On the generalized Snell–Descartes laws, shock waves, water wakes, and Cherenkov radiation

Abstract

The modification of light’s trajectory after refracting through a boundary separating two media is a ubiquitous phenomenon in nature. The laws governing such refraction/reflection, known today as the Snell–Descartes laws of reflection and refraction, were established over four centuries ago and have since become foundational to the field of classical optics. Presently, with the advent of nano-photonic technology, a generalized version of these laws has been developed and implemented, vastly broadening the breadth of light manipulation methods. Despite their popularity, however, a simple and accessible derivation of the Snell–Descartes laws is still lacking, and their generalization is still largely missing from the physics curricula. Here, we use simple analogies between light’s refraction and reflection and other a priori unrelated radiating wave systems, namely, shock waves, water wakes, and Cherenkov radiation to derive both the classical and generalized Snell–Descartes laws, relying solely on simple and intuitive arguments. The basis of the derivation considers the excitation of a surface perturbation, induced by light incident at an angle on a boundary, that propagates at a velocity exceeding the phase velocity of light in the medium. The perturbation thereafter acts as a radiative source that reflects and refracts light away from the interface, at angles satisfying the classical Huygens interference condition. These derivations are meant to be accessible to a broad range of readers, including students of all levels, middle/high school teachers, and beyond.