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 d Ω′ (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 d Ω′ 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.
M. Lax and D. F. Nelson, "Radiance theorem and optical invariants in anisotropic media," J. Opt. Soc. Am. 65, 668-675 (1975)