
of propagation of the wave, it follows that the medium must be
one, which is capable of propagating waves of transversal vibrations
unaccompanied by waves of longitudinal vibrations. Green there-
fore assumed that the medium possessed this property, and
investigated the relations which must exist between the coefficients,
in order that this might be possible.
247. The equations of motion of the medium are
d?udP dU dT
with two similar equations, where P=dW/de &c. Substituting
the values of P, Q ..., the equations of motion become
d?u F d*u d*u d*u , d*v ,/ F /, m ^^
P ~HZ & ~T~~ + v T~T + & TT + (Or + O) ^ i f-(J? + x>) ; =-
r dtf dx- dy- dz 2 ' dxdy J dxdz
d 2 v n d 2 v ^d 2 v A d 2 v ,, A . d 2 w
P -ji = C j~ + F j + -4 -T- O + (E ' + A ) -3-3-
r dp da? dy dz- ' dydz
,.
(3).
v
d 2 w _
Differentiate with respect to x, y, z and add, and we obtain
v d 2 c
-
' dxdy
If in this equation we put
v-r-^j W '
it becomes p -^ = jj% 2 8 (5).
Hence the relations between the coefficients which are given
by (4), are the conditions that a longitudinal wave may be capable
of being propagated through the medium, unaccompanied by
256 GREEN'S THEORY OF DOUBLE REFRACTION.
transversal waves; and therefore if these conditions are satisfied,
longitudinal waves will be propagated through the medium with a
velocity (p/p)*.
By means of (4), the equations of motion may now be written
d-v
from which we deduce,
.(7),
where
dx dy dz"
and the expression for the potential energy becomes
2W=p(e +f+g)* + A (a 2 - 4fg) + B( - 4ge) + C(c?-
The stresses are given by the equations
(8),
...(9).
,(10).
2 (Be + Af)
T=Bb, U=Cc
248. Equations (6) and (7) show, that the special kind of
seolotropic medium considered by Green, is capable of propagating
two distinct types of waves, viz. dilatational waves, whose velocity
of propagation has been shown to be equal to (/V/o)*, an d dis-
tortional waves, whose velocity of propagation is determined by (7).
We shall presently show, that the velocity of propagation of the
distortional waves, is determined by the same quadratic equation
as in Fresnel's theory ; but previously to doing this, it will be
desirable to consider a little more closely the properties of the
medium.
PROPERTIES OF THE MEDIUM. 257
249. In a crystalline medium, which possesses three rectangular
planes of symmetry, the shearing stress across any plane which is
not a principal plane, will in general be a function of the exten-
sions as well as of the shearing strain parallel to that plane. It is
however possible for a medium to be symmetrical, as regards
rigidity, with respect to each of the three principal axes: in
other words, the medium may be such, that if any plane be drawn
parallel to one of the principal axes (say x), and Si, di be the
shearing stress and strain parallel to that plane and perpendicular
to the axis of as, then Si = Aa^ We shall now show, that when a
medium possesses this property, the relations (4) must exist
between the coefficients.
Let Ox, Oy, Oz be the axes of crystalline symmetry; and let
BC be the intersection of any plane parallel to Ox with the plane
yz ; and consider a portion of the medium, which is bounded by
the plane BO and two fixed rigid planes perpendicular to Ox.
Draw Oy lt Ozi respectively perpendicular and parallel to BC, and A j 4
let the suffixed letters denote the values of corresponding quantities
referred to Ox, Oy l} Oz as axes. -z.
If be the angle which Oyi makes with Oy, then .
Si = S cos 20 + 1 (R - Q) sin 20.
Oj = A3 CUS ZiC7 -f 2 \H ~ V/ S1U ^ U " / ^T^ /
Also, since the medium is supposed to be boundmHby two
rigid planes perpendicular to Ox, there can be no extension nor
contraction parallel to Ox, whence
Q^Ff+E'f,, R = E'f+Og;
accordingly,
But, if m = cos 6, n = sin 0, $ *t e -
f=
= m 2 /,
also g = n?fi + m 2 gi +
Again,
dw dv
a = , + -=--
dy dz
f d d\ , /d d\ ,
= ( m -j- - n , (nvj. + mw^) + In - -, -4- m ^ (mv l - n
\ dy^ dzj \ dy dzj
= a,i cos 26 + (/ - gi) sin 20.
B. O. 17
258 GREEN'S THEORY OF DOUBLE REFRACTION.
Substituting in (11), we obtain
8 1 = {a^A cos 2 20 + J (G + F- 2E') sin 2 2<9j
+ H/i - ffi) [A-i(G + F- 2^1 sin 40
It therefore follows that if
we shall have S l = Aa l .
In a similar way it can be shown, that in order that T^ =
and U l = Cc^ we must have
which are equivalent to (4).
If therefore a portion of the medium considered by Green,
which is bounded by two fixed planes perpendicular to any one of
the principal axes, be subjected to a shearing stress whose
direction is perpendicular to that axis, and which lies in any plane
parallel to that axis, the ratio of the shearing stress to the shearing
strain is equal to the principal rigidity corresponding to that axis.
Moreover a crystalline medium which possesses this property, also
possesses the property of being able to transmit waves of trans-
versal vibrations unaccompanied by waves of longitudinal vibrations.
Hence the relations which Green supposed to exist between the
nine constants, are not mere adventitious relations, which were
assumed for the purpose of obtaining a particular analytical result,
but correspond to and specify a particular physical property of the
medium.
250. We shall now show, that the velocity of propagation of
the distortional waves is determined by Fresnel's law.
To satisfy (7) let u, v, w be those portions of the displacements
upon which distortion depends ; let /, m, n be the direction cosines
of the wave-front, and X, /A, v those of the direction of vibration.
Then we may assume that
u S\, v = SJA, w = Sv,
when S = 6 t/c (fo+ wj 2/+ n2r - vt)
, From these equations combined with (7) of 187, we obtain
+ % = ilc ( mv ~ 7?
ELOCITY OF PROPAGATION.
259
If we put
\' = mv-nfj,, fi=n\ lv, v' = l/ji-m\ ...... (12),
so that X', //, v are the direction cosines of the rotation, and thei
substitute the values of , 97, f in (7), we obtain
(p V s -A)\' + (Al\' + Snip! + Gnv') I =
(13),
(p F 2 - C) v' + (AIM + BnifA + CW) n =
P m 2 . n
J) F 2 2. /oF 2 -
u From (12) we at once deduce
whence
It follows from (14), that the velocities of propagation
waves within the crystal are determined by the same quadratic as
in Fresnel's theory, and that the wave surface is Fresnel's. \ .
^ ^-s**j y a^ erj t rr ^+ <*-^ 4*? t *X* - st****. ** - ***V*L &&****
251. From (12) it follows, that the direction of displacement ^
and rotation are in the front of the wave, and also that these ^
directions are at right angles to one another. / / _/ -
Multiplying (13) by X', //, v and adding, we obtain V^ & ~ (,{
which shows that the velocity of propagation of either wave, is
inversely proportional to the length of that radius vector of the
ellipsoid
which is parallel to the direction of rotation.
We also obtain from (13)
C r'triAJi'a'
^C
='(/i^
-6V> ...... (17).
252. Writing a* = A/p, b* = B/p, c*=C/p, we see from (19) of
109, that if oc, y, z be the coordinates of the point of contact of the
tangent plane to the wave surface, which is parallel to the wave- /> ^
(pY-^0* , i >
a! = IV (r 2 - A/p)/(V* - A/p),
front, then
and therefore by (17)
(r*-A/p)\'/x = (r*-B/p)
2 - C/p) v'jz ...... (18),
172
260 GREEN'S THEORY or DOUBLE REFRACTION.
from which it follows, as in 112, that the direction of rotation in
any wave, coincides with that of the projection of the ray on the
tangent plane to the wave surface, which is parallel to the wave.
Now the direction of displacement is perpendicular to that of
rotation, and therefore Green's theory requires us to suppose, that
the vibrations of polarized light are parallel to the plane of
polarization.
From (11) we deduce
I- 1 (pV*-A) (mv - ??/0 = m- 1 (p F 2 - B) (n\ - Iv)
= n
and since l\ + rnp + nv=Q
we obtain .(B - C) + -(C-A) + -(A - B) = ..... (19),
A, // V
which detei mines the direction of vibration.
253. The theory of Green, although dynamically sound,
renders it necessary to suppose that the vibrations of polarized
light are parallel to the plane of polarization, which is one
objection ; also if we disregard this difficulty, another difficulty
crops up in applying the theory to crystalline reflection and
refraction, owing to the necessity of making some assumption,
involving relations between the physical constants of isotropic and
crystalline media.
To investigate this point, let us consider the reflection and
refraction of light at the surface of a uniaxal crystal, whose face
is perpendicular to the axis. In order that the incident light
should give rise to an extraordinary wave, it is necessary on this
theory, to suppose that the incident vibrations are perpendicular
to the plane of incidence.
In the first medium, the equation of motion is
and in the crystal,
'
where we have written a 2 , c 2 for A and C.
Let W =- 4 l "
where K sin i = /^ sin r, xV=K l V l (22).
REFLECTION AND REFRACTION. 261
From (20) we obtain V 2 = n/p,
and from (21) Ff = (a 2 sin 2 r + c 2 cos 2 r)//^ .
The surface conditions for continuity of displacement and stress
give
dw _ 3 di^
dx dx '
when # = ; whence A + J.' = A 1 ,
Kii (A A') cos i = ^.xC'^i cos r,
the last of which, by (22), becomes
A ' A ^ ^ an *
1 ?i tan r '
., . n tan r c 2 tan i
whence A = A- - (23),
n tan r + c 2 tan i
^ = 24ntanr
/t tan r + c 2 tan i '
We have hitherto avoided assuming, that any relations exist
between the physical constants of the two media; but, in order
that these results should be consistent with those which the theory
furnishes for isotropic media, it would be necessary to suppose
that n = c 2 , and the formulae then show that the amplitudes of the
reflected and refracted light would be the same as if the crystal
were an isotropic medium. Since the wave whose velocity is c is
refracted according to the ordinary law, the assumption that n = c 2
might at first sight appear to be a plausible one in the case of
uniaxal crystal ; but, if we attempt to apply the theory to biaxal
crystals, there is no valid reason why n should be assumed to be
equal to one of the three principal rigidities, rather than to either
of the other two.
If we adopt the assumption of MacCullagh and Neumann, that
p=pi, the intensities will be proportional to the square roots of
the amplitudes, and we shall obtain
(a 2 sin 2 r + c 2 cos 8 r) sin 2i + c 2 sin 2r '
. 2 A (a. 2 sin 2 r + c 2 cos 2 r) sin 2i
(a 2 sin 2 r -f c 2 cos 2 r) sin 2i + c- sin 2r '
The formulas, as will be shown hereafter, agree with the
expressions found for the intensity on the electromagnetic theory ;
262 GREEN'S THEORY OF DOUBLE REFRACTION.
but Lord Rayleigh has shown, that the assumption that the
densities are equal is not a legitimate one in the case of two
isotropic media, since it leads to two polarizing angles, and there
can be little doubt that, in the case of crystalline media, the same
assumption would lead to a similar result, and would therefore
be one which it is not permissible to make. It thus appears that
Green's theory fails to furnish a satisfactory explanation of
crystalline reflection and refraction.
To work out a rigorous theory of the reflection and refraction
of waves, at the surface of separation of an isotropic medium, and
an a3olotropic medium such as Green's, on the supposition that the
velocities of propagation of the dilatational or pressural waves in
both media, are very great in comparison with the velocities of
propagation of the distortional waves, would be a mere question of
mathematics, and could be effected without difficulty on the lines
of Green's and Lord Rayleigh's investigations, when both media
are isotropic. But the only physical interest of such investigations
lies in their ability (or inability) to explain optical phenomena ;
and therefore, having regard to the failure of Green's theory to
furnish satisfactory results in the case of crystalline reflection
and refraction, it seems scarcely worth while to pursue such in-
vestigations.
254. The theory of Green stands on a perfectly sound
dynamical basis, and the various suppositions which he has made
with regard to the relations between the constants, are not
adventitious assumptions made for the purpose of deducing
Fresnel's wave surface, but correspond to definite physical pro-
perties of the medium. The assumption, that the medium
possesses three rectangular planes of symmetry, is necessary, in
order to account for the fact, that in biaxal crystals, there are three
perpendicular directions, in which a ray of light can be transmitted
without division. Also since the phenomenon of polarization can
only be explained on the supposition, that the disturbance which
produces optical effects is a vector, whose direction is perpendicular
to that of the propagation of the wave, it is necessary to suppose,
that the medium is one which is capable of transmitting distortional
vibrations independently of dilatational vibrations; and the
conditions for this require, that certain relations should exist
between the constants, which are given by equations (4), and
CRITICISMS ON GREEN'S THEORY. 263
which reduce the expression for the potential energy to four
terms. It is no doubt the case, that when waves of light whose
vibrations lie in the plane of incidence, are reflected and refracted
by a crystal, waves of longitudinal vibrations would be excited;
but this difficulty might be evaded, by supposing that //, is very
large compared with A, B and G. The theory accordingly at first
sight appears to be a very promising one ; but, as we have already
shown, there are strong grounds for believing, that the vibrations
of polarized light are perpendicular instead of parallel to the
plane of polarization ; and the circumstance, that Green's theory
requires us to adopt the latter hypothesis, is one of the principal
reasons which has prevented it from being accepted as the true
theory.
255. Attention has been called to the fact, that the potential
energy contains cubic and higher terms, which have been neglected.
Glass, however, and most transparent isotropic media exhibit
double refraction, when under the influence of stress ; and this fact
shows, that the propagation of ethereal waves is modified, when
the medium is subjected to stress. A theory which would take
into account the effect of these external stresses, and might
also throw light on double refraction, could be constructed as
follows.
The quantity e is the extension parallel to x, and to a first
approximation its value is du/dx] if however the approximation
were carried a stage further, it would be found that the strains
contain quadratic terms. Accordingly if the more complete values
of the strains were substituted in (I), they would give rise to cubic
terms in W z . Moreover in this case, it would be necessary to take
W 3 into account ; but in forming the expression for this quantity,
it would be sufficient to take e = dii/da, f= dvjdy, &c. The final
equations of motion would accordingly contain quadratic as well as
linear terms. The solution of these equations would then have to
be conducted on the same principles, as the well-known problem
of the propagation of waves in a liquid, which has a motion
independent of the wave motion. In the first place, let u lt v l} iu l
be the statical portions of the displacements, which depend upon
the external stresses ; and let these quantities be found from the
complete equations of equilibrium. Next let u 2 , V 2) tv> 2 be the
portions of the displacements due to the wave motion, so that
M! + a,, v l + v. 2) Wi + Wt are the total displacements of the medium
264 GREEN'S THEORY OF DOUBLE REFRACTION.
when in motion ; and let these quantities be substituted in the
equations of motion, neglecting quadratic terms of the form w 2 2 &c.
We should thus obtain three linear equations for determining
u. 2 , v 2 , w 2 , into which the external stresses would have been
introduced. So far as I am aware, a theory of this kind has not
been worked out, but it would be interesting to examine the
results to which it leads in some simple case.
CHAPTER XV.
THEORY OF LORD RAYLEIUH AND SIR W. THOMSON.
256. THE theory which we shall now consider, was first
suggested by Rankine, but was subsequently proposed and
developed independently by Lord Rayleigh 1 . The theory might
be regarded as one, which depends upon the mutual reaction of
ether and matter; but inasmuch as it is capable of explaining
several important phenomena, it will be desirable to consider it at
once.
We have already pointed out the unsatisfactory character of
Green's theory, when applied to double refraction. We have
moreover seen, that there are strong grounds for supposing, that
the rigidity of the ether is the same in all isotropic media, and
that reflection and refraction are due to a difference of density.
The properties of an isotropic medium are the same in all directions,
but those of a crystal in any direction depend upon the inclination
of that direction to the axes of symmetry of the crystal. Lord
Rayleigh therefore assumed, that the two elastic constants of the
ether are the same in crystalline as in isotropic media ; but that
owing to the, peculiar structure of the matter composing the
crystal, the ether behaves as if its density were seolotropic.
257. Since the density of every medium is a scalar function,
it might appear that this assumption involves a physical im-
possibility ; but it is easy to give an example of a system which
behaves in this manner. Let an ellipsoid, suspended by a fine
wire, perform small oscillations without rotation in an infinite
liquid. If U, V, W be the velocities of the ellipsoid parallel to
1 Hon. J. W. Strutt, Phil. Mag., June, 1871.
266 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON.
its axes, its kinetic energy will be equal to ^M(U-+ V- 4- W~)\
the kinetic energy of the liquid is equal to \(P'U- + Q'V- + RW*) ;
and therefore the kinetic energy of the solid and liquid will be of
the form %(PU 2 + QV 2 + RW' 2 ). Hence the effect of the liquid is
to cause the ellipsoid to oscillate in the same manner as a particle,
whose density is a function of its direction of motion. We thus
have an example of a system, which behaves as if its density were
sbolotropic.
The point may also be considered from a somewhat different
aspect. In the hydrodynamical problem, the resultant pressure
exerted by the liquid, consists of three components P'U, Q'V,
RW. Lord Rayleigh's hypothesis is therefore equivalent to
the assumption, that the effect of matter upon ether, is re-
presented by a force whose components are p x il, p v v, p z w
parallel to the axes of crystalline symmetry, where u, v, w denote
the displacements of the ether; in other words, these forces are
proportional to the component accelerations of the ether. In a
biaxal crystal, p x , p y , p z are all different; but in an isotropic
medium they are equal.
258. The kinetic energy of the ether may accordingly be
taken to be equal to
%fff(pxtf + pyV~ + pzW*) dxdydz ;
whilst the potential energy is the same as in an isotropic
medium. And by employing the Principle of Least Action, or
the Principle of Virtual Work, the equations of motion will be
found to be
d-u ^ d
where A B = in = k + ^n, B = n, in and n being the elastic
constants in Thomson and Tait's notation.
259. Before entering into any further discussion respecting
this theory, it will be desirable to solve these equations, in order
to find out what they lead us to. We shall accordingly proceed to
determine the velocity of propagation *.
1 Glazebrook, Phil. May. (5), Vol. xxvi. p. 521.
cr ' v
*a J i/" /) f
.. -=- f ^ - / " ^ /* ~ ^LL'i^
^VELOCITY OF PROPAGATION. . rf*J26
Substituting in (1), we obtain 2 / \
Transposing the terms BX &c. in (3) to the left-hand side,
! multiplying by I, m, n, dividing by F 2 a 2 &c., and adding, we ^
obtain Ji (V\* x - /3
7\ \
71 :l\ mil nv\
B ( ; + "IT- + !
(^ - B) (IX + ^ 4 nv) - + -f y . . .(6),
and therefore by (5),
This is a cubic equation for determining the velocity of
propagation, and shows that corresponding to a given direction,
the medium is capable of propagating three waves.
By means of (4), equations (3) may be written in the form
V 2 - a 2 = (A-B) (l\ + nifi + nv) a-l/\B,
|l with two similar equations ; whence we readily obtain
a ? (b 2 - c 2 ) + b ~ m (c 2 - a 2 ) + C - (a 2 - b 2 } = (8).
A, Lb V
= 0.
1 1 /^ 1 . /
268 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON.
This equation determines the velocity of propagation of the
two optical waves. The velocity of propagation of the longitudinal
wave is infinite.
The direction of vibration is determined by (8) together with
the equation
l\ + mp + nv = Q (10).
Equation (9) does not lead to Fresnel's wave surface ; and
inasmuch as the experiments of Glazebrook 1 have shown, that
Fresnel's wave surface is a very close approximation to the truth,
the theory in its present form is unsatisfactory. We shall there-
fore proceed to consider a modification of this theory which has
been proposed by Sir W. Thomson 2 , by means of which Fresnel's
wave surface can be obtained.
261. When a disturbance is communicated to a homogeneous
isotropic elastic medium, two waves are propagated from the centre
of disturbance with different velocities ; one of which is a wave of
dilatation, whose vibrations are perpendicular to the wave-front,
and whose velocity of propagation is equal to (k + Jft)Vf>* ; whilst
the other is a distortion al wave, which does not involve dilata-
tion, and whose velocity of propagation is equal to n*/p*.
In applying the theory of elastic media to explain optical pheno-
mena, it is necessary to get rid of the difficulty which arises from
the fact, that such media are capable of propagating dilatational
waves. This may be done by supposing that the ratio (k + $n)*/n*,
of the velocity of propagation of the dilatational wave to that of
the distortional wave, is either very large or very small; which
requires either that k should be very large compared with n, or
should be very nearly equal to f ft. Green adopted the former
supposition, on the ground that, if the latter were true, the medium
would-be unstable. Sir W. Thomson, however, has pointed out
that, if & is negative and numerically less than f/i, the medium
will be stable, provided we either suppose the medium to extend all
through boundless space, or give it a fixed containing vessel as a
boundary.
Putting U = (k + fri)*/p*, V = w */p*,
it is obvious, that if a small disturbance be communicated to the
medium, U will be real, provided k + ^n be positive, and therefore
1 Phil. Trans. 1879, p. 287 ; 1880, p. 421.
2 Phil Hag. (5), Vol. xxvi. p. 414.
SIR w. THOMSON'S HYPOTHESIS. 209
the motion will not increase indefinitely with the time, but will be
periodic ; but, if k + n be negative, V will be imaginary, in which
case the disturbance will either increase or diminish indefinitely
with the time, and the medium will either explode or collapse,
and will therefore be thoroughly unstable. If k = fw, U will be
zero, and therefore the medium will be incapable of propagating a
dilatational wave. The principal difficulty in adopting this
hypothesis appears to me to arise from the fact, that it requires us
to suppose that the compressibility is negative : in other words,
that an increase of pressure produces an increase of volume. So
far as I am aware, no medium with which we are acquainted
possesses this property ; and it is very difficult to form a mental
representation of such a medium. On the other hand, there does
not appear to be any a priori reason for supposing, that a medium
possessing this property does not exist ; if, therefore, we adopt
Sir W. Thomson's hypothesis, it follows that elastic media may be
classed under the following three categories : (i) media which
contract under pressure, for which k may have any positive value ;
(ii) media which expand but do not explode or collapse under
pressure, for which k may have any negative value which is
numerically less than fw; (iii) media which explode or collapse
under pressure, for which k may have any negative value which is
numerically greater than %n.
262. In order to explain more clearly the necessity of sup-
posing, that the medium extends through infinite space, or
is contained with rigid boundary, we observe that the potential
energy is equal to
i/// K m + *>) & + n ( 2 + V + c 2 ) - 4?? (ef+fg + ge)} dxdydz.
Integrating the last term by parts, it becomes
If the boundary is fixed, or at an infinite distance, u y v, w must
be zero at the boundary, whence the surface integral vanishes;
accordingly
W = i/// {(m + n) & + n (p + tf + f 2 )1 dxdydz.
The value of W is positive when m + n is positive, i.e. when
k > _ i w ; in other words, work will have to be done in order
to bring the medium into its strained condition.
270 THEORY OF LOUD RAYLEIGH AND SIR W. THOMSON.
263. We shall now develop the consequences of supposing
that m + n or A is so exceedingly small, that it may be treated as
zero.
In the first place, the right-hand side of (7) is zero, and
therefore the velocity of propagation is determined by Fresnel's
equation, and accordingly the wave surface is Fresnel's.
Since B is not zero, equations (6) and (7) show that
Equations (3) may be written
(12).
v ( F 2 /c 2 - 1) = - (l\ + m/jL + nv) n }
Multiplying these equations by X/a 2 , /Lt/6 2 , i//c 2 , adding, and
taking account of (11), we obtain
F 2 (X 2 /a 4 + /A 2 /6 4 + ^ 2 /c 4 ) = X-/a 2 + /i 2 /6 2 + *> 2 /c 2 (13),
which gives F in terms of the direction of vibration.
Again from (12), we obtain
(V* - a-) \/a*l = ( V- - 6 2 ) fji/tfm = ( F- - c 2 ) v/c*n = H (say). . .(14),
which determine the direction (X, /-t, v) of the vibrations corre-
sponding to a given wave-front.
Equation (7) with A very small, but not zero, shows that a
quasi-dilatational wave will be propagated, whose velocity is very
small ; if therefore in (14), F denote the velocity of this wave, V
will be very nearly zero, and consequently the direction of vibration
will be approximately determined by the equations
\/l = f jL/m=v/'n (15),
which shows that the direction of vibration in this wave is sensibly
perpendicular to the wave-front.
264. We have also shown, (19) of 109, that in Fresnel's
wave surface
i by (14), with two similar equations, accordingly
'
whence the direction of vibration is perpendicular to the ray.
DIRECTION OF VIBRATION.
271
265. Since the equation IX + m/j, + nv = is not satisfied, the
direction of vibration does not lie in the wave-front. We shall
now show, that it is determined by the following construction.
Let P be the point where a ray proceeding from a point
within the crystal, meets the wave surface whose centre is ; let
PY be the tangent plane to the wave-surface at P, OF the
n. - OR
perpendicular on it from 0, and draw YR perpendicular to OP.
Then RY is the direction of vibration.
To prove this, let Z, M, N be the direction cosines of R Y, then
But OY= V, and OE = OF 2 /OP = F 2 /r; whence
L.
-
A\A^-~
*/>
- < -^
y(y'**y.
/*
A
f5vS";v* '
,..(18).
x*
by (14), whence
In Fresnel's theory, PF is the direction of vibration ; but
although on this theory the direction of vibration is not the same
as in Fresnel's theory, yet it lies in the plane containing the ray
and the wave normal, and therefore the vibrations of polarized
light on emerging from the crystal, are perpendicular to the plane
of polarization.
