Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XI
XI.
ON DOUBLE REFRACTION AND THE DISPERSION OF COLORS IN PERFECTLY TRANSPARENT MEDIA.
[American Journal of Science, ser. 3, vol. xxiii, pp. 262–275, April, 1882.]
1. In calculating the velocity of a system of plane waves of homogeneous light, regarded as oscillating electrical fluxes, in transparent and sensibly homogeneous bodies, whether singly or doubly refracting, we may assume that such a body is a very finegrained structure, so that it can be divided into parts having their dimensions very small in comparison with the wavelength, each of which may be regarded as entirely similar to every other, while in the interior of each there are wide differences in electrical as in other physical properties. Hence, the average electrical displacement in such parts of the body may be expressed as a function of the time and the coordinates of position by the ordinary equations of wavemotion, while the real displacement at any point will in general differ greatly from that represented by such equations.
It is the object of this paper to investigate the velocity of light in perfectly transparent media which have not the property of circular polarization in a manner which shall take account of this difference between the real displacements and those represented by the ordinary equations of wavemotion. We shall find that this difference will account for the dispersion of colors, without affecting the validity of the laws of Huyghens and Fresnel for double refraction with respect to light of any one color.
In this investigation, it is assumed that the electrical displacements are solenoidal, or, in other words, that they are such as not to produce any change in electrical density. The disturbance in the medium is treated as consisting entirely of such electrical displacements and fluxes, and not complicated by any distinctively magnetic phenomena. It might therefore be more accurate to call the theory (as here developed) electrical rather than electromagnetic. The latter term is nevertheless retained in accordance with general usage, and with that of the author of the theory.
Since the velocity which we are seeking is equal to the wavelength divided by the period of oscillation, the problem reduces to finding the ratio of these quantities, and may be simplified in some respects by supposing that we have to do with a system of stationary waves. That the relation of the wavelength and the period is the same for stationary as for progressive waves is evident from the consideration that a system of stationary waves may be formed by two systems of progressive waves having opposite directions.
2. Let be the rectangular coordinates of any point in the medium, which with the system of waves we may regard as indefinitely extended, and let be the components of electrical displacement at that point at the time being the average values of the components of electrical displacement at that time in a waveplane passing through the point. Then are perfectly defined quantities, of which are connected with and by the ordinary equations of wavemotion, while each of the quantities has always zero for its average value in any waveplane. We may call the components of the regular part of the displacement, and the components of the irregular part of the displacement. In like manner, the differential coefficients of these quantities with respect to the time, may be called respectively the components of the regular part of the flux, and the components of the irregular part of the flux.
Let the whole space be divided into elements of volume very small in all dimensions in comparison with a wavelength, but enclosing portions of the medium which may be treated as entirely similar to one another, and therefore not infinitely small. Thus a crystal may be divided into elementary parallelopipeds, all the vertices of which are similarly situated with respect to the internal structure of the crystal. Amorphous solids and liquids may not be capable of division into equally small portions of which physical similarity can be predicated with the same rigor. Yet we may suppose them capable of a division substantially satisfying the requirements.
From these definitions it follows that at any given instant the average value of each of the quantities in an element is zero. For the average value in one such element must be sensibly the same as in any other situated on the same waveplane. If this average were not zero, the average for the waveplane would not be zero. Moreover, at any given instant, the values of may be regarded as constant throughout any element and as representing the average values of the components of displacement in that element. The same will be true of the quantities and
3. Since we have excluded the case of media which have the property of circular polarization, we shall not impair the generality of our results if we suppose that we have to do with linearly polarized light, i.e., that the regular part of the displacement is everywhere parallel to the same fixed line, all cases not already excluded being reducible to this. Then, with the origin of coordinates and the zero of time suitably chosen, the regular part of the displacement may be represented by the equations
(1)  
where denotes the wavelength, the period of vibration, the maximum amplitudes of the displacements and the distance of the point considered from the waveplane which passes through the origin. Since is a linear function of and we may regard these equations as giving the values of for a given system of waves, in terms of and
4. The components of the irregular displacement, at any given point, will evidently be simple harmonic functions of the time, having the same period as the regular part of the displacement. That they will also have the same phase is not quite so evident, and would not be the case in a medium in which there were any absorption or dispersion of light. It will however appear from the following considerations that in perfectly transparent media the irregular oscillations are synchronous with the regular. For if they are not synchronous, we may resolve the irregular oscillations into two parts, of which one shall be synchronous with the regular oscillations, and the other shall have a difference of phase of onefourth of a complete oscillation. Now if the mediimi is one in which there is no absorption or dispersion of light, we may assume that the same electrical configurations may also be passed through in the inverse order, which would be represented analytically by writing for in the equations which give as functions of and But this change would not affect the regular oscillations, nor the synchronous part of the irregular oscillations, which depends on the cosine of the time, while the nonsynchronous part of the irregular oscillations, which depends on the sine of the time, would simply have its direction reversed. Hence, by taking first onehalf the sum, and secondly onehalf the difference, of the original motion and that obtained by substitution of for we may separate the nonsynchronous part of the irregular oscillations from the rest of the motion. Therefore, the supposed nonsynchronous part of the irregular displacement, if capable of existence, is at least wholly independent of the wavemotion and need not be considered by us.
