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

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 <i>a priori</i> 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.