Double Refraction

light plane polarized rays perpendicular crystal ordinary spar reflected waves

DOUBLE REFRACTION. - WO now come to phenomena which cannot be even roughly explained by processes based on the vague analogies of sound and water waves which have hitherto sufficed for our elementary treatment of the subject.

These phenomena were first observed in Iceland spar. They were described in a general way by Bartholinus, who showed that one of the two rays into which a single incident ray is divided by this substance follows the ordinary law of refraction. Huygens, who studied the subject only eight years later, verified the greater part of the results of Bartholinus, and added many new ones. From his point of view it was of course obvious that the ordinary ray is propagated by spherical waves, i.e., its velocity is the same in all directions inside the crystal. To explain the extraordinary ray, he assumed that it was propagated in waves of the form of an ellipsoid of revolution, the simplest assumption he could make. To test its accuracy he first noticed that a rhombohedral crystal of Iceland spar behaves in precisely the same way whichever pair of parallel faces light passes through. Hence he acutely concluded that the axes of the ellipsoids of revolution (if such were the form of the waves for the extraordinary ray) must be symmetrically situated with regard to each of these planes. The only such lines in a rhombohedron are parallel to that which joins those corners which are formed by the meeting of three equal plane angles. In the case of Iceland spar these equal angles are obtuse. Huygens then verified, by experiments well contrived, though carried out by a very rough mode of measurement, the general agreement of his hypothesis with the fact ; and he further tested it by comparing its indications as to the position of the two images for any position of the crystal with the results of direct observation. There can be no question that the whole investigation was, for the age in which it was made, of an exceedingly high order. But it must not be left unsaid that far more accurate measurements than those of Huygens were necessary before it could be asserted that the form of the extraordinary wave is an ellipsoid of revolution, and not merely a surface closely resembling such an ellipsoid. These improved measurements were made 1802 by Wollaston, and they have recently been repeated with far more perfect optical means by Stokes, Mascart, and Glazebrook. The result has been the complete verification of Huygens's conjecture. The generating ellipse of the extraordinary waves is found to have its minor axis, which is that of revolution, equal to the diameter of the corresponding sphere for the ordinary ray. Its major axis is to the minor nearly in the ratio 1.654- :1.483.

We are now in a position to trace the paths of the two rays into which a ray falling in any direction on a surface of the crystal is divided by refraction.

Let fig. 34 represent a plane wave front AB (in air) falling on the surface AC of a piece of Iceland spar cut in any way. The figure is a section perpendicular to the surface, and parallel to the incident ray. The wave-front AB cuts the surface of the spar in a line (not shown) at right angles to the plane of the paper. Draw from A the axis Aa (not necessarily in the plane of the paper) and the sphere and ellipsoid of revolution which have Aa for a common axis. Then, if C be taken such that BC is to Aa sa the velocity of light in air is to that of the ordinary ray in DOUBLE-REFRACTION the crystal, the wave-front of the ordinary ray is found by drawing a tangent plane to the sphere, passing through C and perpendicular to the plane of the paper. This touches the sphere in a point o (in the plane of the paper) and AoO is the ordinary ray.' To find the direction of the extraordinary ray, a plane perpendicular to the paper, and passing through C, must be drawn so as to touch the ellipsoid. Let e be the point of contact, which will in general not be in the plane of the paper unless Aa is in or perpendicular to that plane ; then AeE is the extraordinary ray.

Thus, in general, the extraordinary ray is not in the plane of incidence. Also the ratio of the sines of the angles of incidence and refraction is generally different for different directions of incidence, in the case of the extraordinary ray.

In an elementary article we cannot attempt more fully to study these phenomena ; so we merely state that all the observed appearances, so far as the directions of the refracted rays are concerned, are explained by supposing when both eyes are used, the two images of a plane object made to rotate about a perpendicular to the two faces employed ; the other's position varies as the crystal is turned.