272 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON
266. If L', M' } N' are the direction cosines of PY, then
-L 1 .PY=X-IV -ti.n
F 2 - a 2
= \
by (14); whence
267. The following expression for cos% is also useful. We
have
cos x = cos PYR = LL'+ MM' + NN'
X 2 /(/: J
-(X 2 /a 4
by (18) and (19). But
L )V( ;
r r
J-sr^+jr 2 ?!? ( 2o )-
268. Another point of importance is, that according to this
theory, it is necessary to suppose that the rigidity is the same in
crystalline as in isotropic media ; and therefore that refraction is
due to a difference of density. For if we consider two different
media bounded by the plane x = 0, the displacements u, v, w must
be continuous. Now the continuity of v and w when x = 0,
involves the continuity of dv/dy + dwjdz ; but if k + f n = 0, the
continuity of the normal stress P requires that
fdv , dw\ ,(dv' dw'\
T- + -T = 2n ( j- + -T- .
\dy dzl \dy dz ]'
-T j
dy dzl \dy
when a; = ; and this requires that n = T?/.
269. We must now find an expression for the mean energy
per unit of volume.
The component displacements are
u = S\, v Sfji, w Sv,
2_
where 8 = & cos ,_.- (Ix -f my + nz - Vt).
Hence
cos 2 -y^r- (lac + my -f nz Vt).
CRYSTALLINE REFLECTION AND REFRACTION. 273
Since p x = B/a?, &c., where B denotes the rigidity; the mean
kinetic energy per unit of volume is
r 2 U 2 6 2 c 2
by (20).
The mean potential energy per unit of volume is
{(mv - nrf + (n\ - Iv? + (If* - mX) 2 }.
The quantity in brackets is equal to the square of the sine of
the angle between the directions of propagation and vibration, and
is therefore equal to cos 2 ^ ; whence the mean potential energy is
and is therefore equal to the mean kinetic energy. The mean
energy is therefore
cos^
F 2 T 2
Crystalline Reflection and Refraction.
270. Having discussed the preceding theory, which is due
to the combined efforts of Lord Rayleigh, Sir W. Thomson and
Glaze brook, we shall now consider its application to the problem
of reflection and refraction at the surface of a crystal 1 .
Let i be the angle of incidence; r lt r 2 the angles which the
directions of the two refracted waves make with the normal to the
reflecting surface ; %i, % 2 the angles between the two refracted
rays, and the corresponding wave normals.
The conditions at the surface of separation are
u=u l} v = v lt w = w 1 (21), *
, vVi/M / \ / \AJ\J w M/ \ / / \ \AJ M/i
( + '0^ + (-)(^-+^J = <"' +n ^
B> *
(22),
Proc. Lond. Math. Soc., Vol. xx. p. 351.
18
274 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON.
fo dn_d/v 1 dui
dx dy ~ dx dy "
du dw dih dw l
-j- + j = -= + -j
dz dx dz dx
in which m + n and m' + n are ultimately zero. If, therefore, we
disregarded the dilatational waves altogether, we should have six
equations to determine four unknown quantities. We must
therefore introduce a dilatational reflected and a quasi-dilatational
refracted wave, which must be eliminated, and we shall thus
obtain the correct equations for determining the amplitudes of the
reflected and the two refracted optical waves, and the deviation of
the plane of polarization of the former.
Let the displacements in the four optical waves be
and let the displacements in the dilatational reflected and quasi-
dilatational refracted waves be
(g ^^
also, let / and R be the angles which the normals to these waves
make with the axis of x. Let 0, 6' be the angles which the
directions of vibration in the incident and reflected optical waves
make with the axis of z\ lt # 2 the angles which the projections
upon their respective wave-fronts of the directions of vibration in
the two refracted waves, make with this axis.
NiS ! I
Then, omitting the common exponential factor, and also all
terms involving 2 2 , which are of the same form as those involving
2i, and can therefore be supplied at the end of the investigation,
we have at the surface
u = A cos AP + A' cos AP' + B cos /]
v =A cos BP + .^cos BP' + B sin/[ ......... (25)
w = A cos GP + A' cos GF
for the firstmedium & (^^^ ' ^ , ^ (^v^vJ p
v
A
-<, <^^v^l~ A . ^^ X a~-V^C4 i-v^ f^wo - / 7-v-^>-v-
CRYSTALLINE REFLECTl6N AND REFRACTION. 275
For the second medium
U-L = A-L (cos ^ cos AP 1 sin ^ cos r^) B : cos
v 1 = l cos x\ cos j + sn ^ sn
w x = A 1 cos ^ cos OP l
sn
^<5
...(26)?
Since du/dy = dujdy when # = 0, and rfw/^ = du-^jdz =
! (23) and (24) give .
(A cos BP - A' cos BP) d + B/cy sin / = ~^ s -^
A l (cos ^ cos BP l + sin ^ sin r-^ich + A7i si n -B (27),
Since m + n and m' + ?i are ultimately zero, and dv/dy = dv^dy,
both sides of (22) ultimately become identically equal, and this
equation need not therefore be considered.
Now, if A, AX be the wave-lengths of the waves @, ^ ; U, U l
their velocities of propagation,
%TT T ZTT D
/^7 = T- COS 7 . /C'Vi = . COS -ft,
A A!
27T . , 27T . D 27T . .
/^T^ = -_ sin /as -7- sm .fi = sin i = c.,
A Ai X
, and therefore, since CT, CTj are ultimately zero, A, Aj are also
ultimately zero; whence / = 0, jR=0, and therefore 7, 7! are
ultimately infinite. Also,
7 sin / = - cos / sin / = /cm cos /.
182
/f/3
276 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON.
Writing out the equation u = u l in full, multiplying by /cm, and
subtracting from (27), we obtain
(A cos BP A' cos BP') id (A cos AP + A' cos AP') Kin
= A-L (cos %, cos .BPj + sin ft sin ?'j) /^i
AI (cos ^ cos ^.Pi sin ^ sin TJ) #771 (29).
From the preceding investigation we see, that B and B, are not
zero, but finite, and therefore the existence of the waves @, >i
cannot be entirely ignored ; but, since / = R the terms
involving B, B, disappear from the equation v = v', which gives
A cos BP + A' cos BP' = A 1 (cos ^ cos BP, 4- sin ^ sin r^. . . (30),
and the equation w = w' gives
A cos CP + A' cos CP' = A,cos Xl cos CP, (31).
Equations (28), (29), (30), and (31) contain the complete solution
of the problem. . ^ ^ _ _x
ccr? ftp = -^^ * =- ^. " -Ai*
2/2. ^ NOW i s x % . *.
. * cos J.P = sin i sin 0, cos #P = cos i sin 0, cos CP = cos 0,
"}r
y^ with similar expressions for cos AP lf &c. ; also, ^
cos ^4P' = sin i sin X , cos BP' = cos i sin 6', cosCP / = cos(
1 '^whence (28), (29), (30) and (31) become
(A cosO A' cos 0') cot i = A, cot r x cos ^ cos 0!
(A sin + -<4/ sin 0') cosec i = A, cosec r*! cos ^ sin 0^)
(^1 sin A' sin X ) cos i = A, (cos r x cos %, sin 0j + sin rj sin ^
J. cos + A f cos 0' = ^lj cos %, cos a
(32),
in which equations we are to recollect, that we are to add to the
right-hand sides terms in A a similar to those involving A,.
The preceding equations may also be obtained by a process
which does not involve the introduction of the dilatational waves.
Since the continuity of u, v, w involves the continuity of theii
differential coefficients with respect to y and z^ the first of (21
together with (23) and (24) involve the continuity of the rotation:
f and 77 ; also, since m m = n, both sides of (22) are identically
equal, and therefore this equation disappears; we are thus lef
with the last two of (21). The surface conditions are therefore
v=v l} ww,;
ISOTROPIC MEDIA. 277
which furnish four equations to determine the four unknown
quantities.
Equations (32) determine the amplitudes of the reflected and
refracted waves, but according to 10, the intensity is to be
measured by the mean energy per unit of volume. Accordingly by
209, if /, /', /i, / 2 denote the square roots of the intensities of
the four waves
A A' ^cos !
/ sin i T sin i I I sin r x / 2 sin r 2
whence (32) become
(/ cos + r cos 6') sin i = / x cos l sin r x + / 2 cos 2
(/ cos 6 1' cos 6'} cos i = 1^ cos 1 cos TI 4- / 2 cos 2 cos r 2
/ sin + T sin 6' = I t sin B 1 + / 2 sin 2
(/sin - /' sin 0') sin 2 = J 1 (sin0 1 sin2r 1 +2sm 2
+ / 2 (sin 2 sin 2r 2 + 2 sin 2 r 2 tan ^ 2 )
.(33).
271. We shall hereafter show, that these equations are exactly
the same as those**furnished by the electromagnetic theory, and
we shall postpone the complete discussion of them, until we deal
with that theory ; but it will be desirable to consider the results
to which they lead, when both media are isotropic.
1st. Let the light be polarized in the plane of incidence ;
then = 0' = 0! = ; %i = % 2 = 0, and J 2 = 0, whence
(/ -f /') sin i = I I sin r,
(I /') cos i = /! cos r,
from which we deduce
sin (i - r)
'
/,=
which are the same formulae as those obtained by Fresnel.
2nd. Let the light be polarized perpendicularly to the plane
of incidence ; then = 6' = X = ^TT, and
(/ I') sin 2i = /! sin 2r,
T . T tam(i r)
whence 1=1- ;. r ,
tan (i + r)
/ i =
sin (i + r) cos (i - r) '
278 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON-
which are Fresnel's formulae for light polarized perpendicularly to
the plane of incidence. See 172 and 175.
272. The principal results of this theory are, (i) that it leads
to a wave-surface, which is approximately though not accurately
Fresnel's wave-surface, unless k is absolutely and not approxi-
mately equal to $n; (ii) that although the direction of vibration
within the crystal is not the same as in Fresnel's theory (being
perpendicular to the ray instead of to the wave-normal), yet it
makes the vibrations of polarized light on emerging from the
crystal, perpendicular to the plane of polarization ; (iii) the
equations, which determine the intensities in the case of crystal-
line reflection and refraction are, as we shall hereafter see,
identical with those which are furnished by the electromagnetic
theory ; and when both media are isotropic, the results agree with
those obtained by Fresnel. Also, as soon as the assumptions have
been made, that k is equal, or nearly so to n, and that double
refraction arises from the circumstance, that crystalline media
behave as if they were seolotropic as regards density; results
which can be proved to be very approximately true, are capable
of being deduced without the aid of any of those additional
assumptions, which in many cases are indispensable in order to
obtain a particular analytical result.
Theory of Rotatory Polarization.
273. When bodily forces act upon the ether, the equations of
motion will be of the form
p x d ^ 2 =(A-B)^B^u^X (34).
&c. &c.
Now the photogyric properties of quartz and turpentine must
be due to the peculiar molecular structure of such substances ; and
we may endeavour to construct a theory of rotatory polarization,
by supposing that the effect of the mutual reaction of ether and
matter modifies the motion of ethereal waves in a peculiar manner,
which may be represented by the introduction of certain bodily
forces. The mathematical form of these forces is a question of
speculation, but we shall now, following MacCullagh 1 , show that
1 Trans. Roy. Irish Acad> Vol. xvn. p. 461.
THEORY OF ROTATORY POLARIZATION. 279
rotatory polarization may be accounted for by supposing that
these forces are of the form
d?v d?w ^ d 3 w d?u d?u d?v
274. For an isotropic medium such as syrup or turpentine,
p x =zp y = p z - ) p l = p. 2 = p s . If therefore the axis of z be taken as
the direction of propagation, u and v will be functions of z alone ;
whence (34) combined with (35) reduce to
dhi d*u d 3 v
(d
To solve these equations, assume
*?(*-f\ *?(*-- A
u = Le^ v ', v = Me^ v ' ............ (37),
where L and M may be complex constants, r is the period, and V
the velocity of propagation. Substituting in (36), and putting
U = n/p, p 3 /p = p, we obtain
( F a - 6 72 ) L - (Znrp/ FT) M = 0,
( F 2 - U*) M (Znrp/Vr) L = 0.
Now the rotatory effect, which depends on p, is very small,
whence in the terms involving p, V may be written for F; we
accordingly obtain
..... . ............... (38).
If the incident light be represented by
u L cos STT^/T, v = 0,
arid if F 1} F 2 denote the greater and lesser values of F, we shall
obtain in real quantities
- , 2-7T / Z \ ., 2-7T / Z \
u = \L cos ( Y ~ t) + \L cos I Y ~ t) ,
1 , 2-j-r z \ 2-n-
v = ^Lsm
Accordingly
27T
U- Lcos v
2 7 r
280 THEORY OF LORD RAYLEIGH AND SIR W. THOMSON.
Whence if i/r be the angle through which the plane of polari-
zation is rotated, measured towards the right hand of a person ivho
is looking along the ray
TTZ ( 1 1 \
tan -fy = v/u = tan ( -^ -^ \ .
T \ y 2 y\t
Substituting the values of F^Fj from (38) we obtain approxi-
mately
'
which is the expression for Biot's law. The sign of p is positive
or negative, according as the medium is right-handed or left-
handed.
If the wave were travelling in the opposite direction, the sign
of t would have to be reversed in (37) ; this would not make any
alteration in the form of (39), but since in this case z would be
negative, the rotation would be in the opposite direction. This
agrees with experiment.
275. We must now consider the theory of quartz. Taking
the axis of the crystal as the axis of z, we must put
whence writing
njp x = a 2 , n/p g = c 2 , q = pjn,
and recollecting that A B = n, the equations of motion
become
d*u 2 /_ 2 d8\
-j~ = a 2 V 2 u j- + b/a. When r = b/a, //, = oo and F=0; and
when r< b/a, fj? is negative and F is imaginary. The medium is
therefore incapable of propagating waves, whose period is less
than b/a; and an equation of the form (1) might therefore be
employed to illustrate the action of a medium, which is opaque to
the ultra-violet waves.
303. The theoretical explanation of the absorption of certain
colours, depends upon the dynamical theorem to which allusion
has been made. The molecules of all substances are capable of
vibrating in certain definite periods, which are the free periods
of the substance. The number of different free periods depends
upon the molecular structure of the substance ; and in all proba-
bility, the more complex the substa/ice is, the more numerous are
the free periods of the molecules. If therefore any of the free
periods lie within the limits of the periods of the visible spectrum,
absorption will take place. Suppose, for example, that the velocity
of light in an absorbing medium were given by an equation of
the form
A 1 1 .f 2 ,, 2
_ = __J_ 1 K 2 K! ,n\
F 2 /^-r 2 K 2 2 -r 2 (T'-'-A^HT 2 -/^ 2 )
where A is a positive constant, K I} K 2 are the free periods of the
matter, and K Z >K I . When r > K,, V is real; but when r lies
between ^ and K.,, F will be imaginary; and when Tq r . Since the glass appears to
be blue when the thickness is small, it follows that A b > A r . This
must be interpreted to mean, that out of the rays of different
colours which fall upon the glass, a far greater number of blue
rays are capable of being transmitted than of red. In other words,
cobalt glass transmits a very large portion of blue extremity of the
spectrum and very little of the red, but the coefficient of absorption
of the blue rays is greater than that of the red.
308. In the case of chlorophyll, the coefficient of absorption
is very large for all rays but green and red, and is greater for
green than red ; but this substance is capable of transmitting
a larger portion of the green part of the spectrum, than of the
red.
Anomalous Dispersion.
309. We have already pointed out, that the order of the
colours in the solar spectrum is violet, indigo, blue, green, yellow,
orange, red ; and that the violet is the most refracted, and the red
is the least. There are however certain substances, in which the
order of the colours in the spectrum produced by refracting sun-
light through them, is different from that produced by glass, and
the majority of transparent media. The dispersion produced by
such substances is called anomalous dispersion.
310. Anomalous dispersion appears to have been first observed
by Fox Talbot 1 about 1840, but the discovery excited no attention.
It was next observed by Leroux 2 in 1862, who found that vapour
of iodine refracted red light more powerfully than violet. This
substance absorbs all colours except red and violet, and it was
observed that the order of the colours in the spectrum, beginning
at the top, was red, then an absorption band, and then violet.
The indices of refraction, as determined by Hurion 3 , are
/*. = 1-0205, ^=1-019.
1 See Proc. R. S. E. 1870; and Tait, Art. Light, Encycl. Brit.
2 C. R. 1862; and Phil. Mag. Sept. 1862.
3 Journ. de Phys. 1st Series, Vol. vii. p. 181.
ANOMALOUS DISPERSION.
297
311. Anomalous dispersion is most strongly marked in solu-
tions of the aniline dyes in alcohol. Christiansen 1 discovered in
1870, that it was produced by fuchsine, which is one of the rose
aniline dyes; for when sunlight was passed through a prism
containing a solution of this substance, it was found that the order
of the colours was 2 indigo, green, red and yellow, the indigo being
the least deviated. The amount of anomalous dispersion increases
with the concentration of the solution, as is shown in the following
table of the indices of refraction.
Fuchsine solution
B
C
D
V
G
H
18 '8 per cent.
1-450
1-502
1-561
1-312
1-285
1-312
2-5 per cent.
.
1-384
1-419
1-373
1-367
1-373
From this table, we see that the line D is more refracted than
any of the others, and that the violet is less refracted than the red.
312. Kundt 3 afterwards showed, that blue, violet and green
aniline, indigo, indigo-carmine, cyanine, carmine, permanganate of
potash, chlorophyll and a variety of other substances exhibited
anomalous dispersion. The following table gives some of the
indices of refraction found by him 4 .
Cyanine
1-22 per cent,
solution
A
1-3666
B
1-3691
C
1-3714
D
E
1-3666
F
1-3713
G
1-3757
H
1-3793
Do. a stronger
solution
1-3732
1-3781
1-3831
1-3658
1-3705
1-3779
1-3821
Fuchsine
1-3818
1-3873
1-3918
1-3982
1-3668
1-3759
Permanganate
of Potash
1-3377
1-3397
1-3408
1-3442
1-3477
1-3521
1 Pogg. Ann. Vol. CXLI. p. 479; and Phil. Mag. March, 1871, p. 244.
2 Ibid. Vol. CXLIII. p. 250. See also Wiedermann, Ber. der Sticks. Gesell. math.-
phys. Cl. Vol. i. 872 ; G. Lundquist, Nova acta reg. Soc. Sc. Upsaliensis [3] Vol. ix.
Part n. (1874) ; Jour, de Physique, Vol. in. p. 352 (1874).
3 Pogg. Ann. Vols. CXLII. p. 163; CXLIII. pp. 149, 259; CXLIV. p. 128; CXLV.
p. 164.
4 Pogg. Ann. Vol. CXLV. p. 67.
298 MISCELLANEOUS EXPERIMENTAL PHENOMENA.
The preceding table gives a general idea of the condition of
the spectrum, and shows that for cyanine the line E is the least
refracted. In the case of cyanine and fuchsine. the lower end of
the spectrum is blue, then comes an absorption band, and afterwards
red and orange ; so that the blue is least refracted, the green and
some of the yellow are absorbed, and the orange is the most re-
fracted. In the spectrum produced by permanganate of potash,
there is a slight amount of anomalous dispersion between D and
; for Kundt found, that the indices of refraction for green and
blue were 1'3452 and 1*3420 respectively, showing that the blue
is less refracted, than the green in the neighbourhood of D. In
the region between D and G there are also several absorption
bands.
313. By means of his experiments, Kundt deduced the follow-
ing law :
On the lower or less refrangible side of an absorption band, the
refractive index is abnormally increased; whilst on the upper or
more refrangible side, it is abnormally diminished.
In order to clearly understand this law, let us revert to the
spectrum produced by fuchsine. In this substance the absorption
is very strong between D and F, that is in the green portion of the
spectrum ; and on looking at the table, we see that the red and
orange rays, which lie below the green in the spectrum produced
by a glass prism, lie above it in the case of fuchsine ; whilst the
violet rays lie below the green. The refrangibility of the red and
orange rays is therefore abnormally increased, whilst that of the
violet is abnormally diminished.
Selective Reflection.
314. We have already pointed out, that the colours of natural
bodies arise from the fact that they absorb certain kinds of light ;
there is however another class of substances, which strongly reflect
light of certain colours, whilst they very slightly reflect light of
other colours. The phenomenon exhibited by these substances
is called selective reflection
315. Selective reflection appears to have been first discovered
SELECTIVE REFLECTION. 299
by Haidinger 1 . It was subsequently studied by Stokes 2 ; and the
experiments of Kundt, which have already been referred to 3 , show,
that it is exhibited by most substances which produce anomalous
dispersion. In fact absorption, anomalous dispersion and selective
reflection are so closely connected together, that they must be re-
garded as different effects of the same cause, and consequently
ought to be capable of being explained by the same theoretical
considerations.
316. The properties of substances, which exhibit selective
reflection, may be classified under the following three laws :
I. Those rays which are most strongly reflected, when light is
incident upon the substance, are most strongly absorbed, when light
is transmitted through the substance.
II. When the incident light is plane polarized in any azimuth,
the reflected light exhibits decided traces of elliptic polarization.
III. When sunlight is reflected, and the reflected light is
viewed through a Nicol's prism, whose principal section is parallel
to the plane of incidence, the colour of the reflected light is different
from what it is, when viewed by the naked eye.
Since the reflected light is ellipticaliy polarized, it follows that
selective reflection is accompanied by a change of phase of one or
both the components of the incident light.
317. The properties of substances, which exhibit selective
reflection, resemble those of metals, as will be explained in the
Chapter on Metallic Keflection. For in the first place, metals
strongly absorb light, and powerfully reflect it; and in the second
place light reflected by a metallic surface is always ellipticaliy
polarized, unless the plane of polarization of the incident light is
parallel or perpendicular to the"plane of incidence. The optical
properties of these substances appear to occupy a position, inter-
mediate between ordinary transparent media and metals ; and on
this account, selective reflection is sometimes called quasi-metallic
reflection.
1 Ueber den Zusammenhang der Korperfarben, oder des farbig durchgelassenen,
und der Oberflachenfarben, oder des zuruckgeworfenen Lichtes gewisser Korper.
Proc. Math, and Phys. Class of the Acad. of Sciences at Vienna 1852, and the papers
there cited.
2 Phil. Mag. (4) vi. p. 393.
3 Ante, p. 297.
*"" OF THE
UNIVERSITY
300 MISCELLANEOUS EXPERIMENTAL PHENOMENA.
318. On the other hand, metallic reflection produces very
little chromatic effect, whilst the peculiarities of substances which
produce selective reflection principally consist in chromatic effects.
Moreover reflection from ordinary transparent substances, is con-
siderably weakened by bringing them into optical contact with
another having nearly the same refractive index ; but in the case
of quasi-metallic substances, the colours which they reflect, are
brought out more strongly by placing them in optical contact with
glass or water.
319. That there are certain substances, which strongly reflect
light of the same periods as those which they absorb, is strikingly
exemplified in the case of permanganate of potash. Stokes found 1 ,
that when the light transmitted by a weak solution is analysed by
a prism, there are five absorption bands, which are nearly equi-
distant, and lie between D and F. The first band, which lies a
little above D, is less conspicuous than the second and third, which
are the strongest of the set. If however light incident at the
polarizing angle is reflected from permanganate of potash, and is
then passed through a Nicol, placed so as to extinguish the
light polarized in the plane of incidence, the residual light is
green ; and when it is analysed by a prism, it shows bright bands
where the absorption spectrum shows dark ones.
320. Safflower-red or carthamine is another example of a
substance which exhibits selective reflection. Stokes found, that
this substance powerfully absorbed green light, but reflected a
yellowish green light; and that when red light polarized at an
azimuth of 45 was incident upon this substance, the reflected
light was sensibly plane polarized, but when green or blue light,
polarized in the same azimuth, was substituted, the reflected light
was elliptically polarized. It further appeared, that the chro-
matic effects of this substance were different, according as the
incident light was polarized in or perpendicularly to the plane of
incidence ; for when the incident light was polarized perpendi-
cularly to the plane of incidence, the reflected light was of a very
rich green colour, but when it was unpolarized the reflected light
was yellowish-green.
Similar results were obtained by using a compound of iodine
and quinine called herapathite, which was discovered by Dr
Herapath of Bristol, and which strongly absorbs green light.
1 Phil Mag. Vol. vi. (18) p. 293.
SELECTIVE REFLECTION.
301
321. The effect of bringing a transparent medium into
optical contact with a quasi-metallic substance, may be illustrated
by depositing a little safflower-red upon a glass plate, and allowing
it to dry ; when it will be found that the surface of the film which
is in contact with air, is of a yellowish-green colour; whilst the
surface in contact with glass, reflects light of a very fine green
inclining to blue. Similar effects are produced with herapathite
and platino-cyanide of magnesium. The latter crystal is one of a
class of special optical interest, since it is doubly refracting,
doubly absorbing, doubly metallic and doubly fluorescent.
322. Further experiments upon selective reflection have been
made by Kundt 1 , who found that it was strongly exhibited by the
aniline dyes and other substances, which produce anomalous
dispersion. The following table 2 shows the colour of the trans-
mitted and reflected light, when the latter is viewed with the
naked eye and through a Nicol's prism, adjusted so as to ex-
tinguish the component polarized in the plane of incidence.
Substance.
Transmitted.
Reflected.
Reflected & passed
through a Nicol.
Rose aniline or fuchsine
Rose
Green
Peacock blue
Mauve aniline
Mauve
Apple green
Emerald green
Malachite green
Deep green
Plum colour
Orange gold
Blue aniline
Blue
Bronze
Olive green
Fluorescence.
323. When common light -is incident upon a solution of
sulphate of quinine in water, it is found that the surface of the
liquid exhibits a pale blue colour, which extends a short distance
into the liquid; if however the light which is refracted by the
substance, and has therefore passed through the thin coloured
stratum, is allowed to fall upon the surface of a second solution
of sulphate of quinine, the effect is no longer produced.
The peculiar action which sulphate of quinine, as well as
1 Ante, p. 297, footnote.
2 Glazebrook's Physical Optics, p. 273.
302 MISCELLANEOUS EXPERIMENTAL PHENOMENA.
certain other substances, produces upon light, is called fluorescence.
It was first discovered by Sir David Brewster 1 in 1833, who
observed that it was produced by chlorophyll, and also by fluor
spar. Sir J. Herschel 2 found that fluorescence was produced by
quinine, but the subject was not fully investigated, until it was
taken up by Sir G. Stokes 3 in 1852.
