Notes on waveguide localization by slow-wave regions (a.k.a. “total internal reflection”)

A “waveguide” is a structure that allows waves to propagate along one (or more) direction(s) while confining them in one or more other “transverse” directions. These arise in many forms in many different wave equations, with a variety of mechanisms for the transverse confinement: silica optical fibers, sound waves propagating down hollow tubes, coaxial transmission lines, and so forth. One commonplace confinement mechanism is known as “total internal reflection” or “index guiding” in optics [1], where waves are trapped in a region (“core”) of higher refractive index (e.g. the core of an optical fiber) surrounded by a “cladding” of a lower index. It turns out that this mechanism is not specific to optics, and the general principle is that waves can be trapped in a medium where the wave speed (phase velocity) is slower than in the surrounding medium. Moreover, it turns out that we can prove the existence of such localized solutions quite generally—even in cases that we cannot solve analytically—using a “variational” proof that is closely related to proofs of localization (“bound states”) in quantum mechanics (i.e., the Schrödinger equation) [2–4]. In these notes, we demonstrate this approach in the context of a 2d scalar wave equation, but closely related proofs can be applied to other wave equations (e.g. Maxwell’s equations [5, 6]).