Generalized Theory of Optical Resonator and Waveguide Modes and their Linear and Kerr Nonlinear Coupling

TL;DR

This paper develops a comprehensive, first-principles theory for linear and Kerr nonlinear coupling in dielectric optical resonators and waveguides, applicable to various phenomena like frequency combs and scattering losses.

Contribution

It introduces a general, geometry-independent framework for mode coupling and nonlinear interactions, extending understanding of resonator-waveguide systems and symmetry-breaking dynamics.

Findings

Derived a universal theory for mode coupling and Kerr nonlinearities.

Calculated evanescent coupling strengths for waveguide-resonator systems.

Proved the specific relation between self- and cross-phase modulation in symmetric modes.

Abstract

We derive a general theory of linear coupling and Kerr nonlinear coupling between modes of dielectric optical resonators from first principles. The treatment is not specific to a particular geometry or choice of mode basis, and can therefore be used as a foundation for describing any phenomenon resulting from any combination of linear coupling, scattering and Kerr nonlinearity, such as bending and surface roughness losses, geometric backscattering, self- and cross-phase modulation, four-wave mixing, third-harmonic generation and Kerr frequency comb generation. The theory is then applied to a translationally symmetric waveguide in order to calculate the evanescent coupling strength to the modes of a microresonator placed nearby, as well as the Kerr self- and cross-phase modulation terms between the modes of the resonator. This is then used to derive a dimensionless equation describing the symmetry-breaking dynamics of two counterpropagating modes of a loop resonator and prove that cross-phase modulation is exactly twice as strong as self-phase modulation only in the case that the two counterpropagating modes are otherwise identical.