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

References in corpus (12)

• Monolithic Ultrahigh-Q Lithium Niobate Microring Resonator
• Stably accessing octave-spanning microresonator frequency combs in the soliton regime
• Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators
• Frequency comb generation in quadratic nonlinear media
• Soliton and quasi-soliton frequency combs due to second harmonic generation in microresonators
• A self-starting bi-chromatic LiNbO3 soliton microcomb
• Inter-mode breather solitons in optical microresonators
• A hierarchy of coupled mode and envelope models for bi-directional microresonators with Kerr nonlinearity
• Multistability and coexisting soliton combs in ring resonators: the Lugiato-Lefever approach
• The multi-resonant Lugiato-Lefever model
• Soliton trapping and comb self-referencing in a single microresonator with ${\chi }^{(2)}$ and ${\chi }^{(3)}$ nonlinearities
• Nonlinear solutions for χ^(2) frequency combs in optical microresonators