Coupled-mode theory of the polarization dynamics inside a microring resonator with a uniaxial core

Abstract

The use of a uniaxial birefringent material such as lithium niobate $({\mathrm{LiNbO}}_{3},\phantom{\rule{4pt}{0ex}}\mathrm{LN})$ as the core of a microring resonator leads to coupling between transverse-electric and transverse-magnetic modes. We develop a theoretical framework to study polarization evolution inside such a resonator. We write Maxwell's equations in the form of a Schr\"odinger equation and use it to obtain coupled-mode equations modeling the continuous reorientation of the optic axis as light propagates inside the microring resonator. We show that the mode-coupling problem is isomorphic to a quantum-mechanical two-level system modulated in frequency and driven by a classical optical field. We analyze the polarization coupling by using the well-known techniques of quantum mechanics such as time-dependent perturbation theory, the rotating-wave approximation, and the adiabatic approximation. As an example, we consider a LN ring resonator and describe the evolution of the state of polarization of injected light along the ring's circumference.