LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class Y' v V v 4? y, C0lum6ta LI&HT THE JESUP LECTURES 190a-1909 NEWTON COLUMBIA UNIVERSITY LECTURES LIGHT BY RICHARD C. MACLAURIN, LL.D., Sc.D, PRESIDENT OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY OF THE UNIVERSITY OF Nefo THE COLUMBIA UNIVERSITY PRESS 1909 All rights reserved COPTBIGHT, 1909, BY THE COLUMBIA UNIVERSITY PRESS. Set up and electrotyped. Published June, 1909. J. 8. Gushing Co. Berwick to one another. The case last dealt with, that of plane polarization, FIG. 21 . / . i . ,. ., ; is one of special simplicity and special importance. The vibrating elements are all moving backward and forward along a series of parallel lines. This type of motion is well illustrated by look- ing at a string, AB (Fig. 22 a), which is held taut, with its ends fixed at A and B. If the string be plucked aside very slightly at C, its elements will vibrate to and fro in the plane ACB, the various points moving along lines at right angles to AB. In this case ACB is the plane of polarization, and it should be noted that, as the wave of disturbance progresses along the string, the motion of each point is in the wave-front at right angles (a) A Li ^~ -B to the string, the vibrations r , f,\A _^ _^ C V being of the type described as ^ ' ~* FIG 2 J* transverse vibrations. If the vibrations be longitudinal instead of transverse, we have an- other important type, whose leading features can be illus- trated by the motion of an elastic string, AB, which is kept taut as before. If now a point C be moved to C" (Fig. 22 6) along the string, instead of at right angles to it, POLARIZATION 99 a longitudinal wave will move along the string, and the various elements will vibrate to and fro in the direction AB. These two types of vibrations differ in one very important particular, the ' transverse can be polarized, the longitudinal cannot. The peculiarity of a polarized vibration is that each of the moving elements is con- strained to move in a similar orbit, and it is evident that this can be done with the string vibrating tranversely. With the longitudinal vibrations, on the other hand, only one direction of motion is possible, and if this be stopped, there can be no vibration at all. Hence it follows that if light can be polarized, the vibrations must be of the trans- verse and not of the longitudinal type ; the displacements in the ether must be in the wave-front, and not at right angles thereto. We shall see presently that the experiment made at the outset of this lecture with the Nicol prisms is easily explained if we recognize the possibility of polarization, but otherwise it is inexplicable. It is for this reason that this experiment is crucial in the theory of light. The idea of accounting for optical phenomena by ascribing them to motion in the ether is an old one, but in the earlier days this ether was always thought of as an extremely rare medium, a sort of idealized gas rarer than anything of the kind that we know of by experience. Now a gas can not propagate vibrations except those of the longitudinal type, such as the waves in the air that produce the sensa- tion of sound. To transmit transverse vibrations, a medium must be able to resist certain changes of shape; it must have some rigidity, like a piece of steel. This proved a great stumbling-block to many, even to such leading men of science as Arago and Fresnel, when the phenomena of polarization seemed to force upon them the idea of 100 LIGHT transverse vibrations in the ether. Fresnel admitted that he "had not courage to publish such a conception"; but Young and other men were bolder, so that the idea of transverse ethereal vibrations is now a commonplace, and the notion of an ether with some rigidity has lost its terrors. We have already made use of a vibrating string to illus- trate the meaning of a plane polarized wave, and we may FIG. 23 use it also to throw some light on the experiment with the Nicol prisms at the outset of this lecture. I have here a rope, and as I move one end of it, you observe a wave of disturb- ance passing along the rope, and the rope being quite free, the displacements may be in any directions at right angles to "the rope. Next I pass the rope through this simple wooden structure P of Fig. 23. You will observe that it is a box divided up into narrow compartments by a series of parallel partitions that are just wide enough apart to allow the rope to pass freely between two consecutive partitions. The effect of passing the rope through this apparatus is to polarize the wave of displacement that passes along the rope. If the partitions are vertical, the displacements are all con- POLARIZATION 101 fined to a vertical plane, so that we have a plane polarized wave, the plane of polarization being vertical. Now if I take a second box, A, similar to the first, P, and hold it with its partitions vertical (i.e. parallel to those of the first box), you will observe that when the rope is passed through A as well as P, the disturbance that gets through P is freely transmitted through A also. Suppose, however, I turn A somewhat, so that its partitions are no longer parallel to those of P, then you will observe that A destroys some of the motion in the rope after it has been transmitted through P, and that when A is turned so that its parti- tions are horizontal, and therefore at right angles to those of P, then, however vio- lently I move the end of the rope, there is absolutely no dis- turbance that gets through both boxes. Now we shall see in a later lecture, when dealing with crystals, that a crystal acts upon a beam of light somewhat in the same way that this apparatus acts upon our rope. It will not permit vibrations to pass through it, unless they are confined to one or other of two planes at right angles. The effect is the same as if we had a number of obstacles symmetrically arranged, as are the shaded portions of Fig. 24. Any one setting out from could not proceed along a straight line (such as Oa) for any distance without being stopped by an obstacle, unless they moved along one or other of the two lines Ox and Oy, which are at right angles to one another. A to- and-fro motion along these directions might be maintained FIG. 24 102 LIGHT indefinitely, but in no other direction would it be possible. The Nicol's prism used in our experiment is a simple and ingenious instrument made of the crystal Iceland spar, and so arranged that of the two waves that might be prop- agated (each plane polarized at right angles to the other) one is got rid of by total reflection. Thus a Nicol's prism acts upon light in such a way that the only light that can get through the prism is plane polarized in what is known as the principal plane of the prism. If, then, we hold two Nicols with their principal planes parallel, this corresponds exactly to the case of the two boxes with their partitions parallel, and the light that comes through the first is freely transmitted by the second. On turning the second Nicol, a change is made, and if it be turned so that the two Nicols are crossed, that is, if their principal planes be at right angles, we have a state of affairs similar to that with the boxes, the partitions of one being vertical and of the other horizontal. Under such circumstances we have seen that no disturbance in the rope can be propagated through both boxes, and no light gets through both Nicols. I hope that enough has been said to make clear the mean- ing of polarization, and particularly of plane polarized light. Now when a beam of plane polarized light passes through a solid like glass or a liquid like water, its plane of polariza- tion on emergence is the same as it was at entrance. This, however, is not the case with all substances, a large number being so constituted that the emergent light is polarized in a different plane from the incident. Under such circum- stances the plane of polarization has been rotated through a certain angle, and this phenomenon is consequently spoken of as the rotation of the plane of polarization of light, or more briefly as rotatory polarization. If you care POLARIZATION 103 to see the phenomenon, it is very easily exhibited with the apparatus before you. You will observe that after a little adjustment these two Nicols are now "crossed/' with their principal planes at right angles, so that no light can get through them both. Now I place between the Nicols this plate of quartz, and you see at once that the screen is illuminated. However, on turning one of the Nicols gradually, you see that we reach a position where darkness once more reigns, and a very little consideration will show you that this is what we should expect if the quartz has the power of rotating the plane of polarization of the light that passes through it, and that the amount of this rotation is measured exactly by the angle through which it was necessary to turn the Nicol to produce darkness again after the quartz had been introduced. This phenome- non of rotatory polarization is a very interesting one ; we shall be occupied with it exclusively during the remainder of this lecture. It will be advisable, however, to postpone the consideration of the very important case when the rota- tion is effected by the influence of magnetism, as that case will be taken up more appropriately when we are dealing with the relations between light, electricity, and magnetism in the concluding lecture of this course. Substances that are endowed with the power of rotating the plane of polarization of light may differ as to the direc- tion as well as the magnitude of the rotation that they produce in any given circumstances. Some may rotate the plane as if you were turning an ordinary screw to the right, while others rotate it to the left. Such substances are dis- tinguished by various names, such as right-handed and left- handed, or dextro-rotatory and laevo-rotatory, and wher- ever they have the rotatory power at all, they are spoken 104 LIGHT of as optically active. There are two main classes to be considered : in the first class are certain crystals, and in the second certain organic substances in solution. In both cases experiment proves that the angle of rotation is proportional to the thickness of the active medium traversed by the light, so that the rotation produced by a given thickness may be taken as a measure of the rotatory power of the substance. The most obvious thing about a crystal is that it differs from a non-crystal in having a definite structure. It is not a formless thing like a piece of glass. If you could watch the process of crystallization, you would see a definite form being built up as if by the unerring hand of a skilful artist. You might expect, then, that this fundamental difference between crystalline and non-crystalline media would have something to do with the explanation of rota- tory power. And there can be no doubt that it has, the only doubt being as to the actual arrangement of the mole- cules in any crystal, and the mode in which this arrangement makes the crystal optically active. That there is an inti- mate relation between structure and rotatory power was shown long ago by Sir John Herschel. It was known that some specimens of quartz rotate the plane of polarization to the right, while others rotate it to the left. Herschel found that this difference went hand in hand with certain differences of crystalline form. In the quartz of one class certain facets of the crystal were found on minute examina- tion to lean all in one direction, to the right, say, whilst with the other class the corresponding facets leaned to the left. The first class was dextro-rotatory, the sec- ond IsBVO-rotatory. And had there been any doubts that rotatory power is due to structure, these must have been POLARIZATION 105 removed by the consideration of the fact that the optical activity of a substance disappears when its crystalline structure is destroyed, as happens to quartz when it is fused, or to camphor when it is dissolved. Crystalline structure may produce rotation, but how does it effect it? This is a question not easy to answer Q' FIG. 25 satisfactorily, especially within the limits of such a lecture as this, but perhaps I may give you some glimpses of what has been done to solve the mystery. It is first necessary to realize that what looks like a plane polarized beam of light may really be a combination of two equal and opposite circularly polarized beams, the orbits of the two circles being described in opposite senses. Suppose that we set two particles off from C (Fig. 25 a) with equal speeds in opposite senses in the circle, one going round clockwise and the other counter-clockwise. After a time they will arrive at the points B and A respectively, where B is just as far above the level COC' as A is below it. Now if we raise a point a distance BN by means of one motion, and lower it an equal distance, AN, by means of the other t the effect of the combined motions is to keep the particle at the level N on the line COG'. Thus the combined effect of two equal and opposite circular motions is exactly equiva- 106 LIGHT lent to a vibration along the straight line COC f ; in other words, what looks like rectilinear (or plane) polarization may really consist of a combination of two opposite circular polarizations. Let us suppose, in the next place, that the two particles do not set out simultaneously from (7, but that one starts from C and moves round the circle in a clockwise sense, while the other goes in the opposite sense, and starts from E (Fig. 25 6). It will be seen, as before, that the combination of these two motions is equivalent to a vibration along the straight line QOQ', which is such that OQ bisects the angle EOC. Think next of a right- and a left-handed circularly polarized wave moving through a crystal, and that, owing to the peculiar structure of the crystal, these two waves move through the crystal with different speeds. The two waves will take a different time to traverse a given thickness of the material, so that one will get through the plate faster than the other. A point in the left-handed wave (let us say) will, while the wave has traversed the plate, have made a certain number of complete revolutions and come back to its starting-point C (Fig. 25 b) ; the corresponding point in the other wave will have had more time when that wave emerges from the plate, and will have arrived at E. Once through the plate and into the surrounding non-crystalline medium, the two waves will proceed at equal speeds, so that their combined effect will correspond to that of two circular motions de- scribed in opposite senses at the same rate, one starting from C and the other from E. We have already seen that these two are equivalent to a vibration along a straight line Q'OQ, or to plane polarized light. However, the plane of polariza- tion will now be Q'OQ, whereas it was C'OC on entering the crystalline plate ; in other words, the crystal will have POLARIZATION 107 rotated the plane of polarization through an angle repre- sented by COQ. It remains only to consider what structure would give rise to different speeds for right- and left-handed waves. An almost endless variety of such structures might be suggested ; almost anything would serve the purpose that would present a lack of symmetry to a clockwise and a counter-clockwise circu- lar motion. Suppose that we could watch a crystal being built up, as it is when the solid slowly crystallizes out of the mother liquor. Each molecule, or group of molecules, when it fell -, llG. down, would take up its place on the solid already formed, and it would do this not in a random fashion, but according to some definite rule, as if in obedience to some inexorable law of its being. Thus each group might be shaped somewhat as shown in Fig. 26, and the different groups piled on one another in the fashion there depicted. Under such circumstances the crystal would present a lack of symmetry as regards right- and left-handed rotation. It is easy to imagine a great variety of patterns that would be similarly unsymmetrical, and much ingenuity has been displayed in building up artificial media in some such way as this, and arranging them so as to endow the structure with optical rotatory power. Thus Reusch showed that by superposing thin films of mica according to a simple law, the rotatory power of quartz could be reproduced in all its details. More recently, under the 108 LIGHT guidance of the electric theory of matter, it has become common to estimate the influence, in rotating the plane of polarization, of groups of electrons arranged in an un- symmetrical manner. With certain simple assumptions it is possible to express the ideas in the exact language of mathematics, and so to test the theory in a quantitative way by seeing to what extent it agrees with the most careful measurements of rotation. There are two such tests of any theory : first, it must indicate the relation between the amount of rotation and the thickness of the medium that is traversed; and second, it must show in what way the rotation depends upon the color (or, in other words, the frequency) of the light. Any theory such as has been suggested above shows that whatever be the color of the incident light, the amount of rotation should be propor- tional to the thickness of the active substance through which the light passes, i.e. it should be twice as great for two inches as for one. The following table gives the rota- tion produced by two plates of quartz, one being 1 milli- meter, and two others 7^ millimeters in thickness, the rota- tions being given in degrees for various colored lights : THICKNESS BED CHANGE YELLOW GREEN BLUE INDIGO VIOLET 1 mm. . . . 18 gj| 24 29 31 36 42 7.5 mm. . . . 135 161J 180 217 232 270 315 It will be observed that the amount of rotation with each color follows the law of proportionality to the thick- ness, just as the theory indicates. The theory also shows that the relation between the amount of rotation and the frequency of the vibrations in the light is, for a substance POLARIZATION 109 like quartz, given by a formula of the type R = a/ 2 + J J ~~ when R is the rotation, / the frequency, and a, b, and /! are constants depending on the nature of the sub- stance. How closely this fits the facts is shown in the following table, which compares the theoretical and the observed natures of R for the case of quartz, R denoting the rotation in degrees produced by a plate one millimeter in thickness, and / being the frequency in million millions per second : f 140.12 1.57 1.60 169.41 2.29 2.28 206.80 3.43 3.43 277.65 6.23 6.18 447.01 16.56 16.54 456.89 17.33 17.31 508.82 21.70 21.72 517.85 22.53 22.55 519.74 22.70 22.72 R (theory) . . R (observation) f 549.09 25.51 25.53 589.57 29.67 29.72 609.92 31.92 31.97 624.70 33.60 33.67 687.97 41.46 41.55 740.98 48.85 48.93 871.53 70.61 70.59 1091.7 121.34 121.06 1367.1 220.57 220.72 R (theory) . . R (observation) So far we have been dealing with substances that lose their rotatory power when they are brought into a liquid state by fusion or solution. There exists, however, a large number of substances that have this power although they are liquids, and that retain it even when the liquid is turned into a vapor. Thus, in 1815, Biot discovered that turpentine is optically active, this important discovery, like several others in science, being accidental, as it was made when Biot was searching for something quite different. Two years later the same physicist made a discovery that is still more interesting from our point of view. He looked for rotatory power in the vapor of turpentine, and actually observed it. His most conclusive results were obtained when working with vapor in a tube about fifty feet long, set 110 LIGHT up in an old church at Luxembourg. However, although he saw clearly that there was rotation of the plane of polarization of the light that had passed through this tube, he was prevented from measuring it accurately, as the inflammable vapor ignited and destroyed his apparatus. Science had to wait nearly half a century until the investi- gation was resumed in 1864 by Gernez. He succeeded in determining the rotations produced by various liquids and in showing that their rotatory power is retained when they are transformed into the state of vapor. This is a very striking result, in view of what has been said as to the probable explanation of optical activity. This power has been ascribed to structural arrangement, and yet there seems no possibility of permanent structure with the molecules of a vapor which are in constant motion. This seems to drive us to the hypothesis of structure not of the molecules, but in the molecules themselves. The atoms of which the molecules are built may be arranged in such a way as to produce optical activity, so that the study of our subject lures us into the rich and expansive field of Stereo- chemistry. This is that fruitful department of modern chemistry that concerns itself with the arrangement of atoms in space and seeks to determine how, if you were making a model of a molecule, you would place the different atoms of which it is composed. We have time only for a hurried glance into this field, enough perhaps to stimulate our desire for further knowledge and to break down a por- tion of the arbitrary boundary that has been set up between chemistry and physics. The first epoch-making work in the direction that has just been indicated was that of Pasteur, who, in 1848, took up the study of the rotatory power of different forms of POLARIZATION 111 tartaric acid. He found it possible to separate this acid into four different classes, all with the same chemical constitution (i.e. made up of the same elements), but with different physical structure and different optical power. Two of the classes were optically inactive, and two had rotatory power. One of the inactive acids had the pecul- iarity that when it was crystallized, its crystals, on careful examination, proved to be separable into two distinct types, whereas, with the other inactive acid, the crystals were not thus separable. The two types of crystals that have just been mentioned as constituting together the inactive acid of the first class, differed from one another in a simple yet remarkable manner. Their points of resemblance and of contrast were exactly like those of certain objects and their images as seen in a plane mirror. Your right and left hand have many points of likeness, but yet they are quite differ- ent. They are not superposable ; twist the right hand as you will, and it refuses to fit into a glove made for the left. If you hold the right hand before a mirror, and look at the image, you will see a left hand, so that the relation between the two types of crystals under discussion might be indicated by saying that one type was left-handed and the other right-handed. Pasteur, after separating these types from one another, formed a solution of each. The right-handed type was found to have the same chemical constitution as the original acid of which it formed a part ; but instead of being inactive, it had rotatory power. It rotated the plane of polarization, let us say, to the right, and so was ctoro-rotatory. The left-handed type had also the same chemical constitution, but it, too, was optically active and too-rotatory. The presence of equal quantities of these two types in the original acid explained its inactivity, 112 LIGHT for each neutralized the other, the right-handed rotation produced by the first type being exactly counterbalanced by the left-handed rotation of the second. From the con- sideration of these and similar phenomena, Pasteur was led to make the general statement that all organic com- pounds could be divided into one or other of two groups, according to the form of their molecular arrangement. It will be observed that not every object differs in appearance from its image in a plane mirror ; in the case of a perfectly regular figure, such as a cube or a regular tetrahedron, object and image are superposable. On the other hand, with such objects as a screw, an irregular tetrahedron, or the hand, object and image are not superposable. Pasteur suggested that when the arrangement of the atoms fell into the first of these classes there would be molecular sym- metry, and the substance would be optically inactive ; but a group belonging to the second class would represent mo- lecular asymmetry, and rotatory power would be expected. After Pasteur's researches, the next great impetus to work in this field was given in 1874 by the speculations of Van't Hoff and Le Bel. The fundamental conception here was a definite arrangement of atoms, the so-called tetrahedral molecule, which has formed the basis of the larger part of later speculations in stereochemistry. It is now a commonplace of the text-books, and it would be out of place to discuss it here further than is necessary to give a general impression of the main ideas, in so far as they throw any light on the problem of optical activity. We have already remarked that the fundamental idea of stereochemistry is that, in a given compound, the atoms composing a molecule are arranged in a definite form, and the fundamental problem is to determine that form for POLARIZATION 113 different compounds. Now Van't Hoff's hypothesis is that, in the case of organic compounds, the arrangement is such that the different atoms, or groups of atoms, occupy the four corners ABCD (Fig. 27) of a tetrahedron, the carbon element being in the center. If this were so, it would be convenient to separate carbon compounds into two classes, in the first of which there is only one carbon atom present, and in the second there are two or more such atoms. In the first class, if the four groups at the corners of the tetrahedron were all different, any arrangement and its optical image would be different. Thus every form would have its as- sociate, and as each would be unsym- metrical, they would both be optically active, one rotating to the right and the other to the left. A compound made up of equal parts of these two forms would be neutral, and so optically inactive. Hence in this class we should expect three modifications of any arrange- ment : one dextro-rotatory, another Isevo-rotatory, the third inactive. In the second class, where there were more than one asymmetric carbon atom in the molecule, the number of possibilities would be greater. The case of tartaric acid has already been described. Here we have four different forms one dextro-rotatory, another laevo- rotatory, a third inactive, being compounded of equal pro- portions of the first two, and a fourth also inactive, but for a different reason. This inactive form differs from the other in that it cannot be resolved into active constituents. The molecule is built up of two similar halves, so that there is optical compensation within the molecule itself. 