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 ${\chi ^{(2)}}$ induced cascaded Kerr nonlinearity, in the regime of imperfect phase-matching, puts it above the intrinsic Kerr effect by several orders of magnitude.

中文翻译：

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具有二次非线性的微谐振器的耦合模式理论

我们使用麦克斯韦方程组推导了几个模型，这些模型描述了具有二次非线性和准相位匹配的环形微谐振器中多模基本场及其二次谐波的相互作用。我们演示了如何通过场包络乘积的傅立叶变换来计算进入非线性极化响应的多模三波混合总和。具有任意轮廓的准相位匹配光栅被无缝集成到我们的模型中。我们还介绍了几种近似值，它们使我们能够解决非线性系数的离散问题，并演示如何将耦合模式方程式简化为具有自增强项的包络Lugiato–Lefever方程式。$ {\ chi ^ {（2）}} $的估算值 在不完全相位匹配的情况下，Kerr引起的级联Kerr非线性使它比固有Kerr效应高出几个数量级。