Energy partition at the boundary between anisotropic media; Part one: Generalized Snell's Law
Michael A. Slawinski, Raphaël A. Slawinski
The mathematical description of phenomena related to wave propagation in anisotropic media differs significantly from that for isotropic media. In general, the expressions are more complicated and more difficult for intuitive understanding. In the anisotropic case the relationship between the angles of incident, reflected and refracted rays, i.e., Snell's law, cannot be reduced to such a simple form as in the isotropic case. This paper attempts, with aid of familiar, and thus rather intuitive, notions of vector calculus, to provide a framework for calculating these angles. In the anisotropic case there exist the concepts of both phase and group (ray, energy) velocities. The phase-slowness surface, i.e., the inverse of the phase-velocity surface, can be described as a function of three space variables: x, y, and z. By virtue of the continuity conditions across the planar, horizontal boundary between two media, the horizontal components of phase-slowness must be continuous across this boundary. The knowledge of the expression for phase-slowness surfaces in both the incidence and transmission media, the fact that all phase velocities and thus phase-slowness vectors must be coplanar and the enforcement of the continuity conditions form the core of the Snell's law in anisotropic media. The direction of the actual ray is perpendicular to the plane tangent to the phase-slowness surface at a given point and can be mathematically determined using gradients. One must also stipulate that the incident ray points towards the boundary while the reflected and transmitted rays point away from it. This condition can be mathematically described using the properties of the dot product. Implication for critical angles and Fermat's principle are discussed with the aid of analytically derived expressions.
There exist efficient, numerical schemes for calculating the angles of incidence, reflection and transmission. The merit of the present approach is believed to lie in the clarity of the analytical method. A method allowing the description of phase-slowness surfaces corresponding to various symmetry systems as a function of three space variables x, y and z would extend the usefulness of the presented approach.