Abstract

The ratio of solid angles of a beam on the two sides of a surface separating different media is calculated for arbitrary anisotropy of the media, for a surface of arbitrary orientation, and for the beam in an arbitrary direction. This ratio, needed for comparison of theories of scattering of light inside a crystal with experiments performed outside, is obtained by developing a series of optical invariants for anisotropic media. The development makes use only of Snell’s law and the differential geometry of the ω( k) surface. The results are therefore applicable to sound waves, and to other wave phenomena as well as to light waves. The solid angle of ray vectors dΩr (as distinct from the solid angle of k vectors) changes as the beam crosses a surface in such a way as to preserve the invariant dΩr cosβ/K, where β is the angle between ray and surface normals and K is the gaussian curvature of the ω( k) surface for the k that corresponds to the ray direction. When transmission losses are neglected, another invariant across the surface is LK, where L is the radiance (brightness). Additional invariants are also derived.

© 1975 Optical Society of America

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