## 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*(
$\overrightarrow{\text{k}}$) 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 *d*Ω* ^{r}* (as distinct from the solid angle of
$\overrightarrow{\text{k}}$ vectors) changes as the beam crosses a surface in such a way as to preserve the invariant

*d*Ω

*cos*

^{r}*β*/

*K*, where

*β*is the angle between ray and surface normals and

*K*is the gaussian curvature of the

*ω*( $\overrightarrow{\text{k}}$) surface for the $\overrightarrow{\text{k}}$ 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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