114 LIGHT A great deal of work has been done in this field since 1874, and there has been much to give support to the main lines of the theory here set forth. It has been found that there is no rotatory power in any compound that does not contain an asymmetric carbon atom, and that by introducing or re- moving such atoms from a substance the power of optical activity can be made to come or go. Many of the difficul- ties that early presented themselves have been removed. Thus, at the outset, several substances that contain an asym- metric carbon atom were found to be inactive, contrary to the theory; but later research has shown either that such substances really possess some rotatory power (although a feeble one), or that they consist of mixtures in equal quan- tities of two oppositely rotating constituents, or that they are made up of two similarly constituted halves, which, although not separable, have oppositely rotating powers. Of special importance in this domain have been the re- searches of E. Fischer on the members of the sugar group. If Van't Hoff's hypothesis be right, then it is a simple prob- lem of permutations and combinations to predict the num- ber of different modifications that should be possible with a given group of atoms. If the mathematical problem be too hard or too repellent, you may reach the answer by the aid of models and a little patient trial. You have merely to attach a series of differently colored balls to the corners of a tetrahedron, and see how many different arrangements it is possible to make. In doing this you may learn more than the mere number of different forms in which the com- pound could exist, for, by observing the salient points of resemblance and contrast between different arrangements, you may get hints as to the significance of these for chem- istry and for optics. Thus, when Fischer was working on POLARIZATION 115 the members of the sugar group, he made a careful examina- tion of dextrose, and concluded from its chemical reactions that the atoms in the molecule were arranged in a certain way. The arrangement was not symmetrical, and the substance was optically active of the right-handed rotatory type. Fischer concluded that there might be expected to exist another form, the arrangement of whose atoms would be related to that just mentioned in the same way as an object and its image in a mirror. He succeeded, after care- ful trial, in actually isolating such a form, one that was also optically active, but of the left-handed rotatory type. Perhaps enough has been said to indicate that the hy- pothesis of a definite arrangement of the atoms in the mole- cules of a substance is not a mere idle speculation. It has proved a very useful conception in modern chemistry, but our interest in it here is mainly for the light it throws on the problem of optical activity. We have seen that rotatory power is always associated with an asymmetrical arrange- ment of the atoms, and when dealing earlier with quartz and similar active solids, we remarked that lack of sym- metry would result in right- and left-handed circularly polarized waves traversing the medium with different speeds, and so would account for the rotation of the plane of polarization. . Apart, however, from all such speculations, it may be well to remark that there is no doubt about the fact of rotation, so that, whether these theories find favor or not, we may make use of this fact in any way that seems good to us. The facts that are simplest and most important to bear in mind are as follows: Not every substance is optically active, but many possess this power of rotating the plane of polarization of a beam of light that passes 116 LIGHT through them. The amount of the rotation is found to depend on the temperature, and also on the color of the light that is employed. For light of a definite color (e.g. that of one of the sodium lines), the rotation varies directly as the thickness of the substance traversed by the light. In the case of liquids it depends also very markedly on the strength of the solution, and this has given rise to a very simple and very important plan for estimating the strength of a given solution. It could be carried out with the ap- paratus used in the experiment with which this lecture was begun. Let light from a sodium flame pass through the first of these Nicol's prisms. It issues, as we have seen, as plane polarized light, and we can determine the plane of polarization exactly by noting the position in which the second Nicol must be placed in order to cut off all the light from the screen. Now put a vessel containing an optically active solution between the Nicols, and you will find that the second Nicol must be turned through a certain angle in order once more to completely cut off the light from the screen. If you have a means of measuring carefully the angle through which the Nicol was turned, you know ex- actly the rotation of the plane of polarization that this solution has produced. If, then, by previous experiment, you have determined the strength of solution that produces that amount of rotation, you realize that your problem is solved. I have suggested the use of Nicols, but of course other means of producing polarization may be employed. A great variety of polarizing apparatus has been invented and is constantly being used to determine the strengths of solu- tions of such substances as nicotine, cocaine, starch, and alcohol, and most important of all, considering the magni- tude of the commercial interests involved, of sugar. POLARIZATION 117 It may seem a curious ending to a lecture that deals entirely with what seems almost painfully " unpractical/' polarization, rotatory power, molecular structure, and the like, to refer to means of measuring the strength of alcohol or the value of a cargo of sugar. If, however, you know anything of the history of science, you will not think it strange at all, but will rather be inclined to regard it as typical of almost countless similar cases. No wise man would undertake to draw quite clearly the line between " practical" and "unpractical," between "useful" and "useless," knowledge. By all means let us be practical and useful, but let us use these terms in no narrow sense, nor suppose for a moment that the race will advance most rapidly, even with material things, by sticking closely to what is obviously "practical." If our ancestors had always been sticklers for "practical" knowledge, we should probably still be eating acorns. VI THE LAWS OF REFLECTION AND REFRACTION IN the opening lecture of this course it was remarked that man's knowledge of optical laws might be summed up almost to Newton's day within the compass of a single sentence. Of general principles all that was known was the fact and the law of reflection (as regards direction only), the fact of total reflection, and the fact of refraction. It is difficult for any but a specialist to realize what enormous advances have been made since then, both in observation and in theory. We now have a great variety of instru- ments of precision that enable us to observe most optical phenomena with marvellous accuracy, and a theory has been developed that enables us to group together the whole mass of facts with the utmost simplicity and with almost startling success. Few men are in a position to understand the searching nature of the test that can now be applied to optical theories, and to appreciate how well the modern theory stands the test. Not until you have put yourself in such a position can you understand the confidence of a modern physicist in his theories. He is no longer content with a mere descriptive theory which tells him in a general way that such and such phenomena are to be expected. His theory must enter into the minutest details and predict quantitatively. It must tell him that if he measures this or that with sufficient accuracy, he will find its measure to be so and so. In the case of the modern theory of light, 118 THE LAWS OF REFLECTION AND REFRACTION 119 all the improvements and all the refinements of modern instruments but tend to confirm the correctness of the pre- diction. I have already given you instances of this (for example, when dealing with dispersion) ; but, even at the risk of wearying you with figures and with tables, I must give you more of a similar kind to-night and in later lectures. Let us look first at the simpler and more generally known laws of reflection and refraction. These deal only with the directions of the various rays, and show how to de- termine the directions of the reflected and refracted ray of light when that of the incident ray is given. In Fig. 28 AB represents an incident ray which strikes a reflecting surface BK at the point B, in such a way that part of the light is reflected along the ray BO, and the rest refracted along BH. If EBF be drawn at right angles to the reflecting surface, the plane containing AB and BE is called the plane of incidence. The law of reflection, which determines the direction of the re- flected ray, states that BC is in the plane of incidence and that the angle EBO is exactly equal to the angle EBA, orr = i, with the notation indicated in the figure. The law of refraction (sometimes called Snell's law, having first been laid down by Snell in 1621) states that the refracted ray BH is also in the plane of incidence, and that the angle FBH is connected with the angle A BE by the relation sin i = /A sin r', where /A is a constant depending on the nature of the two media on each side of 120 LIGHT BK, and known as the relative refractive index of these media. (If the space above BK is a vacuum, then n is the absolute refractive index, or simply the refractive index of the medium below BK.) As it is impossible to find an angle whose sine is greater than unity, SnelFs law shows that r' could not be found if the angle of incidence i were such that sin i were greater than n, the relative refractive index. If the first medium be more highly refractive than the sec- ond, for example, if the first be water and the second air, then the relative refractive index //. is less than unity, and the angle whose sine is equal to fi is called the critical angle. Under such circumstances r' would be impossible if the angle of incidence i were greater than the critical angle so that we should expect that there would be no refracted ray, and that all the light would be reflected. This is the phenomenon of total reflection that was brought before your notice in the first lecture, and the point to be noticed now is that Snell's law indicates exactly the conditions under which this phenomenon is to be expected. All these laws with reference to reflection, refraction, and total reflection have been verified experimentally with the greatest precision. Of all the countless experiments that have been made with reflected beams, no careful measurement has ever suggested the slightest departure from the law of equal angles, i = r. And the same may be said of Snell's law of refraction. Of course there is a possible error in all such measurements, for no amount of care can make them absolutely exact. A considerable part of modern science consists in estimating carefully the probable errors of measurement. To test these laws of reflection and refraction, it is necessary to measure cer- tain angles, and this, with the wonderful instruments of THE LAWS OF REFLECTION AND REFRACTION 121 to-day, can be done with great nicety, though of course not with absolute precision. With very great care the angles may be measured accurately enough to insure the correctness of refractive indices to six places of decimals; but even with the care and skill necessary to insure this degree of accuracy, no one has found any departure from Snell's law that was outside the limits of the probable errors of experiment. It will have been observed that these laws of reflection and refraction are merely condensed statements of experi- mental facts. No theory is involved in them ; they simply sum up in a convenient form the results of a large number of observations and so serve one of the great ends of science to save labor and relieve our memories of the burden of too many isolated facts. If, however, we are imbued with the scientific spirit, we cannot rest content with such laws, but must strive to fit them in with our other knowledge and to get a view of optics that is comprehensive enough to take in these laws and all else within the optical field besides. To this end we need a theory of light, and for about a century there has been little doubt as to the gen- eral lines along which such a theory must be developed. We need a wave theory of some kind, that is, we must think of light as due to a periodic disturbance like a wave propa- gated in a medium. Now, if we set out with any such wave theory, and with the conception that a wave travels with a definite speed in one medium (such as air), and with a different speed in another (such as glass), we are led simply and inevitably to just these laws of reflection and refraction of which we have been speaking. These laws are required to secure continuity at the interface between two media; without them there would be a rupture there or a sudden 122 LIGHT break. At present I cannot stop to prove such a statement, although it is very easily proved; I must simply ask you to believe that it is so, and that the relative refractive index of which we have spoken is the ratio of the speeds of the waves in the two media under consideration. As far, then, as the mere directions of the reflected and refracted rays are concerned, almost any wave theory will account for the facts. But other things than these directions must be considered. Suppose that you are studying the effect of waves that you see running across the surface of a lake. You may well want to know more than the mere direction in which they are moving. If you wish to esti- mate the damage that the waves will do when they strike upon some object, you will want to know their height. In an ether wave which, according to our theory, gives us the sensation of light, each element of the ether vibrates to and fro about some mean position. Its greatest displace- ment from this position corresponds exactly to the height in a water wave, and is technically known as the amplitude of the wave. This you will wish to know if you are to meas- ure the intensity of the light, for it may be proved that the intensity depends on the amplitude, and is, in fact, propor- tional to the square of this amplitude. Another important element in a wave of water, or of anything else, is its phase. Watch two waves, similar in height and shape, running side by side along the surface of some water. The crest of one may always be in line with the crest of the other. In this case they could be described as being "in phase," or "in the same phase." More probably, however, the crest of one would lag somewhat behind that of the other. To describe this we should say that there was a " difference of phase" between the waves, and this difference might be THE LAWS OF REFLECTION AND REFRACTION 123 a matter of much import. (As a matter of fact, it would have great importance if we came to consider the effect of combining the two waves, as we shall see in the next lecture on Interference.) What, then, does a wave theory of light tell us of the amplitude (or intensity) and the phase of the reflected and of the refracted beams, and how do the predictions of theory compare with the results of obser- vation? These are very important questions. They are, indeed, crucial in optical theory, for they enable us to dis- tinguish one wave theory from another, and to say which best fits the facts. This, of course, settles the question as to which theory is to be preferred, for the whole end of a scientific theory is to fit the facts ; if it fails to do this, it is probably worse than useless. What, however, do we mean by distinguishing one wave theory from another ? Any theory of light that endeavors to coordinate its phenomena by means of the conception of a to-and-fro motion propagated in the ether may be called a wave theory; but before such a theory can lead us to precise results, we must formulate defi- nite ideas as to the nature of the ether. Here there is room for difference of opinion, and so for different wave theories. In any case the idea of an ether is an abstraction; it is reached by taking away certain properties of ordinary matter and endowing an ideal medium with all that remains. Without such a process the ether could not be thought of at all, for our mental conceptions are necessarily derived, more or less directly, from our experience. Such abstract ideas are common enough in scientific and even in ordinary discussion. Thus we have the idea of an incompressible substance. We observe that air is easily compressed, that bread resists compression more strongly, and that water opposes with tremendous force any attempt to diminish its 124 LIGHT volume. It is an easy matter in thought to carry on the process until we have abstracted completely the power of yielding to compression, and so we reach the abstract idea of an incompressible substance. If we were interested in considering the motion of such a substance, we might well apply accepted dynamical principles to aid us in the dis- cussion, and so we might reason as to its behavior, even although it would be impossible actually to point to a sub- stance that was incompressible. Again, we have the idea of a frictionless fluid. We observe that if w pull a spoon through treacle, the treacle resists the motion, and we have to exercise a considerable force to overcome this resistance, or friction. If we replace the treacle by olive oil, the friction is diminished, while with water it is scarcely perceptible. Here, again, it is not difficult to abstract the viscosity, the power of opposing motion by friction, and so to arrive at the abstract idea of a frictionless fluid. In this case, also, we might apply dynamical principles to aid us in discussing the behavior of such a fluid, and we need not be hampered in that discussion by the fact that no one has ever presented us with a bottle of a frictionless fluid. Now the ether that is spoken of so much in these latter days in various branches of science is a similar abstraction. Let us begin with ordinary matter, a piece of steel or jelly, say. It has a definite density, and definite elastic constants which meas- ure its powers of resistance. It resists a change of volume ; it requires force to compress or expand it. It resists at- tempts to twist it and change its shape. Such powers of resistance can be measured on a definite scale and expressed numerically by means of such elastic constants as com- pressibility and torsional rigidity. It seems natural in a wave theory of light to begin with an ether that has all THE LAWS OF REFLECTION AND REFRACTION 125 these powers, and to see if, by a proper choice of the con- stants representing density, compressibility, and rigidity, it is possible to account for the phenomena of light. This is the famous elastic solid theory of light. If a disturbance is set up in a medium such as has been described, it is easy to show that waves will be propagated with a speed that will depend on the magnitude of the elastic constants. Moreover, in passing from one medium to another with dif- ferent elastic constants, reflected and refracted waves will be set up, and, as has been indicated already, the directions of these will correspond exactly to those laid down by the laws of reflection and refraction that have already been formulated and have been fully verified by experiment. It would thus appear that we are on the right track; but when we come to look carefully at the other features of the waves, their amplitudes and phases, we begin to encounter difficulties. There are other difficulties that I need not refer to ; it will be sufficient to say that the only successful way of overcoming them all is to abstract something from our ordinary elastic medium. We have too much cargo and must lighten the ship. Let us throw over all power of resisting change of shape, except the power of resisting a twist. The medium so obtained will possess the mobility of a fluid with some of the rigidity of a solid. As it does not resist a mere change of shape, it will allow bodies to move freely through it like a fluid; but it objects to twisting of its elements, and so has rigidity. A fluid like water, with a number of little gyrostats spinning in it, and by their momentum opposing any change of spin, might serve as a rough model to bring to mind the peculiar properties of this " rotationally elastic" ether. It might be impossible to construct this model, but there is no great difficulty in 126 LIGHT conceiving of such a medium by the process of abstraction and of reasoning as to its behavior in obedience to general dynamical laws. Such a medium, if disturbed, will transmit the disturbance as a wave (i.e. a periodic displacement), and this wave will not be of the longitudinal type, but of the transverse kind that the phenomenon of polarization demands from any theory of light. The speed with which the wave travels will depend on the rigidity and the density of the ether, and the ratio of the constants representing these quantities must be chosen so as to fit in with the observed value of the speed of light in vacuo where there is nothing but ether to affect the speed. The presence of matter will modify the effective rigidity, so that a wave will travel with a dif- ferent speed in water or glass than in vacuo. In passing from one of these media to the other, there will be reflection and refraction, and provided that we assume that there is no discontinuity of motion at the interface, no rupture at the surface of separation, the general principles of dynamics will enable us to calculate not only the directions of the reflected and refracted waves, but also their amplitudes and phases. When this is done, it becomes at once evident that the condition of the reflected and refracted waves must depend on the state of polarization, as well as on the direc- tion, of the incident beam. Two important cases present themselves : in one the light is polarized parallel to the plane of incidence, and in the other at right angles to this plane these cases being specially important, as the details of all other cases can be immediately deduced from a considera- tion of these two. Then it also appears, as might be ex- pected, that the results depend on the nature of the tran- sition from one medium to the other, from air to water, say. In any case, actually presented in an experiment, this tran- THE LAWS OF REFLECTION AND REFRACTION 127 sition may be absolutely sudden, or it may be more or less gradual. Such a question cannot be decided offhand; to the eye the transition may look quite sudden, but this effect may be due to imperfections of our vision, and if we could see things at close enough range, the idea of an abso- lutely sudden transition might appear illusory. However, the hypothesis of a sudden transition is probably the natural one with which to begin, and it was on this hypothesis that formulae from which to calculate all the details of the re- flected and refracted waves were first obtained. In the present course I have promised to eschew mathematics as much as possible, so that here we must be content with a graphical representation of the formulae. Instead of looking at all the details, let us for a time concentrate our attention on a single one, the intensity of the reflected beam a quantity, as has been remarked, that is proportional to the square of the amplitude of the reflected wave. In Fig. 29 the curves marked R and R' represent the percent- age of the incident light reflected from glass, whose refrac- tive index is /* = 1.52, at different angles of incidence i. The different angles of incidence are indicated by distances measured across the page, and the corresponding percent- age of reflected light by distances at right angles to this. Both curves represent the formulae obtained from theory in the manner just indicated, R' dealing with light polarized parallel to the plane of incidence, and R perpendicular thereto. It is specially worthy of remark that for the latter case the intensity begins to diminish as the angle of inci- dence (i) increases, that it goes to zero at the point marked P, and then rapidly rises. The theory indicates that the position of the point P is determined by the simple formula tan i = /*. At this angle none of the light that is polarized 128 LIGHT perpendicularly to the plane of incidence is reflected, so that all the light that can be reflected at that angle is po- larized parallel to the plane of incidence. This indicates 100 1 90 80 70 60 I I 1 ' / 1 \ I I [ I 40 / / / 1 J ' 1 L J y 1 / j Percentage i 1 """ O C x R/ X / - - - - ' - -~ ^^ - / * 10 20 30 40 50 P 60 70 80 90 Angle of Incidence (t) FIG. 29 that by simple reflection we have a means of producing plane polarized light. We have merely to arrange that light should fall on a reflecting surface at the proper angle. This angle is given by the formula tan i = p, and is called THE LAWS OF REFLECTION AND REFRACTION 129 the polarizing angle. It is just about a century since Malus discovered that light could be polarized by reflection, and a few years later Brewster deduced from a series of experi- ments that the polarizing angle was given by the formula tan i = p. The following table shows how the values of the polarizing angles of different substances, calculated from the theoretical formula tan i = /*, agree or disagree with the angles actually observed : i (theory) 53 7' 53 18' 55 33' 55 37' 58 13' 58 36' i (experiment) . 53 7' 53 18' 55 33' 55 37' 58 12' 58 36' i (theory) . . 59 41' 60 30' 63 33' 67 7' 67 32' 67 40' i (experiment) . 59 44' 60 30' 63 34' 67 6' 67 26' 67 30' It will be seen that the agreement is very close, but not perfect, and we should find results of the same character if we compared the theoretical and observed values of the intensity of the reflected light, and, what are much more readily measured with precision, certain phase relations. In all cases the theory fits the facts very nearly, but not ex- actly. We find, however, that all these minute discrep- ancies disappear when we abandon the hypothesis of an abrupt transition from one medium to another. A com- parison of theory and experiment then gives us the means of estimating approximately the thickness of the surface layer within which the transition takes place. We find in many cases that it is less than one-hundredth of a wave length, and how extremely short that is for ordinary light will be made apparent in a later lecture. Let us see how well our theory fits the facts when we take into account the influence of 130 LIGHT this transition layer. We shall consider first the intensity of the reflected light, although the intensity cannot be meas- ured so accurately as most of the other features with which we have to deal. It is true that there has been a great im- provement in photometric processes of recent years, but these are still far from the stage of precision that has been attained in other departments of optics. The following table gives the percentage of the light reflected at different angles of incidence (i), calculated from theory for the case of glass, and compares the results with the most careful observations of the amount of light actually reflected : i 10 20 30 40 Percentage Reflected (theory) . . 3.78 3.78 3.90 3.92 4.39 Percentage Reflected (experiment) 3.78 3.78 3.77 3.92 4.37 i 50 60 65 70 Percentage Reflected (theory) . . 5.37 8.31 11.28 16.12 Percentage Reflected (experiment) 5.53 8.34 11.16 16.04 Some of you, who find numbers distasteful or hard to com- prehend, may prefer to see these results exhibited in a form that appeals to the eye. For this purpose they are ex- hibited graphically in Fig. 30. As we shall have quite a number of similar figures before our course is run, it may be well to adopt a uniform mode of presentation and explain it once for all here. You should bear in mind, then, that in all such figures, the continuous curve corresponds to the predictions of theory, while the crosses indicate the results of actual experiment. Thus the agreement or disagreement THE LAWS OF REFLECTION AND REFRACTION 131 between theory and observation is measured by the degree of closeness with which the crosses lie along the continuous curve. In this case it will be observed that the agreement 16 14 12 10 ,. 10 20 Angle of Incidence (i) 40 60 Fio. 30 is very close, especially in the region where the incidence is small, in which accurate measurements are most easily made. An inspection of the figure will show that, in the case of the most marked disagreement, it is more probable that the meas- ure of intensity was rather too high, or too low, than that 132 LIGHT the theory is in error. In nearly all cases the differences between theory and observation are well within the limits of the probable errors of experiment. So much for the intensity of the reflected light. Next, let us suppose that matters are so arranged that the in- cident light has equal intensities when polarized parallel and perpendicularly to the plane of incidence, and let us measure the ratio of the intensity of the reflected light that is polarized perpendicularly to the plane of incidence to that of the reflected light that is polarized parallel to this plane. The measurement of this ratio can be made far more accurately than that of the intensity of any light. Its value can be obtained without any photometric processes at all, simply by ascertaining the position of the plane of polarization of the reflected light, and the measurement of the angle determining this position is an operation that can be performed with great delicacy. The table that fol- lows gives us the means of comparing the values of this ratio in the case of reflection from diamond at various angles of incidence and of estimating the degree of accuracy with which the theory fits the facts : i 60 61 62 63 64 65 Ratio (theory) . . .0421 .0324 0234 .0166 .0104 .0056 Ratio (experiment) . .0420 .0312 .0213 .0178 .0102 .0057 i 66 67 67 30' 68 68 30' 69 Ratio (theory) . . .0028 .0009 .0006 .0007 .0013 .0020 Ratio (experiment) . .0030 .0009 .0006 .0007 .0013 .0026 i 7O 71 72 73 74 75 . Ratio (theory) . . .0049 .0103 .0177 .0275 .0399 .0552 Ratio (experiment) . .0054 .0106 .0184 .0296 .0469 .0576 Qi- j nc. UNIVERSITY j OF / OF REFLECTION AND REFRACTION 133 The graphical representation of these results is exhibited in Fig. 31, and from either the figure or the table it will be seen that the agreement between theory and observation is extremely satisfactory. Another quantity that is capable of very accurate meas- urement is the difference of phase between the two reflected 0.05 \ 0.04 0.03 \ \ 0.02 0.01 waves when one is polarized parallel and the other per- pendicularly to the plane of incidence. The results for diamond are shown in Fig. 32, the difference of phase being expressed as a decimal fraction of a wave length, so that for a difference marked 0.5 one wave is half a wave length be- hind the other, and thus the crest of the first is in line with the hollow of the second. It was pointed out earlier in this lecture that, on the theory of an absolutely abrupt transi- tion from one medium to another, the polarizing angle would be given by the formula tan i = /*, and a table was 134 LIGHT made out which showed that this is very nearly true for most of the substances referred to. The examination of the influence of a thin surface layer of transition on the position of the polarizing angle shows that the layer should affect this angle very slightly, and that it might either in- crease or decrease it, according to the nature of the layer. 0.5 0.4 0.3 It ^ 60 62 .Angle of Incidence (i) 64 66 68 FIG. 32 70 72 74 In the case of a certain specimen of glass, for which the theory of the transition layer predicted a polarizing angle of 56 23' 38", the mean of a most careful series of experiments fixed this angle at 56 23' 30". Theory also shows that if the first of the two media in contact with one another have a higher refractive index than the second, the whole of the light will be reflected when the angle of incidence is greater than the critical angle. This is the phenomenon of total reflection already referred to, and here, as elsewhere, the agreement between theory THE LAWS OF REFLECTION AND REFRACTION 135 and observation is as close as could be desired. The following table and Fig. 33 set out a comparison between theory and experiment for the difference of phase (A) between two waves that are totally reflected, one being Difference^/ Phase 0-* p L.w k. o \ S X, 1 j - . ~ -t - ^ i i i H 1 H ^ - tft x 40 42 44 46 48 B0 52 64 6 Angle of Incidence (i) FIG. 33 *> J polarized parallel and the other perpendicular to the plane of incidence. The differences of phase are expressed as decimal fractions of the wave length, and, as before, i denotes the angle of incidence. The substance dealt with experimentally had a refractive index, ^ = 1.619, and a critical angle of 38 9'. i 38 13' 39 58' 41 59' 44 3' 46 4' A (theory) . .488 .457 .379 .364 .358 A (experiment) . . . .489 .457 .377 .364 .359 i 47 54' 49 58' 51 57' 53 58' 55 57' A (theory) ... 356 356 360 363 368 A (experiment) . . . .356 .357 .360 .363 .365 All these tables and figures have reference to reflection from transparent, non-crystalline substances. If the re- flector be a crystal, or if it be more or less opaque, theory 136 LIGHT and experiment agree in showing that the laws of reflec- tion may be considerably modified. The phenomena with crystals will be dealt with in a later lecture, but we shall 100 90 80 70 60 60 #> *! n/\ J H 2 II / / / / S / R S* ^ ^ ^ ? M - . , _ . _ _^- -~^. ft "*""* ^ ^v^ T?