We may go farther in the determination of the quantities For in view of the very finegrained structure of the medium, it will easily appear that the manner in which the general or average flux in any element (represented by ) distributes itself among the molecules and intermolecular spaces must be entirely determined by the amount and direction of that flux and its period of oscillation. Hence, and on account of the superposable character of the motions which we are considering, we may conclude that the values of at any given point in the medium are capable of expression as linear functions of in a manner which shall be independent of the time and of the orientation of the waveplanes and the distance of a nodal plane from the point considered, so long as the period, of oscillation remains the same. But a change in the period may presumably affect the relation between and to a certain extent. And the relation between and will vary rapidly as we pass from one point to another within the element
5. In the motion which we are considering there occur alternately instants of no velocity and instants of no displacement. The statical energy of the medium at an instant of no velocity must be equal to its kinetic energy at an instant of no displacement. Let us examine each of these quantities, and consider the equation which expresses their equality.
6. Since in every part of an element the irregular as well as the regular part of the displacement is entirely determined (for light of a given period) by the values of the statical energy of the element must be a quadratic function of say
and 
(2) 
(3) 
 (4) 
8. The value of may be easily found by integration, but perhaps more readily by Poisson's wellknown theorem, that if is any function of position in space (as the density of a certain mass),
(5) 
(6) 
(7) 
(8) 
Now the expression for the kinetic energy of the irregular part of the flux,
This expression reduces by equations (4) to
(9) 
(10) 
(11) 
etc.,  (12) 
(13) 
This relation between the velocity of the waves and the direction of oscillation is capable of a very simple geometrical expression. Let be the radius vector of the ellipsoid
(14) 

(15) 
11. This relation between the wavelength, the period, and the direction of vibration, must hold true not only of such vibrations as actually occur, but also of such as we may imagine to occur under the influence of constraints determining the direction of vibration in the waveplane. The directions of the natural or unconstrained vibrations in any waveplane may be determined by the general mechanical principle that if the type of a natural vibration is infinitesimally altered by the application of a constraint, the value of the period will be stationary.^{[2]} Hence, in a system of stationary waves such as we have been considering, if the direction of an unconstrained vibration is infinitesimally varied in its waveplane by a constraint while the wavelength remains constant, the period will be stationary. Therefore, if the direction of the unconstrained vibration is infinitesimally varied by constraint, and the period remains rigorously constant, the wavelength will be stationary. Hence, if we make a central section of the above described ellipsoid parallel to any waveplane, the directions of natural vibration for that waveplane will be parallel to the radii vectores of stationary value in that section, viz., to the axes of the ellipse, when the section is elliptical, or to all radii, when the section is circular.
12. For light of a single period, our hypothesis has led to a perfectly definite result, our equations expressing the fundamental laws of double refraction as enunciated by Fresnel. But if we ask how the velocity of light varies with the period, that is, if we seek to derive from the same equations the laws of the dispersion of colors, we shall not be able to obtain an equally definite result, since the quantities etc., and etc., are unknown functions of the period. If, however, we make the assumption, which is hardly likely to be strictly accurate, but which may quite conceivably be not far removed from the truth, that the manner in which the general or average flux in any small part of the medium distributes itself among the molecules and intermolecular spaces is independent of the period, the quantities etc., and etc., will be constant, and we obtain a very simple relation between V and p, which appears to agree tolerably well with the results of experiment.