But we have now to inquire why the incident ray is divided into two, and why one of them follows the ordinary Huygens comes to our assistance. We paraphrase the t author's description : - " I will, before concluding, mention another remarkable pheno- I meson which I discovered after the above was written. For, 8 although I have not yet been able to find the cause of it, I do not wish on that account to refrain from pointing it out, in order that c others may have an opportunity of seeking to explain it. It appears that it will be necessary to make hypotheses additional to those already given, - though these will lose none of their probability, confirmed as they have been by so many tests. The phenomenon is that, taking two fragments of the crystal (Iceland spar) and laying them on one another, or even holding them apart, if all the flees of the one be parallel to those of the other, a ray of light divided into two by the first fragment will not be farther subdivided by the second. The ordinary ray from the first will be refracted ordinarily by the second, the extraordinary ray extraordinarily. And the same thing happens not only in this arrangement but in all others in which the principal sections2 of the two fragments are in the same plane, whether the surfaces turned towards one another be parallel or not. It is, in fact, marvellous that these rays, falling on the second fragment, do not divide like the ray incident on the first. One would say that the ordinary ray from the first fragment had lost what is necessary for the production of extraordinary refraction, and the extraordinary ray that which is necessary for ordinary refraction ; but there is something else which upsets this view. For when one places the fragments so that their principal sections are at right angles, whether the opposed surfaces be parallel or not, the ordinary ray from the first suffers only extraordinary refraction by the second, and vice versa.

" But in all the infinite number of positions other than those named, both rays from the first fragment are divided into two by the second. Thus the single incident ray is divided into four, sometimes equally sometimes unequally bright, according to the varying relative position of the crystals. But all together do not seem to have more light than has the single incident ray.

" When we consider that, the two rays given by the first crystal remaining the same, it depends upon the position of the second crystal whether they shall be divided into two or not, while the incident ray is always divided, it appears that we must conclude that the waves of light which have traversed the first crystal have as fared a form or disposition which in some positions enables them to excite the two kinds of matter which give rise to the two kinds of ,efraction, in other positions to excite only one of them. But I have not yet been able to find any satisfactory explanation of this."

So far Huygens. His statements are perfectly in accordance with fact ; and they were reproduced by Newton3 in very nearly the same form. Newton adds : - " The unusual refraction is, therefore, performed by an original ( property of the rays. And it remains to be enquired, t whether the rays have not more original properties than are yet discovered. Have not the rays of light several sides, endued with several original properties ?"

It is very curious to notice how near each of these great men came to the true explanation, and yet how long time for a host of discoveries in a new and immense field of optics.

In the last-mentioned year Malus, while engaged on the theory of double refraction, casually examined through a doubly refracting prism of quartz the sunlight reflected from the windows of the Luxembourg palace. He was surprised to find that the two rays alternately disappeared as the prism was rotated through successive right angles, - in other words, that the reflected light had acquired properties exactly corresponding to those of the rays transmitted through Iceland spar. Even Malus was so imbued with the corpuscular theory of light that he named this phenomenon polarization, holding it as inexplicable on the wave theory, and as requiring a species of polarity (akin to the magnetic) in the light-corpuscles - a close reproduction of one of Newton's guesses.

But after a short time Hooke's old guess was independently reproduced, and in the hands of Young and others, but most especially of Fresnel, the consequences of the assumption, that the vibrations of the luminiferous medium take place perpendicularly to the direction of the ray, were the almost complete explanation of the cause of double refraction, and the discovery (often the prediction) of a long series of the most gorgeous phenomena known to science.

The real difficulty in the way of this conception probably lay in the fact that most of the familiar forms of wavemotion - such as sound-waves in air or in water, and ordinary water waves - are not of this character. In sound-waves the vibrations are wholly in the direction of the ray, while in surface-waves in water they are partly parallel to and partly perpendicular to the direction in which the wave is travelling. That a body may transmit waves in which the vibration is perpendicular to the direction of a ray, it must have the properties of an elastic solid rather than of a fluid of any kind. And our experience of the almost entire absence of resistance to the planetary motion seems, at first sight at least, altogether incompatible with the idea that the planets move in a jelly-like solid, filling all space through which light can be propagated.

. Without going into difficult dynamical details, we may obtain a notion of the nature of the motion now to be considered, by observing the propagation of a wave when a long stretched wire or string is struck or plucked near one end. Here the line of motion of each part of the wire is almost exactly perpendicular to the direction of the wire, i.e. to the line along which the wave travels. (When the string is extensible there may be another wave, due to extension ; but this, which is analogous to sound, has its vibrations along the string, and it usually travels at a very different rate from the other, so that the t wo are not in any way associated).

