Coupled-mode theory for microresonators with quadratic nonlinearity

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.

11 pages

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• Photon-photon polaritons in chi(2) microresonators
• Mid-infrared frequency combs and staggered spectral patterns in ${\chi }^{(2)}$ microresonators
• Ladder of Eckhaus instabilities and parametric conversion in chi(2) microresonators
• Carrier-resolved real-field theory of multi-octave frequency combs
• Walk-off induced dissipative breathers and dissipative breather gas in microresonators
• Parametric conversion via second harmonic generation and two-hump solitons in phase-matched microresonators
• Modulation instability and frequency-comb generation in hybrid quadratic-cubic resonators with spectral filtering
• Optical-parametric-oscillation-based ${\chi }^{(2)}$ frequency comb in a lithium niobate microresonator