324. To examine the nature of fluorescence produced by
quinine, Stokes formed a spectrum b^ means of a slit and a
prism, and filled a test tube with the solution, and placed it a
little beyond the red extremity of the spectrum. The test tube
was then gradually moved up the spectrum, and no traces of
fluorescence were observed, as long as the tube remained in the
more luminous portion ; but on arriving at the violet extremity,
a ghost-like gleam of pale blue light shot right across the tube.
On continuing to move the tube, the light at first increased in
intensity, and afterwards died away, but not until the tube had been
moved a considerable distance into the invisible ultra-violet rays.
When the blue gleam of light first made its appearance, it ex-
tended right across the tube, but just before disappearing, it was
observed to be confined to an excessively thin stratum, adjacent
to the surface at which the light entered.
325. This experiment shows that in the case of quinine,
fluorescence is produced by violet and ultra-violet light, and also
that it is due to a change in the refrangibility of the incident
light. Stokes also found, that quinine was exceedingly opaque to
those rays of the spectrum which lie above the line H } that is to
those rays by which fluorescence is produced. This explains why
light, which has been passed through a solution of quinine, is
incapable of producing fluorescence, for the solution absorbs the
rays which give rise to this phenomenon.
326. The effect may accordingly be summarized as follows.
Quinine is transparent to the -rays constituting the lower or more
luminous portion of the spectrum, but it strongly absorbs the
ultra-violet rays, and gives them out again as rays of lower
refrangibility. The latter circumstance enables the eye to take
cognizance of the invisible ultra-violet rays ; for if this portion of
1 Trans. R. S. E. Vol. xii. p. 542.
2 Phil. Trans. 1845.
3 Ibid. 1852, p. 463.
FLUORESCENCE. STOKES' LAW. 303
the spectrum is passed through a fluorescent substance, it is
converted into luminous rays, which are visible, and can be
examined by the eye. By this method Stokes was able to make
a map of the fixed lines in the ultra-violet region.
327. Fluorescence is also produced by a number of other
substances, among which may be mentioned decoction of the bark
of the horse-chestnut, green fluor spar, solution of guaiacum in
alcohol, tincture of turmeric, chlorophyll, yellow glass coloured
with oxide of uranium &c. It must not however be supposed,
that the light produced by fluorescence is of the same colour for
all substances, since as a matter of fact, it varies for different
substances. Thus the fluorescence produced by chlorophyll con-
sists of red light, showing that this substance converts green and
blue light into red light.
328. As the result of his experiments, Stokes was led to the
following law, viz. ; When the refrangibility of light is changed
by fluorescence, it is always lowered and never raised.
Whether this law is absolutely general has lately been
doubted; and there appears to be some evidence, that excep-
tions to it exist.
329. We have already called attention to the fact, that the
phenomena of dispersion, absorption and the like, are caused by
the molecules of matter being set in motion by the vibrations of
the ether. Now if the molecular forces depended upon the first
power of the displacements, it would follow from Herschel's
theorem, that the period of the forced vibrations would be equal
to that of the force ; if however the molecular forces depended
upon the squares or higher powers of the displacements, Her-
schel's theorem would be no longer true, and under these circum-
stances Stokes suggested, that fluorescence arises from the fact,
that the forces are such, that powers of the displacements higher
than the first cannot be neglected. We have already pointed out,
that ultra-violet light produces strong chemical effects. Now the
molecules of most organic substances consist of a number of
chemical atoms connected together, and forming a system of more
or less complexity, which is stable for some disturbances but
unstable for others. For instance, an ordinary photographic plate
is fairly stable for disturbances produced by sodium light, but
unstable for those produced by violet light. It is therefore not
304 MISCELLANEOUS EXPERIMENTAL PHENOMENA.
unreasonable to suppose, that the amplitudes of the vibrations
communicated by ultra-violet light to the molecules, and to the
atoms composing them, of a substance like quinine, should be of
such far greater magnitude, than those communicated by light of less
refrangibility, that the molecular forces produced under the former
circumstances cannot be properly represented by forces proportional
to the displacements. If this be the case, the period of the forced
vibrations will no longer be equal to that of the force.
330. When a molecule is set into vibration by ethereal waves,
the vibrations of the molecule will give rise to secondary waves in
the ether. The periods of these secondary waves must necessarily
be the same as those of the molecules by which they are produced ;
for Herschel's theorem applies to vibrations communicated to the
ether, although it does not necessarily apply to vibrations com-
municated to the molecules. And if the periods of the secondary
waves are longer than those of the waves impinging on the mole-
cules, these waves will be capable of producing the sensation of
light, provided their periods lie within the limits of sight, even
though the periods of the impinging waves are too short to be
visible. We can thus obtain a mechanical explanation of the way
in which fluorescence is produced, but at the same time the
following illustration will make the matter clearer.
331. The equation of motion of a molecule, which is under
the action of molecular forces, which are proportional to the cube
of the displacement, and which is also under the action of a
periodic force, is
The particular solution gives the forced vibration, whilst the
complementary function gives the free vibration. The deter-
mination of the particular solution when F=Ae tpt , where A is an
arbitrary constant, would be difficult ; but as the above equation
is given as an illustration, and not for the purpose of constructing
a theory, we shall suppose that the force is represented by
5^7o cos 3pt.
6n*J6
The particular solution will then be found to be
2
CALORESCENCE. 305
From this result we see, that the period of the forced vibration
is three times that of the force ; accordingly the secondary waves
will be of longer period, and consequently less refrangible, than
the impinging waves.
Calorescence.
332. This phenomenon is the reverse of fluorescence, and
consists in the conversion of waves of long period into waves of
shorter period. Calorescence is well exhibited by the experiment
of Tyndall already described under the head of spectrum analysis 1 ,
in which the light from an electric lamp is sifted of the luminous
rays, by passing it through a solution of iodine in disulphide of
carbon, which only allows the infra-red rays to pass through.
333. In order to obtain a mechanical model which will
illustrate Calorescence, we may revert to the differential equation
(1). It can be verified by trial, that the complementary function is
y = aciL(afit + a), & = 2~* (2),
where a and a are the constants of integration. From this result
it follows, that the amplitude of the free vibration is a, and its
period is ^Kj^a, which is inversely proportional to the amplitude.
; Hence the period diminishes as the amplitude increases.
Equation (2) may still be regarded as the complete solution of
1(1), provided we suppose that a and a, instead of being constants,
are functions of the time, and their values might be found by the
I method of variation of parameters. If now, we suppose that the
molecular forces are such, that a increases slowly with the time,
[we may illustrate the conversion of waves of dark heat into waves
of light. When the waves of dark heat first fall on the substance
and the forces begin to act, the amplitude is very small, and
Consequently the period r is very large. On both these grounds
^therefore, the vibrations are incapable of affecting the senses. As
|;bhe forces continue to act, the amplitude increases, whilst the period
i Jiminishes, and the vibrations become sensible as heat ; in other
vords the substance begins to get hot. As this process continues,
.he substance becomes red hot, and then intensely luminous. As
/he amplitudes cannot go on increasing indefinitely with the time,
ve must suppose that after the expiration of a certain period, the
1 Ante, p. 286.
B. o. 20
306 MISCELLANEOUS EXPERIMENTAL PHENOMENA.
condition of the substance changes owing to liquefaction or vapour-
ization, and that the equation by means of which the original state
of things was represented, no longer holds good.
Phosphorescen ce.
334. When light is incident upon certain substances, such as
the compounds of sulphur with barium, calcium or strontium, it is
found that they continue to shine, after the light has been removed.
This phenomena is called phosphorescence.
Phosphorescence is closely allied to fluorescence, inasmuch as it
is usually produced by rays of high refrangibility, and the refrangi-
bility of the phosphorescent light is generally less than that of the
light by which it is produced. The principal distinction between
the two phenomena is, that fluorescence lasts only as long as the
exciting cause continues, whilst phosphorescence lasts some time
after it has been removed.
335. In order to give a mechanical explanation of phosphores-
cence, we shall employ an acoustical analogue, which will frequently
be made use of, and which will be fully worked out in 337.
Let plane waves of sound be incident upon a sphere, whose radius
is small in comparison with the lengths of the waves of sound ;
and let the sphere be attached to a spring, so as to be capable of
vibrating parallel to the direction of propagation of the waves.
Then it is known 1 , and will hereafter be proved, that the effect of
the waves of sound will be to cause the sphere to vibrate. If the
strength of the spring is such, that the force due to it is propor-
tional to the displacement of the sphere, the forced period of the
latter will be equal to that of the impinging waves of sound ; if
the law of force depends upon some power of the displacement, the
forced period of the sphere will be different ; but in either case
secondary waves will be thrown off. These secondary waves will
travel away into space carrying energy with them, which has been
in the first instance communicated to the sphere by the incident
waves, and then communicated back again to the air in the form
of secondary waves. If the cause which produces the incident
waves be removed, the sphere will still continue to vibrate, but it
1 Lord Kayleigh, Theory of Sound, Ch. xvu.
PHOSPHORESCENCE. 307
cannot go on vibrating indefinitely, because the energy which it
possessed at the instant at which the incident waves were stopped,
will gradually be used up in generating secondary waves, and will
be carried away into space by them ; hence the sphere will ulti-
mately come to rest, and no more secondary waves will be
produced.
Now although the molecules of a non-phosphorescent substance
cannot be supposed to come to rest immediately the exciting cause
is removed, yet the time during which they continue to be in
motion is too short to be observed; but owing to the peculiar
molecular structure of phosphorescent substances, the molecules
remain in motion for a longer period. Hence a luminous glimmer
exists for some time after the incident light has been cut off.
202
CHAPTER XVII.
THEORIES BASED ON THE MUTUAL REACTION BETWEEN
ETHER AND MATTER.
336. IN the present Chapter, we shall give an account of some
of the attempts which have been made to explain on dynamical
grounds the phenomena described in the previous Chapter.
It may be regarded as an axiom, that when ethereal waves
impinge upon a material substance, the molecules of the matter
of which the substance is composed, are thrown into a state of
vibration. This proposition is quite independent of any hypo-
thesis, which may be made respecting the constitution of the
ether, the molecular forces called into action by the displacements
of the molecules of matter, or the forces arising from the action of
ether upon matter. It may therefore be employed as the basis of
a theory, in which the ether is regarded, either as a medium
possessing the properties of an elastic solid, or as one which is
capable of propagating electromagnetic disturbances as well as
luminous waves. The difficulties of constructing theories of this
description arise, not only from the fact that the properties of the
ether are a question of speculation, but also because the forces due
to the action of matter upon matter, and of ether upon matter, are
unknown.
During the last five and twenty years, numerous attempts
have been made by continental writers to develop theories of this
description, and an account of them will be found in Glazebrook's j
Report 071 Optical Theories 1 . It cannot, I think, be said that any
of these theories are entirely satisfactory ; but at the same time>
1 Brit. Assoc. Eep. 1886.
VIBRATIONS OF A SPHERE IN AIR. 309
they clearly indicate the direction, in which we must look for an
explanation of the phenomena, which they attempt to account for.
337. As an introduction to the subject, we shall work out and
discuss the problem of the sphere vibrating under the action of
plane waves of sound, which has been referred to in the last
Chapter. The problem itself was first solved by Lord Rayleigh 1 ,
and has been reproduced by myself 2 in an approximate form ; but
there are several additional points which require consideration.
Let c be the radius of the sphere; let K = 2-Tr/X, where \ is
the wave-length ; and let a be the velocity of sound. Then the
velocity potential of the incident waves may be taken to be
'? Kat , where ' = e iKX . Now if //, = cos 6,
6 = 2 "^(c)P(/0,
where P n is the zonal harmonic of degree n, and 3
F(c}= (""O n [i ^ v
1.3.. .(2w -1)( 2 2w + 3 "*" 2 . 4 . 2n + 3 . 2n + 5 '
If (f>e LKat be the velocity potential of the secondary waves, we
may assume 4
< = S-M^nPn,
g MBT
where ijr n = f n (ucr\
and/ n is the function defined by (40) of 231.
If X be the resistance due to the pressure of the air,
X = - 27T/9C 2 I* ((// + <) i/cae tKat cos sin 0d0
Jo
r
J
-i ^
^c) + ^i^i(c)} ** ............ (2),
where M' is the mass of the fluid displaced.
If y e at \) Q the velocity of the sphere, the boundary condition
gives
dp + ^=F.
dc dc
1 Proc. Lond. Math. Soc. vol. iv. p. 253.
2 Elementary Hydrodynamics and Sound, 167.
3 Lord Rayleigh, Theory of Sound, Ch. xvn.
4 Stokes, On the communication of vibrations from a vibrating body to the
atmosphere. Phil. Trans. 1868.
310 DYNAMICAL THEORIES OF DISPERSION. ETC.
Now fa
dc C \K 2 C 2 IKC ) '
whence ^.1^1 = -
z
_ ( v ~ dF ^l dc ) ( 2 + K * c * - IKZ c s c + =p (f -
+ ^f = ...... (4).
To integrate this equation, assume f = Ae 2< - ntlT , then
_ _ _
~ M (r /2 - r 2 ) + M' ( % + i* 2 c 2 ) r' 2 - 2 ' * '
338. From these results we draw the following conclusions.
Equation (4), which is the equation of motion, contains a
viscous term, that is a term proportional to the velocity. This
term arises from the circumstance, that the sphere is continually
losing the energy which it receives from incident waves, by
generating secondary waves, which travel away into space carrying
energy with them. If therefore the supply of energy be stopped,
OPTICAL APPLICATIONS. 311
by removing the cause which produces the impinging waves, the
sphere will" gradually get rid of all its energy, and will ultimately
come to rest. The time which elapses before the sphere comes to
rest, will depend upon the value of the modulus of decay ; if this
quantity is small, the vibrations will die away almost instanta-
neously ; but if the modulus of decay is larger, the vibrations will
continue for a sufficient time to enable our senses to take cog-
nizance of them.
339. Now whatever supposition we make concerning the
mutual reaction between ether and matter, it is practically
certain that the motion of a molecule of matter will be repre-
sented by an equation, whose leading features are the same as (4),
although the equation itself may be of far greater complexity.
We therefore infer, that the molecular structure of non-phos-
phorescent substances is such, that the modulus of decay is so
small as to be inappreciable; whilst the molecular structure of
phosphorescent substances is such, that the modulus of decay is
considerably larger.
340. We must now consider the amplitude of the vibrations
of the sphere, which is given by (5). The density of all gases is
exceedingly small, compared with the densities of substances in
the solid or liquid state ; consequently M f is very small compared
with M. Hence the amplitude of the sphere is exceedingly small
in comparison with that of the incident waves (which has been
taken as unity), unless r and T' are nearly equal. When T = T',
the large term in the denominator disappears, and A is approxi-
mately equal to 3/a. Under these circumstances, the amount of
energy communicated by the incident waves to the sphere, is very
much greater than what it would have 'Been, if the difference
between T and T were considerable.
341. Let us now consider a medium, such as a stratum of
sodium vapour. We may conceive the molecules of the medium
to be represented by a very large number of small spheres, and
the molecular forces to be represented by springs. The medium
will therefore have one or more free periods of vibration. The
interstices between the molecules are filled with ether, which is
represented by the atmosphere. When waves of light pass through
the medium, the molecules will be set into vibration, and a certain
amount of energy will be absorbed by them ; but since the mass of
312 DYNAMICAL THEORIES OF DISPERSION, ETC.
a molecule is exceedingly large compared with the mass of the
ether which it displaces, the amplitudes of the vibrations of the
molecules will be very small, and very little energy will be
absorbed, unless the period of the waves is equal, or nearly so,
to one of the free periods of the system. But in the case of
equality of the free and forced periods, the amplitudes will be so
large, that a great deal of energy will be taken up by the mole-
cules, and of the energy which entered the stratum of vapour,
very little will emerge. Light will therefore be absorbed. The
absorption bands produced by sodium vapour may therefore be
explained, by supposing that sodium vapour has two free periods,
which are very nearly equal to one another, and accordingly
produce the double line D in its absorption spectrum. Hydrogen,
on the other hand, has three principal free periods, which are
separated from one another by considerable intervals.
The occurrence of an imaginary term in the denominator of A
shows, that A can never become infinite for any real value of r.
This remark will be found of importance later on.
Lord Kelvins Molecular Theory.
342. We shall now consider a theory, which was developed by
Sir W. Thomson, now Lord Kelvin, in his lectures on Molecular
Dynamics, delivered at Baltimore in 1884.
The molecules of matter are represented by a number of
hollow spherical shells, connected together by zig-zag massless
springs; and the outermost shell is connected by springs to a
massless spherical envelop, which is rigidly connected with the
ether. The space between any two shells is supposed to be a
vacuum, and transparent and other substances are supposed to
consist of a great number of such shells, which may be imagined
to represent the molecules.
The degree of complexity of the molecule will depend upon
the number of shells which it contains ; and we can by this means
represent chemical compounds of every degree of complexity.
We shall first of all investigate the motion of a single molecule,
on the supposition that the centres of all the spherical shells are
vibrating along a fixed straight line. We shall also suppose, that
the force exerted by the springs joining two consecutive shells is
proportional to their relative displacements; and that the force
LORD KELVIN S MOLECULAR THEORY.
313
exerted by the ether on the envelop is proportional to x ;
where x l is the displacement of the outermost shell, and f may be
regarded indifferently, either as the displacement of the envelop,
or of the ether in contact with it.
343. Let mi/47r 3 be the mass of the i th shell, Xi its displace-
ment, Ci the strength of the spring connecting the i tl * and (i l) th
shells. Then the equations of motion of the system of shells will be
X. 2 = (7 2 #1 - # 2 ) ~ ^3 (#a -
If r be the period, and if we put
a i = m i /T*-Ci-C i+1
the equations of motion will reduce to the form
(7),
(8).
The first equation of the form (8) is
-03=0^ + C& ..................... (9),
so that we may regard # , as equal to f the displacement of the
ether.
If we suppose that the / h shell is attached to a fixed point,
the j tb equation of motion will be
whence C^Cj_! = o^- ........................ (10).
Although it is scarcely admissible to suppose, that the / h shell
is attached to a fixed point, yet if we suppose that the (j + l) th
shell consists of a solid nucleus, whose mass is large compared
with that of the other shells, its motion will be sufficiently small
to be neglected.
There are j 2 equations of the form (8), which are obtained
by putting i = 2, 3, . . . j - 1 ; and these equations together with
(9) and (10) furnish altogether j equations, from which the j 1
quantities # 2 , x s ... Xj can be eliminated, and we shall thus obtain
a relation between x^ and f.
314 DYNAMICAL THEORIES OF DISPERSION, ETC.
344. To perform the elimination, let
JSL ........................ (ii).
Then (9), (8) and (10) become
(13).
Also since Xj +l = ; Uj +1 = oo , and therefore
,-=; .............................. (14).
From these equations we see, that u^ can be expressed in the
form of the continued fraction
Putting for a moment & for d/dr~ z , it follows from (7) that
Sa,i = mi ;
and therefore by (13) and (14),
(7 2 -
whence
^ (16).
\ ,+!+.
C*, . - . .
But from 11)
i & 1 l*CW
also out = - %T 3 -= ,
dui 2 ,
whence -T- = (m^ + m i+1 a; 2 i +1 -I- ... m^) (17).
From this equation we see, that dui/dr is always negative;
and therefore Ui diminishes as T increases.
I
DISCUSSION OF THE CRITICAL CASES. 315
345. When r is sufficiently small, all the us are exceedingly
large positive quantities ; for since Uj +1 = GO , it follows from (7) and
(14) that
so that Uj can always be made as large a positive quantity as we
please, by taking T small enough ; whence it follows from (7) and
(13), that all the u's can be made positive, provided r is small
enough. But if HI is positive, we see from (11), that the signs of
Xi and Xi-i must be different ; accordingly when r is very small,
each shell is moving in the opposite direction to the two adjacent
ones ; also when Ui is large, the numerical value of x^ must be
very much greater than that of #$.
These considerations show, that when the period is exceedingly
small, the vibrations of each shell, and also those of the outer mass-
less envelop, which is supposed to be rigidly connected with the
ether, are executed in opposite directions ; and that the ampli-
tudes of the vibrations of successive shells diminish with great
rapidity, as we proceed inwards into the molecule.
It follows from (7) and (13) that as r increases, ai diminishes,
whilst C 2 i+1 /Ui +l increases; accordingly when T has sufficiently
increased, u^ will be zero. Now when Ui is zero, Ui_^ = oo ; and
will therefore have passed through zero, and have changed sign for
some value of r, less than that for which ui became zero. It
therefore follows, that as T increases from a value for which all
the us are positive, u^ will be the first quantity which vanishes
and changes sign, and that u 2 will be the next and so on.
346. In the problem we are considering, the motion is sup-
posed to be produced by means of a forced vibration of amplitude
f, and therefore when x = oo , Ui = ; but u will also vanish when
f = 0, hence the critical period for which u vanishes is the least
period of the free vibrations of the system, when the massless
envelop is motionless. As soon as T exceeds the first critical
period, y^ will become negative ; and consequently the first shell
and the massless envelop, will be moving in the same directions,
whilst all the other shells will be moving in opposite directions.
If now T be supposed to still further increase, u 2 will diminish and
finally vanish, in which case u : = oo , x = 0. This is the second
critical case ; and the period of vibration is equal to the period of
the free vibrations of the system when m x is fixed, and all the
316 DYNAMICAL THEORIES OF DISPERSION, ETC.
other shells are vibrating in opposite directions ; and this period
is the least period of the possible free vibrations of the system
under these conditions. The remaining critical cases can be
discussed in a similar manner.
347. From (13) and (14), we have
From this equation we see, that I/u>j-i is a fraction, whose
numerator is a linear function, and whose denominator is a
quadratic function of r~ 2 . It therefore follows that 1/X, or
QhJQ-J;* is a fraction whose numerator is a (j l) th , and whose
denominator is a j ih function of r~ 2 . Since z^ is zero, when r is
equal to any one of the j periods of the free vibrations of the
system, when the envelop is held fixed, it follows that the denomi-
nator of 1/M,. is expressible as the product of factors of the form
K\ r 2 , where K I} K 2 ... KJ are the above mentioned free periods.
The value of 1/X may therefore be resolved into partial
fractions, and may accordingly be expressed in the form
1 X, _ gl ?2
^~ " C " ^ 2 /T 2 - 1 + K 2 2 /T 2 - 1 +
where q ly q 2 ... are constants.
Writing for a moment Di for ^/r 2 - 1, (18) becomes
, du,
whence
Now Xi is the amplitude of the i th shell ; if therefore we denote
the actual displacement by #';, we shall have
x'i = Xi sin 27r/r,
provided t be measured from the epoch at which each shell passes
through its mean position. The energy in this particular con-
figuration will be wholly kinetic; whence remembering that the
mass of each shell is equal to ra^Tr 2 , it follows that if E be the
total energy,
MOTION OF ETHER AND MATTER. 317
by (17). Let
R~ l
so that R~ l denotes the ratio of the whole energy of the molecule
to that of the first shell; then (19) becomes
Hence if RI denote the value of R when r KI, we obtain
qi
and (18) becomes
- r
348. Let us now imagine a medium, whose structure is
represented by a very large number of molecules of the kind we
have been considering ; and let us suppose, that the interstices
between the molecules are filled with ether, which is assumed to
be a medium, whose motion is governed by the same equations as
those of an elastic solid. Then if we confine our attention to a
small element of the medium, which contains molecules and ether
surrounding them, and for simplicity consider the propagation of
waves parallel to the axis of x, the equation of motion of a
particle of ether will be
Integrating this equation throughout the volume of the ele-
ment, we obtain
- n
The first surface integral on the right-hand side is to be taken
over the outer boundary of the element, whilst the second is to be
taken over the boundaries of each of the molecules. Let w be the
mean value of w' within the element ; then the values of w' at the
points x + ^Sx and x J&c will be
, dw ~ , , dw ~
w + \ -j- ox and w * ^- ox,
ax ax
so that the first integral reduces to
318 DYNAMICAL THEORIES OF DISPERSION, ETC.
The second integral represents the resultant of the forces,
which each molecule exerts upon the ether ; arid if we represent
the force due to a single molecule by 4m z C (' w') dx f dy' dz' , it
follows that we may represent the resultant force due to all the
molecules within the element by 4<7r 2 C(w)SxSy&z, where f is
the mean value of the displacements of all the molecules. The
equation of motion therefore becomes
349. To solve (23) assume
w== g 2nr/T
=a , i6 2t7r/T
Substituting in (23), we obtain
P ?
Now (n/p)% is the velocity of light in vacuo, whence the left-
hand side is equal to p 2 , where //, is the index of refraction.
Hence if we substitute the value of # x /f from (21), and write q L for
C-iR-^K-^/mu we shall finally obtain
350. This equation determines the index of refraction in
terms of the period. To apply it to ordinary dispersion, we shall
write it in the form
From the manner in which this result has been obtained, it
follows that K I} K Z ... are in ascending order of magnitude. Now
in the case of ordinary dispersion, //, increases as the period
diminishes, whence we must have T > ^ and < K 2 ; also the
quantities q 2) q 3 must be inappreciable, and q l must be very
slightly less than unity. Under these circumstances, we approxi-
mately obtain
If we omit the term (1 q t ) r 2 , this expression is the same
as Cauchy's dispersion formula, which agrees fairly well with
experiment; Ketteler has however shown that for certain sub-
stances, the term (1 ^ ; then when T is excessively
small, yu, 2 will be less than unity ; as T increases, /i 2 will diminish
to zero, and will then become a negative quantity. When p* is
negative, the velocity of the waves will be imaginary, and conse-
quently waves whose periods produce this result, are incapable
of being propagated in the medium, and absorption will take place.
When r = tf 1 , /A 2 = QO ; and as soon as T>K I , //, 2 becomes a very
large positive quantity, and regular refraction begins to take place.
As T further increases, ^ diminishes, until it vanishes and changes
sign. A second absorption band accordingly commences, and
continues until r>/c 2 , when regular refraction begins again, and
so continues until T = oo .
