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 dΩ^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 dΩ^r cosβ/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

Imaging through a surface of an anisotropic medium with application to light scattering

M. Lax and D. F. Nelson
J. Opt. Soc. Am. 66(7) 694-704 (1976)

Fresnel’s and Snell’s laws for the multimode optical waveguide of circular cross section

John D. Love and Allan W. Snyder
J. Opt. Soc. Am. 65(11) 1241-1247 (1975)

Beam wander in a turbulent medium: An application of Ehrenfest’s theorem

Richard J. Cook
J. Opt. Soc. Am. 65(8) 942-948 (1975)

Ewald–Oseen extinction theorem and the diffraction formula for inhomogeneous media, derived from the classical theory of dispersion*

Takashi Suzuki
J. Opt. Soc. Am. 65(6) 679-681 (1975)

Transparency of pair-correlated, random distributions of small scatterers, with applications to the cornea*

Victor Twersky
J. Opt. Soc. Am. 65(5) 524-530 (1975)