Coupled-mode theory for microresonators with quadratic nonlinearity

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 polarization 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 that allow us 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 χ(2) induced cascaded Kerr nonlinearity, in the regime of imperfect phase-matching, puts it above the intrinsic Kerr effect by several orders of magnitude.