Now it is clear that waves of this wholly transverse character can have, in Newton's language, sides. And it is also clear that they cannot interfere so as mutually to ' destroy one another unless their corresponding sides are parallel to one another ; nor can they interfere at all if their sides are perpendicular to one another. Hence a very severe test of the theory will be furnished by examining various cases of interference of polarized light, which ought to present in general marked differences from those of ordinary light. It was by experiments of this kind that Fresnel and Arago first firmly established the bases of the theory of polarization. The important fact discovered by Malus was soon generalized into the following statement :- Light reflected from the surface of substances so different as water, glass, polished wood, &c., at a certain definite angle, which depends on the nature of the substance, is found to possess all the properties of one of the rays transmitted through Iceland spar. If the plane of reflexion is parallel to the axis of the spar, the properties of the reflected light are those of the ordinary ray ; if perpendicular to it, those of the extraordinary ray.

It was reserved for Brewster to discover, as the resul of an extraordinary series of experimental measurements, the very simple law which follows :- The tangent of the polarizing angle is equal to the] refractive index of the reflecting substance.

This may be put in another form, in which its connexion with theory is a little more evident : - When the reflected Tay is completely polarized, it is perpendicular to the refracted ray.

Bearing in mind Huygens's observations on light which has passed through two crystals of Iceland spar, we can now see that a ray of light polarized by reflexion is in general divided into two by a crystal of Iceland spar. But there is only one ray when the principal plane of the crystal is parallel to the plane of reflexion, and none when these planes are perpendicular to one another.

We may now much simplify matters by suppressing the Iceland spar, and using two reflecting plates of glass, so placed that a ray meets each of them in succession at the polarizing angle. It is then found that when the planes of reflexion are parallel the ray is reflected (almost without loss) from the second plate, but when they are perpendicular to one another there is complete extinction. In intermediate positions the intensity was found by Arago to be as the square of the cosine of the inclination of these planes.

This very simple experiment, which any one may easily make for himself, by putting two pieces of glass at the proper angle in the ends of two wooden tubes which fit into one another, enables us to form a general notion of the modification which is called polarization. The " sides " of the reflected ray are obviously in, and perpendicular to, the plane of incidence ; for a ray can be reflected over and over again if the successive planes of incidence are parallel, but is stopped at once if one of them be perpendicular to the others.

Here, however, two new difficulties come in at once :(1) Are the vibrations of the reflected ray in, or perpendicular to, the plane of reflexion ? (2) As ordinary sun or lamp light, reflected at the proper angle from a polarizing surface, shows no variation of intensity when the azimuth of the plane of reflexion is changed, what can be then the direction of its vibrations ? These questions have not yet been answered in a thoroughly satisfactory- manner Many important phenomena are explained in terms quite independent of the proper answer to (1); and, in others which do depend on the answer, the theoretical differences between the results of the two hypotheses are so small as to have hitherto remained undetected. In an important test, suggested by Stokes, the experimental results have been at variance in a way not yet explained. It is quite possible that, as is required by Clerk Maxwell's electromagnetic theory of light (see ETHER), there may be simultaneous displacements, but of different characters, in each of these planes, and then the question would be reduced to - Which of these displacements is the luminous one ? But on this theory, both are probably essential to vision.

