Coupled-mode theory for astrophotonics

Abstract: Coupled-mode theory (CMT) is a powerful tool for simulating near-harmonic systems. In telecommunications, variations of the theory have been used extensively to study waveguides, both analytically and through numerical modelling. Analogous mathematical techniques to the CMT are also widely used in quantum mechanics. The purpose of this work is to collect different formulations of the CMT and their underlying connections to quantum mechanical techniques, and to showcase their utility in modelling slowly varying waveguides including directional couplers and photonic lanterns. My choice of example waveguides is motivated by the astronomical applications of such devices in starlight nulling, wavefront sensing, and high-resolution spectroscopy. I first provide a brief review of the standard form of the CMT, applicable for waveguides with fixed eigenmodes. Next, I show that the CMT also applies for slowly varying waveguides, and demonstrate the close relation between the CMT and several well-known approximation methods from quantum mechanics, as well as concepts like geometric phase. Finally, I present a verification of my analysis, in the form of the numerical package cbeam.