352. The following figure will serve to give the reader a
general idea of the value of /t a . The abscissa represent the
values of r 2 , and the ordi nates the values of /t 2 . The dotted lines
AC, ED, EF are the lines T = /C I , T = tc 2 , yu, = l, and the upper
parts of the curves Pp, Qq represent the visible portion of the
spectrum produced by a prism filled with a substance, which
produces anomalous dispersion, and has an absorption band in the
green.
The portion Ee may be supposed to consist of waves whose
periods are too short to be observed, then comes an absorption
band, and beyond A a region of highly refrangible ultra-violet
light commences. The line H in the spectrum may be supposed
to commence at P; and a band of light accordingly becomes
visible, which continues through the indigo to the blue, and in
320 DYNAMICAL THEORIES OF DISPERSION, ETC.
which the violet is the most refracted and the blue the least.
According to the figure, this ought to be followed by a region of
blue-green light, for which the index of refraction is less than
unity; and we must therefore suppose, that on account of the
narrowness of the region, or the faintness of the ligbt, this region
has either escaped observation, or is incapable of being detected
without more powerful instruments. An absorption band then
follows, and is succeeded by another band of more highly refracted
light, corresponding to Qq, in which the red is the least, and the
orange is the most refracted. Since the value of /x, when r is
slightly greater than /e 2 , is large, it follows that the dispersion
is anomalous; and we thus see why it is, that when there is an
absorption band in the green, orange and red are more refracted
than blue light.
353. If we compare these results with the table on p. 297, it
will be seen, that they give a fairly satisfactory explanation of the
anomalous dispersion produced by fuchsine and cyanine. The five
absorption bands produced by permanganate of potash, could be
explained by taking into account some of the terms in the value
of /u- 2 , which have been omitted.
VON HELMHOLTZ' THEORY. 321
Von Helmholtz Theory of Anomalous Dispersion.
354. The theory proposed by Von Helmholtz 1 , is a theory
relating to the mutual action of ether and matter, of somewhat
the same character as Lord Rayleigh's theory of double refraction ;
but instead of following Von Helmholtz' method, we shall give the
theory in a somewhat extended form 2 .
Let u, v, w be the component displacements of the ether, and
u 1} # 15 w l those of the matter. We shall suppose, that in vacuo
the ether is a medium, whose motion is governed by the same
equations as those of an elastic solid.
When ethereal waves pass through a material substance, the
molecules of the matter will be displaced, and the matter will
acquire potential energy. The proper form of the mathematical
expression for this potential energy is a question of speculation ;
and the first hypothesis we shall make will be, that the molecular
forces are proportional to the displacements of the matter, and
consequently the potential energy W 3 of the matter will be of
the form
where ^,33 ... are constants.
The second hypothesis is, that the potential energy of the
system contains a term, which depends upon the relative displace-
ments of ether and matter. This portion of the potential energy,
which we shall denote by W 2 , is supposed to arise from the mutual
reaction of ether upon matter ; and if we assume tfeat the corres-
ponding forces are linear functions of the relative displacements,
Tfa will be a homogeneous quadratic function of the relative
displacements, so that
(u u-tf + B (v Vj) 2 + C (w w^f + 2A' (v Vi) (w - w^
The third part of the potential energy, which we shall denote
by Wi^ is the potential energy of the ether alone, and is of the same
j form as that of an elastic solid. The total potential energy W of
| the system will therefore be
W^W.+ W.+ W, (3).
1 Pogg. Ann. vol. CLIV. p. 582; Wissen. Abhand. vol. u. p. 213.
2 Proc. Lond. Math. Soc. vol. xxm. p. 4.
B. O. 21
322 DYNAMICAL THEORIES OF DISPERSION, ETC.
If p! be the density of the matter, its kinetic energy will be
In order to introduce Lord Rayleigh's theory, we shall suppose
that the effect of the matter upon the ether, is to cause the latter
to behave as if its density were seolotropic, and the third hypothesis
will therefore be, that the kinetic energy T of the ether is
(5).
355. The equations of motion of the system may now be
deduced by the Principle of Least Action, viz.
JIP (7\ + ^ - W l - W 2 - W 3 ) dxdydzdt = ;
and will be found to be
p d?u_ d$ V2 dW.
dt 2 ~ dx du
with two similar equations, and
__,_,
pl dt?~ du, dii,"
with two similar equations.
When the medium is isotropic,
P = Q = R = P)
A = B = C = a 2 ; A' = B' = C" = 0,
,
and the equations of motion accordingly become
&c. &c.
Pl d ^= , Q = a
whence dp?ld-r* is negative; it therefore follows that p? decreases as
r 2 increases.
Since p is the density of ether when loaded with matter, it
follows that p>po\ hence when T = 0, /A > 1. As r increases, p?
diminishes to unity; it then becomes less than unity, until T
attains a value T 3 , which makes /-t = 0. When T > r 3 , p? is negative;
and consequently at this point an absorption band commences,
which continues until T = T 2 , where r 2 is the value of r which
makes the denominator vanish. When T = r 2 , ft 2 = oo ; and
when T>r 2 , /u, 2 is a very large positive quantity, and regular
refraction begins again. As T still further increases, p? continues
to diminish, until T attains a value T I? such that p? is again zero ;
when T>T!, p? becomes negative, and remains so for all greater
values of T.
The medium is therefore absolutely opaque to waves whose
periods are greater than T X ; it is transparent for waves whose
periods are less than r x and greater than r 2 ; it is ogaque for waves
whose periods lie between r 2 and r 3 , and is transparent for waves
of shorter period.
If we now suppose that r 2 corresponds to the double sodium
line D, whilst r 3 corresponds to the hydrogen line F, we shall obtain
a mechanical representation of a medium, which has an absorption
band in the green ; also the dispersion is anomalous, since the
value of p? when r is a little greater than r 2 , is greater than it is
when T is a little less than r s . A medium of this kind accordingly
represents a substance such as fuchsine, which has an absorption
band in the green, and produces anomalous dispersion.
360. To explain ordinary dispersion, we shall suppose that T
is greater than K ; then if we put
326 DYNAMICAL THEORIES OF DISPERSION, ETC.
(19) may be written
With the exception of the term involving r 2 , this value of
is of the same form as Cauchy's formula
Ketteler 1 has however shown that the term P&V//e 2 is required
to explain the dispersion produced by certain substances.
361. When there are several absorption bands, a molecule of
a more complicated character is required ; and it has been sug-
gested by Von Helmholtz, that a theory might be constructed
by a hypothesis, which practically amounts to assuming that TF 2
and W 3 consist of a series of terms of the form
{(u - u
Selective Reflection.
362. We must now consider the reflection of light at the
surface of a medium, which produces anomalous dispersion.
In forming the equations of motion by means of the Principle
of Least Action, we observe that there are no surface integral
terms, which arise from W z and W s ; it therefore follows, that the
boundary conditions at the common surface of two different media,
are unaffected by the presence of the terms depending on the
action of the matter. These conditions will therefore be, con-
tinuity of the displacements and stresses arising from the action of
the ether. With regard to the physical properties of the ether, I
shall provisionally adopt the hypothesis of Lord Kelvin, that the
latter is to be treated as an elastic medium, whose resistance to
compression is a negative quantity, the numerical value of which
is slightly less than Jrds of the rigidity.
Under these circumstances, the intensities of the reflected and
refracted light will be given by Fresnel's formulae, and so long as
/j, > 1, the reflection takes place in the same manner as from glas?.
If however the incident light is white, and K lies within the visible
spectrum, say between D and F, it follows that for certain rays of
the spectrum fju 1, which is
regularly reflected, and constitutes the remaining portion of the
spectrum. The first portion is by far the most intense, since the
reflection is total ; the third portion is the least intense ; whilst
of the second, for which p < 1, those rays for which the critical
angle is less than the angle of incidence, will be totally reflected,
and those for which it is greater, will be regularly reflected. Now
if the angle of incidence is nearly equal to the polarizing angle,
it follows that the yellow portion of the incident light, most of
which is regularly reflected, will be polarized in the plane of
incidence by reflection, and will therefore be unable to get
through the Nicol ; but the green, and also a portion of the blue
in the neighbourhood of F, will get through. The colour of the
light, when viewed through a Nicol, will accordingly change from
a green to a greenish blue, owing to the absence of the yellow
light.
CHAPTER XVIII.
METALLIC REFLECTION.
366. THE leading experimental facts connected with metallic
reflection may be classified as follows.
(i) Metals are exceedingly opaque to light, but at the same
time reflect a very large proportion of the incident light.
(ii) When plane polarized light is incident upon a polished
metallic surface, the reflected light is always elliptically polarized,
unless the incident light is polarized in or perpendicularly to the
plane of incidence, in which case the reflected light is plane
polarized.
(iii) Metals do not possess a polarizing angle, but there is a
certain angle of incidence, for which the intensity of light polarized
perpendicularly to the plane of incidence is a minimum.
(iv) When the incident light is circularly polarized, there is
a certain angle of incidence, for which the reflected light is plane
polarized.
Whatever the character of the incident light may be, it can
always be resolved into two components, which are respectively in
and perpendicular to the plane of incidence ; and the above
experimental results show, that metallic reflection produces a
change of phase in one or both of these components.
367. The angle of incidence, for which circularly polarized
light is converted into plane polarized light, is called the principal
incidence; and the azimuth of the plane of polarization of the
reflected light, is called the principal azimuth. The principal
azimuth is usually measured from the plane of incidence
330 METALLIC REFLECTION.
towards the right hand of an observer, who is looking at the
point of incidence along the reflected ray. Since the course
of a ray may be supposed to be reversed, it follows that if light
polarized in the principal azimuth, is reflected at the principal
incidence, the reflected light will be circularly polarized.
368. The values of the principal incidence and azimuth
depend not only upon the particular metal of which the reflector
consists, but also upon the transparent medium in contact with it ;
and it has been found by experiment, that the principal incidence
diminishes, whilst the principal azimuth increases with the increase
of the index of refraction of the medium in contact with the
metallic reflector.
369. Although a plate of metal of sensible thickness is
opaque, yet a very thin film of metal is semi-transparent ; and
if white light be incident upon the film, the transmitted light
is frequently coloured. Thus, if sunlight is passed through a piece
of gold leaf, the transmitted light is green. The experiments of
Quincke 1 show, that the phases of both components of the refracted
light are accelerated by transmission ; and Sir John Conroy 2 has
shown, that when light is reflected from a thin metallic film, the
principal incidence and azimuth both increase with the thickness
of the film.
That a thin film should reflect light differently from a thick
plate is to be expected. For since thin films are semi-transparent,
the wave penetrates a sufficient distance to be reflected from the
posterior surface ; whilst when the plate is thick, no second
reflection takes place, owing to the refracted wave being extin-
guished before arriving at the posterior surface. If a perfectly
satisfactory theory of metallic reflection existed, there would be
no theoretical difficulty in explaining the peculiarities connected
with reflection from, and transmission through, thin metallic films ;
all that would be necessary would be, to take into account the
successive reflections and refractions from both surfaces of the
film, in the same way as is done in the ordinary theory of the
Colours of Thin Transparent Plates.
370. The opacity of metals can be partially explained by
supposing, that the index of refraction is a complex quantity ; but
1 Pogg. Ann. vol. cxxix.
2 Proc. Roy. Soc. vol. xxxi. p. 500.
THE PSEUDO-REFK ACTIVE INDEX IS COMPLEX. 331
as this statement is ambiguous, we shall proceed to consider it
carefully.
Let the axis of x be the normal to a metallic reflector in
contact with air; and let the displacements of the incident and
refracted waves be
Since the coefficients of y must be the same in the two waves,
we must have
V sin i
If the second medium were transparent, V 1 would be a real
quantity, and consequently //, would be real ; but in a metal, there
is properly speaking no refracted wave, and therefore F 1? and
consequently p, cannot be real. We must therefore suppose, that
fju is complex.
For these reasons it is often said, that the index of refraction
of metals is a complex quantity. The expression is not however
very happily chosen, since there is no such thing as an index of
refraction in the case of metals. If however we regard the index
of refraction as a convenient name for the mathematical quantity,
which is defined by (2), there will be no danger of any ambiguity.
Putting fjL = Re ia , we obtain
V
Fj = -^ (cos a - i sin a),
and therefore, when the incidence is normal, so that I = ^ = 1,
w : = A e 27r/A ** 8in a cos -^ ( Rx cos a + Vt),
A,
where X is the wave-length in the first medium, which is supposed
to be transparent.
Now x is negative in the second medium ; accordingly sin a
must be positive, otherwise the amplitude would increase as x
increases. Hence the amplitude diminishes very rapidly as the
distance from the surface of separation increases, and at a distance
of a few wave-lengths in air, the refracted wave becomes insen-
sible.
*
332 METALLIC REFLECTION.
Since /* 2 = # 2 (cos 2a + t sin 2a),
and sin a must be positive, it follows that a must lie between
and TT. Hence ft 2 must be a complex quantity, whose imaginary
part must be positive, but whose real part may be either positive
or negative. We shall presently show, that there are reasons for
thinking that for certain metals the real part of jj? must be nega-
tive, in which case a must lie between JTT and JTT.
371. Theories of metallic reflection have been proposed by
MacCullagh 1 , Cauchy 2 and others ; and although these theories in
their original form cannot be said to stand on a satisfactory
physical basis, yet the formula of Cauchy furnish results, which
agree fairly well with experiment, and may therefore be regarded
as an empirical representation of the facts. We have shown in
the previous chapter, that it is possible to construct a dynamical
theory, such that for certain rays of the spectrum, /-t 2 shall be a
real negative quantity ; but from 370, it follows that in the case
of a metal, yu, 2 must be a complex quantity, whose imaginary part
must be positive. Following Eisenlohr 3 , we shall first show how
Cauchy's formulse may be deduced by transforming Fresnel's
formulae for transparent media; and shall afterwards discuss a
dynamical theory, by means of which this transformation may be
justified.
Cauchy's Theory.
372. When the incident light is polarized in the plane of
incidence, Fresnel's formulas for the amplitudes of the reflected
and refracted light are
sin (i + r)
2 sin r cos i
in which the amplitude of the incident light is taken as unity.
We have now to transform these formulas, by supposing that //, is
a complex quantity of the form Re ia .
1 Proc. Eoy. Ir. Acad. vol. i. p. 2.
2 C. E. 1838 and 1839.
3 Pogg. Ann. vol. civ. p. 368.
CAUCHY'S THEORY. 333
Equation (3) may be written
A' _ /* t> o ($\
cos i + fi cos r ' '
where cos r = (1 /jr 2 sinH')* (6).
Since //- is complex, it follows that cos r is complex, and we
shall therefore put it equal to ce tw ; whence we obtain from (6)
c 2 cos 2u = 1 R~ 2 cos 2a sin 2 i ) ,^.
which determine c and ^ in terms of R, a and i. Equation (5)
now becomes
,_ cos i- Rc^ a+u ^
A. r~
R 2 c 2 co$ 2 i + ZiRc cos i sin (a + u)
' + 2Rc cos i cos (a + ?^)
where
(8),
C S ' C S
# 2 c 2 + cos 2 i + 2Ec cos i cos (a + M)
2?re 2J?c cos * sin (a + u)
:
whence the reflected wave, which is the real part of
is & cos ( cos i + y sin i Vt + e),
A.
which shows that reflection is accompanied by a change of phase,
whose value is given by (10).
If we introduce a new angle f, such that
,_ 2Rc cos i cos (a + u)
/= ^ 2 c 2 + cos 2 i
= cos (a + u) sin 2 ftan" 1 -^ J ......... (11),
(9) and (10) become
& 2 = tan(/-i7r) ........................... (12),
tan = sin (a + u) tan 2 1 ......... (13).
Equations (11), (12) and (13) are Cauchy's formulae for light
polarized in the plane of incidence.
334 METALLIC REFLECTION.
373. To find what the refracted wave becomes, we have
from (4)
2 cos i
cos i + fj, cos r
2 cos i [Re cos (a 4- u) + cos i iRc sin (a + u)}
R?c 2 + cos 2 i + 2jRc cos i cos (a + u)
+ sin 2 i
_ B .R 2 c 2 cos 2 1 sin 4 i + 2iRc cos i sin 2 i sin (a + w)
~~ A ' R*c 2 cos 2 i + sin 4 1 + 2 Re cos i sin 2 1 cos (a + u) '
whence
A* ' R 2 c 2 cos 2 i + sin 4 i + 2.fic cos i sin 2 i cos (a + )
2?r , , x 2.fic cos i sin 2 1 sin (a 4- u)
tan - (e e) = ^^ ^ ; ~V^
A, v /t 2 c 2 cos 2 1 sm 4 i
If the incident light is circularly polarized, A = B ; also if the
angle of incidence is such that
.fie cos i* = sin 2 i (23),
it follows from (22), that
whence the reflected light is plane polarized. If ft be the azimuth
of the plane of polarization, we obtain from (21) and (23)
tan 2 ft = = 7 { = tan 2 i (a + u),
1 + cos (a + u)
whence ft = ^(r ji + u) (24).
We have therefore established the fourth experimental law,
which is enunciated in 366, by means of Cauchy's theory.
Accordingly (23) and (24) combined with (7) determine the
principal incidence, and the principal azimuth. It also follows
from the formulae, that if light which is plane polarized in the
principal azimuth be incident at the principal incidence, the ;
reflected light will be circularly polarized. We shall denote the
principal incidence by /.
376. These results enable us to calculate the constants R ;
and a.
Let the azimuth of the plane of polarization be defined to be {
the angle, which this plane makes with the plane of incidence, i
measured to the right hand of an observer who is looking at the j
reflected light through an analyser. Let the light polarized at an
azimuth JTT undergo two reflections at two parallel plates of
JAMINS EXPERIMENTS. 337
metal ; and let the angle of incidence be equal to the principal
incidence.
After undergoing two reflections, the component displacements
perpendicular to and in the plane of incidence will be
|V2 = & 2 cos -^ (x sin / + y sin I- Vt + 20),
A/
77 V2 = 3 2 cos -^ (a; sin / + y sin /- Ftf + 2e + JX) ;
A,
whence the light which has been twice reflected, will be plane
polarized at an azimuth %, where
tan x = 77/f = -i3 2 /^' 2 = - tan2 /3 (25).
Now the principal incidence /, and the azimuth ^, can be
determined experimentally, whence by (23) and (25) the values of
Re and ft can be found; accordingly from (24) and (7) the values of
R and a can be calculated.
377. Jamin 1 has tested Cauchy's formula for the intensities
of light polarized in and perpendicularly to the plane of incidence,
and has found that they agree fairly well with experiment. His
method of procedure was as follows. The quantities R and a were
first determined by experiment, and the amplitudes gfc and i3
were then calculated for different angles of incidence by means of
(9) and (15). This process gives the theoretical values of the ratio
of the intensities of the incident and reflected light.
To obtain the values of the intensities by experiment, a plate
of glass and metal were placed side by side, so as t^be accurately
in the same plane. A pencil of light was then allowed to fall on
the compound reflecting surface, so that part was reflected by the
glass, and part by the metal ; and the two portions of the
reflected light were passed through a doubly refracting prism,
whose principal section was inclined at an angle 7 to the plane of
incidence.
The light on emerging from the prism, thus consisted of
four images, two of which were produced by reflection from the
metal, and the other two by reflection from the glass. Let ^ 2 ,
gl' 2 be the intensities of the light reflected from the metal and the
glass, when the incident light is polarized in the plane of incidence ;
1 Ann. de Chimie et de Physique, Vol. xix. p. 206.
B. o. 22
338
METALLIC REFLECTION.
then on emerging from the doubly refracting prism, the intensities
of the ordinary and extraordinary images will be
Metal
Glass
I '2
cos 4 7 ;
E & 2 sin 2 7, a /2 sin 2 7.
For a certain value of 7, the intensity of the ordinary image of
the metal, will be equal to the extraordinary image of the glass.
For this value,
whence remembering the value of &', we obtain
sin 2 (i - r)
= tan 2 7
.(26).
The value of 7 is determined, by observing the angle at which
the intensities of the two images become equal, and thence ^ 2 can
be found by (2(j).
If the incident light had been polarized perpendicularly to the
plane of incidence, we should have had
but inasmuch as the intensity of the light reflected from the glass,
is exceedingly small in the neighbourhood of the polarizing angle,
accurate results cannot be obtained, when i is nearly equal to this
angle.
The following table for steel, which is taken from Jamin's
paper, shows how far theory and experiment agree.
Steel. Principal incidence 76.
9
33
Angle of
Incidence
i
Observed
Calculated
Observed
Calculated
85
951
977
719
709
75
946
932
566
563
65
898
892
627
599
55
869
856
...
45
818
827
689
701
35
800
804
741
717
25
791
787
769
751
JAMINS EXPERIMENTS. 339
378. Jamin also made experiments upon the difference of the
changes of phase of the two components ; and arrived at the fol-
lowing laws.
(i) The wave which is polarized perpendicularly to the plane
of incidence, is more retarded than that which is polarized in the
plane of incidence.
(ii) The difference of phase is zero at normal incidence, and
increases up to grazing incidence.
From (22), we see that
| when i 0, e' e = ;
when i = I, e' e = JX ;
when i = JTT, e' e = ^X ;
so that e f > e, and their difference gradually increases from i =
to i = JTT.
379. Experiments on the difference between the changes of
phase were made by Jamin by the method of multiple reflections.
When light polarized in any azimuth is reflected m times from
two parallel metallic reflectors, the difference of phase of the
resulting light is m (e r e) ; and if this quantity is equal to a
multiple of JX, the resulting light will be plane polarized. This
will be the case, when the angle of incidence is such that
e'-e = n\/2m (28),
where n is equal to 1, 2, ... m 1. The least angle of incidence
at which this can happen, is given by e e = X/2m ; and the
greatest by e' e = (m 1) X/2m. Hence there are altogether
1 angles of incidence. Now these angles of incidence can be
|i>bserved, and the resulting differences between the changes of
ohase calculated from (22), and compared with (28), and the two
esults ought to agree. For example, let three reflections take
)lace, and let i 1} i. 2 , the least and greatest angles, at which
)olarization is re-established, be observed; then if we substitute
he values of i lt i 2 in (22), the resulting values of e' e corre-
)onding to i lt i 2 ought to be JX and ^X. We have thus a method
testing the formulae for the difference between the changes of
base experimentally; and the experiments of Jamin show, that
lere is a fair agreement between experiment and theory.
380. It has been stated in 368, that the principal incidence
jnd principal azimuth, depend not only upon the nature of the
Iietal, but also upon the medium in contact with it. The values
222
340
METALLIC REFLECTION.
of these angles have been determined experimentally by Quincke 1
for silver, and by Sir John Conroy 2 for gold and silver, when
certain other media are substituted for air. The following table
shows the results obtained by the latter, when the incident light
was red.
Medium
Principal Incidence
Principal Azimuth
Silver in air
in water
in turpentine
74 19'
71 28'
69 16'
43 48'
44 03'
43 21'
Gold in air
,, in water
in carbon disulphide
76 0'
72 46'
70 03'
35 27'
36 23'
36 48'
In a second series of experiments, Sir J. Conroy 3 found th
following values.
Medium
Bed
Yellow
Blue
P.I.
P.A.
P.I.
P.A.
P.I.
P.A.
Gold in air
do. water
do. carbon disulphide
73 57'
70 24'
69 24'
41 52'
42 27'
42 33'
71 43'
67 39'
66 36'
41 14'
41 15'
41 41'
67 10'
63 20'
60 05'
35 40'
36 11';
3657'i
Silver in air
do. water
do. carbon tetrachloride
76 29'
73 55'
72 39'
43 51'
44 02'
44 20'
74 37'
72 15'
71 39'
43 22'
44 09'
43 40'
71 33'
67 26'
66 58'
43 00'
43 26'
44 31'
The following table shows the percentage of light reflected
different angles of incidence from the following mirrors 4 .
Angle of
incidence
Silver
Steel
Tin
Speculum
metal
10
70-05
54-38
39-76
66-13
20
70-06
55-39
40-28
66-88
30
71-35
54-93
44-38
66-87
40
70-87
55-62
44-11
67-26
50
72-49
56-74
47-48
67-26
60
74-19
57-63
50-60
66-32
65
73-58
58-37
52-32
66-53
70
74-63
58-09
54-97
67-65
75
77-25
58-69
58-85
67-43
80
81-19
63-56
65-08
70-17
1 Pogg. Ann. Vol. cxxvin. p. 541.
2 Proc. Roy. Soc. Vol. xxvin. p. 242 ; Ibid. pp. 248 and 250.
3 Proc. Roy. Soc. Vol. xxxi. pp. 490, 496.
4 Proc. Roy. Soc. Vol. xxxv. pp. 31, 32; and Vol. xxxvi. p. 187,
KUNDT'S EXPERIMENTS.
341
381. The ratio of the velocity of light in air to that in
metals, has been investigated experimentally by Quincke, Wer-
nicke, Voigt 1 and Kundt 2 .
The values which Kundt has obtained for this ratio are given
in the following table for red, white and blue light.
Bed
White
Blue
Silver
0-27
Gold
0-38
0-58
1-00
Copper
Platinum
045
1-76
0-65
1-64
0-95
1-44
Iron
1-81
1-73
1-52
Nickel
2-17
2-01
1-85
Bismuth
2-61
2-26
2-13
From this table it appears, that the velocity of light in silver
is nearly four times as great as in vacuo ; but the dispersion was
so small, that it could not be measured. Also in gold and copper,
the velocity is greater than in vacuo, and the dispersion is normal ;
but in the other four metals it is anomalous.
Beer 3 has calculated the above ratio according to Cauchy's
theory, from Jamin's observations on reflection. He found, that
silver exhibited no marked dispersion, and that the mean ratio of
the velocities was 0'25. Copper showed strong normal dispersion,
and for the red rays the ratio was less than unity; iron, on
the contrary, showed anomalous dispersion, giving //, red = 2'54,
iolet = l'47, where //, is the ratio of the velocity of light in air to
that in the metal.