As to the second question, it may be said - first, that, so far as the test of double refraction can inform us, a polarized ray whose plane of polarization is made to rotate rapidly produces precisely the same effects as a ray of ordinary light ; and, secondly, that, so great is the number of vibrations even of red light in one second, it would be impossible to make the plane of polarization rotate fast enough to affect the circumstances of any of the phenomena of interference, even when they take place between two portions of the same ray, one of which is retarded thousands of wave-lengths more than the other. But, thirdly, the fact that, when homogeneous light is used, Newton's rings have been counted up to the 7000th shows that, whatever be the actual nature of the vibrations of unpolarized light, they must for at least 7000 waves in succession be almost precisely similar to one another. Then, for other 7000 waves or so, we may have a totally different type of vibration. But, fourthly, in the course of 4-th of a second, at the very utmost, the vibrations must have been almost uniformly distributed over all directions perpendicular to the ray. Again, however, fifthly, another quite different view may be suggested. All common light has its origin from a practically infinite number of sources, consisting of the vibrating particles of the luminous body. The contributions from each of these sources (so far as one definite wave-length is concerned) may be and probably are at any one point as different in direction of vibration as they certainly must be in phase.' From this point of view, which we cannot develop here, the uniformity of optical phenomena becomes quite analogous to the statistical species of uniformity which is now found to account for the behaviour of the practically infinite group of particles forming a cubic inch of gas. The reader need only think of the fact that, so numerous are those particles, it is practically (though not theoretically) impossible that even a cubic millimetre of air should, even for 1-0-huth of a second, contain oxygen particles alone.

When light is reflected at an incidence either less or greater than the polarizing angle, it behaves as if part of it only were polarized and the rest ordinary light; and it is said to be partially polarized. Tested by a crystal of Iceland spar, it gives two images in all positions of the crystal; but their brightness is unequal except in the special positions where they would be of equal brightness were the ray wholly polarized.

From the fourth of the remarks made above regarding common light, and the facts of double refraction, it follows at once that, when light is to any extent polarized by reflexion, there must be an exactly equal amount of polarized light in the refracted ray, and its plane of polarization must be perpendicular to that of refraction. This was established by experiment soon after Malus's discovery. But as the reflected ray from glass, water, &c., is in general much weaker than the refracted ray, the percentage of polarized light is generally much greater in the former. It was found, however, by experiment that refraction at a second glass plate parallel to the first increases the proportion of polarized to common light in the transmitted ray, and thus that light may be almost completely polarized by transmission, at the proper angle, through a number of parallel plates. The experimental data of this subject were very carefully obtained by Brewster. He has found, for instance, how the angle of incidence for the most complete polarization varies with the number of plates. The plane of polarization of such a bundle is perpendicular to the plane of refraction.

This, however useful on many occasions, is at best a rough arrangement for producing polarized light. By far the most perfect polarizer for a broad beam of light is a crystal of Iceland spar, sufficiently thick to allow of the complete separation of the two rays. But such specimens are rare and costly, so that the polarizer in practical use is now what is called ..Nicol's prism, invented in 1828 (Jameson's Journal, p. 83). By cutting a rhomb of Iceland spar in two, and cementing the pieces together with Canada balsam (after carefully polishing the cut faces), Nicol produced an arrangement in which one only of the two rays is transmitted, the other being totally reflected at the surface of the balsam. The reason is simply that the refractive index of Canada balsam is intermediate to those of the ordinary and extraordinary rays in the spar. The ordinary ray, falling very obliquely on a medium of a smaller refractive index, is totally reflected ; the extraordinary ray, falling on a medium of greater, but very little greater, refractive power, is almost wholly transmitted. The only defect of the Nicol's prism is that, to secure the total reflexion, its length must be considerably greater than its breadth; and thus it necessarily limits the divergency of the beam' it allows to pass.

Certain doubly refracting crystals exert considerable I absorption on one of the two rays they produce, and can i; therefore, when in plates of sufficient thickness, be : employed as polarizers. This is the case with some specimens of tourmaline when cut into plates parallel to the axis of the crystal. It is also found in the flat crystals of several artificial salts, such as, for instance, iodo-sulphate of quinine.

Let us now suppose that by one or other of these pieces of apparatus, say a Nicol's prism, light has been polarized. / If we examine this ray by means of a second Nicol, placed in a similar position to the first, it passes practically unaltered. As the second Nicol is made to rotate, more and more of the light is stopped, till the rotation amounts to a right angle. Two well-constructed Nicols, 'placed in this position, are practically opaque to the strongest sunlight. During the next quadrant of rotation the transmitted ray gradually increases in brightness, until at 180° of rotation it passes practically unaltered. Precisely the same phenomena occur in the same order during the next half of a complete rotation. The reader will observe that this is merely Huygens's original statement, limited to one of the four rays which are produced by passing common light successively through two crystals of Iceland spar.