If we set
(16) 
(17) 
(18) 
and 
(19) 
13. If we now give up the presumably inaccurate supposition that etc., and etc., are constant, equation (19) will still subsist, but and will not be constant for a given direction of oscillation, but will be functions of or, what amounts to the same, of Although we cannot therefore use the equation to derive a priori the relation between and we may use it to derive the values of and from the empirically determined relation between and To do this, we must make use again of the general principle that an infinitesimal variation in the type of a vibration, due to a constraint, will not affect the period. If we first consider a certain system of stationary waves, then a system in which the wavelength is greater by an infinitesimal (the direction of oscillation remaining the same), the period will be increased by an infinitesimal and the manner in which the flux distributes itself among the molecules and intermolecular spaces will presumably be infinitesimally changed. But if we suppose that in the second system of waves there is applied a constraint compelling the flux to distribute itself in the same way among the molecules and intermolecular spaces as in the first system (so that shall be the same functions as before of —a supposition perfectly compatible with the fact that the values of are changed), this constraint, according to the principle cited, will not affect the period of oscillation. Our equations will apply to such a constrained type of oscillation, and etc., and etc., and therefore and will have the same values in the last described system of waves as in the first system, although the wavelength and the period have been varied. Therefore, in differentiating equation (18), which is essentially an equation between and or its equivalent (19), we may treat and as constant. This gives
 (20) 
By means of these equations, the ratios of the statical energy (), the kinetic energy due to the regular part of the flux (), and the kinetic energy due to the irregular part of the flux (), are easily obtained in a form which admits of experimental determination. EquationS (8) and (9) give

(21) 
(22) 
(23) 
14. It remains to consider the relations between the optical properties of a medium and the planes or axes of symmetry which it may possess. If we consider the statical energy per unit of volume () and the period as constant, we may regard equation (2) as the equation of an ellipsoid, the radii vectores of which represent in direction and magnitude the amplitudes of systems of waves having the same statical energy. In like manner, if we consider the kinetic energy of the irregular part of the flux per unit of volume () and the period as constant, we may regard equation (9) as the equation of an ellipsoid, the radii vectores of which represent in direction and magnitude the amplitudes of systems of waves having the same kinetic energy due to the irregular part of the flux. These ellipsoids, which we may distinguish as the ellipsoids ( etc.) and ( etc.), as well as the ellipsoid before described, which we may call the ellipsoid ( etc), must be independent in their form and their orientation of the directions of the axes of coordinates, being determined entirely by the nature of the medium and the period of oscillation. They must therefore possess the same kind of symmetry as the internal structure of the medium.
If the medium is symmetrical about a certain axis, each ellipsoid must have an axis parallel to that. If the medium is symmetrical with respect to a certain plane, each ellipsoid must have an axis at right angles to that plane. If the medium after a revolution of less than 180° about a certain axis is then equivalent to the medium in its first position, or symmetrical with it with respect to a plane at right angles to that axis, each ellipsoid must have an axis of revolution parallel to that axis. These relations must be the same for light of all colors, and also for all temperatures of the medium.
15. From these principles we may infer the optical characteristics of the different crystaDographic systems.
In crystals of the isometric system, as in amorphous bodies, the three ellipsoids reduce to spheres. Such media are optically isotropic at least so far as any properties are concerned which come within the scope of this paper.
In crystals of the tetragonal or hexagonal systems, the three ellipsoids will have axes of rotation parallel to the principal crystallographic axis. Since the ellipsoid ( etc.) has but one circular section, there will be but one optic axis, which will have a fixed direction.In crystals of the orthorhombic system, the three ellipsoids will have their axes parallel to the rectangular crystallographic axes. If we take these directions for the axes of coordinates, will vanish and equation (13) will reduce to
But since the lengths of the axes of the ellipsoid ( etc.) vary with the period, it may easily happen that the order of the axes with respect to magnitude is not the same for all colors. In that case, the optic axes for certain colors will lie in one of the principal planes, and for other colors in another. For the color at which the change takes place, the two optic axes will coincide. The differential coefficient becomes infinitely great as the optic axes approach coincidence.
In crystals of the monodinic system, each of the three ellipsoids will have an axis perpendicular to the plane of symmetry. We may choose this direction for the axis of X. Then will vanish and equation (13) will reduce to
It is evident that in this system the plane of the optic axes will be fixed, or will rotate about one of the lines which bisect the angles made by the optic axes, according as the mean axis of the ellipsoid ( etc.) is perpendicular to the plane of symmetry or lies in that plane. In the first case the dispersion of the two optic axes will be unequal. The same crystal, however, with light of different colors, or at different temperatures, may afford an example of each case.
In crystals of the triclinic system, since the ellipsoids ( etc.) and ( etc.) are determined by considerations of a different nature, and there are no relations of symmetry to cause a coincidence in the directions of their axes, there will not in general be any such coincidence. Therefore the three axes of the ellipsoid ( etc.), that is, the two lines which bisect the angles of the optic axes and their common normal, will vary in position with the color of the light.
16. It appears from this foregoing discussion that by the electromagnetic theory of light we may not only account for the dispersion of colors (including the dispersion of the lines which bisect the angles of the optic axes in doubly refracting media), but may also obtain Fresnel's laws of double refraction for every kind of homogeneous light without neglect of the quantities which determine the dispersion of colors.
But a closer approximation than that of this paper will be necessary to explain the phenomena of circularly polarizing media, which depend on very minute differences of wavevelocity, represented perhaps by a few units in the sixth significant figure of the index of refraction. That the degree of approximation which will give the laws of circular and elliptic polarization will not add any terms to the equations of this paper, except such as vanish for media which do not exhibit this phenomenon, will be shown in another number of this Journal.
 ↑ The fluxes are supposed to be measured by the electromagnetic system of units. It is to be observed that the difference of opinion which has prevailed with respect to the estimation of the energy of electrical currents does not extend to such as are solenoidal, which may be regarded as composed of closed circuits.
 ↑ See Rayleigh's Theory of Sound, voll. i, p. 84.