382. Kundt also found, that there is a close relation between
the velocity of light in metals, and their electrical conductivities.
In the accompanying table, the velocity of light and the electrical
conductivity of silver are both taken to be 100, and the con-
ductivities are taken from Everett's Units and Physical Constants,
p. 159.
1 Wied. Ann. Vol. xxm. pp. 104147; Vol. xxv. pp. 95114.
2 Sitz. der Kon. Preuss. Akad. der Wissen., 1888; translated Phil. Mag. July
1888.
3 Pogg. Ann. Vol. xcn. p. 417.
342
METALLIC REFLECTION.
Metal
Conductivity
Velocity of Light
Silver
100
100
Gold
71
71
Copper
94
60
Platinum
16-6
15-3
Iron
15-4
14-9
Nickel
12-0
12-4
Bismuth
1-1
10-3
With the exception of copper and bismuth, it appears that
there is a fair agreement between the two sets of numbers.
383. Eisenlohr in the paper referred to in 371, has applied
Jamin's experimental results to calculate the quantities R and a
by means of Cauchy's formulas, and some of the values found by
him are given in the following table.
Extreme Bed
Yellow
Blue
Metal
a
log 7?
a
log R
a
logR
Copper
53 37'
4395
39 45'
3962
29 45'
3698
Silver
82 46'
5676
79 31'
4516
77 58'
3374
Speculum metal
57 37'
6111
51 55'
5078
51 33'
4605
Steel
31 29'
6621
32 10'
6102
36 28' ! -5782
Zinc
30 04'
5882
37 38'
5207
44 05'
4589
Now the value of /u, 2 is
.# 2 (cos2a-Msin2a),
from which we see that for silver and speculum metal, jj? must
a complex quantity, whose real part is negative. For steel, the
real part of fj? is positive ; for copper it is negative for red light, and
positive for yellow and blue ; whilst for zinc it is positive for red,
yellow and blue, but is negative for the remainder of the spectrum,
since Eisenlohr found that for indigo a = 46 23', and for the
extreme violet a = 49 08'.
384. The circumstance that Cauchy's formulae lead to the
conclusion, that for certain metals the real part of yu, 2 must be
negative, has led to an important criticism by Lord Rayleigh 1 ,
which we shall now consider.
If we suppose that the opacity of metals can be represented
mathematically by a term proportional to the velocity, the equa-
1 Hon. J. W. Strutt, Phil. Mag. May, 1872.
LORD RAYLEIGH'S CRITICISM. 343
tion of motion within the metal, upon the elastic solid theory, may
be written
, , dw l
where h is necessarily a positive constant.
The equation of motion outside the metal, will be
d 2 w _ fd*w d 2 w\
To solve these equations assume
^- (-xcosr+ysinr-Vj)
W l = A 1 V ^ T
Substituting in (29), we shall obtain
pi ihr
p + 27rp-
Under these circumstances, it follows that y? is a complex
quantity, whose real part is positive ; hence a must lie between
and JTT. Lord Rayleigh's investigation accordingly shows, that
for silver and all metals for which OO^TT, reflection cannot be
accounted for on the elastic solid theory, by the introduction of a
viscous term.
385. When we consider the electromagnetic theory of light,
it will be shown, that if we attempt to explain metallic reflection
by taking into account the conductivity of the metal, we shall be
led to equations of the same form. Hence metallic reflection
cannot be completely explained, upon the electromagnetic theory,
by means of this hypothesis.
386. We shall now show, that the circumstance of the square
of the pseudo-refractive index being a complex quantity, whose real
part is negative, may be explained by Von Helmholtz' theory.
Measuring the axis of z in the direction of propagation, and the
axis of x in the direction of vibration, the equations of motion (11)
and (10) of 357 and 356, are
d*u d 2 u
344 METALLIC REFLECTION.
Since we require a solution in which y? is a complex quantity,
we must not neglect the viscous term, and we shall find it convenient
to conduct the integration of these equations, in a manner some-
what different from that of 357.
Assume u = A e 2t7r / T ^ v ~ *>,
1\ K
-- 2 --- 2
K 2 J l
whence
47T 2
If po be the density of the ether, and U the velocity of light in
free space, n/p U 2 ; also since the pseudo-index of refraction of a
metal is defined to be the ratio of U/V, we obtain from (30)
2 _p_ ctV [ _ a 2 T 2 K 2 _ } ( .
* " fr 4^T { * 47r 2 ^ (* 2 - T 2 ) - (A 2 //?! + a') K 2 T 2 + 467T/^ 2 TJ ( '
Rationalizing the denominator, we see that the imaginary part
of y? is positive, whilst the real part is equal to
{47T 2 /)! (/C 2 - T 2 ) -
387. In order to apply this result to metallic reflection, we
shall suppose that h is a small quantity, whose square may be
neglected, under which circumstances, the real part of yP which we
shall denote by v 2 , becomes equal to
2 p
v =-
This expression is the same as the square of the refractive
index of a substance, which produces anomalous dispersion, and
has a single absorption band ; and it follows from 359, that
it may be negative in two distinct ways.
In the first place, if TJ is the least value of r for which z/ 2
and r 2 is the value of r, which makes the denominator of the
third term zero, the real part of /t 2 will be negative for values
of T lying between TJ and r 2 . In the second place, there is another
value T S of T, which is greater than r 2 , for which z/ 2 = 0; and for
all values of r > r 3 , v 2 is negative. We may therefore explain
A\JL
'
o,
APPLICATION OF VON HELMHOLTz' THEORY. 345
reflection from silver in two distinct ways. In the first place we
may suppose, that the period of the free vibrations is such, that
throughout the luminous portion of the spectrum, and some
distance beyond it and on either side, T lies between TJ and r 2 ;
or in the second place we may suppose, that throughout this
range r > T S . Now metals reflect rays of dark heat 1 in much
the same way as they reflect light; accordingly if we adopted
the first hypothesis, it would be necessary to suppose, that K
the free period of the matter vibrations, lies below the infra-red
portion of the spectrum ; if on the other hand, we adopted the
second hypothesis, it would be necessary to suppose, that K corre-
sponds to a point in or above the ultra-violet portion of the
spectrum. To explain reflection from steel, we must suppose that
K is such, that throughout the luminous portion of the spectrum
and some distance beyond, r is less than TJ, or lies between T 2 and
T S . To explain reflection from copper, we must suppose that
either r x or r 3 corresponds to a point of the spectrum intermediate
between the red and yellow, since in going up the spectrum, the
real part of fj? passes through zero, from a negative to a positive
value. But in the case of zinc, the real part of yu, 2 begins by being
positive, and then passes through zero to a negative value at a
point between the blue and violet. The theory in its present
form is therefore not applicable to zinc. It is however neces-
sary to point out, that the theory which has been developed, only
applies to a medium having a single absorption band; whereas
there is no a priori reason why metals should not possess several.
A theory such as von H-elmholtz' could be extended, so as to
apply to a medium having a number of absorption, bands ; and
there can be little doubt, that the real part of jj? would be given
by an expression of much the same form, as that furnished by Lord
Kelvin's theory.
388. The investigations of the last two Chapters, will give the
reader some idea of the various theories relating to the mutual
reaction between ether and matter, which have been proposed
to explain dispersion and metallic reflection. Further information
upon this subject, will be found in Glazebrook's Report on Optical
Theories' 2 , where a variety of theories due to Lommel, Yoigt,
Ketteler and others are considered. It must however be con-
1 Magnus, Pogg. Ann. Vol. cxxxix.
2 Brit. Assoc. Rep. 1886.
346 METALLIC REFLECTION.
fessed, that most of these theories are of a somewhat tentative
and unsatisfactory character ; and depend to a great extent upon
unproved hypotheses and assumptions made during the progress of
the work, for the purpose of obtaining certain analytical results.
The fundamental hypothesis, first suggested by Stokes 1 , and after-
wards more fully developed by Sellmeier 2 , that these phenomena
are due to the fact, that some of the free periods of the vibrations
of the molecules of matter fall within the limits of the periods of
the visible spectrum, is deserving of attentive consideration and
development. This hypothesis is quite independent of any suppo-
sitions, which may be made respecting the physical constitution of
the ether ; since any medium, which is capable of propagating
waves, would produce vibrations of the molecules of the matter
embedded in it, of the same kind as those we have been dis-
cussing.
389. We shall see in the next Chapter, that the electro-
magnetic theory of light presupposes the existence of a medium
or ether; and that the general equations of the electromagnetic
field show, that the motion of this medium is governed by equations,
which are nearly identical with those furnished by the elastic solid
theory. When electromagnetic waves impinge upon the molecules
of a material substance, the latter are thrown into a state of vibra-
tion, and by making additional assumptions respecting the mutual
reaction of ether and matter, we may translate many of the inves-
tigations based upon the elastic solid theory, into the language of
the electromagnetic theory. Moreover most transparent bodies
are dielectrics, whilst metals are conductors of electricity ; and
certain metals such as iron, cobalt and nickel are strongly magnetic.
We should therefore be led to expect, that there would be a marked
difference between the propagation of electromagnetic waves in
dielectrics on the one hand, and in metals on the other hand.
Unfortunately the electromagnetic theory, in the form in which
it has hitherto been developed, does not readily lend itself to an
explanation of dispersion and metallic reflection ; and it must be
admitted these phenomena have not as yet been satisfactorily
accounted for.
1 Phil. Mag., March 1860, p. 196.
2 Pogg. Ann. Vols. CXLII. p. 272 ; CXLV. pp 399, 520 ; CXLVII. pp. 386, 525.
CHAPTER XIX.
THE ELECTROMAGNETIC THEORY.
390. THE electromagnetic theory of light, which was first
proposed by the late Prof. Clerk-Maxwell, supposes that the
sensation of light is produced by means of an electromagnetic
disturbance, which is propagated in a medium ; and we cannot
do better than to give the fundamental idea of this theory in
Maxwell's own words 1 :
"To fill all space with a new medium, whenever any new
phenomenon is to be explained, is by no means philosophical, but
if the study of two different branches of science has independently
suggested the idea of a medium, and if the properties which must
be attributed to the medium in order to account for electro-
magnetic phenomena, are of the same kind as those which we
attribute to the luminiferous medium in order to^account for the
phenomena of light, the evidence of the physical existence of the
medium will be considerably strengthened.
"But the properties of bodies are capable of quantitative
measurement. We therefore obtain the numerical value of some
property of the medium, such as the velocity with which a
disturbance is propagated through it, which can be calculated
from electromagnetic experiments, and observed directly in the
case of light. If it should be found that the velocity of propa-
gation of electromagnetic disturbances is the same as the
velocity of light, and this not only in air, but in other transparent
media, we shall have strong reasons for believing that light is an
electromagnetic phenomenon, and that a combination of the
1 Electricity and Magnetism, Vol. n. p. 383.
348
THE ELECTROMAGNETIC THEORY.
optical with the electrical evidence will produce a conviction of
the reality of the medium, similar to that which we obtain, in the
case of other kinds of matter, from the combined evidence of the
senses.
391. We shall now proceed to apply the general equations of
the electromagnetic field, to obtain the velocity of propagation of
an electromagnetic disturbance.
The equations of electromotive force are 1
^_
dt dx
_
dt dy
-p
"dt 'dz}
The equations of magnetic induction are,
_dH dG\
dy dz
b _dF_dH
dz dx
(2).
(3)-
dx dy
The equations of the currents are,
dy_d{3}
dy dz
da dy
47TV = -j- - -j 1
dz dx
d/3 da
4i7TW = -j- - -j-
dx dy\
Maxwell's notation being employed.
392. If the medium is magnetically isotropic, the magnetic
force and the magnetic induction will be connected together by
the equations
a = pa, b = /i/3, c = py (4),
where p is the magnetic permeability of the medium.
1 Electricity and Magnetism, Vol. n. Chapter ix.
EQUATIONS OF THE ELECTROMAGNETIC FIELD. 349
393. If the medium were electrostatically isotropic, the
electromotive force in any direction, would be proportional to
the electric displacement in the same direction ; but if the
medium is aeolotropic, the relation between electromotive force
and electric displacement will depend upon the peculiar consti-
tution of the medium. We have already pointed out, that all
doubly-refracting media possess three rectangular planes of sym-
metry ; and we shall now show, that double refraction can be
explained by supposing, that the medium is electrostatically
seolotropic.
If the axes of symmetry are the axes of coordinates, the
equations connecting the electromotive force and electric dis-
placement may be written
P = 4flr//i, Q = 4,7rg/K,, R = 4>7rh/K 3 ......... (5),
where K l} K^, K z are the three principal electrostatic capacities.
If the medium were a conductor, the equations between the
electromotive force and the conduction current would be
p = C 1 P, q = C,Q, r = C 3 R ............... (6),
where C l} C 2 , C 3 are the three principal conductivities. If we
suppose the medium isotropic as regards conduction, the three C's
will be equal.
The equations connecting the true current, with the electric
displacement and conduction current, are
u =f+P> v=9 + w = h + r ............... (7).
*.
Since most transparent media are good insulators, we shall
suppose that the conduction current is zero, which requires that
394. We can now obtain the equations of electric displace-
ment.
From (1) and (2) we obtain
dt dy dz
db dP dR
dt dz dx
dt ~ dx dy
(8).
350 THE ELECTROMAGNETIC THEORY.
From (3) and (7), we obtain
dc_db
with two similar equations. Eliminating a, b, c from (9) by
means of (8) and putting
we shall obtain
&n ^a
(ii).
In proving these equations, we have not as yet made any
assumption respecting the relation between electromotive force
and electric displacement. Let us now substitute the values of
P, Q, R from (5), also let
n _ A* d f , z? 2 d 9 , n* dh n ox
**> -a. -= h Jj 7 r L> -j- (Ao),
dx dy dz
then (11) become
2 + CV .................. (21).
Also from the same equations it can be shown, that
(#-#) + -(C*-4*)+*(4 a -.B 8 ) = ...... (22),
and ( F 2 - A*) \/l = ( F 2 - &) fi/m = ( 7 2 - O 2 ) v/n ...... (23).
Let P be the point of contact of the tangent plane to the wave
surface, and let I, m, n be its direction cosines ; also let F be the
foot of the perpendicular from the origin on to this tangent plane.
Then if L, M, N be the direction cosines of PT, it is known from
the geometry of the wave surface, see (19) of 109 and 112, that
(F 2 - A*)L/l = (F 2 - &)M/m = (F 2 - C*)N/n,
accordingly by (23) we have
which shows that the electric displacement is perpendicular to the
plane of polarization.
By treating equations (15) in the same way as (14), it can be
shown that the magnetic induction lies in the plane of the wave-
front, and that it is propagated at the same rate as the electric
displacement.
398. We must now determine the magnetic force in terms of
the electric displacement.
Substituting the values P, Q, R from (5) in (8), and taking
account of (12), we obtain
at \ dz dy
MAGNETIC FORCE, 353
dq dF nadF A
But ---- -
whence a = 4?r ((7 2 mz> - 5 V ) 8/ V \
)8/V\ ............... (24).
I
y = 4-7T (BHtJL - A*m\)S/V
Multiplying these equations by I, m, n and adding, we obtain
la + m/3 + 717 = 0,
which shows that the magnetic force, and therefore the magnetic
induction, lies in the plane of the wave-front.
Multiplying by X, fj,, v and taking account of (22) we obtain
which shows that the magnetic force is perpendicular to the
electric displacement.
Let X', fi ', v be the direction cosines of the magnetic force, then
I = fjbv' fjiVy m = v\' v'\, n = A// V//-.
Substituting in the first of (24), and taking account of (21), we
obtain
a = 47T { F 2 X' - X (A*\\' + &w' + C7W)} S/V;
but X' = mv n/j, &c. ; whence it follows from (22) that
A*Mf + &pfi' + <7W = 0,
accordingly a = 4<7rVS\' ........................ (25),
and therefore the magnetic force, corresponding to an electric
displacement S, is equal to 47rF
In (24) put 1=1, ra = n = 0; ft = 1, X = z> = ; then it follows,
that when a wave is propagated along the positive direction of the
axis of x, and the electric displacement is parallel to the axis of y }
the magnetic force will be parallel to the axis of z.
399. We shall now find the direction of the electromotive
force.
From (5) and (12) we have
P = 47ryM 2 /; Q = ^irfi^g f R = *TrfrC*h,
where /^ temporarily denotes the magnetic permeability, to distin-
guish it from fju the ^/-direction cosine of the electric displacement.
Let % be the angle between the radius vector and the normal to
the tangent plane at its extremity, then
cos x = V/r, cos (J-7T 4- %) = - sin % = (Xa? + py + vz)jr. . .(26),
B. o. 23
354 THE ELECTROMAGNETIC THEORY.
where x, y, z are the coordinates of the point of contact. Now by
the geometry of the wave surface
-
(r 2 - A 2 ) \jx = (r 2 - 2 ) fi/y = (r 2 - (7 2 ) v\z = - - -- = - r sin % ,
/C
by (21) and (26) ; whence
(r 2 - A*} \jVx = (r 2 - &) fJL/Vy = (r 2 - tf 2 ) v/Vz = - tan x .
, .
But = V*-A* '
whence
(7 2 -^ 2 )X/^ = (F 2 -^ 2 )/ A / m = ( F2 -^)^ = - F2tan %'
and therefore by (19)
AH\ + ^m/Lt + 2 ni; = F 2 tan % .
The component of the electromotive force along the wave
normal is therefore
PI + Qm + En = 4-Tr^ (AH\ + Wm^ + &nv) 8 = 4^^! F 2 S tan x
and the component along the direction of displacement is
PX + Q/i + J?z/ = 4 7 r / i 1 F 2 >Sf ............... (28).
From (27) and (28) we see, that the resultant electromotive
force is equal to 4^^ F 2 $ sec %, and that its direction is perpen-
dicular to the ray.
400. The preceding analysis contains a complete investigation
of the propagation of electric and magnetic disturbances in an
electrostatically seolotropic medium; and we have shown, that both
disturbances are propagated with a velocity, which satisfies the
same mathematical conditions as the velocity of propagation of
light in a biaxal crystal, according to Fresnel's theory; but that
the electric disturbance is perpendicular to what in optical
language is known as the plane of polarization, whilst the magnetic
disturbance lies in the plane of polarization. If therefore the
sensation of light is the result of electromagnetic waves, we
must conclude, that light is the effect of the electric displace-
ment and not of the magnetic displacement (or magnetic induction).
Before however we can decide whether light is an electromagnetic
phenomenon, it is necessary to ascertain whether the electric and
magnetic disturbances are propagated with a velocity, which is
equal to or comparable with that of light.
EXPERIMENTAL VERIFICATIONS.
401. In an isotropic medium, K! = K?> =
whence (14) become
355
and fl is zero;
From these equations we see, that the velocity of propagation
V is equal to (//JT)~*. If the medium is air, and we adopt the
electrostatic system of units, K=l, and /j, = v~*, where v is the
number of electrostatic units in one electromagnetic unit, whence
V = v; or the velocity of propagation of light is equal to the
number of electrostatic units in one electromagnetic unit. If on
the other hand we adopt the electromagnetic system, K = v~ 2 ,
and //, = 1, so that the equation Vv is still true.
402. The methods of determining v are explained in Maxwell's
Electricity and Magnetism, Vol. n. Ch. xix., and are quite inde-
pendent of the methods for determining the velocity of light;
hence the agreement or disagreement of the values of V and v
furnishes a test of the electromagnetic theory of light,
The following table, taken from Maxwell, gives the values of
V and v in c. G. s. units.
Velocity of Light.
Eizeau 31400 x 10 6
Aberration &c., and
Sun's parallax 30800 x 10 6
Foucault 29836 x 10 6
Ratio of Electric Units.
Weber 31074 x 10 6
Maxwell 28800 x 10
Lord Kelvin 28200 x 10 6
From these results we see, that the velocity of light, and the
ratio of units are quantities of the same order ; but none of them
can be considered to be determined with such a degree of accuracy
as to enable us to assert, that one is greater than the other 1 .
403. In all transparent media the magnetic permeability is
very nearly equal to that of air, hence refraction must depend
principally upon differences of specific inductive capacity. Ac-
cording to the electromagnetic theory, the dielectric capacity of a
transparent medium is equal to the square of its index of refrac-
tion. But the index of refraction of light is different for different
colours, being greater for light of short period; we must therefore
select the index of refraction, which corresponds to waves of
longest period, since these are the only waves whose motion can
1 See also the note, p. 379.
232
356
THE ELECTROMAGNETIC THEORY.
be compared to the slow processes, by which the capacity of a
dielectric can be determined.
The square root of the value of K for paraffin 1 is T405 ; whilst
the index of refraction for waves of infinite period is about 1/422.
404. In discussing these experimental results Maxwell concludes
as follows :- " The difference between these numbers is greater
than can be accounted for by errors of observation, and shows
that our theories of the structure of bodies must be much j
improved, before we can deduce their optical from their electrical
properties. At the same time, I think, that the agreement between |
the numbers is such, that if no greater discrepancy were found i
between the numbers derived from the optical and the electrical
properties of a considerable number of substances, we should be I
warranted in concluding that the square root of K, although it j
may not be the complete expression for the index of refraction, is |
at least the most important term in it."
405. In 1873, which was the date of publication of the first
edition of Maxwell's treatise on Electricity and Magnetism, paraffin
was the only transparent dielectric, whose electrostatic capacity
had been determined. Since that date, the capacity of a variety
of other media have been determined, and it has been found that
for many substances, the square of the refractive index differs
considerably from the value of the electrostatic capacity.
The experiments of Hopkinson 2 give the following results forfi
the electrostatic capacity of Chance's glasses.
p
K
Kjp
M
Light flint
3-2
6-85
2-14
1-574
Double extra-dense
4-5
10-1
2-25
1-710
Dense flint
3-66
7-4
2-02
1-622
Very light flint
2-87
6-57
2-29
1-541
In this table, p is the density, K is the electrostatic capacity
and IJL is the index of refraction of the double line D of th
spectrum.
1 Gibson and Barclay, Phil. Trans. 1871, p. 573.
2 Phil. Trans. 1878, p. 17.
HOPKINSON S EXPEKIMENTS.
357
A further series of experiments was made by Hopkinson 1 ,
which gave the following results.
P
K
Double extra-dense \
flint glass J
4-5
9-896
Dense flint
3-66
7-376
Light flint
3-2
6-72
Very light flint
2-87
6-61
Hard crown
2-485
6-96
Plate glass
8-45
Paraffin
2-29
406. In the last paper an account is given of experiments
made upon certain liquids; and the results are shown in the
following table. The value of /u 2 ^ is calculated by means of
Cauchy's formula
*
K
Petroleum spirit (Field's)
do. oil (Field's)
do. common
1-922
2-075
2-078
1-92
2-07
2-10
Ozokerit lubricating oil (Field's)
Turpentine (commercial)
Castor oil
2-086
2-128
2-153
2-13
2-23
4-78
Sperm oil
Olive oil
2-135
2-131
3-02
3-16
Neats' foot oil
2-125
3-07
From these tables it appears, that the vegetable and animal
oils do not agree with Maxwell's theory, but the hydrocarbon oils
do. But in the electrical experiments, the determination was
effected by the charge and discharge of a condenser ; and it must
be recollected, that even when the time of charge and discharge is
only 5 x 10~ 5 of a second, this period is many million times longer
than the period of the waves of any portion of the visible
spectrum,
407. The capacities of Iceland spar, fluor spar and quartz have
been examined by Romich and Nowak 2 , and give results which
1 Phil. Trans. 1881, p. 355.
2 Wiener. Sitzb. vol. LXX. part ii. p. 380.
358 THE ELECTROMAGNETIC THEORY.
are much in excess of the square of the refractive index. On the
other hand, the same observers, and also Boltzmann, obtain for
crystallized sulphur, a value of the capacity in reasonable accord
with theory.
The experimental determinations of electrostatic capacities,
made by Boltzmann for paraffin, colophonium and sulphur 1 , and
also for various gases 2 ; by Silow for turpentine and petroleum 3 ; by
Schiller 4 and Wullner 5 for plate glass, will be found in the papers
referred to below.
Hertz s Experiments*.
408. The rapidity of the propagation of electrical effects across
space or any insulating medium, which has until recently eluded
all attempts at measurement, early suggested to natural philo-
sophers, that it might be connected with the mode of propagation
of light across space. For all kinds of mechanical tremors in the
matter of bodies are propagated comparatively slowly, in the
manner of sound waves ; while the propagation of free gravitation
was shown long ago by Laplace, to be extremely rapid, even
compared with that of light itself.
This suggested connection was enormously strengthened when
Maxwell, who was the first to try to express the known equations
of electrodynamic action in a form, which suggested and implied
propagation across a medium, found that his system gave rise to
electric waves of the same transverse character as waves of light,
whose velocity of propagation is an electric constant, which on
measurement turns out to be for a vacuum the same as the
velocity of light. Nor is the fact that, except for media of simple
and homogeneous chemical constitution, this agreement in velo-
cities is not very generally observed, a serious drawback to the
theory, when we consider the great difficulty of unravelling the
complex effect of the molecules of matter on the propagation of
light, and on the character of electric actions.
1 Pogg. Ann. (1874), vol. CLI. pp. 482 and 531; vol. CLIII. p. 525.
2 Ibid. (1875), vol. CLV. p. 403.
3 Ibid. (1875), vol. CLVI. p. 389; (1876), vol. CLVIII. p. 306.
4 Ibid. (1874), vol. CLII. p. 535.
5 Ibid. New Series, vol. i. pp. 247 and 361.
6 I am indebted to Mr Larmor for 3 408409.
HERTZ'S EXPERIMENTS. 359
409. There remained however another side of the subject to
explore, in the detection and systematic examination of actual
electric vibrations propagated across space. The difficulties in the
way were (i) to obtain a vibrating electric system, with periods
high enough to give waves of manageable length ; (ii) to obtain
some method of detecting their propagation. These difficulties
have been successfully surmounted within the last few years by
Hertz 1 . The vibrations were set up by the snap of an electric
discharge between two conductors, whose capacity and self-induc-
tion were so arranged as to give a wave-length of the order of
magnitude of ordinary waves of sound, or even down to a few
inches. The detector in one form consists of a wire circuit, with
a minute spark-gap in it. When placed in a field across which
waves are travelling, whose period is the same as that of the free
electric oscillations of the circuit itself, the latter acts as a reso-
nator, and reveals the presence of the waves by sparking. It was
found by Hertz, that such a resonator was excited at equidistant
positions in front of the vibrator, corresponding to half a wave-
length ; and that the circumstances corresponded in all respects
to the mode of propagation of the transverse electric waves of
Maxwell's theory. It is now pretty certain, that the radiation
from the vibrator contains a wide spectrum of wave-lengths.