Whatever be the true mechanism of polarized light, there can be no doubt that its vibrations are symmetrical' with respect to the ray, and also with respect to the plane of polarization. Hence we may, for many important purposes, symbolize them by simple harmonic vibrations taking place either in or perpendicular to the plane of polarization. But, if they be supposed to take place simultaneously in these two planes, their quality or nature must be essentially different in the two, else the symmetry above referred to would be violated. Hence it will be sufficient for the present to assume that they take place perpendicular to the plane of polarization. The nature of the resulting effects, so far as the eye is concerned, will not be different for the different hypotheses. Also, as no instance has yet been observed, even with the most intense beams of light, in which the joint effects observed are not those due to simple superposition, we may assume that the elastic force of the luminiferous medium, called into play by a displacement, is directly proportional to the displacement, and therefore that the vibrations for each wave-length follow the simple harmonic law, that of the cycloidal pendulum.

The subject of the composition of simple harmonic motions of equal period falls to be discussed as an important branch of kinematics (see MECHANICS). We will therefore here assume the following results, - referring to the above-quoted article for their proof :- Two simple harmonic motions of the same period, in I lines perpendicular to one another, give, in general, elliptic t motion, which may be in the positive or negative direction 1 of rotation.

The ellipse becomes a straight line, and the resultant motion therefore simple harmonic, when the phases of the components are the same, or differ by an integral multiple of.

Now, suppose a plane polarized ray to fall on a plate of a doubly-refracting crystal (a thin plate of mica or selenite, for instance). Within the plate it will in general be divided into two, which are polarized in planes at right angles to one another. The directions of vibration in these rays are determined by the physical properties of the material. Let them be represented by the lines Ox, Oy in fig. 35. Then, if OA represents the semiampli 105 tude of vibration in the incident ray, it may be looked on by (2) above as the resultant of two simple harmonic motions of the same period, whose semiamplitudes are OM and ON, and which are in the same phase. Each of these will pass through the plate of crystal unchanged. But one will, in general, travel faster than the other ; for the essential cause of double refraction is the difference of velocities of the two rays. The portions of the two rays which simultaneously escape from the crystal, and which travel together outside it, will therefore (lifer in phase. Hence, to find the nature of the transmitted light, we must recombine the vibrations in OM, ON, taking account of this difference of phase. By (1) above the result will be in general elliptic motion. The ellipse will necessarily be one of the infinite number which can be inscribed in the rectangle AA'BB', whose construction is obvious. We have then, in general, what is called elliptically polarized light. This degenerates (by (2) above) into plane polarized 1 light, whose vibrations are along OA or OA' according as the difference of phase is 0, 27r, 47r, &c., or 7r, 37r, 57r, &c. And it will become circularly polarized light if OM = ON (i.e., if Aar= ir) and the difference of phase be an odd multiple of -17r. By (3) above this will be right or left handed, according to the value of the odd multiplier.

This conclusion from the assumption above made is fully borne out by experiment. When a plate of mica, of such a thickness as to retard one of the two rays a quarter of a wave-length more than the other, is interposed between two Nicols, we observe the following phenomena :- If the Nicols were originally placed so as to extinguish the light, the introduction of the mica plate in general partially restores it. Now, let the mica plate be made to rotate in its own plane. The light vanishes for successive positions, differing by a quadrant of rotation, i.e. whenever the directions of vibration in the crystal coincide with the principal planes of the Nicols. In each of those positions the light from tlhe first Nicol passes unchanged through the mica, and is therefore entirely stopped by the second Nicol. Half-way between these positions the light transmitted through the system is at its brightest ; and in these cases it is not altered in brightness by rotating the second Nicol. It is then circularly polarized, and in whatever direction the second Nicol is placed the component of the circular motion which is ready to pass through it is of the same amplitude. Here, then, is a case in which a Nicol (the second) cannot enable us to distinguish between common light and light very seriously modified.

In what precedes, we have assumed that honzogeneousN light was used. In general, a doubly-refracting plate 1i produces a difference of phase in its two rays which will depend on their wave-length ; and thus when white light is used we have a display of colour, sometimes extremely gorgeous, and we may distinguish light thus circularly polarized from common light by slight changes of colour and intensity as the second Nicol is turned.