~ * ****. >^ \ / \ n Percentage Reflectet m ^g 8 % s ** j, ,/ = ) 10 20 30 40 60 60 70 80 90 Angle of Incidence (i) FIG. 34 not have time for more than a passing reference to the laws of reflection from opaque substances, such as metals. In this case what corresponds to the refracted wave is absorbed by the metal, but theory enables us to predict all the details THE LAWS OF REFLECTION AND REFRACTION 137 of the reflected beam. The laws are more complex, but the general character of the results has some resemblance to that for transparent reflectors. This will be made evi- dent by a comparison of Figs. 29 and 34, which represent corresponding quantities for glass and steel. It will be 0.5 n 4 / / / 0.3 0.2 ^0.1' / / x 7 / / / x ^^ ^ f . J -^ 30 40 50 60 70 80 91 ^ Angle of Incidence (i) FIG. 35 observed that in both cases for light polarized parallel to the plane of incidence the intensity of the reflected beam increases steadily with the incidence. With light polarized perpendicularly to the plane of incidence, the intensity in both cases begins by diminishing, reaches a minimum, and then increases rapidly. The main difference is that the amount of light reflected at normal incidence is very much greater for the metal than for the transparent sub- stance, and that even at the angle where the reflection from the metal is a minimum (called the quad-polarizing angle from its resemblance to the polarizing angle of a transparent 138 LIGHT medium), there is still a considerable quantity of light re- flected from the metallic surface. The difference of phase between the two reflected waves is represented graphically in Fig. 35 for the case of reflection from gold. This figure should be compared with Fig. 32, which is the corresponding figure for the case of reflection from a transparent substance. It will be seen that in all cases the crosses lie closely along the curves, indicating on all points an excellent agreement between theory and observation. The numbers corre- sponding to these figures are set out in the following table: i R (THEORY) R (EXPERIMENT) R' (THEOBY) R' (EXPERIMENT) A (THEORY) A (EXPERIMENT) 30 50.5 50.1 60.4 60.7 .028 .032 40 46.5 46.2 64.0 64.2 .052 .056 50 41.0 41.0 68.9 69.4 .088 .088 60 33.9 34.1 74.8 74.5 .135 .130 70 26.7 26.5 82.1 82.3 .211 .210 75 25.4 25.5 86.1 86.1 .265 .265 80 29.5 27.5 90.4 90.3 .331 .324 I hope that by this time enough has been said to show you that modern optical theory gives a completely satisfactory account of reflection and refraction, telling us all that we can want to know with the utmost precision, and agreeing in its predictions on every point with the most accurate measurements of the best experimenters. We may thus feel that we have our feet on solid ground when we set out to apply this theory to aid us in the solution of any problem that may present itself. In the time that remains of this lecture I wish to speak mainly of the application of the THE LAWS OF REFLECTION AND REFRACTION 139 laws of reflection and refraction to the design and con- struction of optical instruments. Clearly, if I am to do this at all effectively, I must limit myself strictly. The number and variety of optical instruments is enormous, and it requires not a little thinking to suggest many instruments of great precision that do not involve some optical principle. Optics has been called the " directing science" of modern times, because the principles that have been developed in its study have formed the basis of many of the most far- reaching speculations in modern science. It deserves the name perhaps even more truly for another reason. The advancement of science depends largely on the precision with which its researches can be conducted. Optical prin- ciples enter into nearly all instruments of precision, and thus the whole army of science is interested in these principles, and should realize that it is under a deep obligation to those men who have established them so firmly. The laws of reflection and refraction that are most fre- quently made use of in the design of optical instruments are those simpler ones that deal with the directions of the rays. These laws have already been stated and discussed, but perhaps you will bear with me if I call your attention to a different mode in which they may be presented. We have seen that if a ray of light proceeding from A (Fig. 36), strike a surface BC so as to be reflected to E, the lines AB and BE will be equally inclined to the reflecting surface. Suppose, now, that we endow a ray of light with intelligence, and set it this problem: to start from A, strike the re- flecting surface somewhere, and be reflected to E, and to choose its path so that it will reach E as quickly as possible. If you have any skill in elementary geometry, you will be able to prove that B, the point of striking the reflector, 140 LIGHT must be chosen so that AB and BE make equal angles with BC ; in other words, the law of reflection must be obeyed. Similarly, if you take the corresponding problem in re- fraction, and ask the ray to set out from A (Fig. 36), be refracted into another medium, and reach a point E in the shortest possible time, you will find that here, again, the law of refraction will have to be obeyed. In both cases you can prove the statements by showing that the time of pass- BQ Reflection FIG. 36 age along ABE is less than that along any other route, such as ACE, and in the second problem you must bear in mind that the velocity in any medium is inversely proportional to the absolute refractive index of that medium. It would thus appear that rays of light always try to reach their destina- tion as quickly as possible a curious principle. It would be very interesting to trace its development, its limitations, and its applications to a variety of problems. However, there is no time for this now, nor can we do more than refer to the fact that this principle has suggested to science a much more far-reaching law, what is known as the Principle of Least Action, the greatest generalization of modern science. The principle was first enunciated a century and a half ago by Maupertuis, then president of the Berlin Academy. THE LAWS OF REFLECTION AND REFRACTION 141 He laid it down because, in his judgment, it was eminently in accord with the wisdom of the Creator. More modern men of science do not often feel so confident about sharing in the secrets of Providence; but they find the principle none the less useful in making for the great end they have in view to comprehend all knowledge in a single law. To return to the application of the laws of reflection and refraction, I repeat that it is necessary to limit myself very strictly. You will find large treatises on Geometrical Optics, which are taken up wholly with applications of the simplest of these laws, and whole books that treat of their bearing on the construction of special instruments. In the short time that remains in this lecture, it is obviously impossible to cover so much ground. I must select a single illustrative example, and deal with one optical instrument, and even with that in a very cursory manner. What is this instru- ment to be? The one most generally interesting would be the human eye, for there we have an optical instrument that we must all use. Apart from that, it is extremely interesting merely as an illustration of optical principles. It is truly a wonderful instrument or combination of instru- ments. It is at the same time a microscope, a telescope, a range-finder, a stereoscope, a photometer, a kinemato- graph, and an autochrome camera. An instrument that serves so many purposes can scarcely be expected to be free from imperfections, and the eye is not without its defects. At the same time, to the student of optics it is as interesting in its defects as in its strength. How to cure these defects or how to minimize their evil conse- quences is a human problem that requires for its successful solution an intimate knowledge of the scientific principles here discussed. However, the eye is not an instrument 142 LIGHT that we have completely under our control ; at the best we can supplement it. So it will be better for our present pur- pose to take another instrument, where there are no such limitations on our actions, and endeavor to indicate how our knowledge of the laws of light may be employed to make it as effective as possible. To this end, let us select the Astronomical Telescope, the purpose of which is simple and well known, to enable us to see distant objects as clearly as possible, and in some cases to photograph their details or their relative positions. I need not spend time in emphasizing the fundamental idea, which is to get an image of the object near at hand, and look at this image through a magnifier. You are all doubtless quite familiar with the idea of the image of an object. You can obtain this by reflection from a mirror which is either plane or curved. The plane mirror is the only perfect optical instrument, in the sense that it forms an image that is absolutely faithful to the original, free from all distortion or other defects. A curved reflector, as you know, produces a certain amount of distortion, which is very marked if the object be a large one, such as the human figure. You can also get an image by refraction, as with a pair of spectacles, a hand magnifier, or a photo- graphic lens with one or other of which every one is more or less familiar. However, although the fact of an image being formed in some such way is well known, you may not have thought of the mode in which this image is produced, or of the bearing of optical principles upon its formation. Here we have time only for the briefest out- line. The fundamental principle, as usually stated, is that the image of a point is a point. Each point of an object has its image, and the whole collection of such points forms THE LAWS OF REFLECTION AND REFRACTION 143 a picture more or less like the original. Is it true, however, that the image of a point is a point ? Yes, absolutely so, if the reflector be a plane mirror ; but not so for any other case. In all such cases, if we take a series of points in the object, the rays from any one of them will, in general, after reflection or refraction, or both, at best pass only approxi- mately through a corresponding point in the image. They may all pass very near indeed to this point in the image, but again, many of them may pass some distance away, and the clearness of the image will depend on how close to a point the rays from any point of the object converge. If we follow out the consequences of the laws of reflection and refraction, we find that the rays from a point converge more nearly to some other point if they all strike the reflect- ing or refracting surfaces very nearly at right angles than if they strike it at oblique and widely varying angles. The latter, at best, will give a blurred image; the former will make for clearness. Hence, in our telescope we must ar- range that all the reflecting and refracting surfaces are "square on" to the impinging rays, and we must choose the form of these surfaces so that a slight departure from the perfect square will introduce as little indistinctness as possible. In the case of a reflecting telescope the form of the re- flecting surface is easily determined. The telescope being used for astronomical purposes, the incident rays come from extremely distant points, so that we have practically to deal with a series of parallel rays striking the reflector. It is a simple problem of geometry to prove under such circumstances that the form of reflector that will give the clearest image is a paraboloid the surface formed by re- volving a parabola about its axis. Figure 37 represents a 144 LIGHT portion of a parabola of which AX is the axis, S the focus, AS the focal length, and BE' the aperture. The geometrical property of the parabola, which makes it useful for this B optical purpose, is that if P be any point on the curve, and PR be drawn paral- lel to the axis, then the lines SP and PR make equal angles with PG, which is at right angles to the curve at P. Hence a ray of light that comes from a distant point in the di- rection PR will be reflected to the focus S, wherever be the point P. So much for the form of the reflector; what next as to its material? We want to have the image as bright as possible, so we must have a surface with a high reflecting power. Theory and ob- servation agree in indicating some of the metals as the FIG. 37 THE LAWS OF REFLECTION AND REFRACTION 145 best reflectors. In the earlier reflectors speculum metal was commonly employed as being a fairly good re- flector and not too expensive. Silver, however, is a much better reflector, and any objections to its use have been overcome by the discovery in recent times of thoroughly satisfactory methods of depositing it chemically upon glass. The thin film of silver is not expensive, and the glass supporting it, if carefully made, is fairly rigid, and so not very easily distorted. Freedom from distortion is extremely important where good results are required, for a slight change from the paraboloidal form will give different images in different parts of the reflector and a consequent blur. In fact, in the best modern reflectors, the greatest care is taken to preserve their form ; they are kept as free as possible from changes of temperature, and the system of support is planned with the utmost thoroughness. It was mainly through lack of such precautions that the great reflectors of the past proved, in many ways, so disappoint- ing. Consider next the problem of the size of the reflector. What is to be its aperture BB', and its focal length AS ? A considerable increase in aperture will make the instru- ment more cumbrous and greatly add to its cost. Its countera vailing advantages are mainly two. In the first place, a larger aperture collects more light, and so gives a brighter image. This may be a matter of great im- portance, if we wish to see or to photograph very faint objects. The brightness of the image depends upon the area of the aperture, and is therefore proportional to the square of the diameter BB'. Thus, if BB' be doubled, the brightness of the image will be increased fourfold. The second important advantage of a large aperture 146 LIGHT will be more fully appreciated after we have dealt with Diffraction. In the lecture on that subject it will be shown that the image of a point is not a point, but a disk whose di- ameter depends upon the size of the aperture, being smaller for large ones than for small. If you are looking at two distant objects (e.g. a double star) through a telescope, each point will appear as a disk, and the smaller are the disks the less will they tend to overlap and produce a blurred effect. Hence, if great resolving power is required, the disks must be as small as possible, and this demands a large aperture. The first great reflector (that of Lord Rosse) was made more than half a century ago, and was 6 feet in diameter. After a time a reaction set in against reflectors, but they have come into prominence again of late, and now such an instrument, with the enormous aperture of 100 inches, is being made for the Mt. Wilson Solar Observatory. As to focal length, the advantage of increasing this is that the size of the image is magnified in proportion. If you double AS, you double the image, but there is a correspond- ing disadvantage in greater length of telescope, and so greater inconvenience and expense. Lord Rosse's telescope had a focal length of 54 feet and was exceedingly cumbrous. Having considered such questions as the size, form, and material of the reflector, you may look for a moment at the problem of making the glass support for the reflecting film of silver. The glass must be as free as possible from flaws or strains, so as to minimize the danger of a change of shape, and to obtain a suitable disk of glass proves, in the case of a very large reflector, a very arduous process. Once this has been secured, the front surface of the disk is made concave by means of a tool of suitable curvature. It is important to avoid differences of curvature in different THE LAWS OF REFLECTION AND REFRACTION 147 parts of the surface, and any errors of this kind can be detected with extraordinary nicety merely placing the finger on the glass will cause a swelling of the surface that can easily be detected. After a uniform curvature has been obtained, the next step is to set to work in the process of polishing to hollow out the surface in the center so as to produce an exact paraboloidal form, any departure from this form being readily discovered by a simple optical device. FIG. 38 Then the surface is silvered by one of those exceedingly ingenious devices of modern times designed for this end, and, let us hope, an almost perfect reflector is the result. Now, if such an instrument were turned toward a star or other heavenly body, it would produce an image of the object in the neighborhood of the focus S. With great focal length this image might be fifty feet or more away from A, and so would be inconveniently placed for purposes of close inspection. This inconvenience may be avoided by intercepting the rays as they converge toward S, and reflecting them backward so as to converge to a point C behind the large reflector (Fig. 38). It is a simple problem of geometry to determine, by the aid of the laws of reflection, the form of the reflecting surface that will produce this result. The surface must be a hy- 148 LIGHT perboloid formed by the revolution about its transverse axis of a hyperbola whose foci are S and (7, and it is made by the same general processes as are employed in forming the paraboloid. In order that the rays should reach C. a hole would have to be made in the center of the large reflector in the neighborhood of A. This has been done in some telescopes, but the plan has many disadvan- tages, and these may be avoided by again intercepting the rays as they converge toward C by a small plane mirror at D, and reflecting them to one side so as to form an image at F. A reflecting apparatus made in this way will give very good definition near the optic axis, but if the rays are oblique to this axis the images will be indistinct. It is therefore important that the instrument should be kept in almost perfect adjustment, and the greatest care must be employed to secure this, if the best results are to be obtained. Thus far, in considering the formation of an image of an object* we have supposed that this is achieved by means of reflection. You need scarcely be reminded that the same end can be reached by means of refraction. You must all be more or less familiar with the action of a lens in bringing the rays of light from a point to a focus, and it is not difficult to investigate the features of the image thus formed, by aid of the laws of refraction. What has already been said as to the size of the aperture and the focal length of the lens applies to a refractor just as to a reflector. The form of the refracting surfaces is determined mainly from the consideration that two defects must be specially guarded against, these being known technically as chro- matic effects and spherical aberration. The latter defect has already been referred to, without the name. It has been remarked that if rays from a point strike a surface obliquely, THE LAWS OF REFLECTION AND REFRACTION 149 they are not brought (either by reflection or refraction) to the same focus as when they strike the surface almost at right angles. If, then, we have a number of rays striking such a surface, some of them nearly normally and others much more obliquely, there will be no definite image of a point, and the whole image will be blurred and indistinct. FIG. 39 Investigation shows that it is possible to lessen this indis- tinctness by increasing the number of refracting surfaces. A common arrangement is to have a double objective, such as is illustrated in Fig. 39 a. This is made of two lenses of glass of different refractive powers, one (Lj) of flint and the other (L 2 ) of crown glass. The curvatures of the various surfaces are arranged so as to make the defect due to rays striking one surface obliquely counterbalance that due to the other surfaces, and in every objective of any value this is done with great precision. 150 LIGHT In constructing the lenses it is important to avoid hav- ing different curvatures in different belts of the lenses, as this will inevitably introduce aberration and cause a blur. Also, as the curvature of each surface must be maintained constant, care must be taken to avoid its change due to fluctuations of temperature, to flexure from the weight, or any other causes. The danger of flexure is of course much less serious with small lenses than with the large ones such as those of the great 40-inch refractor of the Yerkes Observatory. To avoid flexure with such large lenses, we must have considerable thickness in the center of the lens, and this introduces a serious defect when the telescope is to be used for photographic purposes. The amount of light absorbed in passing through a considerable thickness, even of the clearest glass, is far from negligible, and it in- creases enormously for those light-waves of high frequency which play the leading part in photographic work. Hence, if we wish to take a photograph of a faint object, so that we cannot afford to lose much light, a thick lens is objec- tionable. This is one of the reasons why refractors are being replaced by reflectors for some of the work in modern astronomy. Perhaps a stronger reason, however, is that the reflector avoids entirely the serious difficulties due to chromatic or color effects. The law of reflection is the same for all colors, so that the position of the image after any number of reflections is quite independent of the color of the light, and no chromatic effects can possibly be introduced by the process of reflection. With refraction, however, it is very different. The laws of refraction show that the position of the image depends on the refractive index of the refractor, and this again depends on the color of the incident light. White light, as we have seen, is a THE LAWS OF REFLECTION AND REFRACTION 151 composite of many colors, and the image formed by each of its constituents will be in a different place, and of a different size. Clearness and precision in the image thus appear to be impossible, and the question must arise can this be avoided ? The answer is that it can, at least in a partial manner. If we have two lenses, one may be made of such a form and material that it will throw the red image farther away than the blue, while the other reverses things by throwing the blue farther away than the red. In com- bination it may be arranged that they throw the two images together. This is one reason why the objective of an astronomical telescope is always made up of at least two lenses, such as the double objective depicted in Fig. 39 a. It is an easy matter to calculate how to arrange two lenses of given materials so as to combine any two given colors. If the telescope is to be used for visual work, it is natural to combine two colors to which the eye is most sensitive, such as green and yellow. This combination, however, will be of little use for photographic work, as the rays that are most important in that field have been neglected. To improve matters, we may do as Ritchey did with the Yerkes refractor, and put in a yellow screen to cut out the blue and violet; but of course we do this at the expense of the light that is most effective photographically, and it is often our great end to conserve what little light there is. In any case, when two differently colored images have been com- bined in this way, the other colored images are not, as a rule, combined. Their presence in different positions gives rise to what are called secondary spectra, which not only produce indistinctness, but cause a considerable loss in the light that contributes effectively to the brightness of the image. Thus it has been calculated that the loss from 152 LIGHT this cause with the Lick 36-inch refractor was about one- quarter of the whole light, and with other refractors of shorter focal length the loss was considerably greater. Modern researches and experiments in the manufacture of glass have made it possible to select two glasses that in combination avoid these secondary spectra almost entirely. Unfortunately, however, the objectives so made have not been wholly free from defects, the most important arising from a lack of permanence in the quality of the glass. With a triple objective consisting of three lenses, Z/ 1; L 2 , Z/g, such as is depicted in Fig. 39 b, the chro- matic effects can be avoided almost perfectly, but as yet no very large refractors have been equipped in this fashion. The cost, of course, is greater, and the extra lens in- volves more loss of light by absorption, a serious thing, as we have seen, especially in the photography of faint objects. So far we have been occupied entirely with the design and construction of the objective, which forms the image of an object, whether by reflection or refraction. We have still to inquire what means are employed to get a close view of the image so formed. For this purpose an eye-piece is used, and this is designed to magnify the image, just as when you take up a hand magnifier to look closely at a small object. Our time is too far exhausted to enable us to go into details as to the arrangement of the parts of this eye-piece. Suffice it to say that, as a rule, it consists of two lenses so constructed and placed as to diminish as much as possible the defects due to spherical aberration and the chromatic effects. It is not completely achromatic, but an effort is made to bring it about that the different colored images that are formed should have the same THE LAWS OF REFLECTION AND REFRACTION 153 apparent size, as this avoids the indistinctness due to a series of colored images overlapping one another. The point that I have hoped to make clear to you in the latter part of this lecture is that the design and construction of a modern optical instrument is no haphazard process, guided by rule of thumb. On the contrary, every detail is carefully planned and calculated with the aid of the funda- mental laws of reflection and refraction. Calculation, on the basis of these principles, determines the size and the form of the various parts; calculation determines their relative position, calculation determines even the ma- terials of which they are made. Absolutely nothing is left to chance or guesswork; everywhere law and intelli- gence are supreme. VII THE PRINCIPLE OF INTERFERENCE THUS far, in dealing with the theory of light, we have emphasized the idea of a periodic disturbance propagated through a medium, and we have emphasized this because the idea of periodicity is the fundamental one. Any such periodic disturbance may be called a wave, and the theory a wave theory of light; but it will be well to guard your- selves against being misled by following too closely the analogy presented by the familiar phenomena of water waves. Here, too, you have the fundamental idea of a periodic disturbance propagated with a definite velocity, and it is doubtless because of this that the phrase a wave of light is so generally employed. The analogy, however, is not complete, and must not be pressed too hard; for one reason, water waves, as ordinarily observed, are sur- face phenomena; if you could see what goes on below the surface, the analogy would be much more instructive. At the same time there is much that arises in discussing light that can be most conveniently spoken of in language that is suggested by the familiar phenomena of water waves. A common term when dealing with such matters is the wave-length. In the case of waves in water this is some- thing that you can readily see and measure, the distance from crest to crest of consecutive waves. Suppose that you are watching a swarm of corks floating on the water, and ob- serve how they rise and fall as waves pass over them. If 154 THE PRINCIPLE OF INTERFERENCE 155 you fixed your attention on two successive corks, each of which was at the highest point of its path, they would, of course, be each on the crest of successive waves. After a definite interval of time (the period) they would each once more be on the crest of a wave, having fallen and risen in the interval as the wave-form advanced. The wave-length is the distance from crest to crest, and you can see that there must be a relation between the wave- length (X), the velocity of the wave-form (v), and the period (p = l// ; where /is the frequency). This relation is, in fact, \ = vp = v/f. In the case of light the velocity is always the same where there is no matter (that it changes with the frequency in the presence of matter was explained at con- siderable length in the lecture on Dispersion). Hence, as the frequency (/) changes with the color of the light, so must the wave-length (X). In other words, differently colored waves have different lengths. Waves of high fre- quency, such as violet waves, are short; waves of lower frequency, such as red waves, are longer ; what the actual lengths are and how they are measured will appear in a later lecture. So far we have spoken of a single train of waves ; but what if more than one train moves across the same space ? It is an interesting and instructive thing to observe, if a sheet of water be at hand. Throw in two stones at A and B respectively. You will see a wave-form running outward from each of these centers, and in due time the two trains of waves will cross one another's path. A curious pattern will be the result, and you may learn much by trying to account for its leading features. The clue to everything here is the Principle of Superposition of Small Motions, or the Principle of Interference as it is usually called when its 156 LIGHT applications to optical phenomena are under considera- tion. The principle lays down a rule for determining the effect of combining two small displacements due to dif- ferent causes. It states that each cause produces the same effect as it would were the other cause absent, and that in computing the displacement