The vibrator being worked by a rapid torrent of sparks from an
induction coil, each spark sets up an electric vibration swaying in
it, which is very rapidly damped by radiation, even in a very few
swings. The succession of sparks thus sends out a succession of
disturbances, which have no single definite period, but are capable
of being decomposed in Fourier's manner into a whole spectrum
of simple waves travelling out into the medium. Of these the
resonator takes up the appropriate one, and reinforces it ; thus
the observed wave-length corresponds to the period of the reso-
nator, and is in fact different for different resonators. This mode
of explanation appears to require, that when an electric vibration
is started in a resonator, it persists sensibly over the period
between two successive sparks of the primary; and therefore
that the resonator should present a small surface for radiation.
At any rate, Hertz's experiments have firmly established that
electric radiation does exist; and that its properties are exactly on
the lines indicated by the appropriate a priori electric theories.
1 Wied. Ann. vols. xxxi. to xxxvi.
360 THE ELECTROMAGNETIC THEORY.
Thus we can experiment with electrical waves of sensible length,
and thereby check theoretical developments ; and we can push on
the correspondence in properties between such waves and the
waves of light, which are of very minute length. And it hardly
admits of doubt, that in the case of a vacuum, where the compli-
cation of ponderable molecules with their disturbing free periods
does not come in, absolute continuity will be found to exist in the
transition from the one class to the other. But in the case of
ponderable media, the two classes of waves will be influenced by
free molecular periods of wholly different orders; so that any
minute numerical correspondence is perhaps not to be anticipated.
410. By means of his experiments, Hertz proved the inter-
ference, reflection and polarization of electromagnetic waves; arid
from certain calculations based upon the results of his experi-
ments, he has shown that the velocity of electromagnetic waves is
approximately the same as that of light. Trouton 1 has further
proved experimentally, that if electromagnetic waves are incident
at the polarizing angle upon a bad conductor, the waves are not
reflected when the direction of magnetic force is perpendicular to
the plane of incidence ; but when the direction of the former is
parallel to the latter, reflection takes place at all angles of inci-
dence. This experiment confirms Fresnel's hypothesis, that the
vibrations of polarized light are perpendicular to the plane of
polarization ; and that upon Maxwell's theory, the disturbance
which gives rise to optical effects is represented by the electric
displacement.
A complete discussion of the experiments of Hertz, and of the
various other theories on the connection between light and
electricity, belongs rather to a treatise on electromagnetism than
to one on light. The reader, who desires further information upon
these matters, is recommended to consult the original memoirs,
and also Poincare's Electricite et Optique, Part II., in which a very
full account of Hertz's experiments is given.
1 Nature, 22nd Aug. 1889 ; see also Fitzgerald, Proc. Eoy. Inst. March 21st, 1890.
INTENSITY OF LIGHT. 361
Intensity of Light.
411. The intensity of light is usually measured on the electro-
magnetic theory, by the average energy per unit of volume.
In a doubly refracting medium, the electrostatic energy per
unit of volume is
i (Pf+ Qg + M) = 2ir/*i (AW +
by (21).
The electrokinetic energy is
by (25).
It therefore follows that in any medium, the electrostatic and
electrokinetic energies are equal.
Let E be the total energy, and let
S= & cos -^ (Ix + my + nz Vi),
A
E = ^TrpffiV* jl + cos ~ (las + my + nz- Vt)\ .
( A, )
then
The energy therefore consists of two parts, one of which is a
constant term, and the other is a periodic term. The first term is
the average energy per unit of volume ; and consequently the
intensity of light on the electromagnetic theory, is proportional to
the product of the magnetic permeability of the medium, the
square of the amplitude, and the square of the velocity of propa-
gation in the direction in which the wave is travelling.
Conditions to be satisfied at the Surface of Separation
of Two Media.
412. The conditions of continuity of force require, that the
electric and magnetic forces parallel to the surface of separation
should be the same in both media. These conditions furnish four
equations.
As regards the conditions to be satisfied perpendicular to the
surface of separation, Maxwell has shown, Vol. I. 83, that if P, P'
be the normal components of the electromotive force at the surface
362 THE ELECTROMAGNETIC THEORY.
of separation of two media, whose specific inductive capacities are
K, K', then
whence the components of the electric displacement perpendicular
to the surface of separation must be the same in both media.
Again, if fju, // be the magnetic permeabilities of the two media,
and a, a! the normal components of the magnetic force, Maxwell
has shown Vol. 11. 428, that
fjia. p!a! = ;
whence the components of the magnetic induction perpendicular
to the surface of separation must be the same in both media 1 .
413. These conditions furnish altogether six equations, but
we shall presently show that they reduce to only four; inasmuch
as it will be proved later on, that the condition, that the electric
displacement perpendicular to the surface of separation should be
continuous, is analytically equivalent to the condition, that the
magnetic force parallel to the line of intersection of the wave-front
with the surface of separation should be the same in both media ;
and that the condition, that the magnetic induction perpendicular
to the surface of separation should be continuous, is analytically
equivalent to the condition, that the electric force parallel to the
line of intersection of the wave-front with the surface of separation
should be the same in both media.
414. The equations of motion (29) of an isotropic medium are
of the same form, as those furnished by the elastic solid theory
when 8 is absolutely zero; for since there is no accumulation of
free electricity df/dx 4- dg/dy 4- dhjdz is always zero. We may
therefore explain a variety of optical phenomena relating to
isotropic media, by means of the electromagnetic theory just as
well as by the elastic solid theory. There is however one impor-
tant distinction between the two theories, viz. that the supposition
that 8 = 0, requires that m = oo , and accordingly w8 may be finite;
and in studying Green's theory of the reflection and refraction of
light, we saw that under these circumstances it was necessary to
1 The continuity of electric displacement and magnetic induction can be at once
deduced from the condition, that these quantities both satisfy an equation of the
same form as the equation of continuity of an incompressible fluid. This equation
is likewise the condition, that the directions of these quantities should be parallel to
the wave-front.
REFLECTION AND REFRACTION. 363
introduce a pressural wave. Nothing of the kind occurs in the
electromagnetic theory, and accordingly we are relieved from one
of the difficulties of the elastic solid theory. We shall now proceed
to consider the reflection and refraction of light at the common
surface of two isotropic media.
Reflection and Refraction 1 .
415. Instead of beginning with the case of an isotropic
medium, we shall suppose that the reflecting surface consists of a
plate of Iceland spar, which is cut perpendicularly to its axis ;
so that we can pass to the case of an isotropic medium by putting
a = c.
The wave surface consists of the sphere
and the planetary ellipsoid
416. Let A, A', A-^ be the amplitudes of the incident, reflected
and refracted waves ; and let us first suppose that the incident
light is polarized in the plane of incidence, so that the refracted
ray is the ordinary ray.
The condition that the electric forces parallel to the plane of
incidence should be continuous, gives
V 2 (A + A')=c'*A l ..................... (30).
The condition that the corresponding components of magnetic
induction should be continuous, gives
(A -A') Vcosi = A^cosr ................. (31).
But if /, /', /! be the square roots of the intensities, it follows
from 411, that
1 JL Ji
AV~ A'V~ A^
whence (30) and (31) become
(/ + /') sin i = I l sin r,
(/ I') cos i = /! cos r ;
1 J. J. Thomson, Phil. Mag. Ap. 1880 ; Lorentz, Schlomilch Zeitschrift, vol.
xxn. ; Fitzgerald, Phil. Trans. 1880 ; Lord Eayleigh, Phil. Mag. Aug. 1881.
364 THE ELECTROMAGNETIC THEORY.
/sin(i-r)
,
accordingly / = -
/sin
which are the same as Fresnel's formulae. Since the light is
refracted according to the ordinary law, these formulae are true in
the case of two isotropic media.
417. In the next place, let the light be polarized perpen-
dicularly to the plane of incidence, so that the refracted ray is an
extraordinary ray.
The conditions that the electric displacement perpendicular to
the reflecting surface should be continuous, gives
(A +4 / )sini = 4 1 sinr (33).
The condition that the electric forces parallel to this surface
should be continuous, gives
(34).
Now if F! be the velocity of the extraordinary wave
/ /' /!
AV~ A'V~ AJfS
al so F/ F! = sin I'/sin r,
and Fi 2 = c 2 cos 2 r + o? sin 2 r =p 2 ,
where p is the perpendicular from the point of incidence, on to the
tangent plane to the ellipsoid, at the extremity of the extra-
ordinary ray.
Equations (33) and (34) accordingly become
/+/=/
j- j, _/ 1 c 2 sin2r
T/ I ( p 1 sin 2i - c 2 sin 2r) )
whence / = , :
p 2 sin 2i + c 2 sin 2r
The formulae are the same as those furnished by MacCullagh's
theory ; see 253.
CRYSTALLINE REFLECTION AND REFRACTION. 365
When the second medium is isotropic, a = c=p, whence (35)
become
tan (i + r*)
/sin ti I ............... < 36 >>
sin (i + r) cos (i r)
which are the same as Fresnel's formulae for light polarized
perpendicularly to the plane of incidence.
418. Returning to (35), we observe that the intensity of the
reflected light vanishes, when
p 2 sin 2i = c 2 sin 2r,
and since p sin i = Fsin r,
this becomes F 2 cot i = c 2 cot r,
whence eliminating r, we obtain
which determines the polarizing angle.
419. Let us now suppose, that light polarized perpendicularly
to the plane of incidence, is internally reflected at the surface of
the crystal in contact with air. Then
a 2 sin 2 i + c 2 cos 2 i = F 2 sin 2 i cosec 2 r,
c 2 cos 2 i ( F 2 a 2 ) sin 2 i
whence cos 2 r = . \ . ,
a 2 sin 2 1 + c 2 cos 2 1
and therefore since V>a, cosr will become imaginary, when
The right-hand side of this inequality determines the tangent
of the critical angle, for light polarized perpendicularly to the plane
of incidence.
Under these circumstances, the right-hand sides of (35) become
complex, and it can be shown by the same methods as have
already been employed, that the reflection is total, and is accom-
panied by a change of phase e, whose value is determined by the
equation
tan Tre/X = F{(F 2 - a 2 ) tan 2 ; - c'f/c 2 .......... (38).
366
THE ELECTROMAGNETIC THEORY.
When the second medium is isotropic, so that a = c, we fall back
on Fresnel's formulae.
The corresponding results for a uniaxal crystal cut parallel to
the axis, may be obtained by interchanging a and c.
Crystalline Reflection and Refraction 1 .
420. We shall now pass on to the general case in which the
first medium is isotropic, whilst the second medium is a biaxal
crystal.
Let be the point of incidence, and let the axis of x be the
normal to the reflecting surface, and let the axis of z coincide with
the. trace of the waves. Let COP be the front of the incident
wave, OP, OQ the directions of the electric and magnetic dis-
placements ; also let COP = 0, so that COQ = \TT + 0. Let COP',
(70P 1} COP 2 (not drawn in the figure) be the fronts of the
reflected and the two refracted waves ; i, r l} r 2 the angles which
the normals to the incident and two refracted waves make with
the normal to the reflecting surface ; also let OP', OP l , OP 2 be
the directions of the electric, and OQ', OQ ly OQ 2 be those of the
magnetic displacements in these waves.
Since the terms involving the suffix 2 are of the same form as
those involving the suffix 1, they may be omitted during the work,
and can be supplied at the end of the investigation.
1 Glazebrook, Proc. Camb. Phil. Soc. vol. iv. p. 155.
CRYSTALLINE REFLECTION AND REFRACTION. 367
The continuity of electric displacement along OA, gives
A cos AP + A' cos AP' = A 1 cos AP 1 (39).
The continuity of electric force parallel to OB and 00, give
V*A cos BP + V 2 A' cos BP' = V l *A l cos BP l + F 2 Aj tan ^ sin n
(40),
and
i ^ * i \ /'
where % : is the angle between the refracted ray and the wave
normal.
The continuity of magnetic induction along OA, gives
VA cosAQ + VA' cos AQ' = Fj^cos AQ l (42).
The continuity of magnetic force, parallel to OB and 0(7, give
VAcosBQ + VA'cosBQ' = V.A^osBQ, (43),
and VA cos CQ + VA' cos CQ' = V,A, cos CQ l (44).
V V
Now -, = -.
also cos AP = sin i sin ft cos CQ = sin 6 ;
whence (39) and (44) reduce to
(A sin + A' sin 6') sin i = A sin ^ sin ft (45),
which proves the equivalence of (39) arid (44).
\ Again cos OP = cos ft cos AQ =sin i cos 6 ;
whence (41) and (42) reduce to
(A cos d 4- A' cos 0') sin 2 1 = ^1 1 cos ft sin 2 ^ (46).
Since cos BP = cos i sin ft cos BQ = cos i cos ft
(40) and (43) become,
( A sin # A' sin #') sin 2 i cos i = A l (cos r x sin ft + sin 2 r x tan ^) sin 2 r
(47),
(A cos A' cos 0') sin i cos i = A t sin ^ cos n (48).
Recollecting that if /, /', 7 a are the square roots of the
intensities,
/ /' = /.
A sin i A' sin i A l sin r x '
368 THE ELECTROMAGNETIC THEORY.
and restoring the terms in A^ we finally obtain from (46), (48),
(45) and (47)
(7 cos + 1' cos 0') sin i 7 X cos ft sin r x + 7 2 cos ft sin r 2
(7 cos I' cos 0') cos % = 7j cos ft cos r + 7 2 cos ft cos r 2
7 sin + I' sin 0' = ^ sin ft + 7 2 sin ft \ (49).
(7 sin 7 sin 6') sin 2i = 7 t (sin ft sin 2^ 4- 2 sin 2 ^ tan ^,
+ 7 2 (sin ft sin 2r 2 + 2 sin 2 r 2 tan ^
When the angle of incidence is given, ft, ft, r 1} r 2 , ^ 1? ^ 2 are
known from the properties of the wave surface, hence these
equations are sufficient to determine the unknown quantities 7',
/!, 7 2 and &.
421. Equations (49) are the same as those obtained by means
of Lord Kelvin's modification of Lord Rayleigh's theory see
(33) of 270 ; and it is also remarkable, that they are the same
as those obtained in 1835 by MacCullagh 1 by means of an
erroneous theory. MacCullagh discussed these equations, and
compared the results obtained from them with the experiments of
Brewster 2 , and found that they agreed fairly well. Accordingly,
although we cannot at the present time accept the assumptions, upon
which MacCullagh based his theory, as sound, yet most of the
results of his first paper, with certain modifications necessitated
by his having supposed that the vibrations of polarized light are
parallel to the plane of polarization, are applicable to the electro-
magnetic theory ; and thus MacCullagh's investigations regain
their interest.
422. The discussion of (49) may be facilitated by a device
invented by MacCullagh 3 .
When polarized light is incident upon a crystalline reflecting
surface at a given angle, it is known both from experiment and
theory, that it is always possible by properly choosing the plane
of polarization of the incident light, to make one or other of the
two refracted rays disappear. The two directions of vibration for
which this is possible, are called by MacCullagh uniradial di-
1 Trans. Roy. Irish Acad. vols. xvm. p. 31 and xxi. p. 17.
2 Phil. Trans. 1819, p. 145 ; Seebeck, Pogg. Ann. vol. xxi. p. 290; xxn. p. 126;
xxxvm. p. 230; Glazebrook, Phil. Trans. 1879, p. 287; 1880, p. 421.
3 Trans. Roy. Irish Acad. vol. xvin. p. 31.
UNIBADIAL DIRECTIONS.
369
rections. In the figure, let CA be the line of intersection of the
plane of incidence with the plane of the paper, and let CO be the
direction of vibration of the incident light, when the ordinary ray
alone exists, and CE the corresponding direction when the extra-
ordinary ray alone exists ; also let CO', CE' be the directions of the
vibrations in the reflected waves corresponding to CO and CE.
Now whatever may be the character of the incident light, the
vibrations may always be conceived to be resolved altihg the
two uniradial directions CO, CE ; and these two vibrations will
give rise to the vibrations CO', CE' in the reflected wave. If the
incident light is plane polarized, the vibrations GO, CE, and also
the vibrations CO', CE' will be in the same phase, and therefore
the reflected light will be plane polarized, although its plane of
polarization will not usually coincide with that of the incident
light. If however the incident light be not plane polarized, the
phases of the vibrations CO, CE, and therefore of CO', CE' will be
different ; hence the reflected light will not usually be plane
polarized. It is however usually possible by properly choosing
the angle of incidence, to make the two reflected vibrations CO',
CE' coincide ; and whenever this is possible, the reflected light will
be plane polarized, and the angle of incidence at which this takes
place is therefore the polarizing angle. These considerations, as
we shall presently show, greatly simplify the problem of finding
the polarizing angle.
423. The four vibrations CO, CE, CO', CE' do not usually lie
in the same plane ; we can however show, that when one of the
refracted rays is absent, the lines of intersection of the planes of
polarization of the three waves with their respective wave-fronts lie
in a plane.
B. o. 24
370 THE ELECTROMAGNETIC THEORY.
Let CP, OP, CP l be the lines of intersection of the three
planes of polarization with their respective wave-fronts ; X, //,, v ;
X', fjf, v ; \, fii t vi the direction cosines of CP, CP, CP^
Then
X = cos 6 sin i, /x = cos 6 cos i, v = sin 0,
X' = cos 0' sin i, /JL = cos 6' cos i, i/ = sin 0',
Xj = cos 0! sin n , /*! = cos a cos r lt *>i = - sin X .
Putting 7 2 = in the first three of (49) and substituting, we
obtain
whence
\ V
A,, A, ,
/A, /*',
= 0,
which is the condition that CP, CP', CP^ should lie in the same
plane.
The preceding theorem is a modification of one due to
MacCullagh.
424. Let us now suppose, that the reflecting surface is a
uniaxal crystal, and let the suffixes 1 and 2 refer to the ordinary
and extraordinary rays respectively; then %i = 0. If we suppose
that the ordinary ray alone exists, 7 2 = 0, and we easily obtain
from (49) the equations
tan = cos (i r^ tan 0^
tan 6' = - cos (i + n) tan 0J
Since the angle of incidence is supposed to be given, r^ and O l
are known; and therefore (50) determine 0, 6' which give the
directions of vibration in the incident and reflected waves.
Again, suppose that the extraordinary ray alone exists ; putting
/! = 0, and writing , ' for 0, 7 , we obtain
tane= cos (I- r^tan ft + si ^ tan X
cos 2 sin (i + r,)
-
cos 2 sin (i - f 2 )
which determine B and '.
POLARIZING ANGLE. 371
Now we have shown that in order that the reflected light
should be plane polarized, it is necessary that the two directions
CO', CE' should coincide, in which case & ' ; we thus obtain
from (50) and (51),
cos (i + r a ) tan 6, - cos (i + r,) tan (9 2 -f ^ 2 _ = (52),
cos 2 sm (i r 2 )
which determines the polarizing angle i.
425. A very elegant formula is given by MacCullagh for the
polarizing angle, when the plane of incidence contains the axis of
a uniaxal crystal, which is most simply obtained directly from (49),
by determining the angle of incidence at which the intensity of
the reflected light vanishes, when the incident light is polarized
perpendicularly to the plane of incidence.
Wehave /' = I, = 0, 6 = 0'= 0. 2 = JTT;
also if &> is the angle which the extraordinary wave normal makes
with the axis of the crystal
_ (a 2 c 2 ) sin &> cos a> _ (a 2 c 2 ) sin &> cos a> sin 2 i
Jnx2 ~ 2 ~ 2 2 ~ F 2 sin 2 r d)j
where c and V are the velocities of propagation of the ordinary
wave within the crystal, and in the medium surrounding the
crystal, and r = r 2 .
From the third of (49) we obtain / = I 2 , and from the last
sin 2i sin 2r = 2 sin 2 r tan ^ 2
= F~ 2 (a? -c 2 ) sin 2&> sin 2 i ..... ....(54),
by (53).
If X be the angle which the optic axis makes with the re-
flecting surface, &> + \ = %TT - r; whence multiplying (54) by
tan r, we obtain
sin 2 r = sin i cos i tan r - F~ 2 (a 2 c 2 ) sin (r + X) cos (r + \) siu 2 i tan r.
But sin 2 r = F~ 2 sin 2 i {c 2 + (a 2 - c 2 ) cos 2 (r + X)}.
Equating these two values of sin 2 r, and reducing we shall obtain
a 2 cos 2 X + c 2 sin 2 X
_
(a 2 -c 2 )sinXcosX'
Substituting in (54), and reducing, we finally obtain
F 2 (F 2 - a 2 cos 2 X - c 2 sin 2 X)
which is the formula in question, which determines the polarizing
angle i.
242
372 THE ELECTROMAGNETIC THEORY.
Reflection at a Twin Plane.
426. We shall conclude this Chapter by giving an account of
a peculiar kind of reflection, which is produced by iridescent
crystals of chlorate of potash.
The phenomena exhibited by the crystals in question were
first examined by Sir G. Stokes 1 , and the experimental results at
which he arrived may be summed up as follows :
(i) If one of the crystalline plates be turned round in its own
plane, without altering the angle of incidence, the peculiar re-
flection vanishes twice in a revolution, viz. when the plane of
incidence coincides with the plane of symmetry of the crystal.
(ii) As the angle of incidence increases, the reflected light
becomes brighter, and rises in refrangibility.
(iii) The colours are not due to absorption, the refracted light |
being strictly complementary to the reflected.
(iv) The coloured light is not polarized. It is produced
indifferently, whether the incident light be common light, or light |
polarized in any plane ; and is seen, whether the reflected light be
viewed directly, or through a Nicol's prism turned in any way.
(v) The spectrum of the reflected light is frequently found to
consist almost entirely of a comparatively narrow band. When
the angle of incidence is increased, the band moves in the direction
of increasing refrangibility, and at the same time increases rapidly
in width. In many cases the reflection appears to be almost total.
427. Sir G. Stokes has shown that the seat of the colour is a
narrow layer about a thousandth of an inch in thickness, and he
suggested that this layer consists of a twin stratum. The subject
was subsequently taken up by Lord Rayleigh, who attributed
the phenomena to the existence of a number of twin planes in
contact with one another; and he has accounted for most of the
phenomena by means of the electromagnetic theory of light. He
has also shown, both from theory and experiment, that when the
angle of incidence is sufficiently small, and the planes of incidence
1 On a remarkable Phenomenon of Crystalline Keflection, Proc. Roy. Soc., Feb.
26, 1885; see also Lord Eayleigh, Phil. Mag. Sep. 1888, p. 256; Proc. Roy. Insti-
tution, 1889.
REFLECTION AT A TWIN PLANE. 373
and symmetry are perpendicular, reflection at a twin plane
reverses the polarization ; that is to say, if the incident light is
polarized in the plane of incidence, the reflected light is polarized
in the perpendicular plane and vice versa. This very peculiar law
was not even suspected, until it had been obtained by theoretical
considerations.
428. The easiest way of understanding what is meant by a twin-
crystal, is to suppose that a crystal of Iceland spar is divided into
two portions by a plane, which is inclined at any angle a to the
optic axis, and that one portion is turned through two right angles.
The optic axes of the two portions will still lie in the same plane,
but instead of being coincident, they will be inclined to one
another at an angle 2a. Crystals whose structure is of this
character, are called twin-crystals ; and it is evident, that a crystal
may possess more than one twin layer.
429. We shall now consider Lord Rayieigh's theory 1 .
When the plane of incidence contains the optic axes of .the
two portions, and the light is polarized in the plane of incidence,
the wave surfaces in both crystals are spheres of equal radii ; and
therefore the crystal will act like two isotropic media, whose
optical properties are identical. Hence no reflection can take
place, and the wave will pass on undisturbed.
430. We shall in the next place suppose, that the light is
polarized perpendicularly to the plane of incidence.
Let the axis of x be normal to the twin plane, and let the
plane xy contain the optic axes of the two portions. Let Oy' be
the axis of the upper portion, and let Ox be perpendicular to Oy'
in the plane xy. Let x'Ox = a ; let /', g' be the electric displace-
ments along Ox, Oy' ; and let /, g, h be the displacements along
Ox, Oy, Oz.
In the upper portion, the wave surface for the extraordinary
ray consists of the planetary ellipsoid
#^c 2 + (y' 2 -M 2 )/a 2 = 1.
Accordingly by (5) we have
P = 47TC 2 /' cos a + 4>7ra?g' sin a)
Q = - 47TC 2 / sin a + 4?ray cosa> ............ (56).
E = ^Trtfh
1 Phil. Mag. Sep. 1888, p. 241.
374 THE ELECTROMAGNETIC THEORY.
But /' =/cosa-<7sina, g' =/sin a + ^cosa,
whence if A = (a 2 sin 2 a + c 2 cos 2 a),
C = (a 2 cos 2 a 4- c 2 sin 2 a),
B = (a 2 c 2 ) sin a cos a,
D = a 2 ,
(56) become
P = 4-7T (4/4- %), Q = 4-7T (JB/ + CV/), U = 4nrDh . . . (57).
The equations of electric force for the lower medium are
obtained by changing the sign of a, whence
Since none of the quantities are functions of s, it follows that
if we substitute these values in (11), and put ^= 1, and recollect
that h = h l = 0, we obtain
(59)
Let the displacements in the incident, reflected, and refracted
waves be
/=?, ff = -p8
f = qA'8 t g' = -p'A'S'[ ............... (60)
where S =
and ^S' and $ x are obtained by changing p into p' and p l respect-
ively. Since p and q are proportional to the direction cosines of
the incident wave, these equations satisfy the conditions that the
displacement lies in the front of the wave.