Hitherto we have spoken of the polarizing angle for light I reflected in air from bodies such as glass, water, &c., which t have a higher refractive index than air, and we have seen that an equal amount of light is polarized in the refracted beam. But what if there be no refracted beam ? This is the case of total reflexion inside the denser body. Fresnel discovered that in this case the two kinds of polarized light (in planes at right angles to one another) co-exist in the totally reflected ray, but that they differ in phase, and therefore in general recombine into elliptically polarized light. Guided by peculiar theoretical considerations, lie was led to construct a piece of glass (Fresnel's rhomb),: inside which light is twice totally reflected at a certain I angle with the result that, if it be originally polarized in a plane inclined at 45° to the plane of reflexion, the emergent light is circularly polarized.

Reflexion from the surface of metals, and of very highly refractive substances such as diamond, generally gives at 1 all incidences elliptically polarized light. Attempts have' been made to determine from such effects the refractive indices of metals and other opaque substances. These are all based upon theory, and cannot as yet command much confidence. With certain doubly-refracting substances the light reflected at a definite angle is differently polarized, and sometimes even differently coloured, for different azimuths of the plane of incidence.

When a thin plate of doubly-refracting crystal, which gives a bright colour when placed between two Nicols, is slightly inclined to the ray, the colour changes as the difference of phase of the two refracted rays is increased. If, now, we take a plate of Iceland spar cut perpendicularly to the axis, no colour will be produced by parallel rays passing through it perpendicularly, because both rays have a common velocity parallel to the axis ; but, if divergent light be used, there is a gorgeous display of circular coloured rings surrounding the axis, which depends upon the increasing retardation of the ordinary ray behind the extraordinary as their inclination to the axis increases. When the principal planes of the Nicols are at right angles, this system of rings is intersected by two black diameters, in these planes respectively. When the second Nicol is turned through a right angle, we have exactly the complement of the former appearance, i.e., a figure such that, if superposed on the former, it would give an uniform field of white light.

It is to be noticed that none of these phenomena can be observed without the use of the second Nicol. This arises from the fact that, where the vibrations in any direction interfere so as to destroy one another, those in the direction perpendicular to the former interfere so as to strengthen one another. The second Nicol enables us to select one of these portions, and examine it independently of the other.

The only double refraction we have considered particularly is that of Iceland spar, where everything is symmetrical about the axis of the crystal. Such crystals, and they include as a rule all those of the second and third systems in CRYSTALLOGRAPHY (q.v.), are called uniaxal. Crystals of the first system are not doubly refractive. But it was one of the most valuable of Brewster's discoveries that the great majority of non-isotropic substances are doubly refracting, and in general are biaxal, i.e., have two equally important axes inclined to each other at angles of all values from 0' to 90°. The form of the wave-mface in such bodies was, at least very approximately, assigned by Fresnel. This forms one of the most brilliant of his many grand discoveries ; and it led to Hamilton's prediction of the existence of the two species of conical refraction, which was experimentally verified by Lloyd.

Fresnel also made the striking discovery that glass and other simply refracting bodies are rendered doubly refracting when in a state of strain. To this Brewster added the observation that the requisite strain might be produced by unequal heating instead of by mechanical stress, and also that unannealed glass is usually doubly refractive. Clerk Maxwell in 1873 (Proc. Roy. Soc.) showed that shearing stress in viscous liquids, such as Canada balsam, renders them temporarily doubly-refractive. This subject has been elaborately investigated by Mundt (Fogg. Ann., 1879).

The details of these subjects, with those of the polarization of light reflected from small particles, the rotatory polarization produced by quartz, sugar, transparent bodies under the influence of magnetism, S c., must be deferred TO OPTICS (PHYSICAL).

There is, however, one elementary point which must not be omitted here, as it is intimately connected with the wave-theory,---that is, the alteration which light undergoes in consequence of the relative motion of the source and spectator in the line of vision.

When a steamer is moving in a direction perpendicular to the crests of the waves, she will encounter more of them in a given time if her course is towards them than if she were at rest, while, if she be moving in the same direction as the waves, fewer of them will overtake her in a given time than if she were at rest. The same thing is true of sound-waves. When an express train passes a level crossing at full speed, the pitch of the steam whistle is higher during

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