due to the combina- tion of both causes we have merely to add together the displacements due to each separately, of course taking account of the direction of the displacements in the process of addition. Thus, if the motion due to one cause would raise a point an inch, and that due to another would raise it half an inch, then the point would be raised 1 + J, or an inch and a half, under the combined influence of both causes. If, on the other hand, one cause would raise a point an inch and the second depress it half an inch, the combination would raise it 1 J, or half an inch. Sup- pose that we accept this principle and apply it to two trains of water waves of the same height and length, and moving in the same direction. What would be the com- bined effect of two such waves? The answer would de- pend entirely on their relative phase. If crest corresponded to crest so that the waves were "in phase," the two would combine into a single wave of double the height of each. If, however, crest corresponded to furrow, so that there was a difference of phase of half a wave-length, then the com- bination would produce no wave at all, but absolute rest. The crest of one would just fill up the furrow of the other, and the two waves might be said to interfere with one an- other. It is on this account that the principle is com- monly spoken of as Interference. It is a principle that was well known to Newton, and was applied by him to explain certain phenomena of the tides. However, it was THE PRINCIPLE OF INTERFERENCE 157 reserved for another great Englishman, Thomas Young, to realize that the same principle is applicable to light and to use it as a means of overcoming most of the obstacles that had retarded the progress of the science of optics. Young's is one of the very greatest names in science, although almost wholly unknown to the man in the street. He was endowed, according to Helmholtz, with "one of the most profound minds that the world has ever seen." His application of the Principle of Interference to light was only one of his strokes of genius ; but it was far-reach- ing in its consequences, and made Young in a sense the father of the wave theory of light. It was he, more than any one else, who, in the early days, just a century ago, turned men's speculations along the track that has led to so much in more recent times. Perhaps you would like to hear how he expressed himself, as it is always interest- ing to listen to an original thinker when he is expounding his own ideas. Here, then, is a brief extract from his writings on the subject of Interference: "It was in May, 1801, that I discovered, by reflecting on the beautiful experiments of Newton, a law which ap- pears to me to account for a greater variety of interesting phenomena than any other optical principle that has yet been made known. I shall endeavor to explain this law by a comparison. Suppose a number of equal waves of water .to move upon the surface of a stagnant lake with a certain constant velocity, and to enter a narrow channel leading out of the lake. Suppose, then, another similar cause to have excited another equal series of waves, which arrive at the same channel with the same velocity and at the same time with the first. Neither series of waves will destroy the other, but their effects will be combined; 158 LIGHT if they enter the channel in such a manner that the eleva- tions of one series coincide with those of the other, they must together produce a series of greater joint elevations; but if the elevations of one series are so situated as to correspond to the depressions of the other, they must exactly fill up those depressions, and the surface of the water must remain smooth; at least I can discover no alternative, either from theory or from experiment. Now I maintain that similar effects take place whenever two portions of light are thus mixed, and this I call the general law of the Interference of Light. I have shown that this law agrees most accurately with the measures recorded in Newton's " Opticks, " relative to the color of transparent substances, observed under circumstances which had never before been subject to calculation, and with a great di- versity of other experiments never before explained." I shall direct your attention in a moment to some ex- periments designed to test or illustrate the Principle of Interference, but before doing this I should perhaps state explicitly that in applying it to the explanation of optical phenomena you are not restricting yourself to any special form of the wave theory of light. It is a principle that is applicable to displacements of any kind, and its most im- portant consequence for our present purposes is that an upward displacement in the ether due to one cause may be exactly counteracted by an equal and opposite down- ward movement due to some other cause, and that this will inevitably be the case if there be a certain phase rela- tion between the two periodic movements. It is in this way that two lights may produce darkness in certain places, although it may at first seem paradoxical that a combination of lights should produce darkness. Further- THE PRINCIPLE OF INTERFERENCE 159 more, if you are to understand the experiments that are about to be referred to, you should call to mind that white light is of a composite character, and that, by suppressing some of its constituents, color effects are produced. At one place blue may be suppressed by interference, at an- other green, and at another red, so that interference phe- nomena should be characterized by bands of color wher- ever white light is employed in producing them. E FIG. 40 One of the most famous of Young's experiments to test his theory of interference is, in principle, as fol- lows. Light is allowed to stream through, say, a verti- cal slit the position of which is indicated by S (Fig. 40), and to fall on two other vertical slits, A and B, which are very close to one another in a screen parallel to that containing the first slit S. The light is intercepted on a vertical screen indicated by the section ECPF in the figure. Now if we consider a point such as P on this screen, it will be observed that it is illuminated by light that comes from two sources, A and B respectively. As things are arranged, the light that sets out from A at any moment will be in the same phase as that which has B as its starting-point; but that which travels to P along AP will reach P in a different phase than the light from B, for the two started together and moved at the same rate along roads of different lengths. If the point P be so 160 LIGHT situated that the difference between AP and BP be half a wave-length, or any odd multiple thereof, the two lights reaching P will interfere and nullify one another. Hence the screen EF will not be uniformly illuminated, but there will be a series of dark vertical lines, or of colored bands, according as the incident light is homogeneous or other- wise. If one of the slits (A or B) be covered, the bands should disappear. All these phenomena may be observed, and the position of the bands and the arrangement of the colors are found to conform in the closest manner to the predictions of the theory thus sketched. Figure 41 gives an indication of the alternations of light and shade on the screen EF. Unfortunately, I am not able to show you Young's experiment, owing to the difficulty of exhibiting the phenomena to a large audience, but you will find it easy to make the experiment for yourselves. One method of proceeding is to rule two narrow lines very close to one another on a photographic plate that has been developed, and then to look through the slits so formed at the light that shines through a slit in front of a bright light, such as the electric light. A still simpler procedure is to make two pinholes close to one another in a card, and look through them at the light streaming through another hole. With a little care you will see the interference fringes quite distinctly. Another simple experiment designed to show interference is due to Fresnel, one of the great names in the development of the theory of light. He made light THE PRINCIPLE OF INTERFERENCE 161 from a slit S (Fig. 42) fall on two mirrors, A and B, that had their edges parallel to the slit and their planes in- clined at a very small angle. After reflection from these two mirrors, the two streams of light were in a condition to interfere with one another, and a series of bands similar to those just described made their appearance on a screen PE. With this experiment, as with Young's, it is difficult to arrange things so as to exhibit the phenomena to many FIG. 42 persons at once, but you can repeat FresnePs experiment for yourselves. Take two pieces of the same glass, blacken them on the back, and lay them on a board that is covered with a black cloth. Raise the edge of one strip of glass very slightly, and adjust the slit so as to be parallel to the common edge of the pieces of glass. With proper care in the adjustment, you will get interference fringes exhibited to the eye properly placed to receive the light, and you will find that these fringes disappear if one of the reflected beams is suppressed by blackening one of the mirrors. A modification of FresneFs experiment, due to Lloyd, should perhaps be mentioned. Take a strip of plate-glass blackened at the back, and allow light to fall upon it at nearly grazing incidence, as in Fig. 43. Light from a slit S reaches a screen at P by two paths, one directly along SP, and the other along SBP after reflection at the mirror. These two 162 LIGHT beams, the one direct and the other reflected, may inter- fere and give rise to fringes, as before. In this, as in all the experiments referred to recently, considerable care must be exercised in the adjustments, otherwise no results or spurious results will be obtained. FresneFs device, and Lloyd's modification of it, consist in producing interference between the two parts of a beam that have been separated by reflection. Fresnel also arranged to split the beam by means of refraction. To do this he employed a biprism, consisting of a piece of glass made in the form of two FIG. 44 prisms of very small angles placed back to back. In Fig. 44, the shaded portion represents a biprism, S a source of light, PE a screen. The light from S that falls upon the upper portion of the prism is bent downwards and made to pro- THE PRINCIPLE OF INTERFERENCE 163 ceed as if from S^ while that which falls upon the lower portion is bent upwards and proceeds as if from S 2 . Thus Si and $ 2 correspond to A and B in Fig. 40, representing Young's experiment, and the explanation of the interfer- ence fringes is the same as was there indicated. All these experiments are specially designed to exhibit interference fringes and to test the explanation by a comparison between theory and observation as to the exact position of the bands and the arrangement of their colors. Kindred phenomena, however, are obtained in almost countless other ways, many of them far more striking and beautiful than those to which reference has just been made. It has been explained that in order that waves of light may interfere, they must set out simul- taneously from the same source and meet with such dif- ferent treatment that one wave becomes half a wave- length in phase behind the other. Now think of a beam of light falling on a thin film of any material. Some of the light will be reflected at the first face, while some will penetrate the film, be reflected at the second face, and, after emerging from the film, be in a condition to interfere with what was first reflected. If the film be of the proper thickness, this interference will be inevitable, and as a consequence some of the light will be suppressed in places, so that we shall see alternations of light and darkness, or variations of color, according as the incident light is homogeneous or composite. You must all have observed the brilliant colors produced in this way by a thin film of oil on the surface of water. You may see the same thing here by looking at the beautiful color on the wall, pro- duced by reflecting light from the surface of the water in this hand tray after a drop of turpentine has been allowed 164 LIGHT to fall upon the water. Much more beautiful effects, due to similar causes, are obtained with a soap-film, as every- body knows who has seen a soap-bubble. Fortunately, youth is not a question of age, and the blowing of such bubbles has afforded interest and amusement to genera- tions of young people between seven and seventy. Nor have philosophers been ashamed to enter into the game and to discuss the phenomena in their grave way. If it be true that to the poet's mind " the meanest flower that blows can give thoughts that do often lie too deep for tears/' then it need cause no surprise that so common a thing as a soap-bubble has engaged the serious attention of the greatest men of science, such as Boyle and Newton of olden times, Stokes and Kelvin of our own day, to select only a typical few. All the gorgeous phenomena of color exhibited by soap-bubbles are explicable by means of the principle of interference. The color that is suppressed by interference varies with the thickness of the film, its re- fractive index, and the angle of incidence of the light that falls upon it. Theory enables us to calculate all the details and to predict what will happen with a film of given thickness when the wave-lengths corresponding to the different colors have been determined. How these wave-lengths may be measured will be indicated in a later lecture on Diffraction. Meanwhile, as the phenomena of the soap-bubble are somewhat complicated by the curva- ture of the surface, it may be well to show you similar color effects with a flat film. I dip this ring into a solu- tion of soap, fix it in a vertical plane, and by means of a lens bring the light reflected from the film to a focus on THE PRINCIPLE OF INTERFERENCE 165 the screen. The exact interference effects depend on the thickness of the film, and therefore change as the thick- ness alters while the liquid streams down. The image on the screen is inverted by the lens, so that everything appears upside down. The upper part of the film is seen at the bottom of the picture on the screen, and you will observe that the liquid seems to be streaming upwards. Notice the changing color as the liquid thins away from the top of the film. First you see a bright green, then it changes gradually until now you have a deep red. Now, again, in this part it is blue, now violet, now quite black, and now the film has broken, having become too thin to bear the strain of its weight. An interesting modification of this experiment is to arrange things so that the light comes from a narrow slit, and after reflection, as before, from the film, passes through a prism before falling on the screen. If light were re- flected from a single surface and treated in this way, the prism would separate the different colors and produce the familiar spectrum. With the film, however, there will be places where the light is cut out by interference, so that, as the film thins, dark bands will be seen to travel across the spectrum. You can see them distinctly in the ex- periment that Mr. Farwell is now conducting. You observed in these experiments with films that just before the film broke it looked quite black at the thinnest part. This is a curious fact, and one that seemed para- doxical for a time. Newton observed the same thing with an ordinary soap-bubble, and you can easily repeat the observation. Blow such a bubble, and cover it with a glass to screen it from air currents, and so prevent its breaking too soon. As the liquid drains downwards, the 166 LIGHT film gets thinner at the top, and just before it breaks this part looks quite black. At first sight this seems contrary to what might be expected. As this portion of the film is extremely thin, it takes practically no time for light to travel across it and back to the upper surface, so that you might expect the light that has made this short pass- age to be in the same phase as the light that was reflected at the first surface. If this were so, the two waves should reinforce one another instead of interfering, so that we should have brightness instead of darkness. However, on examining the matter by the aid of theory, it appears that at one of the reflections, but not at the other, there should be a change of phase of half a wave-length in the very act of reflection, and this completely accounts for what is observed. All the bands of color produced by interference that you have seen to-night have been arranged in straight lines, but it is easy to get them in other forms. Here, for example, is a simple modification of our experiment with the flat film. With these acoustical bellows I pro- duce a slight blast and direct it almost tangentially on the surface of the film. This sets the liquid in the film in motion, and arranges it in regions of varying thickness, producing, as you see, brilliant curves of color. In one case you have a series of concentric circles, such an arrange- ment of color as is found in the famous phenomena of Newton's Rings. These Newton studied with great care, the second book of his "Opticks" being almost wholly devoted to a discussion of their features. Newton's ar- rangement for producing these rings is extremely ingen- ious, because extremely simple and extremely effective. It consists in pressing together two pieces of glass, one or THE PRINCIPLE OF INTERFERENCE 167 both of them being slightly curved (Fig. 45 a). When light is allowed to fall on this and to be reflected, a beauti- ful series of colored rings is seen arranged in concentric circles round a central spot. At all points such as P (Fig. 45 b) on a horizontal circle of which is the center, the thickness of the air-space between the two pieces of glass is the same, and equal to PN. Thus waves that pass to and fro in this region have to traverse an air film of this thickness (PN). If, then, PN be of the length necessary to produce the requisite phase difference for (a) FIG. 45 waves of a given length, there will be interference, and the corresponding color will be absent from this region. Thus, we should expect to see a series of colored rings if the incident light be composite like sunlight, and there is no great difficulty in predicting the main features from theory and verifying the correctness of this theory by careful observation of what actually takes place. The most important laws were discovered by Newton by in- duction from his experimental results. Thus, he found the law of the radii of the rings, viz. that at a given angle of incidence the radii of the different rings are proportional to the square roots of the numbers 1, 2, 3, 4... (These different rings are spoken of as rings of different orders.) He found also in what way the radii varied with the angle of incidence, and verified his law with wonderful accuracy, considering the rough instruments of measurement at his 168 LIGHT disposal. The following table compares the radii of a ring of a given order for different angles of incidence on the glass, and shows how Newton's law and Newton's experiments agreed with one another. Moreover, by means of a prism Newton analyzed the light before it fell INCIDENCE o 10 20 30 Radius (law) 1 10077 1032 1075 Radius (experiment) . . 1 1.0077 1.033 1.075 INCIDENCE 40" 50 60 70 Radius (law) . ... 1.142 1.247 1.415 1.71 Radius (experiment) . . 1.140 1.250 1.4 1.69 upon his ring apparatus, and so was enabled to inves- tigate the phenomena when employing light of a single color, and to see in what way a change of color affected the size of the rings and their distinctness. "I found/' he says, "the circles which the red light made to be mani- festly bigger than those which were made by blue and violet. And it was very pleasant to see them gradually swell or contract according as the color of the light was changed." As the radii of the rings depend on the color, the larger (red) rings of one order will tend to overlap the smaller (blue) rings of the next higher order. This overlapping will produce indistinctness, so that it will be difficult to see the rings of high order when the incident light is white. If, however, homogeneous light be em- ployed, there is no possibility of overlapping, so that far- more rings may be seen. "I have sometimes," says New- ton, "seen more than twenty of them" (when working THE PRINCIPLE OF INTERFERENCE 169 with a prism to produce homogeneous light), "whereas, in the open air" (without the prism), "I could not discern above eight or nine." Instead of using a prism, we may get what is very nearly homogeneous light by interposing colored screens in front of the powerful electric light in the lantern. These screens cut off a good deal of the light, so that the phenomena, as you see, are not so brilliant as before; but if you look carefully, you will have no diffi- culty in making out the main features. Now there is a red screen and you see the red rings (of course no other color is possible with this arrangement) ; now we have a blue screen, and you notice the blue rings distinctly smaller than the red ones that you have just been looking at. Since Newton's day there have been many modifications of his experiments and many new phenomena of a kindred character discovered; but there is nothing that is not completely accounted for, down to the minutest detail, by means of the principle of interference coupled with the known laws of reflection and refraction. All these examples of interference have been produced by apparatus that has been specially designed to exhibit this effect. Not infrequently, however, we meet with similar phenomena where no such pains has been taken to produce the result. In such cases the design, if design there be, is not of man's contrivance. Thus you have all observed that polished steel becomes colored when it is exposed to the air. A thin film of oxide is formed on the surface, and produces interference effects by reflection like any other film. Antique glass, especially when it has long been buried, becomes coated with a thin layer that shows beautiful interference colors. The wings of a butterfly owe their color to their delicate ribbed structure and the 170 LIGHT interference that this produces. The gorgeousness of a peacock's tail is due to the same cause. You will observe that the color of this feather is not intrinsic ; it changes with the incidence of the light, as you see when I turn it in the lime-light. The changing colors of opals are ex- plained in the same way, and so are those of mother-of- pearl. If you examine such an object closely with a microscope, you will find that it is made up of layers, and that the surface cuts across these layers, and so presents a series of minute grooves. The lights that are reflected from opposite edges of these grooves are in the condition to interfere, and you can easily see that the color changes with the incidence of the light that falls upon the surface. None of this beautiful color is really in the shell. Brewster showed this conclusively when he stamped the shell on black wax, thereby reproduced the grooves, and obtained the same colors from the wax as from the original shell. Before bringing this lecture to a close, there is just time to refer, all too briefly, to an ingenious application of the principles of interference to the problem of color pho- tography. This was first made in 1891 by Lippmann, but since that date considerable improvements have been effected in the practical application of Lippmann's ideas. The theory of the process is not without its difficulties, but the broad lines of the explanation, as suggested by its author, are easily seen. The first matter that must be firmly grasped is that there is an intimate relation between the intensity of light reflected from a very thin film and its thickness. If the thickness be altered, so will the brightness of the reflected beam. We saw a short time ago that for a film so thin that it can scarcely be said to have any thickness, there is no light reflected at all. Start- THE PRINCIPLE OF INTERFERENCE 171 ing with this, let us imagine the thickness to increase gradually, and consider the effect on the intensity of the reflected light. For simplicity we shall suppose that the light is incident normally and not obliquely. The re- flected light will grow in intensity until the thickness of the film is exactly half a wave-length of the light that is used. (That length will depend, as has been seen, upon the color of the light and upon the refractive index of the film.) After this thickness of half a wave-length has been reached, less light will be reflected, and this diminu- tion will continue until a thickness of a wave-length has been attained, when once more there will be no reflected light. This variation of intensity is all accounted for by the principle of interference. We are thus led to the important conclusion that when dealing with thin films less than a wave-length in thickness, we immensely in- crease their reflecting power if we make their thickness half a wave-length of the light that we wish to reflect. Let us suppose that X R is the wave-length of red light for the material of which the film is composed, and that we make a film of thickness X R , and observe the light that it reflects from a landscape or a picture. It will be much more effective in reflecting red than any other color, and its power of selective reflection will be greatly im- proved if we back it by several parallel films of the same thickness. With such an arrangement we shall practi- cally see nothing but the red parts of the picture. With other films of thickness % X G , where X G is the wave-length for green light, we shall similarly pick out the green por- tions, and with films of thickness X v (where X v is the wave-length of violet light) the violet portions of our picture. If, now, we have any means of combining these 172 LIGHT three colored reflections, we shall have a faithful repre- sentation of the original, according to the explanation set forth in the earlier lecture on color photography. The practical difficulty in carrying out such a process that will probably first present itself to your minds will be that of obtaining films of the right thickness. The actual lengths of some waves of light will be set forth in the lecture on Diffraction, and if you have any conception of their minuteness, measured by any ordinary standard, you will realize that it is quite hopeless by any mechanical process to produce a film whose thickness is exactly J X R , or any of the other quantities that have been specified. And yet such films can be manufactured quite accurately by optical means. The device for doing this is, of course, an essential feature of the Lippmann process; but the same principle was employed a little earlier by Wiener. It is another simple application of the Principle of Inter- ference. Suppose that we have two series of waves mov- ing through a medium, and that they are similar in every other respect except that they are moving in opposite direc- tions. These waves will be in a condition to interfere with one another, and there will be a series of points N 1} N 2 , N B ... at each of which the upward displacement in one wave is exactly counteracted by the downward dis- placement in the other wave that is moving in the opposite direction. At such points, which are called nodes, the displacement due to the two waves will be zero. At inter- mediate points, L v L 2 , Z/ 3 ..., the two displacements will be in the same direction, and will reinforce one another, and these points, where there is a maximum of displacement, are called loops. Investigation shows that the positions of these nodes and loops are stationary, that they do not THE PRINCIPLE OF INTERFERENCE 173 change from moment to moment. The aspect of this combination of two trains of waves is thus very different from that of either taken separately. The nodes always remain at rest, and halfway between these points (at the loops) the crests of the waves rise and fall periodically. There is no moving of the wave-form in one direction or the other, but a mere gradual change of height. Such a set of waves are consequently called stationary waves. They have often been set up in air and water; but the difficulties of producing them with light-waves in the ether and of demonstrating their existence were not suc- cessfully overcome till 1890. In that year Wiener set up these stationary waves by reflecting light from the silver coating of a plate of glass, and proved their existence by their effect on a thin film of sensitized collodion super- posed on the glass. We should expect the photographic action to be different at the nodes, where there is no dis- placement, than at the loops where the displacement is greatest, and Wiener succeeded in showing that the pho- tograph was crossed by bright and dark bands at regular intervals, and thus in affording another ocular demon- stration of the soundness of the Principle of Interference. Now there is one feature of these stationary waves that has not yet been mentioned and that is specially impor- tant for the purpose that we have in hand. The distance between successive loops, as well as that between successive nodes, is exactly half a wave-length. You will realize the significance of this at once. It gives us an optical means of producing a film, or a series of parallel films, whose thickness is half a wave-length of any color that we wish to use. Set up these stationary waves with red light, and they will so act on the sensitive emulsion as to arrange it 174 LIGHT effectively in layers whose thickness is X B , and when this is afterwards viewed by reflection it will send back practi- cally nothing but red light. Do the same with the other colors, and this part of your problem is solved. You will probably see, too, that it is not really necessary to have these different films and to devise a means of combining the pictures that they present by reflection. All the work can be done by the same material. Each color that strikes it will build up a little film by means of stationary waves acting on the sensitive emulsion with special force at regular intervals of half a wave-length, and this film