Substituting from the first of (60) in (59), we find that both
equations lead to
s* = Aq 2 -2Bpq+Cp' 2 .................. (61),
which is a quadratic equation for determining the two values of p
corresponding to a given value of s.
Changing the sign of B, we find for the second medium
s^Aqt + ZBpy + Cp* .................. (62).
Equating the two values of s, we obtain
(63).
REFLECTION AT A TWIN PLANE. 375
We have now to express the boundary conditions.
The condition of continuity of electric displacement perpendi-
cular to the twin surface gives
l + A' = A,.
The condition that the electric forces are continuous gives
Bq-0p + A' (Bq - Cp') = -A, (Bq + %)
Eliminating A lf we obtain
by (63); whence -4' = 0.
431. This result shows, that when the light is polarized per-
pendicularly to the plane of incidence, the amplitude of the
reflected light is zero. Accordingly when light of any kind is
incident upon the crystal, the light reflected at the twin-plane
vanishes, when the plane of incidence is a plane of symmetry.
Accordingly under these circumstances, the reflected light is
entirely produced by the outer surface of the crystal, and is
unaffected by the existence of the twin plane.
432. We shall next consider the case ; in which the plane of
incidence is perpendicular to the plane of symmetry.
In this case, none of the quantities are functions of y, and we
may accordingly write for the incident wave
f=S\, g = Sit, h=Sv,
where 8=? ( **+r*-*t). 9
but since \p + vr = 0, these may be written
f,g, h = (r, p,-p)S.
By (5) and (57), the equations of motion for the upper medium
are
Substituting the above values of/, g, h, the first and third give
s* = r(Ar + Bfju)+p 2 D .................. (64),
whilst the second gives
(65).
376 THE ELECTROMAGNETIC THEORY.
433. These equations determine p and /x, when r and s, which
are the same for all the waves, are given; accordingly if we
eliminate //, from (64) and (65) we shall obtain a quadratic in > 2 ,
the roots of which may be written p lt p 2 , where p lt p 2 are
positive quantities. If therefore a wave polarized in any azimuth
be incident from an isotropic medium upon the upper face of the
first twin, the two refracted waves will be determined by the
values of p l} p. 2) and their direction cosines by the equations
II/PI = n i/ r , k/pz = n z /r. These two waves, when incident on the
twin plane, will give rise to two reflected waves in the first twin,
and two refracted waves in the second ; and the directions of the
two former are given by l ] ^/p l = njr and Z 2 /p 2 = n.Jr. If the
azimuth of the plane of polarization is such, that there is only one
refracted wave, say p lt in the first twin, there would still be two
reflected waves, whose directions are determined by the preceding
equations ; and it is worthy of notice that the angle of reflection
of one of these waves is not equal to the angle of incidence, but is
equal to the angle of incidence of the other wave p.,. We shall
also denote the values of p corresponding to p lt p. 2 by ft, /n 2 .
With regard to the two refracted waves in the second twin, we
see from (57) and (58), that the sign of B must be changed. This
will make no difference in the values of p lt p. 2 , but will change the
signs of /*!, y^ 2 ; so that in the second twin, the values of fju corre-
sponding to p lt p 2 are - ft, - ft.
434. We are now in a position to find the intensities of the
light reflected at the surface of separation.
Let the incident wave be
f,g,h = (r, ft, - Pl ) l * K = (r, ft,pj 4V<-ft* +
and the refracted waves
fi,ffi, h = (r,-ft,-pi) A l *te0+'"*-* + (r, - ft, -p. 2
The continuity of/ at the twin plane requires that
1 + 2 + 4 / + 4" = 4 1 + 4. (66).
The continuity of electric force parallel to z and y give
p^+p, 2 -p 1 A / -p. 2 A' / =p 1 A 1 +p,A, (67),
and
REFLECTION AT A TWIN PLANE. 377
Since dc/dt dQ/dx, the continuity of magnetic force parallel
Jto z requires, that dQ/da should be continuous ; whence
p l (Br +CX) (! -A' + A^ +p. 2 (Br +(7/* 2 ) ( 2 A" + A. 2 ) = 0. . .(69).
If V be the velocity of any wave,
and therefore by (65)
F> = Br +
Writing p,fp l = vr, fji, F 2 2 /^ F x 2 = a
equations (66) to (69) become
! -f OT 2 - ^l 7
+ A'
Solving these equations we obtain
A "=
These equations are perfectly general, but when the doubly-
refracting power of the twin is small, they may be simplified. In
this case p lt p. 2 , V 1 , F 2 are very nearly equal, and we may write
iff = 1, or = fa/Pi, and (70) and (71) become
(72),
435. We shall now suppose, that the amplitudes of the two
components of the light incident upon the first face of the crystal
are M, perpendicular to, and N in the plane of incidence. Then
since the doubly-refracting power is supposed to be small, it
follows, that as the right-hand sides of (72) and (73) involve the
factor pi p-2, we may neglect the slight loss of light due to
refraction, and take
378 THE ELECTROMAGNETIC THEORY.
Similarly if M', N' be the amplitudes of the emergent vibra-
tions, we may take
M' = frA + ^A" \ ,,_ .
N' = (p* + r 2 )* (A -f .4")) '
If the twin stratum is thin, we may, as a first approximation,
substitute the values of A, A" from (72) and (73) in (75), and we
shall obtain
and N , = _,- l ..............
Eliminating 5 between (64) and (65), we obtain
Br^ + {(A - C)r* + (D-
from which we obtain
From (64) we also obtain
whence
Accordingly (76) and (77) become
M= 2 2 ^f t (78) -
N == ~~ ^
These equations show, that the intensity of the light which
emerges from the upper surface, after having been reflected at the
twin plane, is proportional to that of the incident light, without
regard to the polarization of the latter. If the incident light is
unpolarized, which occurs when M and N are equal, and without
any permanent phase relation, so is also the emergent light ; also
if the incident light is polarized in or perpendicularly to the plane
of incidence, the emergent light is polarized in the opposite manner.
436. The preceding results are true only as a first approxi-
mation, when the double refracting power is small, and the twin
stratum is thin; and by proceeding to a higher degree of approxi-
METALLIC REFLECTION. 379
mation, Lord Rayleigh has shown, that the reversal of the polari-
zation will only take place, when the angle of incidence is small.
This agrees with experiment.
Metallic Reflection.
437. We stated in Chapter XVIII., 385, that metallic re-
flection could not be satisfactorily accounted for on the electro-
magnetic theory, by taking into account the conductivity.
When the conductivity is introduced, we obtain from (6), (7)
and (5)
u=f+CP =/+ faCf/K ;
accordingly for an isotropic medium, the general equations of
electric displacement become
.__
dt* "*' K dt~ J '
These equations are of a very similar form to the equations of
motion of an elastic medium, into which a viscous term has been
introduced, and by integrating them in the usual manner, it can
be shown that the square of the pseudo-refractive index (that is
sin 2 1'/sin 2 r) must be a complex quantity, whose real part must be
positive.
NOTE TO 402.
Mr Larnior has pointed out the following additional results :
Velocity of Light. Ratio of Electric Units.
Cornu (1878) 29985 x 10 6 Hioistedt (1888) 30076 x 10 6
Michelson (1882) 29986 x 10 6 Klemengic (1887) 30150 x 10 6
Newcomb (1882) 30040 x 10 6 Lord Kelvin (1889) 30040 x 10 6
See also J. J. Thomson, Proc. Roy. Soc. 1890.
CHAPTER XX.
ACTION OF MAGNETISM ON LIGHT.
438. THE electromagiietic theory of light, so far as it has been
developed in the preceding chapter, depends upon the hypothesis
that a medium exists, whose special function is to propagate electro-
static and electromagnetic effects ; and that when electromagnetic
waves, whose periods lie between certain limits, are transmitted
through the medium, the sensation of light is produced. If there-
fore light is the effect of an electromagnetic disturbance, the
natural inference is, that an intimate connection exists between
electricity and light ; and that when a wave of light -passes through
an electromagnetic field, it ought to experience certain modifica-
tions during its passage, and to emerge from the field in a different
condition from that in which it entered.
439. The conviction that a direct relation exists between
electricity and light, led Faraday to attempt many experiments
for the purpose of discovering some mutual action between the
two classes of phenomena; but it was not until 1845, that he
made the important discovery, that a field of magnetic force
possesses the power of rotating the plane of polarization of light.
During recent years, much attention has been devoted to this
subject, and numerous experiments have been made by Kerr,
Kundt and others, which have greatly extended our knowledge.
We shall therefore commence by giving an account of the principal
experimental results, and then proceed to enquire, how far they
may be explained by theoretical considerations.
FARAD AYS DISCOVERY. 381
Faraday s Experiments.
440. The first experiment described by Faraday 1 , consisted in
placing a plate composed of a variety of heavy glass, called silicated
borate of lead, between the poles of an electromagnetic ; and he
found that when a ray of plane polarized light was transmitted
through the glass in the direction of the lines of magnetic force,
the plane of polarization was rotated in the same direction as that
of the amperean current which would produce the force.
441. Further experiments upon a variety of other transparent
media led to the following law: In diamagnetic substances the
direction of rotation of the plane of polarization is positive ; that is
to say, it is in the same direction as a positive current must circulate
round the ray, in order to produce a magnetic force in the same
direction, as that which actually exists in the medium.
The amount of rotation depends upon the nature of the
medium and the strength of the magnetic force. No rotation
has been observed, when the magnetic force is perpendicular to
the direction of the ray.
442. Verdet 2 however discovered, that certain ferromagnetic
media, such as a strong solution of perchloride of iron in wood
spirit or ether, produced a rotation in the opposite direction to
that of the current, which would give rise to the magnetic force.
Kerr's Experiments.
443. Between the years 1875 and 1880, two very important
series of experiments were made by Dr Kerr of Glasgow, upon the
connection between light and electricity. The first series relate
to the effect of electrostatic force, and the second to magnetic
force.
Experiments on the Effect of Electrostatic Force.
444. In these experiments 3 , a transparent dielectric was
subjected to the action of electrostatic force, and the effect of
the latter upon light was observed.
1 Phil. Trans. 1845, p. 1 ; Exp. Res. XlXth series 21462242.
2 Ante, p. 159.
3 Phil. Mag. Nov. 1875, p. 339.
382 ACTION OF MAGNETISM ON LIGHT.
We shall first consider the case in which the dielectric is a
plate of glass placed in a vertical plane, and shall suppose that the
electrostatic force is horizontal.
Polarized light was transmitted at normal incidence through
the plate of glass, and the analyser was placed in the position of
extinction ; and it was found, that when an electrostatic force was
made to act upon the dielectric, the light reappeared, and dis-
appeared after the force was removed. The effect was most
marked, when the plane of polarization of the incident light was
inclined at an angle of about 45 to the force ; but when the
incident light was polarized in or perpendicularly to the direction
of the force, no effect was observed.
The light restored by electrostatic action was elliptically
polarized, and could not therefore be extinguished by any rotation
of the analyser.
It was also found that the optical effect was independent of the
direction of the force ; that is to say, its intensity remained un-
changed when the direction of the force was reversed.
The optical effect did not acquire its maximum intensity at the
instant the force commenced to act, but gradually increased
during a period of about thirty seconds, at the end of which it
attained its full effect. Also when the force was removed, the
effect did not immediately disappear, but faded away at first
rapidly, and then more gradually to perfect extinction.
445. It is known that compressed glass acts like a negative
uniaxal crystal, whose axis is parallel to the direction of compres-
sion ; whilst stretched glass acts like a positive uniaxal crystal,
whose axis is parallel to the direction of extension. Accordingly
Kerr introduced a slip of glass, called a compensator, and found
that when the slip was compressed in a direction parallel to the
lines of electrostatic force, the optical effect produced by the latter
was strengthened, but when the slip was stretched in that direction,
the effect was weakened.
From these experiments Kerr concluded, that the effect of
electrostatic stress on glass is to transform it into a medium, which
possesses the optical properties of a negative uniaxal crystal,
whose axis is parallel to the direction of the force. Under these
circumstances, it ought to follow, that glass, when under the
action of electrostatic force, should be capable of producing the
KERRS EXPERIMENTS. 383
rings and brushes of nniaxal crystals, but no experiments elucidat-
ing this point appear to have been performed.
446. From experiments made on resin, it appeared that the
effect of electrostatic force upon this substance was to convert it
into a medium, which is optically equivalent to a positive uniaxal
crystal.
447. A few experiments were made on a plate of quartz,
whose axis was perpendicular to the direction of the force ; and
these experiments indicated, that the optical effects were of a
similar kind to those produced upon glass. It is to be hoped, that
more elaborate experiments upon quartz will be attempted ; for if
the optical effects are of a similar kind to those produced upon glass,
it would follow that the effect of electrostatic force would be to
convert a plate of quartz, whose axis is perpendicular to the force,
into a biaxal crystal, which is capable of producing rotatory polari-
zation. Since the principal wave velocity in the direction of the
force, is capable of being varied at pleasure by increasing or
diminishing the force, we should anticipate that some very curious
phenomena would be observed in connection with coloured rings, and
also possibly in connection with conical refraction.
448. Experiments made on liquids 1 showed, that disulphide of
carbon, benzol, paraffin and kerosin oils, and spirits of turpentine
act, when subjected to electrostatic force, like a positive uniaxal
crystal, whose axis is parallel to the direction of the force ; whilst
olive oil acts like a negative uniaxal crystal. Turpentine, as is
known, produces rotatory polarization, some specimens being
right-handed and others left-handed; and therefore in experiment-
ing upon this liquid, two samples of contrary photogyric power were
mixed together, in such proportions as to destroy the rotatory
properties of the mixture.
449. Further experiments 2 led to the following law : The
effect of electrostatic force upon an isotropic transparent dielectric,
is to render it optically equivalent to a uniaxal crystal, whose axis
is parallel to the direction of the force ; and the difference between
the retardations of the ordinary and extraordinary rays, is pro-
portional to the product of the thickness of the dielectric, and the
square of the resultant electric force.
1 Phil. Mag. December, 1875, p. 446.
2 Ibid. March, 1880, p. 157.
384 ACTION or MAGNETISM ON LIGHT.
Kerr's Experiments on Reflection from a Magnet.
450. Shortly after the experiments described in the preceding
sections had been made, Kerr commenced a series of experiments
upon light reflected from an electromagnet. In the first series
of experiments, the light was reflected from the pole of the
magnet; whilst in the second series, a bar of soft iron was laid
across the poles of an electromagnet, so that the lines of magnetic
force were parallel to the reflecting surface.
451. We shall now describe the first series of experiments,
and the apparatus employed 1 .
An electromagnet M, consisting of a solid core of soft iron
surrounded by a wire making 400 turns, was worked by a Grove's
battery of six cells ; and the poles of the magnet were carefully
polished, so as to form a good reflecting surface. The source of
light was a narrow paraffin flame L, which was polarized by a
Nicol N t and the reflected light was analysed by a second Nicol N'.
A wedged-shape piece of soft iron B with a well-rounded edge,
called a submagnet, was placed in close proximity to the reflecting
surface, with its rounded edge perpendicular to the plane of
incidence, so as to leave a space of about ^th of an inch between
the two. The object of the submagnet was to intensify the
magnetic force in the neighbourhood of the mirror, when the circuit
was closed ; and Kerr found that without it, he never obtained any
optical effect. The preceding arrangement was employed, when
the angle of incidence lay between 60 and 80 ; but when the
incidence was perpendicular, a different arrangement was adopted,
which will be explained later on.
1 PUl. Mag. May, 1887.
KERRS EXPERIMENTS.
385
452. Experiment I. Light polarized in or perpendicularly to
the plane of incidence is allowed to fall on the pole of the electro-
magnet, and the analyser is placed in the position of extinction.
When the circuit is closed, so that the reflector becomes magnet-
ized, the light immediately reappears ; when the circuit is broken,
the light disappears, and again reappears when the current is
reversed.
The light reflected whilst the circuit is closed is elliptically
polarized, since it cannot be extinguished by rotating the analyser.
Experiment II. The arrangements are the same as in the last
experiment, and the analyser is turned from the position of ex-
tinction, through a small angle towards the right hand of an
observer, who is looking through it at the point of incidence,
giving a faint restoration of light. When the circuit is closed, so
that the reflector becomes a negative pole, the intensity is in-
creased ; but when the current is reversed, so that the reflector
becomes a positive pole, the intensity is diminished. The weaken-
ing effect of the second operation is always less than the strengthen-
ing effect of the first, and its effect diminishes as the angle through
which the analyser is turned is diminished.
In these two experiments the angle of incidence lay between
60 and 80 ; and Kerr does not appear to have observed the effect
produced, when the angle of incidence lay between 80 and 90.
See 463.
453. We must now describe the arrangements, when the
incidence is perpendicular.
Al'
M
B. O.
25
386 ACTION OF MAGNETISM ON LIGHT.
Instead of employing a wedge for the submagnet, Kerr sub-
stituted a block of soft iron BB, rounded at one end into the
frustum of a cone. A small boring was drilled through the
block, narrowing towards the conical ends, and the block was
placed next the magnet M. The surface of the boring was well
coated with lampblack. Above the block, a thin sheet of glass C
was placed at an angle of 45 to the horizon, which received the
horizontal beam from the first Nicol N, and reflected it downwards
through the boring, perpendicularly to the surface of the reflector.
The reflected beam then proceeded back again through the thin
sheet of glass and the second Nicol N', which served as the
analyser.
454. Experiment III. The polarizer and analyser are first
placed in the position of extinction, and the analyser is then turned
through a small angle, towards the right hand of a person who is
looking through it at the point of incidence, giving a faint restora-
tion of light. The circuit is now closed ; and it is found, that
when the reflector is negatively magnetized, the intensity is
increased ; and that when it is positively magnetized, the intensity
is diminished.
When the analyser is turned towards the left, the results
are the same, provided the operations of positive and negative
magnetization are reversed.
455. As the result of his experiments, Kerr deduced the
following general law, viz. ;
When plane polarized light is reflected from the pole of an
electromagnet, the plane of polarization of the reflected light is
turned through a sensible angle, in a direction contrary to that of
the amperean current, which would produce the magnetic force ; so
that a positive pole of polished iron acting as a reflector, turns the
plane of polarization towards the right hand of an observer, looking
at the point of incidence along the reflected ray.
456. In the preceding experiments, the reflector was supposed
to be magnetized perpendicularly to its surface. We shall now
describe the second series of experiments made by Kerr, for the !
purpose of investigating the effect of a reflector, which is magnet- \
ized parallel to its surface 1 .
1 Phil. Mag. March, 1878,
KERBS EXPERIMENTS.
387
457. The electromagnet stands upright upon a table, and a
rectangular prism of soft iron, one of whose faces is carefully
planed and polished, lies. upon the poles of the magnet, with its
polished face vertical. The two Nicols N, N', and the lamp L,
stand upon the same table as the magnet, and at the same height
as the mirror.
' A \
v_y
(V
V S
y
f
The arrangement is shown in the figure ; AB is the reflector,
E is the eye of the observer, and the dotted lines represent the
poles of the magnet. P is a metallic screen, containing a slit Jth
of an inch wi-de, placed between the first Nicol and the lamp.
In the above arrangement, the magnetic force is very nearly
parallel to the reflector, and may be conceived to be produced by
currents circulating spirally round the prism AB from one pole to
the other. Such a current will be considered right-handed, when
its direction is towards the right hand of an observer viewing it
from F\ and a rotation of the analyser N' t which is in the direction
of the hands of a watch, when viewed from E, will be considered
right-handed.
458. In the following two experiments the plane of incidence
is parallel to the direction of magnetization.
Experiment IV. The incident light is polarized in the plane
of incidence, and the analyser is initially placed in the position of
extinction, and is then turned through a small angle. The circuit
is now closed ; and it is found, that the light restored from ex-
tinction by a small right-handed rotation of the analyser, is always
strengthened by a right-handed magnetizing current, and always
weakened by a left-handed current. Conversely the light restored
by a small left-handed rotation, is always weakened by a right-
handed current, and strengthened by a left-handed one.
The intensity of these optical effects of magnetization varies
with the angle of incidence. At an incidence of 85 the effects
252
388 ACTION OF MAGNETISM ON LIGHT.
are very faint; at 75 they are stronger; at incidences from 65
to 60, they are clear and strong; at 45 they are fairly strong,
though fainter than at 60 ; at 30 they are again very faint, and
much the same as at 85.
Experiment V. The incident light is polarized perpendicularly
to the plane of incidence, and the arrangements are the same as in
the last experiment. At an incidence of 85, the light restored
by a right-handed rotation of the analyser is strengthened by a
ri^ht-handed current, and weakened by a left-handed one ; and the
effects are undistinguishable from those of the fourth experiment,
except that they are considerably weaker. At 80, the effects are
of the same kind, but a good deal fainter; and at 75 they dis-
appear. At 70, they reappear faintly, but the phenomena are
now of a contrary character; for the light restored by a right-
handed rotation, is now weakened by a right-handed current, and
strengthened by a left-handed one. At incidences of 65, 60, 45,
30, the effects are of the same kind as at 70; and at 60 they are
comparatively clear and strong, though sensibly fainter than those
obtained in the last experiment at the same incidence. At 30
they are faint, but stronger than the contrary effects obtained
at 85.
459. The results of the last two experiments may be summed
up as follows :
(i) When the incident light is polarized in the plane of in-
cidence, the plane of polarization of the reflected light is always
rotated in the opposite direction to that of the amperean current,
which would produce the magnetic force.
(ii) When the incident light is polarized perpendicularly to
the plane of incidence, the rotation of the plane of polarization of
the reflected light is in the opposite direction to that of the current,
so long as the angle of incidence lies between 90 and 75 ; and in
the same direction, when it lies between 75 and 0.
460. Experiment VI. In this experiment, the plane of inci-
dence was perpendicular to the direction of magnetization ; and it
was found that no optical effect was produced by magnetization.
Experiment VII. In this experiment, the incidence was-
normal, and the inclination of the plane of polarization to the
direction of magnetization was varied from to 90 ; and it was
found, that no optical effect was produced by magnetization.
KUNDT'S EXPERIMENTS. 389
461. Dr E. H. Hall 1 of Baltimore has examined the effects pro-
duced, when the electromagnet is composed of nickel and of cobalt ;
and he found that in both metals, the sign of Kerr's effect was
the same as in iron.
462. The experiments of Kerr would lead us to anticipate,
that when light is reflected from a conductor, which is strongly
charged with electricity, the reflected light would experience
certain modifications ; but no experiments of this character appear
to have been performed.
Kundt's Experiments.
463. The experiments of Kerr were repeated by Kundt 2 , and
were completely confirmed by the latter with one exception, viz.
that when light polarized perpendicularly to the plane of incidence
is reflected at the pole of an iron electromagnet, the direction of
rotation is reversed at an incidence of about 82 ; that is to say,
the rotation is in the contrary direction to that of the current so
long as the angle of incidence lies between and 82, and in the
same direction when it lies between 82 and 90.
464. When light was reflected from the pole of a nickel
electromagnet, it was found that the rotation was more feeble
than that produced by iron. When the light was polarized
in the plane of incidence, the rotation was always negative (that
is in the contrary direction to that of the current) ; but when the
light was polarized perpendicularly to the plane of incidence, the
rotation was negative from to 50, and changed sign between
50 and 60.
465. Kundt also made experiments upon the rotation pro-
duced, when light is transmitted through films of iron, cobalt and
nickel, which were so thin as to be semi-transparent ; and he
found, that all these metals, when magnetized perpendicularly to
the surface of the film, produced a powerful rotation of the plane
of polarization of the transmitted light; and that the rotation
takes place in the direction of the magnetizing current. The
rotation produced by iron upon the mean rays of the spectrum, is
1 Phil. Mag. Sep. 1881, p. 171.
2 Berlin Sitzungsberichte, July 10th, 1884 ; translated Phil. Mag. Oct. 1884,
p. 308.
390 ACTION OF MAGNETISM ON LIGHT.
more than 30,000 times as great as that produced by glass of
equal thickness ; that produced by cobalt is nearly the same ;
whilst that produced by nickel is decidedly weaker, being only
about 14,000 times greater than that produced by glass.
466. All these metals exhibited rotatory dispersion. The
dispersion produced by cobalt and nickel was feeble, whilst that
produced by iron was much more powerful, and was anomalous ;
for iron was found to rotate red light to a greater extent than
blue.
467. Kundt also made the following experiments upon
magnetized glass, which are of some importance, inasmuch as
they afford an experimental test of the theory, which will after-
wards be proposed.
The poles of a large electromagnet were adjusted at a distance
of about 3 cms. apart. A glass plate, the sides of which were not
quite accurately parallel, so that the rays reflected from the
posterior surface were well separated from those reflected at the
anterior surface, was laid upon the poles of an electromagnet.
The lines of magnetic force were accordingly parallel to the re-
flecting surface ; also the plane of incidence was parallel to the
lines of magnetic force, and the polarizing angle of the glass was
66*tf.
The light which had been twice refracted at the anterior
surface and once reflected at the posterior surface, was examined
on emergence ; and it was found, that when the incident light
was polarized in the plane of incidence, the plane of polarization
of the emergent light was always rotated in the positive direction ;
but that when the light was polarized perpendicularly to the plane
of incidence, the rotation was negative from normal incidence up
to the polarizing angle, and positive from the polarizing angle to
grazing incidence.
When the glass plate was magnetized perpendicularly to the
reflecting surface, it was found that when the incident light was
polarized in the plane of incidence, the rotation was always
positive; but that when it was polarized perpendicularly to the
plane of incidence, the rotation was positive from normal incidence
to the polarizing angle, and negative from the polarizing angle
to grazing incidence.
HALL'S EFFECT. 391
It thus appears, that with regard to the reflected light, the
glass plate behaves in an opposite manner to that of iron, nickel
and cobalt. With respect however to the transmitted light, glass
behaves in the same manner as these metals.
