Snell's law revisited and generalized via Finsler Geometry

Abstract

We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted -- broken -- as they pass through the interface, due to the discontinuity of their velocities. This "breaking" must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell's law. In the same way, optimal paths bouncing off the interface -- without crossing into the second domain -- provide the generalized law of reflection. The classical Snell's and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci -- stemming from the self-intersection of the wavefronts -- which typically appear when they approach a high-speed profile domain from a low-speed profile.