will be of just the right thickness to reflect that particular color most copiously. The form of the object will be produced, just as in ordinary photography, by the grada- tions of light and shade over different portions of the plate, the color by the thickness of the different films beneath the several portions. VIII CRYSTALS TO-NIGHT we are to deal with some of the optical properties of crystals. It has been remarked in an earlier lecture that the distinguishing feature of a crystal is its structure. Its parts are not thrown together at random, but are built one upon another according to some definite plan. The result is that a crystal does not seem the same when looked at from different directions. If you could imagine yourself moving through water or glass (which are not crystals), it would make no difference to your rate of progress whether you went north, south, east, or west. In a crystal, however, it might well be different; the structure might be so arranged as to make progress easier in one direction than in another. In optical problems we are interested especially in the propa- gation of waves, the speed of which for a medium like the ether depends on the rigidity of that medium. Here we need not stop to inquire exactly how the presence of matter modifies the effective rigidity of the ether contain- ing it ; but owing to the structure of a crystal it is natural to suppose that its presence in the ether will modify the rigidity differently in different directions. If we apply general dynamical principles to the discussion of the propagation of waves in such a medium (that is, a medium with different rigidities in different directions), the first 175 176 LIGHT striking result that we reach is that, as a general rule, for a wave traveling in any given direction there are two speeds with which the wave can travel. We may express this by saying that two waves can travel through a crystal in any given direction, and that in general these will travel with different speeds. As there is a ray of light corresponding to each wave, we see that when a ray strikes a crystal it will give rise not to one, but to two different rays within the crystal. This prediction from theory cor- (o) FIG. 46 responds to the well-known fact of double refraction pro- duced by a crystal. Here are two double prisms of the same size and shape. They are represented in section in Fig. 46 a and b. The first is made of two prisms, ABD and BDC, of the same non-crystalline material, glass. The second (Fig. 46 b) is made in exactly the same way, but is of crystal, Iceland spar. Now observe the difference of behavior when a ray of light falls perpendicularly on a face of each double prism and is afterwards received on a screen behind the prism. With the glass the ray goes straight through as indicated in Fig. 46 a and forms a single patch at P on the screen. With the crystal the ray splits into two on crossing B'D', each of these rays is further bent on passing out of the prism, and on the screen we see two widely separated spots of light, one at P l and CRYSTALS 177 the other at P 2 - You see, then, that this double refraction is no dream of the theorists, but an actual fact. Theory, however, does much more than predict that we should find two waves traveling in a given direction with different speeds. It indicates, further, and this is very important, that these two waves will be differently polarized. When each is plane polarized, the planes of polarization for the two waves are at right angles to one another. This deduction from theory is amply verified by experiment, and the use of crystals to produce or to test plane polarized light is one of the regular resources of an optical laboratory. You may remember that in introducing the subject of Polarization we employed a Nicol's prism to produce plane polarized light. Its power of doing this depends entirely on its crystalline structure, and you should have no difficulty in understanding its action if you bear in mind two facts. The first is the one just referred to, that for a given ray in a crystal the vibra- tions must be confined to one or other of two planes at right angles, say a vertical and a horizontal plane. It appears that the molecules of a crystal are so arranged that the ether cannot continue to vibrate to and fro along any arbitrary direction, but must confine its movements to one or other of two directions at right angles to one another. A mechanical analogue was suggested on p. 101 and illustrated in Fig. 24 ; but it may not be out of place to repeat that this is merely an analogy, and that it is not suggested that the figure depicts the actual arrange- ment of the molecules. The second fact to remember in dealing with Nicol's prism is the fact of total reflection when the angle of incidence exceeds the critical angle. NicoFs prism is made by cementing together two prisms 178 LIGHT of Iceland spar, as indicated in Fig. 47. When a ray AB strikes the prism, it is split into two by double refraction, and the two rays in the crystal EG and BF are differently polarized, BC horizontally (say) and BF vertically. The angles of the prisms are so arranged that BF strikes the thin layer of cement between the prisms at an angle greater than the critical angle. Thus, the ray BF is totally re- flected along FH, and so does not emerge from the face D. FIG. 47 The other ray, BC, passes over into the second prism and emerges at D, polarized in a horizontal plane. It has been stated above that, in general, two different waves may be propagated in any direction. Theory, how- ever, indicates, and experiment verifies, that there must always be one, and in some cases two, directions in which only a single wave can pass. Those directions are called the optic axes of the crystal, and crystals are classified into uniaxal and biaxal, according as they have one or two of such axes. In the first case the arrangement of the molecules of the crystal must be perfectly symmetrical round the axis ; in the second case there is no such perfect symmetry about any line. Theory, moreover, does much more than indicate these general features ; it enables us to calculate all the details of the wave-motion. Thus we can compute exactly the speeds with which waves will travel in any given direction. It is convenient to express the CRYSTALS 179 speed in terms of the refractive index, it having been ex- plained before that the speed is obtained by dividing a known constant by the refractive index. The results can be exhibited in a geometrical form by drawing lines from a point 0, the directions of the lines indicating the direc- tion in which the wave is traveling and the length of the line measuring its refractive index. If lines are drawn in all directions in this way, their ends will all lie on a surface, which is called the Index Surface. Theory pre- dicts the precise form of this. In the case of uniaxal FIG. 48 crystals, where there is perfect symmetry about an axis, the index surface consists of a sphere and a spheroid, with the optic axis as a common diameter. A spheroid is an egg-shaped surface with perfect symmetry about an axis, so that you may think of the index surface for a uniaxal crystal as being made up of an egg and a sphere. You will realize at once that the surface might have two dis- tinct forms : the sphere might be inside the egg (Fig. 48 a), or the egg might be inside the sphere (Fig. 48 b). The crystals will have different optical properties in the two cases, and those of the first type are called positive crystals, those of the second negative crystals. You will see from 180 LIGHT the figure that a line drawn in any other direction than the optic axis OA will cut the index surface at two dif- ferent distances from the center 0, when it crosses the sphere and the spheroid respectively. These two distances represent the refractive indices (and so measure the speeds) of the two waves that, we have seen, can be propagated in any direction. You will notice that the law which connects the refractive index with the direction of propa- gation is quite different for the two waves. With the sphere the radius is everywhere the same, so that for the corresponding wave the refractive index is the same in all directions. This is the case with ordinary non-crystal- line substances, so that the ray obeys the ordinary laws of refraction already discussed, and is consequently called the ordinary ray. This deduction from theory, according to which one of the rays in a uniaxal crystal should obey the ordinary laws of refraction, has been completely verified by experiment. Very careful estimates of the refractive indices have been made for waves in all directions, and it is found that the refractive index is absolutely constant, or more strictly, the variations in its measurement never exceeded 0.00002, a variation well within the limits of the probable errors of the experiments that were made. So much for one of the waves within a uniaxal crystal. With the other wave and its corresponding ray the law of refraction is less simple, and as the ordinary law is not obeyed, the ray is called the extraordinary ray. As you see from Fig. 48, the length of the line drawn from the centre to the surface of the spheroid varies with the direction of the line, so that the refractive index varies with the direction of propagation of the wave. It is a simple problem of geometry to compute its value for any CRYSTALS 181 direction, making a known angle 6 with the optic axis. The following table shows a comparison between theory and observation for the refractive index (n) correspond- ing to different directions (6}. It will be seen that the agreement is excellent, the differences being of the order of the probable errors of experiment : n (THEORY) n (EXPERIMENT) 9 n (THEORY) n (EXPERIMENT) 2' 40" 1.66779 1.66780 46 46' 2" 1.56645 1.56653 4 19' 58" 1.66660 1.66663 49 23' 10" 1.55861 1.55876 7 51' 58" 1.66387 1.66385 52 42' 6" 1.54902 1.54914 11 23' 12" 1.65967 1.65978 58 39' 10" 1.53303 1.53312 17 8' 26" 1.64987 1.64996 61 39' 33" 1.52570 1.52573 20 26' 1" 1.64279 1.64287 63 9' 6" 1.52228 1.52241 23 50' 45" 1.63451 1.63455 66 14' 27" 1.51579 1.51571 25 49' 35" 1.62934 1.62930 72 18' 55" 1.50476 1.50475 29 18' 42" 1.61965 1.61974 75 36' 18" 1.50009 1.50005 34 48' 0" 1.60336 1.60336 79 6' 26" 1.49612 1.49610 35 58' 47" 1.59048 1.59058 80 14' 4" 1.49507 1.49507 40 49' 21" 1.58478 1.58487 87 6' 40" 1.49112 1.49114 45 45' 57" 1.57000 1.57014 89 49' 6" 1.49074 1.49074 These results have reference to uniaxal crystals which are perfectly symmetrical about a line. With biaxal crystals there is no such symmetry, and the optical proper- ties are consequently more difficult to deal with. How- ever, the same general principles lead to a complete solu- tion of the problem, although the results are much less simple. The index surface no longer consists of a sphere and a spheroid, but of two sheets that are less familiar in form. Its geometrical properties can be investigated mathematically and the values of the refractive indices for waves in any given direction easily computed. The 182 LIGHT following table, corresponding to that just given for a uniaxal crystal, compares theory and experiment for a number of different directions in a biaxal crystal. In this table r&! is the refractive index corresponding to the inner sheet of the index surface, while n 2 represents the same quantity for the outer sheet: f "i (THEOEY) i (EXPERIMENT) 1 , (THEOEY) 2 (EXPERIMFNT) 1.68103 1.68099 1.68533 1.68526 3 12' 50" 1.67714 1.67721 7 9' 10" 1.68465 1.68454 13 6' 20" 1.66298 .66300 17 2' 40" 1.68445 1.68448 21 4' 30" 1.64607 .64603 25 0' 50" 1.68443 1.68452 28 14' 10" 1.62824 1.62807 32 10' 30" 1.68443 1.68447 35 29' 20" 1.60900 .60897 38 27' 30" 1.68444 1.68453 45 14' 50" 1.58363 1.58365 49 13' 0" 1.68445 1.68457 60 1' 30" 1.55154 .55157 63 59' 30" 1.68447 1.68452 69 37' 40" 1.53784 1.53774 73 35' 50" 1.68448 1.68444 When we wish to estimate the velocities of the waves that can travel through a crystal in any direction, it is convenient, as has been seen, to know something of the form of the Index Surface for the crystal in question. There is, however, another surface which is referred to perhaps even more frequently in discussions of the optical properties of crystals. This surface is known as the Wave Surface, and we must try to realize what is its significance. If you throw a stone into a pool and watch the waves spreading outward, you will have no difficulty in observ- ing the position of the crest of the moving wave at the end of any time, such as a second. As the wave moves out with the same speed in all directions, the crest will form a circle round the original point of disturbance as CRYSTALS 183 center. If, instead of dealing with surface waves, you had waves that spread out in all directions with equal veloci- ties, then it is clear that after a second the crests would all lie on a sphere. This, then, is the wave surface for a uniform medium, the surface that contains the crests of all the waves that have been moving outwards for a given time, such as a second. In a crystal the waves move with different speeds in different directions, so that the wave surface is no longer spherical. Its form can be de- termined from theory, and its geometrical properties dis- cussed as fully as may be desired. As there are two waves in any given direction, the wave surface consists of two sheets, as does the index surface, and, just as with that surface, its form is specially simple for a uniaxal crystal. In that case the wave surface is made up, like the index surface, of a sphere and a spheroid, as shown in Fig. 48, with the difference, however, that (a) is the wave surface for a negative, and (6) for a positive crystal. A knowledge of the form of the wave surface is very helpful when dealing with the optical behavior of a crystal. It enables you, for example, to determine the directions of the rays corresponding to waves in a given direction and to exhibit by a simple geometrical construction the directions of polarization in the two waves. The theory shows that a ray is represented by the line drawn from the center of the wave surface to the point of contact with this surface of a plane which touches it, and is parallel to the front of the advancing wave. I hold in my hand an apple, and will suppose for the sake of illustration that it represents a wave surface. In my other hand I have a sheet of paper, and I shall take this to represent the front of a wave of light moving through the crystal. The direc- 184 LIGHT tion of this wave-front being known, the problem before me is to determine the direction of the corresponding ray of light. Move the sheet of paper parallel to itself until it touches the apple at P; then, according to the theory, if be the center of the apple, corresponding to the center of the wave surface, OP is the direction of the ray. In reality, of course, the wave surface differs very + / g ~~ obviously from the sur- face of an apple. It has pj symmetry about a point 0, its center, and it con- sists of two sheets, so that planes in a given direc- tion (both perpendicular to a given line ON) will touch it at two points, P 1 and P 2 (one on each F IG< 49 sheet), on the same side of the center (Fig. 49). In this figure the curves A 1 P 1 B 1 and ^L 2 P 2 J5 2 represent portions of plane sections of the two "sheets," as they are called, of the wave surface. OP 1 and OP 2 represent the two rays for waves propagated in the direction ON, and it should be understood that the three lines ON, OP 1} and OP 2 are not, in general, in the same plane. If you will return for a moment to the case of this apple, you will see that as I move the sheet of paper in different directions it touches the apple, as a rule, just in one point. However, there is one striking exception to this general rule. Now I hold the paper at right angles to the stem of the apple, and you observe that it touches the apple CRYSTALS 185 not in a single point, but in an infinite number of points encircling the stem. The apple, as has already been re- marked, is very different in form from the wave surface, but the two surfaces have some points of similarity. If you were to make a model of the wave surface, you would find that it has four points that closely resemble that on an apple near the stem (singular points is their technical name), and that a plane that touches the surface in the neighborhood of one of these points touches it not at an isolated point, as is the general rule, but at an infinite number of points forming a circle round the singular point. It should still be true that a line drawn from the center of the wave surface to any one of these points where the plane touches the surface should represent a ray of light corresponding to a wave-front parallel to the plane in question. All the lines drawn thus from the center to the various points of contact will form a cone, so that we should expect that if we could get a wave of light to travel in the right direction in a crystal, we should see not two rays only, as in the ordinary case of double refraction, but a whole cone of rays. That this phenome- non was to be expected was first suggested from theo- retical considerations such as have just been indicated. The theory was developed by Sir W. Hamilton, and, at his instigation, was put to the test of experiment by Lloyd. Knowing what to look for, Lloyd had not much difficulty in observing this phenomenon, and it is now well known under the name of Conical Refraction. With the crystal used by Lloyd, Hamilton's theory indicated that the angle of the cone of rays formed in this way should be 1 55', and Lloyd's measurements made it 1 50', the agree- ment being as close as could be expected in the determina- 186 LIGHT tion of such a quantity. Here, then, we have an example of something whose existence had never been suspected until the theory of light suggested the search for it. Much has been made of this prediction from theory, perhaps too much. We have already seen far more wonderful agreement between theory and observation in other fields of optics, the only peculiarity of this case being that the theory came before the observation and not vice versa. How- ever, it should be remembered that the one aim of the theory is to fit the facts, and it makes little difference to the value of the theory whether the facts happen to have been previously observed or not. This may be largely a matter of accident, and the only advantage that can be claimed for a theory that predicts the unknown is that its power to do so should inspire extra confidence, seeing that the theory cannot have been suggested by this fact that is being explained, as is often the case with "explanations." In a previous lecture we saw how successfully the theory of light can deal with the problem of reflection and refrac- tion at the surface of a non-crystalline medium such as glass or water. It is equally successful in its treatment of crystals. Once the general laws of wave propagation in such media are understood, there is no special diffi- culty in proceeding by means of dynamical principles to calculate the amplitudes and phases, as well as the direc- tions and velocities, of the various waves that may arise. Of course, the mathematical processes are more complex than when we are dealing with non-crystalline substances, but all the difficulties that present themselves have been overcome. Just a few of the results may be referred to here, in so far as they can be tested by experiment. CRYSTALS 187 We have seen that, with a non-crystalline substance, if light be incident at a certain angle, an angle that goes by the name of the polarizing angle, the reflected light has the peculiarity of being plane polarized. The position of this angle for any substance is easily determined from the simple law, due to Brewster, that the tangent of the angle is equal to the refractive index of the substance. In the case of crystals, theory indicates that there will also be a polarizing angle, but that the law from which it may be computed is less simple. With crystals the refractive index is not a constant, but depends on the direction in which the wave is being propagated and the nature of its polarization. We should expect, therefore, that the polarizing angle would depend on these things, and in this theory and observation agree. The following table gives a comparison between theory and observation as to the values of the polarizing angle under different circum- stances of reflection from a uniaxal crystal. The angle is different according as the plane of incidence is parallel or perpendicular to the plane containing the optic axis of the crystal. These two cases are distinguished by sub- scripts; thus, P 1 and P 2 . The angle is the angle that the optic axis makes with the reflecting face of the crystal. The results are shown graphically in Fig. 50 : e .0 25' 27 2' 45 29' 64 1' 30" 89 47' Pi (theory) . . . 54 3' 55 25' 57 25' 59 25' 60 41' PI (experiment) . 54 12' 55 26' 57 22' 59 19' 60 33' P 2 (theory) . . . 58 55' 59 17' 59 48' 60 23' 60 41' P 2 (experiment) . 58 56' 59 4' 59 48' 60 75' 60 33' 188 LIGHT In the case of reflection from a non-crystalline substance we have seen that at the polarizing angle the reflected light is polarized in a plane parallel to the plane of inci- 56 ^Q 10 20 30 40 50 60 70 80 90 FIG. 50 dence. With a crystal, however, this is not the case. The plane of polarization deviates from that of incidence, being inclined to it at a small angle, A, which can be cal- culated from theory. The values of A obtained from theory and experiment were as follows, for the case of reflection from a uniaxal crystal whose optic axis was parallel to the reflecting surface. The angle a denotes the angle between the optic axis and the plane of incidence. a 23 30' 45 67 30' 90 A (theory) .... 2 46' 3 54' 2 46' A (experiment) . . 2 46' 3 57' 2 43' When dealing with ordinary reflection, we made a com- parison between theory and observation as to the differ- ence of phase between two reflected waves which are CRYSTALS 189 polarized respectively parallel and perpendicular to the plane of incidence. The corresponding problem for crystal- line reflection is more complex, but the general character of the results is the same. This will be seen at once by comparing Fig. 51, which shows how the difference of 0.5 0.4 0.3 0.2 I 55 57 c 58 c 59 FIG. 51 62 phase depends upon the angle of incidence in reflection from a crystal, with Fig. 32 of the earlier lecture. In this, as in some other lectures, I have brought before your notice a number of tables and figures that will prob- ably prove attractive or repellent according to the degree in which you realize their significance. Their object in all cases is to show how well, or how ill, the theory fits the facts, and I hope that by this time their cumulative effect will have convinced you that the modern theory of light keeps always very close to the solid ground of fact. Such things are full of interest to a serious scientist, as they 190 LIGHT give him what, above all, he is anxious to have, a search- ing test of his theories; but the optical effects with which they deal do not make a very wide appeal. They would not usually be described as beautiful, and few men, out- side the narrow circle of the physicists, would display much enthusiasm over tables of refractive indices, polariz- ing angles, and the like. It happens, however, that with crystals we can produce effects that are generally recog- nized as extremely beautiful, and that the careful observa- tion of some of these also serves, in a measure, as a test of the accuracy of our theory of the propagation of light in a crystal. You are aware, 'perhaps, that if you make a solution of tartaric acid, pour it over glass, and evaporate the water by means of a steady heat, you may, with proper precautions, get a film of minute crystals of the acid de- posited on the glass. Here is a glass disk upon which is such a deposit. I place it between these two NicoFs prisms, and allow the bright light from the lantern to shine through the apparatus. If you direct your attention to the screen, you will admit, at any rate, that the colors are very gorgeous, and probably that the picture is a beautiful one. Its beauty is enhanced by its irregularity, and this is due to the fact that the little crystals on the glass pre- sent their facets to the light at angles of all sorts. There is thus a total absence of that mathematical precision which is the only objection that can be brought against the claim of beauty made on behalf of the phenomena with which we are to be occupied during the remainder of this lecture. These phenomena are all produced by interposing a thin crystalline plate between two Nicols. The effect of each Nicol is to confine the vibrations to a definite plane, CRYSTALS 191 so that the light that gets through a Nicol must be plane polarized in a direction that depends on the way in which the Nicol is turned. The effect of the plate of crystal is to split up the incident wave of light into two waves, moving forward with different speeds. By the time that these two waves have traversed the crystal, they will have FIG. 52 got out of phase, and if their difference of phase be of the right amount, then, as we saw in the last lecture, they will interfere and in combination produce darkness. If we arrange things so that at different points of the plate the difference of phase between the emerging waves is dif- ferent, then there will be interference at some points and not at others, so that we shall get alternations of light and darkness. It is easy to see that if the incident beam be parallel, we shall have no such alternations, but uniform brightness (or darkness) over the plate. If AB (Fig. 52) represent the front of a wave falling on the plate, this will set up two waves moving with different velocities, and these will emerge from the plate with a difference of phase represented by ef in the figure. If A'B' and A"B" 192 LIGHT be other wave-fronts, the phase differences on emergence will be e'f and e"f f respectively. Now it is obvious that if A'E' be parallel to AB, as will be the case if the incident beam be parallel, then e'f will be equal to ef. Thus the phase difference in the neighborhood of C r will be the same as that at C, and it will be equally bright at these two FIG. 53 points. If, then, we want alternations of light and dark- ness, we must abandon a parallel incident beam. , If we arranged that the incident beam should diverge from or converge to a point, as indicated in Fig. 53, then any two wave-fronts would not be parallel, but would be inclined to one another, as represented by AB and A"B" in Fig. 52. The differences of phase on emergence would be ef and e"f" , and as these are different it might be bright at C and dark at C". We shall suppose that things are arranged to avoid a parallel beam, and that the incident pencil of light is of a diverging or converging character, with OQ for its axis. We shall then examine the simplest case that can be pre- sented, that of perfect symmetry, where we have a uni- CRYSTALS 193 axal crystal, the plate of which is cut at right angles to the optic axis, the direction of this axis coinciding with that of the incident pencil. If we were to look down upon the plate in the direction of the axis, and observe a plan of the mechanical analogue referred to on pp. 101 and 177, then, as there must be perfect symmetry about the axis, the arrangement of the obstacles would be that repre- sented in Fig. 54 a. At any point, P ly the vibrations (a) FIG. 54 must be confined to one or other of two directions, PR and P!$, at right angles to one another. Of these the first, PiR, is along the direction QPi, and the second is at right angles to this. Now suppose that the first Nicol is so placed that it stops all vibrations except those parallel to QN l (Fig. 54 6). Then PjT 7 , which is parallel to QN 1} may represent a displacement in the incident wave as it strikes the crystal plate. Before considering what happens to the 'wave within the crystal, it is convenient to " re- solve" the displacement P 1 7 7 into its equivalent, P^R com- bined with P!$, or what is the same thing, PR combined with RT. It will be seen that RT is equal and parallel to P!