468. Kundt sums up the facts connected with the electro-
magnetic rotation of the plane of polarization of light as follows.
(i) Most isotropic solid bodies, fluids and those gases, which
have been examined, rotate the plane of polarization of the trans-
mitted light in the positive direction.
(ii) A concentrated solution of perchloride of iron produces a
negative rotation.
(iii) Oxygen, which is comparatively powerfully magnetic,
produces positive rotation.
(iv) When light is transmitted through a thin film of iron,
cobalt or nickel, the rotation is positive.
(v) When light is reflected at normal incidence from a mag-
netic pole of iron, cobalt or nickel, the rotation is negative.
(vi) Upon passing through, as well as upon reflection from,
iron, the rotatory dispersion of the light is anomalous ; the red
rays being rotated more powerfully than the blue.
Hall's Effect.
469. Before we proceed to the theoretical explanation of these
phenomena, we must refer to a very important experimental fact,
which was discovered by Dr E. H. Hall 1 of Baltimore. He found
that, when an electric current passes through a conductor, which is
placed in a strong field of magnetic force, an electromotive force is
produced, whose intensity is proportional to the product of the
current and the magnetic force, and whose direction is at right angles
to the plane containing the current and the magnetic force.
Hence if a, 0, 7 be the components of the external magnetic
force, u, v, w those of the current, and P, Q', E', those of the
additional electromotive force, we shall have
-j3w), Q'=-C(aw-yu), R'=
1 Phil. Mag. March, 1880.
392
ACTION OF MAGNETISM ON LIGHT.
470. The constant G is a quantity, which depends upon the
physical constitution of the medium through which the current
is flowing. We shall refer to it as Hall's constant, and to the
additional electromotive force as Hall's effect.
471. Let the conductor consist of a plane plate, which will be
chosen as the plane of xy ; let the magnetic force be in the positive
direction of the axis of z, and let the primary current flow along
the positive direction of the axis of y. Then
The additional electromotive force will therefore act in the
positive or negative direction of the axis of x, according as C is
negative or positive. We may express this by saying, that Hall's
effect is positive, when Hall's constant is negative.
472. Various experiments have been made for the purpose
of determining the magnitude and sign of Hall's effect, the de-
scription of which more properly belongs to a treatise on Electro-
magnetism than to one on Optics 1 . It will however be desirable
to call attention to the experiments of Von Ettinghausen and
Nernst 2 , who found the following values for Hall's effect, its value
for tin being taken as unity.
Copper
-13
Nickel
-605
Silver
-21
Antimony
+ 4800
Gold
-28
Carbon
-4400
Cobalt
4-115
Bismuth
-252,5000
Iron
+ 285
Tellurium + 13,250,000
They found in addition, that the effect was positive in steel,
lead, zinc, and cadmium ; but negative in all the other metals
which they examined.
1 Phil. Mag. Sept. 1881, p. 157 ; Ibid. (5) xvn. pp. 80, 249, 400.
3 Amer. Journ. of Science, (3rd Series), xxxiv. p. 151 ; and Nature, 1887, p. 185.
THEORY OF MAGNETIC ACTION ON LIGHT. 393
Theory of Magnetic Action on Light.
473. In the experiments of Kerr and Kundt, on reflection
from magnets, and transmission through thin magnetized films,
the substance experimented upon was a metal. It is therefore
hopeless to attempt to construct a theory, which will furnish a
complete explanation of these phenomena, until a satisfactory
theory of metallic reflection has been obtained. The theory of
magnetic action on light, which we shall now consider 1 , only
applies to transparent media, and depends upon the experimental
result discovered by Hall, which has been discussed in the
preceding sections.
Now Professor Rowland 2 has assumed, that this result holds
good in a dielectric, which is under the action of a strong magnetic
force ; if, therefore, we adopt this hypothesis, we must substitute
the time variations of the electric displacement for the current, and
the equations of electromotive force become
where a, /3, 7 are the components of the total magnetic force.
When the magnetic field is disturbed by the passage of a
wave of light, a, @, 7 may be supposed to have the same values
as before disturbance, since their variations when multiplied by
f, g, h are terms of the second order, which may be neglected.
Since we shall confine our attention to the propagation of light
in a uniform magnetic field, a, & 7 may be regarded as constant
quantities.
We shall therefore assume, that when light is transmitted
through a medium, which, when under the action of a strong
magnetic force, is capable of magnetically affecting light, the
1 Phil. Trans. 1891, p. 371. For other theories, see Maxwell, Electricity and
Magnetism, vol. n. ch. xxi.; Fitzgerald, Phil. Trans. 1880, p. 691.
2 Phil. Mag., April, 1881, p. 254.
394 ACTION OF MAGNETISM ON LIGHT.
equations of electromotive force are represented by (1), where C
is Hall's constant. Since we shall require to use the letters
a, ft, 7 to denote that portion of the magnetic force which is due
to optical causes, we shall write these equations in the form
dG
-di
dH
where p 1 = GoL, &c.
All the other equations of the field are the same as Maxwell's,
with the exception that we do not suppose that
dFldx+dG/dy
s zero.
474. In order to obtain the equations of electric displacement,
let us consider a medium which is magnetically isotropic but
electrostatically seolotropic. Let k be the magnetic permeability;
K 1} K Z) K 3 the three principal electrostatic capacities ; also let
i 1 ^ ^i -= -4- jy = -4- C/ ~~^ . ( o )
dx dy dz
d d d d
-7- =P! -j- + PS -y- + Ps -j (*)
ao> aa? flty tt*
From the last two of (2) we obtain
da df d/_ d^_ dh_dQ_dR
dt dy dz dy dz dz dy
Substituting the values of P, Q, R from the equations
= IrnflKi, &c., and recollecting that
weobtai,
with two similar equations.
EQUATIONS OF MOTION.
Now 4t7rkf= ^irkii = -^- ( ^-
dt \dy dz
substituting the values of d, b, c from (6) we obtain
V _ A 0772 f d>Q 1 d fdg dh\ >
'<& J dx 4<7rk dco \dz dy)
df
395
(7),
V*/***rf _. VV * \AJtV \
dy 4f7rk dco \d% dz
-
dz
.(8).
These are the equations satisfied by the components of electric
displacement.
475. We shall now confine our attention to isotropic media.
In this case A=B = C= U, where U~ 2 = kK; hence (6) becomes
Let =
,,
then -5- =
whence
i-Vt)
. , a
= - (iifju mv) >S -=-
>SX,
a =
47TZ7 2
- mv) S
.
np s )
Accordingly if |^ denote the component of the external mag-
netic force perpendicular to the wave-front, the equations of
magnetic force become
7 =
where C is Hall's constant.
,
- Iff) ~
(9),
396 ACTION OF MAGNETISM ON LIGHT.
Propagation of Light.
476. We are now prepared to consider the propagation of light
in a magnetized medium.
Let us suppose that plane waves of light are incident upon the
surface of separation of air and a magnetized medium. Let the
axis of x be the normal, and be drawn into the first medium, and
let the axis of z be perpendicular to the plane of incidence ; also
let the direction of magnetization be parallel to the axis of x.
Then j9 2 =^> 3 = 0, and none of the quantities are functions of z\
whence the equations of motion become
p d z h
dxdy
d*h _ mfw _ ^
dx \dv dx
where p is written for p t .
Let
f=A'S, g = A"S, h =
Substituting in (10), we obtain
From these equations we deduce
172 _ 772 j. J?:
J Wr
whence A' = imA, A" = + dA.
Hence, if F l5 F 2 denote the two values of V corresponding to
the upper and lower signs, we see that two waves are propagated
with velocities V lf V 2 .
It is important to notice, that the directions of the two refracted
waves corresponding to an incident wave are in general different.
PROPAGATION OF LIGHT.
397
To see this, let the suffixes 1 and 2 refer to the two refracted
waves, and let the incident wave be
^ _ 6 2i7r/ FT . (lx +my- Vt)
then the displacements in one of the refracted waves will be
where & = e 2l7r / F
and the displacements in the other wave will be obtained by
changing the suffix from 1 to 2, and changing the signs of f lt g^.
Now, if r lt r 2 be the angles of refraction, m : = sin rv, m 2 = sin r 2 ;
and, since the coefficient of y must be the same in all three waves,
we must have
v = v, _ v,
sin i sin r^ sin r 2 '
which shows that r\ is different from r 2 .
cc
Let 23j, 33 2 be the component displacements in the plane z = 0,
then since 1 1 = cos r lt it follows that
33 X =/! sin n + ^ cos n = IA&.
Similarly 33 2 = ^ 2 fif 2 .
The component displacements perpendicular to the wave-fronts
are evidently zero ; whence, in real quantities, the displacements in
the two waves are
9
/ij = A l cos ^FF- (l
V iT
i = - A l sin
and
= cos
27T
= sn
398 ACTION OF MAGNETISM ON LIGHT.
and, consequently, the two waves are circularly polarized in opposite
directions.
477. The results of the last article will enable us to explain
the rotation of the plane polarization, when light is propagated
through a magnetic field parallel to the direction of the lines of
magnetic force. In this case
Jj = 1 2 = 1, m 1 =m z = 0,
whence putting k = 1, since the field is a transparent dielectric, we
obtain from (11),
accordingly if the waves are travelling along the negative direction
of the axis of x,
^ = 4! cos &+*]-, <7i = % = ^2 sin -^ (w-M
T \ r i / T \ r 2
= 4 a cos -M 2 = 23 2 = - A sin - + tj .
We shall hereafter show, that the amplitudes are not quite
equal to one another, but are of the form P + Q and P Q re-
spectively, where Q is a quantity which depends upon the magnetic
force. Since the magnetic effect is very small in transparent
dielectrics, we may as a first approximation neglect the difference
between A 1 and A 2) whence dropping the suffixes, the vibrations in
question become
2?r / x \
g = Asw U^ + n-
T \r i /
2-rr I x
sm
T \^2 /
1 1
Whence if >|r be the angle through which the plane of polari-
zation is rotated, measured towards the right hand of an observer
who is looking along the direction of propagation of the ray,
TTX / 1 1 \
(-~\
i \ r 2 * V
ROTATORY POLARIZATION. 399
Expanding Fj" 1 and V~ l in powers of p, and putting p = Cat, we
obtain
JL 1 c *
F 2 F!~ 20V
accordingly ^ = 7rxCa/2 U 3 r 2 (12).
Since the wave is travelling in the negative direction of the
axis of a?, it follows that, if T be the thickness of the medium
traversed by the wave, x = T\ whence (12) becomes
^ = 7rTCa/2U*T* (13),
which shows that the plane of polarization of the emergent light is
rotated, and that the direction of rotation depends upon that of
the magnetic force.
478. It appears from Faraday's experiments, that the direction
of rotation is the same as that of the amperean current, which
would produce the magnetic force. Now a is measured along the
positive direction of the axis of so, whence the amperean current
circulates from the right hand to the left hand of an observer who
is looking along the direction of propagation ; accordingly C must
be negative for glass, whilst for a medium such as perchloride of
iron C must be positive.
From these results we draw the following conclusions.
(i) The magnitude of the rotation is directly proportional
to the magnetic force, and also to the thickness of the medium
traversed ; and it is inversely proportional to the square of the
period of the light. Hence the rotation is greater for violet light
than for red light.
(ii) The direction of rotation is the same as that of the
amperean current which would produce the magnetic force, for
media for which Hall's constant is negative ; and in the opposite
direction for media for which Hall's constant is positive.
(iii) When the direction of propagation is perpendicular to
that of the magnetic force, it follows from (8), that the magnetic
terms are zero ; hence the magnetic force produces no optical
effect. These results are in accordance with experiment ; subject
to the limitation, that the effect of rotatory dispersion is only
approximately expressed by the first statement.
400 ACTION OF MAGNETISM ON LIGHT.
The Boundary Conditions.
479. When light is reflected and refracted at the surface of
separation of two isotropic or crystalline media, the boundary
conditions are, (i) that the components of the electromotive and
magnetic forces parallel to the surface of separation must be
continuous ; (ii) that the components of electric displacement and
magnetic induction perpendicular to the surface of separation
must likewise be continuous. We have, therefore, six equations
to determine four unknown quantities; but inasmuch as two
pairs of these equations are identical, the total number reduces
to four, which is just sufficient to determine the four unknown
quantities. If, however, we were to assume these six conditions in
the case of a magnetized medium, we should find that we should
be led to inconsistent results, and we shall, therefore, proceed to
prove the boundary conditions.
Since the electric displacement and the magnetic induction both
satisfy the equation
df dg ,dh_
dx + dy* dz~ U)
which is an equation of the same form as the equation of continuity
of an incompressible fluid in Hydrodynamics ; it follows that the
components of the electric displacement and magnetic induction
perpendicular to the surface of separation must be continuous.
To obtain the other conditions, let us suppose, as before, that
the plane # = is the surface of separation, and that the plane
z = contains the direction of propagation. Then, since the
coefficients of y and t in the exponential factor must be the same
in all four waves, djdy and d/dt of any continuous function will
also be continuous, and conversely. Since none of the quantities
are functions of z
_
which shows that 7 is continuous.
Since the continuity of 7 follows from that of /, the conditions
of continuity of both these quantities will be expressed by the
same equation.
Q . dH
Since a = - T - ,
dy*
THE BOUNDARY CONDITIONS. 401
it follows that H is continuous, whence if the accents refer to the
second medium, we obtain from (2)
This equation shows that the electromotive force parallel to z
is discontinuous. This circumstance may, at first sight, appear
somewhat strange, and may perhaps be regarded as an objection to
the theory ; but since the p's are exceedingly small quantities, the
discontinuity is also very small. We have, moreover, assumed that
the transition from one medium to the other is abrupt, whereas,
if we were better acquainted with the conditions at the confines
of two different media, we should probably find that this was not
the case ; but that there would be a rapid but continuous change
in the component of the electromotive force parallel to the
boundary, in passing from one medium to the other.
We have, therefore, as yet, only obtained two independent
boundary equations. Now, we shall presently see that when plane
polarized light is reflected and refracted at the surface of a mag-
netized medium, the reflected light is elliptically polarized ; whilst,
as we have already shown, the two refracted waves are circularly
polarized in opposite directions. We have, therefore, four unknown
quantities to determine, viz., the amplitudes of the two components
of the reflected vibration, and the amplitudes of the two refracted
waves. We, therefore, require two more equations. To find a
third equation, we shall assume, that the component of magnetic
force parallel to the axis of y is continuous. A fourth equation
will be obtained from the condition of continuity of energy ; for
since there is no conversion of energy into heat, or any form of
energy other than the electrical kind, it follows that the rate of
increase of the electrostatic and electrokinetic energies within any
closed surface must be equal to the rate at which energy flows in
across the boundary.
480. We must now obtain an expression for the energy.
It is a general principle of Dynamics, that if equations are
given which are sufficient to completely determine the motion of
a system, the Principle of Energy can be deduced from these
equations. The proper form of the Principle of Energy in the case
of a dielectric medium is this : Describe any closed surface in the
medium, then the rate at which energy increases within the surface,
is equal to the rate at which energy flows in across the boundary.
B. o. 26
402 ACTION OF MAGNETISM ON LIGHT.
If E be the electric energy per unit of volume, the rate at which
energy increases within the surface is fff Edxdydf, and, conse-
quently, this quantity must be capable of being expressed as a
surface integral taken over the boundary ; and any form of E
which is not capable of being so expressed must certainly be
wrong. If the medium were a conductor, in which there is a
conversion of energy into he&t, fff Edxdydz would not be expressible
in the form of a surface integral 1 , but this case need not be con-
sidered, since we are dealing with a transparent dielectric.
Since P = 4*7rf/K l = 4>7rkA*f, equations (6) may be written in
the form
da^dQ_dR_df_
dt dz dy dw'
Multiply this equation and the two corresponding ones by
a, /3, 7 ; then add and integrate throughout any closed surface, and
we shall obtain
dg c
Let W = 2irk llJ(A 2 f 2 + y + C-h?) dxdydz . . .(16),
then -=- = 47T& I (I (A 2 ff+ B 2 gg -f &hh) dxdydz
Cut J J J
rrr
= (Pf+ Qg + Rh) dxdydz.
Jjj
Substituting the values of /, g, h in terms of a, fi, 7, and
integrating by parts, we obtain
dW _ 1 [[
(Jtv \r77" / J
dy
See Poynt.ing, Phil. Trans., 1884, p. 343.
dP\ fdP dQ\\ ,
EXPRESSION FOR THE ENERGY. 403
If in the identity
+ 9 (M ~ PJ) + h (P*f-Pti) = 0,
we substitute the values off, g, h from the equations of
47r/= dy/dy - dj3/dz, &c.,
in the coefficients of the terms in brackets, and integrate by parts,
we shall find that the last volume integral in (15) is equal to
- // P {(pj- Pig) ^ - (pJi - Psf) 7}
+ m {(p# -pji) 7 - (p,f- pj) a]
+ n {(pji -p 3 f) a - (ptf -pji) {3}] dS . . .(18).
Accordingly (15) becomes on substitution from (16), (17) and
(18)
HI fes (a2
+ m{(P+ p.ff -pi) y-(R +p-p$) a]
+ n {(Q + pji -pj)a- (P+p,g-pji)j3}} dS ...(19).
The physical interpretation of this equation is, that the rate at
which something increases within the closed surface must be equal
to the rate at which something flows into the surface. This
cannot be anything else but energy; we are therefore led to
identify the expression
as representing the energy of the electric field per unit of volume.
The first term represents the electrokinetic energy, and the second
term the electrostatic energy.
The above expressions are the same as those obtained by
Maxwell by a different method, and it thus appears that the
expressions for each species of energy are not altered by the
additional terms, which have been introduced into the general
equations of electromotive force.
The right-hand side of (19) represents the rate at which work
is done by the electric and magnetic forces, which act upon the
surface of 8.
404 ACTION OF MAGNETISM ON LIGHT.
481. In the optical problem which we are considering, the
bounding surface is the plane x = ; whence if the quantities in
the magnetized medium be denoted by accented letters, the
condition of continuity of energy becomes
R/3-Qy = (R'+pj 1 - p,ff') 0' - (Q' +M' -K/') /.
Since /3 = fi' and 7 = 7', it follows from (14) that this equation
reduces to
Q = C'+M'-W" (20),
which shows that the component of the electromotive force in the
plane of incidence is also discontinuous.
The boundary conditions are therefore the following; (i)
continuity of electric displacement perpendicular to the reflecting
surface, which is equivalent to continuit}' of magnetic force parallel
to z\ (ii) continuity of magnetic induction perpendicular to the
reflecting surface, which is also equivalent to equation (14) ;
(iii) continuity of magnetic force parallel to y ; (iv) equation (20),
which follows partly from (i), (ii) and (iii), and partly from the
condition that the flow of energy must be continuous.
We have therefore four equations, and no more, to determine
the four unknown quantities.
Reflection and Refraction.
482. We shall now calculate the amplitudes of the reflected
and refracted waves, when light is reflected and refracted at the
surface of a transparent medium which is magnetized normally,
so
Let A t B be the amplitudes of the two components of the
incident light perpendicular to, and in the plane of incidence;
then the displacements in the four waves may be written
h = AS, 33 ==-#$, incident wave ;
h' = A '', 3&' = &&, reflected wave ;
h 1 = A 1 S 1 , 33j = lAA, 1st refracted wave ;
h 2 = A, t S 2) 23 2 = iA 2 S 2 , 2nd refracted wave.
Also
I 1 = cosr 1 , 2 = cosr 2 ,
REFLECTION AND REFRACTION. 405
The boundary conditions (i), (ii), (iii), (iv), of 481, furnish the
following equations :
(B + R)V=i(A 1 V 1 -A,V t ) ............... (21),
(A + A') V* = U*k (A, + A,) - j- r (A, cos r, - A 2 cos r a ) (22),
(A - A') Fcos i = U* (4- 1 cos r, + 4? cos r a
Vri "2
(B - B') V* cos i = iU'k (A cos r, - A, cos n) - ^ (A, + A,) (24),
ZT
where p is written for p l .
We shall now simplify these equations by introducing an
auxiliary angle R, such that
(25).
.
sin i sin r x sin r 2 sin R
Hence, R is the angle of refraction when the second medium
is unmagnetized ; and accordingly r and i\ will differ from R by a
small quantity which depends upon p. Since the magnetic effects
are small, we shall neglect squares and higher powers of p, and we
may, therefore, in the terms multiplied by p, put r x = r 2 = R.
Let q=pl&kr, then from (11)
ii
and also from (25) and (26) we obtain
^ D / __.
cos r x = cos ./ + ~ a , cos r 2 = cos /t -* ,.. a . . .(27).
Substituting these values in equations (21) to (24) and
reducing, they finally become
(B - B'} V 2 cos i = i U 2 k (A l - AJ cos R + iqk (sin 2 ^ - 2) (^ + J 2
(A + A') F = U''k (A, + A 2 ) - 2qk (A^A,) cos R
(28).
These equations determine the amplitudes of the reflected and
refracted waves, when the magnetization is perpendicular to the
reflecting surface.
406 ACTION OF MAGNETISM ON LIGHT.
From the first two of (28) we get
+ B'V{ Uk (2 - sin 2 R) + Fcos i cos R}.
Substituting in the last two, we obtain
(A A')cosi
( uk cos - F cos i)
L 9*
- q C R [Uk(2- sin 2 R)-V cos i cos R} 1
_ ,' f^cosE ( TO cos R + F cos A
L qkAVcosi
TOcos R + Vcos i U(UkcosR+Vcosi)(Ukcosi+VcosR)
(32). '
483. We shall now discuss these results.
Equations (31) and (32) give the amplitudes of the two com-
ponents of the reflected light, and we see that the magnetic terms
vanish at grazing incidence, but do not vanish for any other
incidence.
REFLECTION AND REFRACTION.
407
The equations may be written in the form
A' = Aa
(33).
.(34),
In the figure let / be the point of incidence, 10 the normal to
the reflected wave, and let be the observer; also let OA, OB be
drawn at right angles to 01, perpendicular to and in the plane of
incidence respectively. Let f, 77 be the displacements along OA,
OB ; also let (/> = (2?r/X) (x cos i + y sin i - Vt). Then by (33)
f = AOL cos (f) qB{3 sin
77 = By cos qAfi sin
which shows that the reflected light is elliptically polarized.
Let us first suppose that the incident light is polarized in the
plane of incidence, so that B = 0, and let the principal section of
the analyser coincide with OB. Then the intensity of the reflected
light after it has passed through the analyser is proportional to
A^fi^cf, and is therefore independent of the direction of the
magnetizing current, and vanishes when the current is cut off.
Secondly, let the analyser be turned through a small angle 6
towards the right hand of the observer. From (34) we see that
the intensity of the reflected light after emerging from the analyser
is proportional to
from which it appears that the effect of the current is always to
increase the intensity, and that the intensity is independent of the
direction of the current.
The first result is in accordance with the first of Kerr's experi-
ments, but the second is not ; since he found under these cir-
cumstances, that if a current in one direction strengthened the
reflected light, a current in the opposite direction weakened it.
We must however recollect, that in Kerr's experiments a polished
plate of soft iron was employed, and consequently his results were
408 ACTION OF MAGNETISM ON LIGHT.
affected by the influence of metallic reflection ; it is therefore
hopeless to attempt to construct a theory which will furnish a
theoretical explanation of Kerr's experiments, until a satisfactory
electromagnetic theory of metallic reflection has been obtained.
484. When light is reflected or refracted at the surface of a
transparent medium, which is magnetized parallel to the reflecting
surface, the problem can be worked out in a similar manner to that
employed in 482 ; but for this the reader is referred to my ori-
ginal paper 1 . It will be found that in this case also, the intensity
of the reflected light is independent of the direction of the magnetic
force ; whereas Kerr's experiments show, that the reverse is the
case when the reflector is a metal. When however the plane of
incidence is perpendicular to the lines of magnetic force, or when
the incidence is normal, magnetization produces no optical effect.
This result follows from equations (8), and agrees with the ex-
periments of Faraday, Kerr and Kundt.
485. The experiments of Kundt described in 467, in which
light was incident upon a magnetized plate of glass, furnish a
means of subjecting this theory to an experimental test ; and we
shall therefore consider the case in which the lines of magnetic
force are perpendicular to the faces of the plate, and the incidence
is sensibly normal, and shall calculate the intensity of the light
which has undergone two refractions at the anterior surface, and
one reflection at the posterior.
Let the incident vibration be/=0, # = 0, h = 2Ae- 2t7T * /r ; we
shall find it convenient to resolve this into the two circularly
polarized waves,
(35),
(36),
where S = 6~ 2int/T .
Since we are dealing with glass, we may put k = 1 ; conse-
quently for normal incidence we obtain from (31), (32), and (35)
A' = A(a-q/3), B' = -iA'
where = ^, /^ T ^ F)2 ............... (37).
1 Phil Trans., 1891, 1214.
KUNDT'S EXPERIMENTS. 409
If, however, the incident wave is polarized in the opposite
direction, and is therefore represented by (36), we shall obtain
in which the sign of q is reversed.
In order to calculate the intensity of the refracted light, let us
first confine our attention to the incident wave given by (35).
Then if we put i = R = in (28), we shall obtain A 2 = 0,
A ^2AV^ I g(2ET+7)l
If, however, we considered the other wave (36), we should
obtain A l = 0,
u \u+v u*
in which the sign of q is again reversed.
In order to calculate the intensities, when light propagated in
glass is reflected at the surface in contact with air, we may use
Stokes's Principle of Reversion 1 , and apply it separately to each of
the two circularly polarized waves, into which the incident wave
may be conceived to be resolved. Let Ab, Ac be the amplitudes
of the reflected and refracted waves, when the wave (35) passes
from air into glass ; and let Ae, Af be the amplitudes when the
wave passes from glass into air, then
Also if we denote by accented letters the corresponding quan-
tities for the other wave (36), the values of b', c', e',f will be
obtained from those of 6, c, e,fby writing q for q.
By (37) and (38), the values of b, c, e,/are
U-V
2V*
+imm + 77) sin 77
g = 2^X cos (