A Fundamental Property of Anisotropic Media
Until now we have confined our attention to the propagation of light in isotropic media, i.e., substances whose optical properties are the same in all directions. Liquids, as well as amorphous solid substances such as glass and plastics, are usually isotropic because of the random distribution of the molecules. In many crystals, the optical as well as the other physical properties are different in different directions. This optical anisotropy, often referred to as double refraction, or birefringence, is due to the particular arrangement of the atoms in the crystalline lattice and, is found to produce many curious and interesting phenomena, which we propose now to investigate.
We start with a simple experiment. A parallel beam of monochromatic light passes through a polariscope formed, for example, by two sheet polarizers, and then falls upon a screen. We rotate the analyzer until the light spot on the screen disappears. The transmission axis of the analyzer is then perpendicular to that of the polarizer i.e., the polarizer and the analyzer are crossed). Between the analyzer and the polarizer we now insert a thin, plane-parallel plate cut from a birefringent crystal obtained by cleavage. The light on the screen will, in general, reappear. The analyzer being rotated, the light intensity will change periodically between a maximum and a minimum, but will not become zero for any position of the analyzer. We thus conclude that the light emerging from the plate is no longer linearly polarized.
After removing the plate, we again place the analyzer and the polarizer in the crossed position, reinsert the birefringent plate, and rotate it in its own plane. For each complete turn, we find four positions, at 90° to one another, for which the light spot on the screen disappears. We conclude that the light now emerging from the plate has the same linear polarization as the light incident upon the plate. We can check this conclusion by rotating the analyzer and noting that the corresponding variation of the transmitted light intensity follows the law of Malus. It is thus possible to trace on the plate two mutually perpendicular lines such that a linearly polarized light wave vibrating in a direction parallel to either line traverses the plate without changing its state of polarization. We сall these lines the axes of the plate.
By generalizing this result, we can describe the fundamental property of optically anisotropic medium as follows: for every direction of propagation there are only two waves vibrating in one or the other of two mutually perpendicular planes that preserve their state of polarization while traveling through the medium.
Consider now a wave which, upon entering the plate, is linearly polarized, but does not vibrate in either of the two preferred directions. We may regard the incident wave as the superposition of two linearly polarized waves vibrating in the two preferred directions. If the velocities of propagation of these two waves were the same, the two component waves after traversing the plate would recombine into a linearly polarized wave with the same plane of vibration as the incident wave. Since we know from experiment that this is not the case, i.e. since we know the state of polarization of the wave to change on traversing the plate, we conclude that the velocities of propagation in an anisotropic medium of the waves vibrating in the two preferred directions are different. We can, of course, check this conclusion directly by measuring (e.g. with an interferometer) the velocities of propagation through a birefringent plate of the two waves whose planes of vibration contain one or the other of the two axes of the plate.
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