$, so that these two lines represent displacements of the same magnitude and in the same direction. [This 194 LIGHT " resolution" of a displacement P^T into two displace- ments, P^R and RT, according to the " triangle law/' is really a very simple matter, it being obvious that the final displacement is the same, whether you go direct from P l to T, or by two stages from P l to R, and then from R to T.] Instead, then, of saying that in the incident wave there is a displacement P{F parallel to QN lf we may say that there are two displacements, one, P-JI, being in the direc- tion QP V and the other, RT, at right angles to this. Dis- placements in the first of these directions are characteristic of one of the waves that the crystal can transmit, while displacements in the other characterize the second wave. These two waves, as we have seen, traverse the crystal with different speeds, and emerge with a difference of phase. What this phase difference is will depend, as ap- pears from Fig. 52, on the angle at which the incident wave strikes the face of the crystal. This will be the same for all points P l that are at the same distance from the axis Q, but will be different at different points along the line QP. Let us suppose that the point PI is so placed that the phase difference is one wave-length for the color under consideration. How will the two waves of light of this color combine after they pass through the second Nicol? That, of course, will depend upon the position of this Nicol. Let us suppose that the two Nicols are " crossed," so that the second Nicol stops all vibrations except those in the direction QN 2 (Fig. 54 b), where* A^QA^ is a right angle. Draw R U (Fig. 55) parallel to QN 2 and therefore perpendicular to P^T or QNi. The displacement represented by P-JH is equivalent to a combination of two displacements, represented in magnitude and direction by P a 7 and UR respectively. The displacement Pj7 is, CRYSTALS 195 however, annulled by the second Nicol, which will not permit a wave to pass unless the displacements therein are parallel to QN 2 . We see then that, while PiR repre- sents the displacement in one of the waves that emerges from the plate of crystal, after this wave has traversed the second Nicol the displacement is changed to UR. The displacement in the other wave, represented byRT, may be dealt with similarly. It is equivalent to two displacements, RU and UT, and of these the second is annulled by the Nicol, so that the displace- ment in this wave as it comes through the Nicol is RU. Now we have supposed that P 1 is so situated that the difference of phase between the two waves is exactly a wave-length, and this/ as far as optical effects are concerned, is the same as if the waves were in the same phase. We have thus to combine two waves that arc in the same phase, the displacements in which are so related that one is represented by UR and the other by the exactly equal and opposite RU. Clearly these two displacements annul one another, so that the color cor- responding to this particular wave-length is totally absent from Pj. As everything is symmetrical round the axis, this absence of color will apply to all points on a circle whose radius is QP l and center is Q, so that there will be a dark circle round the axis. The argument will apply FIG. 55 196 LIGHT FIG. 56 equally well to a point P 2 so chosen that the phase differ- ence is two wave-lengths, or to P 3 , where it is three wave- lengths, and so on. Thus there will be a whole series of circles round the axis, which will be dark as far as the color corresponding to this wave-length is con- cerned. These concentric circles will not, however, be the only dark regions of the field of view. Consider any point P (Fig. 54) on the line QN V The only displacement at such a point that the first Nicol will per- mit to pass must be in the direction QN^ and as this is at right angles to QN 2 , the corresponding wave will not be able to get through the second Nicol. There must, therefore, be com- plete darkness at P, and so for any other point on lines in the directions QN l and QN 2 . Hence, we should expect to see a series of dark circles round the axis, with a black cross whose arms are parallel to the di- rections QN 1 and QN 2 such as is represented in Fig. 56. The difference of phase for the two waves that traverse FIG. 57 CRYSTALS 197 the crystal depends on the velocities of these waves, and so is different for waves of different length and color. Thus, the points P-f^.. will have slightly different positions for the different colors that go to make up white light, and if the incident light be of this character, the rings will be colored, giving the beautiful effect that you now see on the screen. Figure 57 is from a photograph of what actually appears, unfortunately, however, robbed of all the beauty of color. You will observe that the darkest part of the field corresponds exactly with the cross and rings predicted from theory and indicated in Fig. 56. We have been dealing with the case in which the two Nicols are " crossed." Suppose, now, that we turn the second Nicol through a right angle, so that QN 2 of Fig. 54 coincides with QN l} and consider in what way this should modify the results. As before, the displacement repre- sented by P!# is equivalent to a combination of P-JJ and UR (Fig. 55), but of these it is the second that is now annulled by the second Nicol, so that when this wave gets through the apparatus the displacement in it is repre- sented in magnitude and direction by Pfl. In the other wave the displacement RT is again equivalent to RU combined with UT, and the first of these is annulled by the second Nicol, so that UT represents the displacements in the emergent wave. Thus the displacements in the two waves are in the same direction, and being effectively in the same phase, their combined effect is additive, and instead of darkness we have brightness. Thus, where formerly we had a series of dark rings we should now expect a complementary series of bright ones. The cross, too, instead of being black, will be bright. For if we take 198 LIGHT a point such as P on QN^ the first Nicol confines its displacements to the direction QN^ and these pass freely through the second Nicol, and this is true also for a point on a line at right angles to QN r Thus we have brightness all along two lines at right angles, as you see from Fig. 58, which represents the actual state of affairs, except, as before, for the color. In dealing with these phenomena of rings and crosses, I have attempted merely to indicate the general character of the results that are to be ex- pected and that are ac- tually found to occur. With the aid, however, of the theory of wave prop- agation in a crystal, it is not difficult to predict more of the details of the phenomena, such as the size and relative intensity of the rings as well as their form. A very great number of arrangements of the crystalline plate and the Nicols have been examined, both from the theoretical and the experimental point of view, and the agreement between the two is thoroughly satisfactory at all points. We have dealt only with the simplest case that can present itself, that of a uniaxal crystal cut at right angles to its optic axis, with the axis of the incident light in the same direction as the optic axis. The re- sults, of course, are more complex with crystals of a less simple form, and it may suffice to refer very briefly to a few other cases. FIG. 58 CRYSTALS 199 Let us take first the case of two thin plates of the same material and thickness, both cut with their faces parallel to their optic axes, and held together with their axes at right angles to one another. Theory shows that in this case we should again see a dark cross, and that the other dark lines in the field should be a series of rectangular hyperbolas such as are represented in Fig. 59 a. This figure shows the darkest portion of the field according to the theory, while Fig. 59 b is from a photograph of what is really seen. Consider next a thin plate cut from a biaxal crystal, with its faces at right angles to the bisector of the angle between the optic axes. Put this plate between a pair of crossed Nicols, and turn it so that the line joining the ends of the optic axes is parallel or perpendicular to the " principal planes " of the Nicols, i.e. is parallel to lines such as QN 1 or to QN 2 of Fig. 54 6. If a beam of light like that used before now fall upon the plate, we should expect alternations of light and darkness. Investigation shows that a black cross is to be looked for in this case 200 LIGHT as before; but the other dark lines in the field will no longer be circles or hyperbolas. Their form is easily deter- mined from theory, and it appears that they belong to a class of curves known as lemniscates, whose foci are at the ends of the two optic axes. Figure 60 a shows the lines drawn through the darkest part of the field, according to the predictions of theory. Figure 60 b is from a photo- graph of what is actually seen (except, once more, for the CRYSTALS 201 color), and by comparing these two figures you will see that there is an excellent agreement. Lastly, let us turn the crystalline plate through an angle of 45 from its last position and see the change that takes place. The lemniscates should appear, as before, but turned through half a right angle; and there should be no black cross, its place being taken by a dark hyper- bola going through the ends of the optic axes. The darkest part of the field, according to theory, should appear as in Fig. 61 a, and this should be compared with the neighboring figure (61 b) from a photograph of the actual appearance. These various figures give no idea of the beauty due to the scheme of color; but they may serve their purpose of bringing home to you with what accuracy theory enables us to foretell what is to be expected under any given circumstances, and to account for all the de- tails of the phenomena that have been observed. IX DIFFRACTION SUPPOSE you throw a stone into water at S (Fig. 62), and watch what happens. A circular wave will travel out over the surface of the water in all directions. At one time the crest of the advancing wave will be at OP; a little later it will have moved on to EQ. How \E is this effect produced? The impact of the stone causes a disturbance at S, an up-and-down motion of the water there, and this is com- municated to the neigh- boring particles. Each particle hands on the disturbance to its neighbor; but in what way does it hand it on? It looks somewhat as if the disturbance could be passed along only in one direction. P seems to pass on its motion only in the direction Pa, forwards towards Q, and not backwards toward S, or laterally along Pb or PC or Pd. If this were so, it would appear to explain some of the phenomena. Thus, it would explain why there is no disturbance behind OP (in the direction toward 'S), and in the case of waves of light and it is, of course, with the analogous problem in light 202 FIG. 62 DIFFRACTION 203 that we are interested why, if we put a screen in the position OF, we appear to get a sharply defined shadow extending to E, where SOE is a straight line. This familiar phenomenon of shadows gave rise to the idea that light moves in straight lines, and, as we have seen, the law of the rectilineal propagation of light was one of the few general laws of optics that were known to the world in OCA pre-Newtonian days. The only objection to the law is ^ that it is not true. Light does not move in straight lines, and the shadow of an obstacle is not sharply defined by drawing straight lines from the source of light to the edge of the obstacle. Closer examination reveals the fact that light bends round a corner. This phenomenon was some- times spoken of as the inflection of light, but is now always referred to as diffraction. When we look into the question of the amount of bend- ing round a corner, and discuss it by means of the prin- ciples to be referred to later, we find that the bending depends very largely on the length of the wave, short waves being much less bent than long ones. This dependence of the bending on the wave-length is so important that it may be well to make an experiment to bring it home to you. In ordinary speaking you set up waves in the air, and you know that these must bend freely, as you can easily hear a person who is speaking round a corner. The length of the waves that are thus set up by speech de- pends on the pitch of the voice, but we may say that normally they are four or five feet long. In my hand I have a whistle that will produce very much shorter waves than that. As it is now arranged it sends out waves about four inches long, and by altering its mechanism I can make the waves shorter and shorter. If one of you 204 LIGHT were to go behind that large screen, and listen carefully while I sound the whistle, you might be able to determine whether there is anything resembling a sound shadow or not. It will be better, however, to arrange things so that all can observe the phenomena together. We can do this easily by aid of this sensitive flame that you can all see, and that will serve just as well as an ear (indeed, better than that in some respects) to detect the presence of a wave in the air. The manner of producing this sensitive flame need not concern us at present, all that need be known being that it is sensitive you observe that the flame ducks when I blow this whistle and the disturbance in the air strikes it. For our purposes this flame has the im- portant advantage over the ear that it can be made sensi- tive to waves that are too short to produce the sensation of sound. It has already been pointed out that the eye is sensitive only to waves whose lengths lie within a cer- tain range, and the same is true of the ear. Very high notes and very low notes cannot be heard, as they do not affect the ear in the right way, and it should be remem- bered when watching this experiment with the whistle and the flame that very high notes are very short in wave length. As I alter the effective length of this whistle, you can hear that the note it emits gets shriller, and now that the wave is so short that you hear no note at all, there is, as you see, a somewhat sharply defined shadow of the screen. You observe that when I move the whistle very slightly to the right or to the left of a line joining the sensitive flame to the edge of the screen, there is a perceptible difference in the effect. In one case the flame is inside the shadow, and is unaffected of the waves in the air; in the other it responds to the action of these waves. You see, then, DIFFRACTION 205 that although an ordinary sound-wave bends readily round a corner, there is scarcely any bending perceptible when the length of the wave is sufficiently short. This problem of the bending of light round a corner has presented difficulties almost from the beginning of modern science. Newton knew some of the phenomena quite well, but he did not observe them closely enough to grasp all that was significant, and his failure to do so led him seriously astray. He knew that the shadows of bodies are bordered with colored fringes. He knew also that if the light from a small source falls upon a body, the shadow is not exactly coincident with the geometrical shadow, as it is sometimes called, i.e. the figure formed by drawing straight lines from the source of light past the edge of the opaque body and observing where these lines are interrupted by the plane on which the shadow is cast. Thus, in his first observation on "The Inflexions of the Rays of Light," in the third book of his " Opticks," he tells us that he let light stream through a pinhole in a piece of lead and fall upon various objects, and he then observed that "the shadows were considerably broader than they ought to be, if the rays of light passed on by these bodies in right (i.e. straight) lines. And particularly a hair of a man's head, whose breadth was but the 280th part of an inch, being held in this light, at the distance of about 12 feet from the hole, did cast a shadow which at the distance of 4 inches from the hair was the 60th part of an inch broad, that is, about four times broader than the hair." In this case there seems to be a bending away from the shadow and not into it. This puzzled Newton, and seemed to him so in- compatible with a wave theory of light that he rejected that theory. Listen to what he says in one of his famous 206 LIGHT queries at the end of his book on optics. "Are not all hypotheses erroneous in which light is supposed to con- sist in motion propagated through a fluid medium ? If it consisted in such motion, it would bend into the shadow. For motion cannot be propagated in a fluid in right lines beyond an obstacle which stops part of the motion, but will bend and spread every way into the quiescent medium which lies beyond the obstacle. The waves on the surface of stagnating water passing by the sides of a broad obstacle which stops part of them, bend afterwards, and dilate themselves gradually into the quiet water behind the obstacle. The waves of the air, wherein sounds consist, bend manifestly, though not so much as the waves of water. But light is never known to follow crooked pas- sages nor to bend into the shadow. For the fixed stars, by the interposition of any of the planets, cease to be seen. And so do the parts of the Sun by the interposition of the Moon, Mercury, or Venus. The rays which pass very near to the edges of any body are bent by the action of the body ; but this bending is not towards but from the shadow, and is performed only in the passage of the ray by the body, and at a very small distance from it. So soon as the ray is past the body it goes right on/ 7 Had Newton varied his experiments, and observed carefully enough, he could have found a bending towards the shadow, as we shall see later. Here, then, we have a striking case of a very great scientist being led astray, and, as we now see it, very seriously astray, by experimental evidence. Snares seem to be laid along every path, and we may be en- trapped by experiment just as well as by theory. There are so many warnings up along the latter road that there is not the same excuse for falling. And yet men fall, as DIFFRACTION 207 Brewster did not so very long ago, if Tyndall reports him fairly. " In one of my latest conversations with Sir David Brewster, he said that his chief objection to the wave theory of light was that he could not think the Creator guilty of so clumsy a contrivance as the filling of space with ether in order to produce light." Such a high a priori road is probably the most dangerous of all. To return to the problem of diffraction, there is by this time not the slightest doubt that light does bend round a corner. As we shall see presently, we have many care- ful determinations of the amount of bending and of vari- ous details of the phenomena. With the refinements of modern instruments at our disposal, it is comparatively easy to deal with these matters experimentally ; but when we come to examine them from the standpoint of theory, a number of difficulties arise. The form that the problem takes in the mind of a mathematical physicist is something as follows. A given disturbance is set up in the ether by the presence of a source of light. This spreads out in a known manner, and there is no special difficulty in calcu- lating and predicting all the details of the phenomena to be observed, provided that no opaque obstacle is present. Suppose, however, an opaque body is put in the way of the waves in the ether. How does this affect the motion of the waves in the region beyond the body ? The physical and mathematical conditions to be satisfied are easily stated. We know the disturbance in the neighborhood of the source, and we know the conditions to be satisfied at all points of the boundary of the opaque obstacle. It looks as if everything that we want should be within our powers of computation, and in other fields many similar problems have been successfully attacked. In the case of 208 LIGHT optics, however, peculiar difficulties present themselves owing to the extreme shortness of the waves of light, and these difficulties have not as yet been successfully over- come, except in a few very special cases. In general, the complete solution of the optical problem of diffraction still awaits us. I trust that you do not misunderstand me here. It is not the case that there is any special difficulty with the general theory, nor any apparent discrepancy be- tween theory and observation. The difficulty that I speak of is purely one of mathematical analysis, and arises en- tirely from the limitations of our skill in that branch of art. Doubtless it will be overcome in time. Meanwhile we are constantly reminded that " Nature is not embarrassed by difficulties of analysis/' and that, in our interpretation of Nature, we must not allow such difficulties to embarrass us unduly. Thus, in the present case, although a rigorous mathematical solution is as yet unattainable in general, it may be possible to get an approximately accurate solution which is good enough for practical purposes. As a matter of fact, this has already been done, and the results are found to be as accurate as we need in the present state of our experimental skill. Thus the difficulties of mathematical analysis to which reference has been made may very properly be handed over to our successors, whose finer instruments and more accurate observations may demand a correspondingly re- fined analysis. The method that is generally adopted in dealing with such problems to-day is to make use of what is known as the Principle of Huyghens. Let us look once more at Fig. 62, with which we dealt at the outset of this lecture. A disturbance was set up at the point S, and from this point waves traveled out in all directions. If this DIFFRACTION 209 be true of the point S, we should expect it to be true for any other point that is disturbed; whether the initial dis- turbance be set up by a stone or some other agent can make no difference, and there is nothing peculiar to S, except that it happened to be the point that was disturbed first. Hence, any other point, such as P, must be re- garded as a center of disturbance from which waves pro- ceed in all directions. The Principle of Huyghens merely states that each point of the front of an advancing wave may be regarded as a center from which secondary waves spread out, not in one direction, such as Pa, but in all directions. What will be the effect of the combination of all the secondary waves thus set out is a question to be answered by the help of the Principle of Interference, which makes it clear that the effect will depend on the amplitudes and phases of the various secondary waves that have to be considered. To determine exactly what is the law governing these features of the secondary waves is a difficult problem. It was attacked by Stokes in a famous memoir "On the Dynamical Theory of Diffraction." In this the problem was to determine what must be the amplitudes and phases of the secondary waves, so that in combination these waves would give the actual disturb- ance in front of the advancing wave and no disturbance at all behind it. Interesting and instructive as was Stokes' s discussion of this problem, his solution has not escaped criticism. Amongst other things it has been pointed out that the problem is really an indeterminate one. The question asked is one that can have several answers, like the question, What two integers, when added together, make 6 ? and there is nothing to determine which of the answers is to be preferred. Various laws have been sug- 210 LIGHT gested other than the one that Stokes arrived at, and it should be noted that, while differing in other respects, they agree as to the disturbance produced by the secondary waves in the only region where these waves are really effective, i.e. in the neighborhood of the direction PQ. The waves that travel in all other directions have their influence neutralized through interference with other waves. If, then, we wish to estimate the effect of all the secondary waves that pass over Q, it appears that we need consider those only that set out from the wave-front OP in the neighborhood of P. It is for this reason that PQ is sometimes spoken of as the path of the effective disturb- ance that passes from P to Q, and this effective disturbance constitutes the ray of light. Let us suppose, now, that a screen OF is interposed so as to interfere with the advance of the waves in the ether. How will this affect the propagation of the waves and the optical phenomena in the region beyond the screen? This is a hard question to answer, owing, as has been ex- plained, to the mathematical difficulties that arise in its discussion. These difficulties have been completely over- come only in one or two special cases, but an approxi- mate solution has been reached in many others. To ob- tain this, an assumption is made that is certainly not justified if we insist on absolute rigor and exactness throughout. Such a lofty attitude, however, makes prog- ress impossible, and as practical men we prefer to make some advance, even by means of an unjustifiable assump- tion, provided we have reason to suppose that this assump- tion will not lead us too far astray. The assumption made is that the effect of the screen is merely to destroy the secondary waves that, but for its presence, would be DIFFRACTION 211 propagated from the various points of its surface, while all the other secondary waves from points not on the screen go forward just as if the screen were away. It is easy to see that this cannot be quite strictly true. Con- sider the simple case of a stream of water flowing in a closed space between two horizontal boards represented in section by AB and CD in Fig. 63 a. Each particle would move horizontally, along lines such as the dotted ones D (a) (6) FIG. 63 of the figure. Now put in an obstacle, such as OF in Fig. 63 b. This would do more than merely stop the onward rush of the drops of water that struck the obstacle. It would affect the motion in the neighborhood of F, and the motion below and to the right of that point in Fig. 63 b would not be just the same as at the corresponding point of Fig. 63 a. In the case of waves it appears, however, on investigation, that the error introduced by this assumption is inappreciable except within a few wave- lengths of the edge of the obstacle. We shall see before the close of this lecture that there are something like 50,000 wave-lengths of light to the inch, and owing to this extreme shortness, the region of error due to the false assumption is so small as to be practically negligible. Proceeding, then, with this assumption, we are able to com- pute all the essential details of the optical phenomena in a large number of interesting cases, and in many of these to test our (admittedly imperfect) theory by comparison 212 LIGHT with the most careful measurements that are available. This was first done by Fresnel in a classical memoir on Diffraction that was crowned by the French Academy in 1819. Fresnel considered the case of light falling on an opaque screen with a straight edge. If the light were propagated strictly in straight lines, there would be a sharply defined shadow, coinciding with the geometrical shadow, the contour of which is determined by drawing FIG. 64 straight lines from the source of light to the edge of the screen. Inside this shadow there would be absolute dark- ness, and outside it uniform brightness. The curve of in- tensity would then be the dotted curve of Fig. 64. In this represents the position of the edge of the geometrical shadow, the shadow being to the left of 0. OP represents the intensity of the incident light, as well as that in the bright part of the field at some distance from the edge of the shadow. Fresnel showed that the theory that has just been sketched would lead us to expect a distribution of light that is indicated by the continuous line of the figure (abode...). It will be observed that there is no longer complete darkness to the left of 0, but that the light fades away rapidly as we go into the shadow. Perhaps the most striking result of the investigation is that outside the shadow (to the right of in the figure) the intensity of DIFFRACTION 213 the light is not uniform, but that there are a series of bright bands where the intensity is much greater than the average (c, e, #...), alternating with bands where it is much less (d, /, h...). The theory indicates, and this is fully confirmed by experience, that the exact distribution of light depends on the wave-length of the incident beam. Fresnel calculated the details for red light of wave-length 0.000638 millimeters or 0.000025 inches. The intensity of the light at the brightest bands, corresponding to c, e, g... of Fig. 64, when expressed in percentage of the intensity of the incident light, was found to be 137, 120, 115, 113, 111, 110, 109..., and that at the intermediate darks bands (d, f, h...) to be 78, 84, 87, 89, 90, 91, 92... Owing to the difficulty of making very accurate measure- ments of intensity, it was not easy to make a searching test of the theory by comparing these results with those derived from experiment. There is, however, another fea- ture that is more easily measured with accuracy, and that is the distances of the different fringes from the edge of the geometrical shadow. The following table gives the posi- tions, obtained from theory and also from observation, of the first five dark bands for the red light used by Fresnel. The results are set out for three different distances (d) of ct=100 d = 1011 d=6007 Theory Observation Theory Observation Theory Observation First Fringe . . 2.83 2.84 2.59 2.59 1.14 1.13 Second Fringe . 4.14 4.14 3.79 3.79 1.67 1.67 Third Fringe. . 5.13 5.14 4.69 4.68 2.07 2.06 Fourth Fringe . 5.96 5.96 5.45 5.45 2.40 2.40 Fifth Fringe . . 