Coupled-mode theory for microresonators with quadratic nonlinearity

TL;DR

This paper develops coupled-mode models for quadratic nonlinear microresonators, incorporating quasi-phase-matching and dispersion effects, and estimates the strength of cascaded Kerr nonlinearities induced by second-order interactions.

Contribution

It introduces comprehensive models for quadratic nonlinear microresonators with arbitrary quasi-phase-matching profiles, including approximations and reduction to envelope equations.

Findings

Models incorporate arbitrary quasi-phase-matching profiles.

Coupled-mode equations can be reduced to Lugiato-Lefever-like equations.

Cascaded Kerr nonlinearity exceeds intrinsic Kerr effect significantly.

Abstract

We use Maxwell's equations to derive several models describing the interaction of the multi-mode fundamental field and its second harmonic in a ring microresonator with quadratic nonlinearity and quasi-phase-matching. We demonstrate how multi-mode three-wave mixing sums entering nonlinear polarisation response can be calculated via Fourier transforms of products of the field envelopes. Quasi-phase-matching gratings with arbitrary profiles are incorporated seamlessly into our models. We also introduce several levels of approximations allowing to account for dispersion of nonlinear coefficients and demonstrate how coupled-mode equations can be reduced to the envelope Lugiato-Lefever-like equations with self-steepening terms. An estimate for the $\chi^{(2)}$ induced cascaded Kerr nonlinearity, in the regime of imperfect phase-matching, puts it above the intrinsic Kerr effect by several orders of magnitude.