Refraction and Double Refraction

It is hardly too much to say that there is no theory of the optics of metals. There is a general resemblance between their optical and electrical qualities in that the best conductors are the best absorbers, and therefore the best reflectors. But in all cases K is greater than n, and this implies that the dielectric con stant E is negative, which has no meaning in electrical theory.

Other substances are opaque besides metals, quite apart from the opacity due to the repeated scattering of light. Indeed or dinary transparent substances are always opaque for light of some part of the spectrum, and for such light they behave much like metals. In particular, light which is strongly absorbed will be strongly reflected. Rubens took advantage of this fact in his study of "rest-rays," which consist of light in the extreme infra red. For example rock-salt absorbs light of wave-lengths round 5o A, and so reflects it strongly, although it is transparent to other wave-lengths. If then the light from a lamp emitting all wave-lengths is reflected to and fro several times by rock-salt mirrors, the other wave-lengths will be eliminated, and the re flected light will be nearly pure. After the last reflection its wave-length is determined by means of a grating. Unlike the case of metals, here the process of absorption has been fairly completely explained with important consequences for the theory of the solid state.

Double Refraction.

In crystals the atoms are packed to gether in a regular manner, and this packing implies that they will fall into rows in certain directions. Consequently the physical characters of the crystal will differ for different directions, and it is said to be anisotropic. The geometrical theory of crystallo graphy only permits of certain definite types of packing, and these are classified according to the types of symmetry they possess. For purposes of electricity and optics we need to know the way in which the anisotropy will affect the relation between electric force and dielectric displacement. It can be shown that in general the displacement need not be in the same direction as the force, but that there must always be three mutually perpendicular directions in the crystal for which they are in the same direction. We take these directions for our axes, and have But the crystal symmetry may make a further restriction. Thus in the regular system of crystals, the three mutually perpen dicular axes are equivalent to one another, so that all physical properties in these three directions must be the same and there fore El, €2, E3 must be equal. For electrical and optical purposes

therefore, though not for others, the regular system is isotropic. In the hexagonal, tetragonal and trigonal systems there is an axis of 6-, 4- or 3-fold symmetry, and, if this is taken as the z-axis, it follows that El= E2, though they need not equal E3. Calcite and quartz both belong to this type. In all other crystal classes all three E's may be different. We thus have three types of crystal, the regular, the uniaxial and the biaxial. The regular behaves for light as though it were isotropic and we shall deal with the uni axial as a special case of the biaxial (see CRYSTALLIZATION).

For a transparent crystal the electromagnetic equations assume the form together with €,E„, D,= The whole question can be discussed with either E, H or D as the primitive quantity, and of course exactly the same results would emerge, but it is most convenient to take D. This is the light vector used by Fresnel in his original theory, before it was given an electrical meaning. The process of solution consists first in eliminating E and H in terms of D, and then fitting a plane wave of arbitrary direction so as to satisfy the equations for D. If /: m:n are the direction cosines of the wave front and L:M:N those of the light-vector, and if the wave-velocity is V, the wave will be of the form In giving the results of the substitution we shall write 0'2, y2 for Then it is found that the wave-velocity V must satisfy the equation This is a quadratic equation in and we conclude that for a given direction of the wave-front there are two wave-velocities. Associated with each of these values there are definite values of L:M :N, and these determine the polarizations of the two waves. They are at right-angles to one another and to the direction of the wave. A simple example is given by a wave going along the direction of z, where the two velocities are a and and the di rections of polarization x and y. Another example is given by a uniaxial crystal where a= j3. The wave-velocities are then given by Thus one wave has velocity independent of the direction; this is the ordinary wave, and its light-vector lies in the plane per pendicular to the axis. The other, the extraordinary wave, is polarized in a direction contained by the axis and the wave direction, and its velocity depends on the wave direction and ranges between y and a.