6.68 6.68 6.11 6.10 2.69 2.69 214 LIGHT the screen from the source of light. All the distances are given in millimeters, and you will remember that one milli- meter is equal to 0.03937 inches. It will be seen that the agreement between theory and observation is excellent. The position of the bright bands depends upon the wave- length, and so on the color of the light. If, then, we em- ploy a mixture of colors such as constitute white light, we shall get a series of colored fringes in slightly different posi- tions. These will tend to overlap one another, and the overlapping will make it difficult to distinguish the outer bands of color. Hence, for accurate measurements designed to test any theory, it is expedient to use homogeneous light, and so have only a single wave-length to deal with. The same general method will enable us to discuss the phenomena to be looked for in various other circum- stances. Thus, instead of dealing with a single straight edge, we may wish to examine the effect of two parallel edges close together constituting a narrow slit. The simplest apparatus to use for such experimental purposes is your own hand. Hold two fingers together so that they are very nearly closed, and look through the narrow opening at a distant bright object. You will see a number of colored fringes, but if you wish to investigate the phe- nomena carefully, it will be better to take a little more trouble and proceed as follows. Cut a slit about an eighth of an inch wide in a black card and fix this in front of a bright light. Look at this slit through the narrower slit made by drawing with the point of a needle a straight line on a piece of blackened glass, and hold the two slits parallel to one another. You will at once observe a series of colored spectra. If you make the light homogeneous by interposing, say, a red glass between the light and the DIFFRACTION 215 first slit, you will see a series of bright red bands, R 1 R%... (Fig. 65), on each side of the central image R, and you will notice that their intensity diminishes as you get further away from the central band. On replacing the red by blue, you will observe a similar effect, but the bright blue bands will be narrower and closer together than were the red, as is indicated roughly in the figure by the posi- tions of the rectangles B, B v .. You see from this that if both colors are present together, the different bands will overlap, and you will understand the various spectra that R ^ PI p R R 3 *i R *, R 2 FIG. 65 are seen when white light is employed. By fixing a nar- row slit on the end of your opera glasses, you can readily see these spectra and examine their details at your leisure. Closely allied to the case of a narrow slit is that of a narrow obstacle placed in the path of a beam of light. Reference has already been made to Newton's experiment with a human hair, which exhibits the phenomena. You can easily see this for yourselves by partially closing your eyes and looking at a bright light through your eyelashes. A fine wire is now placed in front of the lantern, and you observe that the dark shadow on the screen is bordered with colored fringes, and now that a mesh of wires is sub- stituted for the single wire, you see that the color effects are quite gorgeous. The effects to be expected from apertures and obstacles of various forms have been care- fully examined both theoretically and experimentally, and 216 LIGHT the agreement between theory and observation is, on all points, most satisfactory. We have time only to select a single example, that of a circular aperture (and the cor- responding case of a circular obstacle). This is especially important, owing to the fact that most optical instruments (telescopes, microscopes, and the like) are arranged so that the light passes into them through a circular aperture. The mathematical analysis of the case is long and some- what complex, but the fundamental principles employed are the same as those that have already been explained. The investigation shows that where light shines through a circular aperture upon a screen, the screen is not uni- formly illuminated, but that there are marked variations in the intensity at different portions of the circular patch of light. The points where the brightness is least con- stitute a series of concentric dark rings whose radii can be determined from theory, and, of course, observed ex- perimentally. The sizes of these rings depend on the color of the light, so that when white light is employed, the screen exhibits a series of colored rings. Lommel made careful determinations of the radii of these rings for various colors, and compared his observations with the deductions from theory. The results for the first four dark rings are set out in the table below for a few cases, but Lommel dealt with over 180 such cases, and in all of these the agreement between theory and observation was as good as in those here selected. The different colors used were red, orange, green, and blue, corresponding to the spectral lines known as C, D, E, and F and to wave- lengths of 0.0006562, 0.0005889, 0.0005269, and 0.0004861 millimeters. The radius of the aperture was 0.28, the distance of the edge of the aperture from the source of DIFFRACTION 217 light was 2120, and of the source of light from the screen, 2210.4. These numbers and those in the table all repre- sent millimeters. BLUB GREEN ORANGE RED Theory Observation Theory Observation Theory Observation Theory Observation 0.015 0.032 0.034 0.057 0.056 0.082 0.082 0.156 0.158 0.179 0.175 0.308 0.305 0.237 0.231 0.254 0.254 0.275 0.276 0.403 0.406 0.343 0.338 0.333 0.333 0.361 0.361 0.449 0.451 Theory also enables us to calculate the intensity of the light at different positions on the screen. It thus appears that about 84 per cent of the whole light is inside the first dark ring. The central spot is not, however, of uniform brightness, but shades off as we proceed outwards from the center, the intensity halfway between the center and the first dark ring being about 37 per cent of that at the center. Perhaps the most important result to bear in FIG. 66 mind is that the image of a point is not a point, but a complicated system of rings of the kind indicated roughly in Fig. 66 a. Figure 66 b represents the image of two points close together, and shows how the two images tend 218 LIGHT to overlap and produce a blurred effect. Fortunately, most of the light is confined within the first ring, so that we do not go far wrong in supposing that the image of a point as seen through a telescope is a small disk. Do what we will, however, we cannot make this disk shrink to a point, and when we take a photograph we set the light the diffi- cult problem of drawing a clear picture with a blunt pencil. The bluntness of the pencil depends upon the diameter of the little disk of light, and to sharpen it as much as possible we must increase the size of the aper- ture and use light of the shortest wave-length that can be employed. Fortunately, for photography, the short waves have great actinic power; but the other require- ment, that of a large aperture, is not so easily satisfied and, as it involves great size, adds seriously to the cost of the best telescopes used for astronomical purposes. The corresponding problem presented by the shadow of an opaque disk was also solved by Lommel. Here, too, we have a series of alternations of light and darkness, giving the appearance of a number of concentric rings with their center at the center of the shadow. The table below gives the radii of the first four dark rings in a few cases; but Lommel dealt with over sixty cases in all, and found the same good agreement between theory and observation. BLUE GREEN ORANGE BED Theory Observation Theory Observation Theory Observation Theory Observation 0.088 0.090 0.096 0.096 0.109 0.113 0.119 0.124 0.200 0.197 0.219 0.220 0.242 0.237 0.268 0.265 0.307 0.310 0.335 0.333 0.369 0.367 0.403 0.400 0.406 0.406 0.438 0.440 0.478 0.479 0.525 0.525 * OF THE UNIV DIFFRACTION 219 i^C> Here the radius of the disk was 0.32, and the distance of its edge from the source of light was 1485. Observations were made with the four colors previously mentioned, blue, green, orange, and red, the distance of the screen from the source of light being 1639.9, 1642.6, 1643.3, and 1643.2 for the different colors. All the numbers represent millimeters. The different radii for the different colors give an idea of the amount of overlapping when white light is employed, and of the arrangement of the colors in the fringes. In this case also theory enables us to calcu- late the intensity of the light at different points in the shadow. In this connection one feature may be pointed out, as it is probably unexpected. It appears from the investigation that at the very center of the shadow there should be a bright spot, and that this should be just as bright as if there were no disk present to cut off the light. This deduction seemed so absurd when it was first an- nounced that it was regarded as a serious objection to the wave theory. A little care, however, in experiment showed nevertheless that, however unexpected or seemingly im- possible, it was none the less perfectly true. If you have the resources of a physical laboratory at your disposal, you will find no great difficulty in trying this for yourself. You will need a carefully made circular disk that is not too large, and you will need to make the necessary adjust- ments with some precision. I will modify the experiment so as to exhibit the result to the whole audience and deal with sound-waves rather than with light, so as to work on a larger scale. The mathematical analysis is very similar in the two cases, but the sound-waves have the advantage of being much longer, so that we do not need the same refinement. Introducing you once more to this 220 LIGHT whistle and sensitive flame, I fix a circular disk of glass, about a foot in diameter, between the two. By moving the whistle into different places, you observe that there is a marked sound shadow behind the disk; but now that, after some adjustment, I have got the whistle so that it is exactly opposite the center of the disk, you see that the flame ducks, and by doing so indicates the presence of a considerable disturbance in the air. The applications of the theory of diffraction to the construction of optical instruments and to the explanation of various optical phenomena are so numerous that it would be impossible in the time at our disposal to deal with them at all adequately. In the short time that re- mains to me for this lecture I shall endeavor to explain very briefly how it is that the principles of diffraction enable us to measure the lengths of different waves of light and to measure them with wonderful accuracy. Several methods may be employed for this end, but I shall confine myself to what is the simplest for purposes of exposition. This measures the wave-lengths by the aid of a diffraction grating, an extremely simple instrument as far as its appearance is concerned. It is made by rul- ing a great number of very fine parallel lines on speculum metal or glass. The former is viewed by reflection, as the metal reflects a large proportion of the incident light, and is called a reflection grating. The glass reflects some light, and usually transmits more. If viewed by transmission, i.e. if the incident light be allowed to stream through the grating and the transmitted beam be then examined, the arrangement is described as a transmission grating. In either of these cases the effect of the grooves made by ruling is to scatter irregularly the light that falls on them, DIFFRACTION 221 so that the grooves behave as if they were opaque and destroyed the light that strikes them. Let us consider light falling normally on a transmission grating, a cross- section of the surface of which is represented in Fig. 67. The thick lines in this figure, such as x^, z 2 a 3 ..., show the positions of the grooves, which, as we have just seen, FIG. 67 practically stop all the light that falls on them. Waves enter the instrument through the portions a-^x^ a%x 2 ..., and if we consider a^ as the front of an entering wave, then every point on this front is to be regarded (as was stated earlier) as a center of disturbance from which waves, and therefore rays, spread out in all directions. If the front of the incident wave were complete, that is, if there were no obstructing grooves, the waves that spread out laterally in any such direction as a^ would be nullified by inter- ference with the waves that proceed from other portions of the wave-front, and it would only be directly in front of ! that the effective disturbance would be appreciable. The grooves, however, cut out some of the waves that 222 LIGHT would contribute to this interference. This must modify the results, so that it may well be that there is an ap- preciable disturbance in some such direction as a^. To investigate this matter more fully, we must bear in mind the fundamental idea that lies at the root of the Principle of Interference; namely, that two waves that are similar in all other respects, but that differ in phase by half a wave-length, or any odd multiple thereof, will interfere and produce darkness, while if they differ in phase by a whole wave-length, or any multiple thereof, they reinforce one another and give greater intensity of light. Let us consider all the secondary waves that travel outward from the various points of the incident wave- front in a given direction, such as a^ (to which a z n 2 and a 3 n 3 in the figure are parallel). The difference of phase between the waves from a l and a 2 is represented by a^', where a^ is perpendicular to a^n^ This will also be equal to the difference of phase between the waves from &! and & 2 , provided a 2 6 2 be equal to ajb^ Now if the grooves be of exactly the same width, and the spaces between them be equal, it will be possible to divide all the spaces into the same number of equal parts, so that a 1 & 1 = 6 1 c 1 = ... = a 2 6 2 = & 2 c 2 = ...=a 3 & 3 = & 3 c 3 = .... We shall also have a 1 n 1 = a 2 n 2 = a 3 n 3 = ..., and the difference of phase between the secondary waves from a l and a 2 will be the same as between those from &! and 6 2 or from c l and c 2 , or from a 2 and a 3 or from 6 2 and b 3 , and so on. Let us suppose, further, that the incident light is homogeneous, i.e. all of the same wave-length, and that a^i is half this length. Then if all the secondary waves could be brought together without relative change of phase, the wave from a 1 would interfere with that from a 2 , the wave from 6j would interfere with that from 6 2; and DIFFRACTION 223 so on, thus producing darkness in the direction The combination of the different secondary waves is simply effected by means of a lens, which bends the rays so as to bring them to a focus, and alters the direction of the wave-motion without changing the rela- tive phase of the different waves. Let, then, OA in Fig. 68 represent one of the lines in the grating, and OB a line drawn at right angles to the plane of the grating to meet a screen, on which the light falls, at B. If OD 1 be drawn in the di- rection represented by a^ in Fig. 67, and DjZy be drawn on the screen parallel to OA, then from what has been said it will be seen that Di/Y will coincide with a dark line on the screen. This, however, will not be the only dark line, for the same interference will take place when a-pii is equal to any odd multiple of half a wave-length as when it is simply half a wave-length. If OZ) 2 , OZ) 8 ,... be the directions corresponding to phase difference of three half wave-lengths, five half wave-lengths, and so on, then there will be dark lines D 2 D 2 ', D 3 D 3 ',..., all parallel to OA. We have dealt with the case where a^ is half a wave-length or any odd multiple thereof. Let us sup- pose next that a^ is a wave-length, or any exact number of wave-lengths. Then the various secondary waves, in- stead of interfering, will reinforce one another, and the corresponding portions of the field will be unusually bright. We shall thus have a series of bright lines, such 224 LIGHT as #!#/, # 2 ZY>--> m Fig. 68. On comparing the triangle a^aj of Fig. 67 with OBB l of Fig. 68, it is seen that these triangles have equal angles. The angle a^n^ is equal to the angle OBB l} as each is a right angle, and the angle n^aj is equal to the angle BOB V since OB is perpendicular to tt^, and OB l is parallel to a^n^ and therefore perpendicu- lar to n^. As the two triangles have equal angles, it follows geometrically that they must be similar triangles differing only in scale. Hence the ratio of a^ to a x a 2 must be equal to the ratio of BB l to OB lt or, in algebraic symbols, ^1 = - 1. Thus we have a^ = a^ x 1. Now . On 1 can be measured accurately by counting the number of grooves in a given distance. Thus, if on the grating there are twenty thousand lines to the inch (there are more than this on many good gratings), then a^ is one twenty-thousandth of an inch. The distances BB 1 and OB 1 might be measured directly, but it is only their ratio that is wanted, and this can be determined most simply and accurately by the aid of trigonometry, once the angle BOB l has been measured. The measurement of this angle can be made with great precision and then, from T>T> the equation ohtti = a^ x J = a^ sin BOB l} the quantity is readily calculated. It has been indicated, however, that this quantity a^ is the wave-length of the light with which we are dealing, so that the problem of deter- mining the wave-length has been solved. If the process thus sketched be carried out carefully with a good grating, the wave-lengths may be determined with marvellous accuracy. There are several ways of testing the results. Thus, if we deal with the bright line DIFFRACTION 225 B^BI, the calculated value of a^ should be the wave- length. If we make similar measurements with B^B^, then the corresponding value of a^ should be twice the wave-length; with B B B B ' it should be thrice this length, and so on. The consistency of the various estimates of the wave-length thus obtained will enable us to form an estimate of the accuracy of our results. Then, too, we need not confine our attention to the case of light that strikes the grating at right angles to its surface. This case has been dealt with and illustrated in order to simplify the mathematical discussion as much as possible; but it requires a very slight effort to extend the argument to the more general case of oblique incidence and to obtain a corresponding formula for the wave-length. By making observations at various angles of incidence and computing the wave-length, we have other means of testing the con- sistency and accuracy of our results, and when all pre- cautions are taken, it is found that these results are marvellously concordant. For this end, of course, a good grating is indispensable, and a good grating is an instru- ment that requires great care and skill in the making. The rulings must be made with almost perfect accuracy, for the argument supposes that the distance between the various grooves and their width is uniform throughout, and if this be not the case errors will inevitably creep in. There are other methods of measuring wave-lengths than the one here described, but time will not permit us to discuss them. Suffice it to say that few things can now be measured with such wonderful precision as the length of a wave of light. Such is the accuracy that has been attained, that it has been seriously proposed that the length of a wave of light emitted under certain condi- Q 226 LIGHT tions from a specified substance should be adopted as the standard of length. This standard would have some ad- vantages over any that are now in use, for all these are subject to slow and uncertain changes, and the one thing to be required of a standard above all else is that it should not change. The length of a wave of light emitted by a substance depends on properties of the ether and of the atom that, there is reason to believe, are invariable, so that this length seems capable of serving as a true standard. With this end in view Michelson devoted himself for some time to the problem of determining the length of the standard meter in wave-lengths. For this purpose he em- ployed certain radiations from cadmium, which were chosen on account of their simple character. He found that the number of light-waves in the standard meter in air at 15 C and normal pressure was 1,553,163.5 for the red waves from cadmium, 1,966,249.7 for the green, and 2,083,372.1 for the blue, and that the measurements could be made so accurately that he could safely say that the errors were less than one part in two millions. The following table gives some details with reference to wave-lengths and frequencies of the waves corresponding to different parts of the spectrum. The letters ABC... are the names by which these lines in the solar spec- trum are known, and an indication of their color is given. The wave-lengths are expressed in millionths of a meter, and the frequencies in million millions per second. DIFFRACTION 227 LINE OF SPECTRUM WAVE-LENGTH IN MlLLIONTHS OF A METER NUMBER OF WAVES TO THE INCH FREQUENCY IN MILLION MILLIONS PER SECOND A 0.75941 33,447 395 B 0.68G75 36,986 437 C (red) 0.65630 38,702 457 D (orange) 0.58930 43,102 509 E (green) 0.52697 48,200 569 F (blue) 0.48615 52,247 617 G (violet) 0.43080 58,960 696 H 0.39715 63,956 755 Before closing this lecture there is one feature of the phenomena observed when using a grating that must not be overlooked. We have seen that the position of the bright lines, such as B-JZi, depends upon the length of the wave employed. It follows that, if the incident light be white, the bright lines corresponding to its various colored constituents will have different positions, so that instead of a single bright white line at BiBi, there will be a whole series of such lines forming a continuous spectrum, with all the colors of the rainbow, in the neighborhood of #!#/. There will be a similar spectrum near J2 2 -B 2 ', an d so for the other lines, and to distinguish these various spectra from one another they are spoken of as spectra of the first order, second order, or third order, and so on, as the case may be. It was mentioned in the lecture on Spectroscopy that a prism was not the only means of producing dispersion and obtaining a spectrum, and we see now how this can be done by means of a diffraction grating. The spectrum produced by a grating has one great advantage over that formed by a prism in that the distances of the various colored lines from a certain fixed 228 LIGHT line are proportional to the wave-lengths, as the above investigation shows. There are other advantages that cannot now be discussed, but we may say that for many purposes of accurate measurement, where a spectrum is involved, it is better to produce this spectrum by diffrac- tion rather than by means of a prism. X I LIGHT AND ELECTRICITY SCIENCE has a vaulting ambition. It views the whole field of human knowledge and strives to possess it all. It sets about this tremendous task, however, in a business- like way and recognizes clearly that, for practical effec- tiveness, the beginning of wisdom is limitation. To at- tempt too much is to court failure, and, to avoid this, barriers have been placed across the field of knowledge, and individuals are advised to work strenuously within a little fenced-in portion of the whole field. It is well, how- ever, occasionally to reflect that all the fences are artificial, and that they have been put up for practical purposes and for reasons that may not appeal to the more mature judg- ment of later generations. It is natural and convenient to fence off from one another things that seem to have little or nothing in common, but a deeper insight may reveal the fact that there is the closest relationship be- tween what are apparently quite different things. The great divisions of natural science into Physics, Chemistry, and Biology are proving, after all, to be entirely artificial, and barriers between them are being broken down almost daily. And if this be true of the great divisions, it is true even more obviously of the subdivisions. In some cases it is difficult to find any traces to-day of barriers that in earlier ages seemed natural and inevitable. Thus, in one field you had to deal with what affects the ear and goes by the name of sound; in another your problem was to 229 230 LIGHT discuss the mechanical properties of gases and the laws of motion within them. In the field of sound you learned to distinguish one sound from another by its intensity, by its pitch, and by its quality, and in process of time you established various laws of sound that enabled you to foretell the intensity, pitch, and quality of sounds emitted under various conditions. In the other field you found that waves could be set up in gases, and that these could be distinguished by their amplitude, by their frequency, and by their form, and you learned, by the aid of me*- chanical principles, to calculate the amplitude, frequency, and form of the waves set up in given circumstances. In time it seemed expedient to knock down the fence be- tween the fields, for by postulating a relation between the intensity and the amplitude, the pitch and the frequency, the quality and the form, it was possible to explain all the peculiarities of sound on mechanical principles, and to test the theory by experiment in the most rigorous fashion that could be demanded. Thus to-day the problem of sound is regarded as a small part of the wider subject of vibrations (in air and other media), the vibrations being confined to narrow limits determined by the structure of the ear. No two physical sciences seem, at first sight, more widely separated than light and electricity. My aim in this lecture is to show that they are in reality most in- timately related. To this end let me begin by reminding you of the broadest outlines of the theory of light. We have seen that in order to coordinate the great mass of phenomena that have been observed in the field of optics, it is necessary to postulate the existence of a medium that we call the ether, and that we endow with definite LIGHT AND ELECTRICITY 231 and peculiar properties. This ether is capable of trans- mitting disturbances by means of waves that travel through it with a speed that is determined by the properties of the ether, but that have frequencies depending entirely on the source of the disturbance. If the frequency be within cer- tain limits that are determined not by the source of the dis- turbance but by the structure of the eye, the waves will produce the sensation of light. If, however, the frequency be higher than the limit set by the eye, then no light is seen; but the waves may show their presence in other ways, e.g. by their influence on a sensitive photographic plate. On the other hand, if the frequency be somewhat lower than this limit, the waves will produce the sensation of radiant heat, and if it be very much lower, they will give rise to electrical phenomena. Thus, from this point of view, the distinction between photographic action, light, radiant heat, and electricity is mainly a question of frequency, and light is seen to be only a small portion of the problem presented by the propagation of waves in the ether. As a matter of history the science of electricity was built up quite independently of that of light. It soon appeared that to account satisfactorily for electrical phe- nomena it was necessary to postulate the existence of an ethereal medium, and in the process of time it became evident that exactly the same ether, with just the same peculiar properties, was required for electricity as for light. The idea of some such medium is a very old one in scientific speculation, but it was not until about seventy years ago that the great electrical researches of Faraday placed it as a leading article of faith in the creed of the scientist. About thirty years later came the epoch-making work of Clerk Maxwell. He was deeply imbued with Faraday 's 232 LIGHT ideas, but had the great advantage of being a skilled mathematician as well as a physicist. He set himself the problem of considering minutely the manner in which a disturbance would be propagated in the ether. Waves would be set up and would travel with a certain velocity, carrying certain electromagnetic effects along with them. In free space, where there is no matter and nothing but ether, this velocity would be independent of the frequency, and Maxwell showed that, if his theory were correct, the velocity could be expressed in terms of certain quantities that could be determined by electromagnetic measurements. This ve- locity (v) is the ratio of the electromagnetic to the electro- static unit charge of electricity. Maxwell's electromagnetic theory of light consists in the statement that light-waves are merely electromagnetic waves that have a frequency lying within certain limits determined by the structure of the eye. If this be true, the velocity of light (V) in free space should be equal to the quantity that we have de- noted by v. V and v can be measured by direct experi- ment. Here are some of the results expressed in millions of centimeters per second, with the names of the experi- menters responsible for them. The variations in the table F (OPTICAL) v (ELECTRICAL) Foucault . 29,836 Ayrton and Perry . 29 600 Cornu . 29,985 Klemencic . . 30 150 Michelson . ... . 29 976 Rosa . . . 29 993 Newcoml) . . . . . 29 962 Thomson and. Searls 29 955 show that it is difficult to measure these quantities with very great precision, but there is no evidence that shows LIGHT AND ELECTRICITY 233 that one is bigger than the other. The presumption is, therefore, that they are equal, and this is the corner- stone on which the electromagnetic theory of light is based. After Maxwell, the next great step was made by Hertz about twenty years ago. He succeeded in setting up electric waves (some of them about a foot in length, others a yard or more), and investigated their properties. His famous experiments furnish, perhaps, the most strik- ing evidence that can be adduced in support of Maxwell's theory, as he showed that these electric waves obeyed exactly the laws of light, as Maxwell had predicted they should. He found that they were reflected so that the angle of reflection was equal to that of incidence. He passed them through a large prism of pitch about a yard and a half high, and showed that they were refracted ac- cording to SnelFs law. He found, too, that, just as with light- waves, he could get polarization and also diffraction. The determination of the speed with which the waves were propagated was a matter of some difficulty, and at first it appeared that they did not travel with the same velocity as does light, but later researches have shown conclusively that they do. Much has been done since Hertz's first experiments to clear away doubts and diffi- culties, and now an almost complete analogy between electrical and optical phenomena has been proved by ex- periment. Perhaps I should say in passing that these electric waves that Maxwell saw with his powerful mind, and whose properties he predicted, and that Hertz made a commonplace in every physical laboratory, are the same waves that we have all heard so much about in more recent times as employed in wireless telegraphy. They are 234 LIGHT popularly associated with the name of Marconi, whose important discovery of the influence of a " grounded wire/' immensely extended the range of their effective- ness. Let us turn now to other evidences of a relation between light and electricity. It has been stated more than once that in free ether ; where there is no matter, the velocity of all waves must be the same, whatever be their frequency, and we have already seen that there is a good agreement between theory and observation as to the magnitude of this velocity. When, however, matter is present, the speed of the wave varies with the frequency, as was pointed out at some length in the lecture on dispersion. In that lec- ture a formula was given connecting the refractive index (ri), which determines the speed, and the frequency (/), and if we refer to that formula (p. 66), we see that if / is zero, so that there are no vibrations at all, and everything is steady, we then have n 2 =JL Now the electromagnetic theory indicates that, under these circumstances, K should be what is known as the specific inductive capacity of the substance that is dealt with. This can be determined from purely electrical measurements, and it is important to see how this determination agrees with its value obtained, in accordance with our theory, from optical observations. The two substances that have been most carefully ex- amined from this point of view are Rock-salt and Fluorite. The values of K obtained by different observers from electrical experiments on Rock-salt were as follows (the name being that of the experimenter quoted) : Curie, 5.85; Thwing, 5.81; Starke, 6.29; the mean being 5.98. The corresponding numbers for Fluorite are : Curie, 6.8 ; Romich, 6.7; Starke, 6.9; of which the mean is 6.8. The values of LIGHT AND ELECTRICITY 235 K, calculated from optical experiments in the two cases, are 5.9 and 6.8, so that we have the following comparison : SUBSTANCE K (OPTICAL) JT (ELECTRICAL) Rock-salt 5.9 5.98 Fluorite 6.8 6.8 Theory also indicates that there is a relation between the reflecting power of a metal and its electrical conduc- tivity, and shows that the reflecting power must depend on the frequency. By observing the electrical conduc- tivities of different metals, we are able to predict what their reflecting powers should be for a given frequency, and to test the theory by actual measurements of these reflecting powers. The following table sets forth some of the results, the numbers expressing the percentage of the incident light that is reflected. The word " light" is used in rather a wide sense, for the frequencies fi and / 2 that are dealt with are so low that the waves are far outside the range of the visible portion of the spectrum. The fre- quencies are expressed in million millions per second. It METAL /i- 35 /- i OPTICAL ELECTRICAL OPTICAL ELECTRICAL Silver 98.85 98.7 98 87 98.85 98.4 98 6 98 83 98 73 Zinc * . 