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.

Interpretation Methods for Seismic Downhole Test in Inclined Boreholes

“…The DM, IM, and MIM methods assume that the seismic ray path is in a straight line through the bedded medium and that the horizontal distance from the seismic source to the receivers (along the borehole) is much smaller than the depth of each receiver. However, seismic waves generated at the surface travel through soil strata with different stiffnesses and therefore refract at each interface according to Snell's law and Huygens' principle [20]. Therefore, many researchers recommend caution when using these methods in heterogeneous media or with a higher impedance contrast [21].…”

Numerical MCRT Simulation and Inter-Symbol Interference Analysis of Laser Multipath Transmission in Airborne Air-to-Air Channels

“…Combined with Snell's theorem, this gives [18] Figure 3. Schematic diagram of photons' interaction with turbulent bubbles.…”