97 73 97.73 97 45 97.47 96.5 96 5 97 18 97 04 Nickel 95.9 96 4 96 80 96.84 236 LIGHT will be observed that the agreement between the optical and electrical estimates of the reflecting power is better for / 2 than for / x . The explanation of this is that the theoretical formula employed in the computation is only approximately true, the approximation being closer for small frequencies than for large ones. The various types of evidence for an intimate relation between light and electricity that have so far been referred to are all of a somewhat indirect character, and it would seem reasonable to suppose that there should be some phenomena that would prove more directly that light and electricity have something in common. I have now to direct your attention to evidence of this class. The most elementary knowledge of the science of electricity will make it clear that there is a very close relation between electricity and magnetism. It shows, for example, that an electric current gives rise to a magnetic field, and that a piece of iron can be magnetized by passing an electric current round it. If, then, light and electricity are in any sense one, we should expect a magnetic field to have some influence on light, and one of Faraday's epoch-making discoveries proved that this is the case. Faraday found that when such a uniform transparent substance as glass or carbon bisulphide is placed in a powerful magnetic field, and a beam of plane polarized light is made to traverse the field in the direction of the lines of magnetic force, the plane of polarization is rotated. When we dealt in an earlier lecture with a kindred phenomenon exhibited by quartz, solutions of sugar, and other optically active media, we saw that the rotation could be explained once we un- derstood why a wave circularly polarized in the clockwise sense should move through the medium with a different LIGHT AND ELECTRICITY 237 speed than one polarized counter-clockwise. The same problem presents itself in the explanation of the Faraday effect; but the solution must be quite different, for here we have no peculiarities of structure to deal with that can distinguish the right hand from the left when rotations are concerned. In this case the explanation is afforded by the application of certain well-known laws of electro- magnetism which deal with the mutual influence of a cur- rent and a magnetic field, and show that different effects are produced by currents flowing in opposite senses, clockwise and counter-clockwise. A precise form is given to the investigation by the adoption of the electron theory, which has already been referred to. According to this the atoms of a substance are made up of groups of electrons, which constitute small charges of electricity, and, when moving round an orbit, have some of the characteristics of an electric current. A careful analysis shows that a right-handed circularly polarized beam should cross the magnetic field at a different rate than a left-handed one, so that a rotation of the plane of polarization is to be expected. It also appears that the amount of the rotation is directly proportional to the length of the field traversed, a law that is similar to that which governs the behavior of optically active media, and one that, like it, has been amply verified by experiment. The theory also indicates that the amount of the rotation depends upon the frequency, so that we have rotatory dispersion, as in the case of active media. The following table records the rotations produced by creosote and car- bon bisulphide for different lines in the spectrum, and compares the observed values with the predictions of theory : 238 LIGHT CREOSOTE CAKIJON BISULPHIDE T SPECTRUM ROTATION ROTATION ROTATION ROTATION (theory) (observation) (theory) (observation) c 0.573 0.573 0.592 0.592 D 0.744 0.758 0.760 0.760 E 0.987 1.000 0.996 1.000 F 1.222 1.241 1.225 1.234 G 1.723 1.723 1.704 1.704 Wood gives the following results for the rotations pro- duced by sodium vapor for different frequencies in the neighborhood of the natural frequencies of sodium. The frequencies (/) are given in million millions per second, and the rotations (R) are those observed, or calculated, to the nearest degree : / R (THEORY) R (OBSERVATION) / R (THEORY) R (OBSERVATION) 501 5 5 510 93 90 504 10 10 511 43 43 505 23 20 512 41 40 506 38 40 513 20 20 507 59 66 514 9 10 508 89 90 516 5 5 Theory also indicates, and experiment verifies, that rotation is in the same absolute direction when the light is travelling from A to B as from B to A. Thus if, to a person at A looking towards B, the rotation appears to be clockwise when the light goes from A to #, then if the light be reflected from B so as to return to A } the rotation LIGHT AND ELECTRICITY 239 will still appear to A to be clockwise. This leads to a somewhat curious result. The path of a ray of light, no matter how crooked it may be, is usually reversible. If A can see B, then B can see A, and this is true whether they look at one another directly, or whether the light be reflected and refracted at various points in the passage from one to another. You & may not be able to see a per- { son directly, and yet you may have a clear view of him by reflection in a mirror ; but if this be so, you know that he also can see you in the mirror. Thus, by no ordi- nary optical device can A see B without B being able also to see A. However, by util- izing this power of rotating the plane of polarization possessed by a magnetic field, it is possible to think of an arrangement by means of which B could see A, while A could not see B. Take two Nicol prisms and set them with their principal planes ON 1 and ON 2 (Fig. 69) inclined at an angle of 45. Place them in a medium in a magnetic field that has just the necessary strength to turn the plane of polarization counter-clockwise, say, through an angle of 45. The light that goes from A passes through the first Nicol and then is plane polarized, the plane of its polarization being parallel to ON^ After passing across the magnetic field, this plane is rotated through 45, and so is parallel to ON%. The light is thus polarized just in the right plane to pass freely through the second Nicol, so that it reaches B, who will therefore FIG. 69 240 LIGHT have no difficulty in seeing A. Now think of the light that sets out from B towards A. On passing through the first Nicol that it reaches, it will be plane polarized, with the plane of its polarization parallel to ON 2 . It then enters the magnetic field and, in crossing it, has the plane of its polarization rotated 45 in the direction indicated in the figure. Thus the light is polarized in the direction Oa, which is at right angles to ON V so that the light cannot get through the Nicol to reach A. Hence B sees A with- out A seeing B. The Faraday effect with which we have been dealing was the first thing of the kind discovered that exhibited a direct action of magnetism on light, but there have been several similar discoveries since. Thus, about thirty years ago, Kerr found that plane polarized light is con- verted into elliptically polarized light when it is reflected from the polished pole of an electromagnet, under circum- stances in which this change could not occur if the field were not magnetic. A few years earlier the same experi- menter had discovered another interesting relation between optical and electrical phenomena. He found that a dielec- tric, like glass, or even a liquid, such as carbon bisulphide, behaves quite differently when in a powerful electric field than when it is not so placed. It acquires the doubly refracting properties of a crystal. This seems to indicate that the electric field has the effect of arranging the elec- trons in order, and so of producing something like the definite structure that gives a crystal different optical qualities in different directions, and accounts for its doubly refracting power. Even more interesting than the Kerr effect is that of Zeeman, discovered in 1896. He found that a magnetic LIGHT AND ELECTRICITY 241 field could alter the positions and the character of certain lines in the spectrum. This is a very significant fact, if you bear in mind what has been said as to the position of a line in the spectrum and its relation to the frequency of the vibrations going on within the atom. To alter the frequency, you must interfere with the mechanism of an atom, and Zeeman's discovery proves that you can do this merely by placing the atom in a strong magnetic field. In view of the well-known influence of magnetic forces on electric currents, we may find in the Zeeman effect a powerful support for the electric theory of matter that is a leading feature of recent speculation, and it is mainly because of this that the phenomenon has received so much attention from the world of physical science. Let us see, somewhat more clearly, what the Zeeman effect is (at least in its simplest aspect), and then consider the general outlines of the explanation that has been sug- gested. We have been reminded that when light from a luminous body in the form of vapor is viewed through a spectroscope, the spectrum is crossed by certain bright lines which have definite and fixed positions for any given substance in a given condition. So well fixed and well known are these lines that, as has been seen, we may readily determine the nature of a substance by noting carefully the positions of these lines. Zeeman's striking discovery was that, when the luminous body was placed in a strong magnetic field, a single line was replaced in some cases by two lines, one on each side of the position of the original line ; in other cases by three lines, one in the position of the original, and one on each side thereof. Later researches have revealed more complicated cases, but we shall confine our attention to those that are simplest. 242 LIGHT Without entering too much into technicalities, let me indicate the explanation afforded by Lorenz of the simplest case of the Zeeman effect. The fundamental idea is that which lies at the root of the electric theory of matter. It supposes that, in its last analysis, an atom of matter would be found to consist of a number of moving charges of electricity, which now usually go by the name of elec- trons. Theory indicates and experience proves that a FIG. 70 charge of electricity moving rapidly round a closed orbit has an effect similar to that of an electric current flowing in the same circuit. Now it is one of the most widely known and most firmly established laws of electromag- netism that a current is affected by the presence of a mag- netic field, so that we have good reason to suppose that an electron moving in an orbit would be affected by such a field. Moreover, certain laws of electromagnetism that are well grounded in experience enable us to predict how the electron would be affected in any given circumstances. Consider the simple case of an electron moving steadily in a circle, say in the plane of this paper round as a center. (Fig. 70.) As the electron might move round in two senses, clockwise or counter-clockwise, there will be two cases to deal with, and we may distinguish these electrons by the letters E 1 and E z . If the magnetic force is at right LIGHT AND ELECTRICITY 243 angles to the plane of the paper, it follows from the laws of electromagnetism that E 1 will be driven along OE lt away from the center, while E 2 will be pulled along E%0 towards the center. It is a simple deduction from this that the frequency of the vibrations of E 2 will be increased, while that of E l will be diminished. You know, doubt- less, that if you make a stone describe a circle by whirling it round at the end of a string, the force with which you have to pull the string is greater, the greater the number of revolutions per minute, i.e. the greater the frequency. Thus an increased force towards the center means a greater frequency and a diminished force towards the center means a smaller frequency. Now when there are no external magnetic forces present the electrons E 1 and E% are drawn towards with a certain force that depends on the distri- bution of the electrons in that neighborhood. The pres- ence of a magnetic field adds a new force away from in the case of E 1} and towards in the case of E z , so that the total force towards is diminished for E 1 and in- creased for $2, and thus the frequency is diminished for E 1 and increased for E 2 . It thus appears that the effect of placing a number of rotating electrons in a mag- netic field would be that those electrons whose planes of motion were at right angles to the lines of magnetic force would have their frequencies increased or diminished according to the sense (clockwise or counter-clockwise) in which their orbits were described. Thus the original single line in the spectrum would be replaced by a doublet, the members of which would be on opposite sides of the original line. At the same time those electrons that were mov- ing in the same plane as the lines of magnetic force would not be affected, so that their frequency would be un- 244 LIGHT changed. Not only does Lorenz's explanation account for the main feature of the phenomenon, that is, for the pro- duction of two or of three lines from a single line, accord- ing to the direction of the lines of magnetic force, but it also indicates the state of polarization of the different lines. It shows that the two lines of a doublet should be circu- larly polarized, one being right-handed and the other left- handed. It shows also that with a triplet the middle line should be polarized in a plane perpendicular to the direc- tion of the magnetic force, and the two outer lines polar- ized in a plane parallel to that direction. All these de- tails with reference to the nature of the polarization of the different lines were first predicted by Lorenz's theory, and later observation proved them to be correct. It should perhaps be stated that later researches have proved that in many instances the influence of a magnetic field on the character of the spectral lines is much more complex than what was first observed by Zeeman. There are many indications that if an atom be rightly regarded as a group of electrons, the distribution and motion of these must constitute a mechanism that is far from simple, and the complexity of certain aspects of the Zeeman effect is what might well be expected. It would be out of place to enter into such questions here, but before taking leave of the Zeeman effect I should like to call your attention to a very interesting application of the theory that has been made quite recently by Hale. It has long been known that there is an intimate relation between electricity and magnetism. You have been reminded within the last few minutes of the influence on a current of a magnetic field, and you know probably that a current by itself sets up a magnetic field, that is, that there are certain mag- LIGHT AND ELECTRICITY 245 netic effects due merely to the presence of an electric current. If, then, a moving charge of electricity is, under any circumstances, equivalent to a current, it should also give rise to a magnetic field, as Maxwell anticipated and as, in fact, Rowland proved by experiment as long ago as 1876. Now the ingenious device of Hale referred to on p. 88 of the lecture on Spectroscopy, by means of which he takes photographs of the Sun with light from a single line in the spectrum, e.g. one of the lines of hydrogen, quickly led in his hands to many interesting discoveries. It made it clear, amongst other things, that there are numerous vortices or whirlwinds in the solar atmosphere, and such is the detail in some of the photographs that it seems possible to determine, from the form of the streamers round the whirlwind, in what sense (clockwise or counter- clockwise) the vortex is rotating. These whirlwinds are characteristic of Sun-spots, and it seems probable that all such spots are vortices spinning in the solar atmosphere. We know from numerous terrestrial experiments that at high temperatures carbon and many other elements that occur in the Sun send out large numbers of corpuscles charged with electricity. It is natural to suppose that the same thing will happen under similar circumstances in the Sun. Let us suppose, further, that in any region near a Sun-spot a preponderance, say, of negative charges exists. These will be whirled round in the vortex, and as they move round will constitute effectively an electric current, and so give rise to a magnetic field. We should expect, then, that if our hypotheses be justifiable, a Sun-spot should be characterized by the presence of a magnetic field. One way of detecting this presence is to make care- ful observations of the features of the lines of the spectrum 246 LIGHT in this region, and see if we can find evidence of the Zee- man effect. It had been known for some time that the spectrum of a Sun-spot differs in several respects from the ordinary solar spectrum. Amongst the peculiarities of a Sun-spot spectrum are two that are specially significant in view of the Zeeman effect : in the first place, a large num- ber of doublets, or double lines, exist ; and secondly, many of the lines are unusually broad. These are just the features that we should expect, from our knowledge of the Zeeman effect, provided we see the force of the reasons that have been adduced, or of any other reasons, for ex- pecting a strong magnetic field near a Sun-spot rather than in other regions of the Solar atmosphere. Prompted by some such reasons as these, Hale recently devoted the resources of the Mt. Wilson Observatory to the problem of examining the spectral lines in Sun-spots, keeping an especially sharp lookout for evidences of the Zeeman effect. He found that the light from the two edges of certain lines was circularly polarized in opposite direc- tions. He found that right- and left-handed polarizations were interchanged in passing from a vortex spinning clock- wise to one spinning in the opposite sense. He found also that the displacements of the widened lines had just the same features as those detected by Zeeman. With the caution of a man of science he concluded that the existence of a magnetic field in Sun-spots was "probable." By ex- perimenting in his laboratory on the strength of field necessary to produce a shift of the spectral lines of the same amount as those observed in the Sun-spots, he was enabled to form some estimate of the strength of the magnetic field in these spots. Here, then, we have a striking example of the breaking LIGHT AND ELECTRICITY 247 down of barriers that earlier thinkers have set up be- tween different fields of knowledge. Astronomy, chemis- try, electricity, magnetism, and light have each had fences raised around them. In these researches of Hale's yoiT have observations that seem to deal only with light, obser- vations, namely, of the varying intensity of the light in different places. Some portions of the field of view are very bright, others seem relatively dark, and present the appearance of dark lines of different widths in different positions. From these you are enabled to determine cer- tain facts of astronomy, to learn something definite as to the physical condition of the Sun. You also learn some- thing of chemistry, for you can tell, with practical certainty, that you are looking at iron, or chromium, or manganese, or vanadium. Electricity, too, is brought before your view, for you are forced to consider the effect of electric charges caught up in the whirl of the great solar vortices. Finally, these observations on light lead you inevitably into the field of magnetism, and even enable you to esti- mate the strength of the magnetic forces that play about the surface of the Sun, although they are nearly a hundred million miles away. Thus science is, after all, a unity, and in this key I may appropriately bring this course of lectures to a close. Science strives to bring all things, with whatever names they may have been labelled in the past, into harmony with some all-pervading principle or law. "Give me extension and motion/' exclaimed Descartes, "and I will construct the world !" "Give me ether and electrons and the fundamental laws of mechanics," says the modern physicist, "and I will give you a picture of the world that is beautiful in its simplicity and in its faithfulness. I will 248 LIGHT not, however, pretend to explain the world, and I will leave questions of reality and of purpose for others to dispute over." Perhaps it should be remarked that this method of the modern man of science differs essen- tially from what is sometimes called the metaphysical method. I have no intention of saying anything against metaphysics. It would be an impertinence to do so, and I am ready to admit that the remarks of scientists about metaphysicians are often quite as valueless as those of metaphysicians about science. All that need be said is that physicists do not even attempt to evolve the laws that govern the world from their own consciousness. Their knowledge is strictly empirical, their hypotheses and "laws" are valued only so far as they harmonize experi- ences and fit the facts together. Everywhere these laws must be put to this test, and if they fail to satisfy it, they must be ruthlessly abandoned. My aim throughout has been to show you how well the modern theory of light serves its purpose and actually fits the facts, and I hope that I have succeeded in giving you some conception of its comprehensiveness and power, even if I have not revealed its true nature as a noble work of art. INDEX Aberration, spherical, 148. Absorption, 48-58; of energy, 53; in lenses, 150; dark lines due to, 80 ; spectra, 81 ; Hartley on, 83. Abstractions, 123-125. Amplitude, defined, 26, 122 ; and in- tensity, 127. Arago, 99. Aristotle, 9. Art and Science, 1, 4-6, 45, 46, 248. Atom, nature of, 53; vibrations within, 55, 79; arrangement in space, 83, 110-113. Autochrome plate, 42. Balmer, 56. Biot, 109. Brewster, law as to polarizing angle, 129 ; objection to wave theory, 207. Bunsen, spectrum analysis, 77. Cauchy, on dispersion, 62-65. Chromatic effects in telescopes, 148, 151. Color, relation to frequency, 26 ; equa- tion, 27 ; primary, 27 ; vision, 28- 32 ; photography, 33-46, 170-174. Conical refraction, 185. Critical angle, 16, 120. Crookes, spectrum analysis, 177. Crystal, index surface in, 179; opti- cal properties of, 175201 ; optic axis of, 178; ordinary and ex- traordinary rays in, 180; positive and negative, 179; rings and crosses with, 190-201 ; rotatory power, 104 ; structure of, 104, 107. Dark lines in spectrum, due to ab- sorption, 54, 55. Darwin, 4, 93 ; his Origin of Species, 91. Descartes, 247. Diffraction, 202-228 ; dynamical theory of, 209; Fresnel on, 212, 213; Lommel on, 216-218; grat- ings, 220. Dispersion, meaning of, 47; theory of, 59-69; anomalous, 69; appli- cations of, 88. Doppler's principle, 84-87. Double refraction, 176. Doublets, 57. Elastic solid theory, 125. Electricity, its relation to light, 229- 248. Electric waves, 232. Electrons, 53, 242, 247. Energy, 53. Ether, nature of, 99, 100, 121, 124, 125 ; in a crystal, 175 ; in electrical science, 231. Evolution, 91-93. Extraordinary ray, 180. Eye, as optical instrument, 141. Eye-piece, 152. Faraday, 23; electrical researches, 231 ; effect of magnetism on light, 236, 237. Fischer, on optical activity, 114, 115. Fraunhofer, 80. Frequency, defined, 26; relation to color, 26 ; natural, 48-51 ; forced, 4851 ; of vibrations within atoms, 55 ; relation to period and wave length, 155; limits of eye's sensi- tiveness, 70. Fresnel, idea of ether, 99, 100; on interference, 161, 162; biprism, 162; on diffraction, 212, 213. Geometrical shadow, 205. Gernez, on optical activity, 110. Greek science, 7-9. 249 250 INDEX Hale, method of photographing prom- inences and flocculi, 88-91 ; his spectroheliograph, 88; his investi- gations on Sun-spots, 244-247. Hamilton, on conical refraction, 185. Hartley, 83. Helmholtz, 23; on color vision, 28- 31 ; on Young, 157. Hering, on color vision, 28, 31. Herschel, on rotatory power, 104. Hertz, on electric waves, 233. Huggins, on nebulae, 76 ; method of viewing solar prominences, 87. Huyghens principle, 208, 209. Index surface, 179. Intensity of reflected light, 127; ratio of intensities, 132 ; influence of layer of transition on, 130, 131 ; from metals, 136 ; reflecting power and electrical conductivity, 235. Interference, Principle of, 34, 154- 174; Young on, 157-160; Fresnel on, 161, 162; Lloyd on, 161; color due to, 163, 166-170; stationary waves produced by, 172, 173; ap- plication to diffraction, 209; to gratings, 222. Kayser and Runge, 56. Kerr effect, 240. Kirchhoff, 80, 81. Least action, 140. Le Bel, 112. Lippman, color photography, 34, 170-174. Lommel on diffraction, 216-218. Lorenz, on Zeeman effect, 242-244. Lumiere, color photography, 42-44. Magnetism and light, Faraday effect, 236-239; Kerr effect, 240; Zee- man effect, 240-246. Marconi electric waves, 234. Maxwell, 23; theory of color vision, 28; color photography, 34; on Saturn's rings, 87; electromagnetic theory of light, 231, 232 ; on mag- netic effects of a moving electric charge, 245. Method of science, 10, 13, 14, 206, 207, 247. Michelson, spectroscope, 75; visi- bility curves, 78, 79 ; wave-lengths as standards, 226. Morse, 58. Nebula, nature of, 76. Newton, on color, 9; greatness of, 13, 21-23; his "Opticks," 13, 20, 166, 205 ; his method, 13, 14 ; his experiments, 14-21 ; on soap- bubbles, 164; his rings, 166-169; on diffraction, 205, 206; on shadows, 205, 206; objection to wave theory, 205. Nicol's prism, explanation of its action, 177, 178; used, 95, 99, 102, 103, 116, 177, 191. Objective, 149, 152. Optical activity, 104-117; Biot, 109; Gernez, 110; Pasteur, 110; Van't. Hoff and Le Bel, 112; Fischer, 114, 115. Optic axis, 178. Ordinary ray, 180. Pasteur, on optical rotation, 110. Period, 155. Phase, meaning of, 122; difference of, 122; for reflection from trans- parent substances, 134 ; for total reflection, 134, 135; for reflection from metals, 137; influence on interference, 156. Photography, ordinary, 35-37 ; color, 33-46; direct methods, 33; in- direct methods, 33; Maxwell on, 34; difficulties of, 41; Lippman's process, 170-174; Lumiere 's process, 42-44; defects of, 44, 45; relation to art, 45, 46. Pickering, 57. Plato, 8. Polarization, 95-117; different kinds of, 97, 98; mechanical analogues, 100, 101, 177 ; effect on intensity of reflection, 127 ; plane of, in crystals, 188; rotatory, 102-117, 236-239. Polarizing angle, 129 ; Brewster's law, 129 ; quasi, 137 ; in crystals, 187. INDEX 251 Rays in crystals, 184. Reflection, fact of, 10, 11; laws of, 10, 11, 118-153; from metals, 136; illustrating principle of least action, 139, 140; total, 10, 11, 120, 135. Refraction, fact of, 10, 11; laws of, 9, 11, 118-153; illustrating princi- ple of least action, 139, 140; double, 176. Refractive index, meaning of, 120 ; in crystals, 179-182; measured electrically, 234. Reversal, 81. Rings and crosses formed with crystals, 190-201. Rotatory polarization, structural, 102, 103; magnetic, 236-239. Rowland, grating, 75; map of spectra, 80; on magnetic effect of moving electric charge, 245. Ruskin, 4-6. Saturn's rings, 87. Science, aim of, 118, 121, 123, 186, 248; alleged inhumanity, 3; rela- tion to art, 1, 4, 5, 6; early, 7-12; language of, 24 ; divisions of, 229 ; method of, 10, 13, 14, 206, 207, 247. Secondary spectra, 151. Shadows, geometrical, 205; Newton on, 205, 206. Singular points, 185. Snell, 9. Soap-bubbles, 164, 165. Solar prominences, 85, 87. Solar vortices, 245. Spectrograph, 76. Spectroheliograph, 88, 89. Spectroscope, 74, 75. Spectroscopy, 70-94. Spectrum, 15; dark lines in, 54-57, 80; bright lines in, 71, 74; con- tinuous, 72, 73, 76; band, 57; produced by gratings, 227. Spherical aberration, 148. Stellar evolution, 91-93. Stokes, 23 ; on absorption lines, 80 ; theory of diffraction, 209. Sun-spots, 85, 245, 246. Superposition, principle of, 155, 156. Telescope, purpose of, 142; form, 143; material, 144; size, 145; arrangement of parts, 147 ; defects, 148-150; objectives, 149, 152; eye-piece, 152. Transition from one medium to an- other, 126, 127, 129, 134. Turner, 5. Van't Hoff, 112. Vibrations, transverse, TS, 99; longi- tudinal, 98, 99 ; confined to wave- front, 97, 99. Visibility curves, 78, 79. Vision, limits of, 27, 30, 70; theory of color, 28-32. Watts, Marshall, 58. Wave, 25; length, 154, 220-228; front, 96; theory, 121, 205, 207; surface, 182-185 ; different theories, 123, 154 ; stationary, 173 ; electric, 232-234. Wood, 238. Wordsworth, 21. Young, on color vision, 28-31 ; on interference, 157-161. 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