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We theoretically study the generation of optical frequency combs and corresponding pulse trains in doubly resonant intracavity second-harmonic generation (SHG). We find that, despite the large temporal walk-off characteristic of realistic cavity systems, the nonlinear dynamics can be accurately and efficiently modeled using a pair of coupled mean-field equations. Through rigorous stability analysis of the system's steady-state continuous-wave solutions, we demonstrate that walk-off can give rise to an unexplored regime of temporal modulation instability. Numerical simulations performed in this regime reveal rich dynamical behaviors, including the emergence of temporal patterns that correspond to coherent optical frequency combs. We also demonstrate that the two coupled equations that govern the doubly resonant cavity behavior can, under typical conditions, be reduced to a single mean-field equation akin to that describing the dynamics of singly-resonant-cavity SHG [F. Leo et al., Phys. Rev. Lett. 116, 033901 (2016)]. This reduced approach allows us to derive a simple expression for the modulation instability gain, thus permitting us to acquire significant insight into the underlying physics. We anticipate that our work will have a wide impact on the study of frequency combs in emerging doubly resonant cavity SHG platforms, including quadratically nonlinear microresonators.

We derive a time-domain mean-field equation to model the full temporal and spectral dynamics of light in singly resonant cavity-enhanced second-harmonic generation systems. We show that the temporal walkoff between the fundamental and the second-harmonic fields plays a decisive role under realistic conditions, giving rise to rich, previously unidentified nonlinear behavior. Through linear stability analysis and numerical simulations, we discover a new kind of quadratic modulation instability which leads to the formation of optical frequency combs and associated time-domain dissipative structures. Our numerical simulations show excellent agreement with recent experimental observations of frequency combs in quadratic nonlinear media [Phys. Rev. A 91, 063839 (2015)]. Thus, in addition to unveiling a new, experimentally accessible regime of nonlinear dynamics, our work enables predictive modeling of frequency comb generation in cavity-enhanced second-harmonic generation systems. We expect our findings to have wide impact on the study of temporal and spectral dynamics in a diverse range of dispersive, quadratically nonlinear resonators. DOI: 10.1103/PhysRevLett.116.033901 The generation of frequency combs in high-Q microresonators has attracted significant attention over the last decade [1]. These combs originate from the third-order optical (Kerr) nonlinearity [2]: four-wave mixing drives modulation instability (MI), leading to the growth of signal and idler sidebands [3]. In the time domain, such "Kerr" combs can correspond to temporal dissipative patterns or localised structures-temporal cavity solitons [4][5][6].Certain microresonators exhibit a weak second-order χnonlinearity, which may lead to the intracavity conversion of the Kerr comb to shorter wavelengths [7,8]. But recent experiments in free-space resonators have remarkably demonstrated that frequency combs can also arise entirely through χ ð2Þ effects. On the one hand, it is known that cascaded second-harmonic generation (SHG), subject to large phase mismatch, can give rise to an effective Kerr nonlinearity [9,10], and comb generation has indeed been observed under such conditions [11,12]. But on the other hand, frequency combs have recently been observed also for phase-matched cavity-enhanced SHG [13].In cavity-enhanced SHG, comb formation initiates from the spontaneous down-conversion of the second-harmonic field [13], a process widely investigated in the context of internally pumped optical parametric oscillators (OPOs) [14][15][16][17]. However, theoretical analyses have hitherto been limited to a very small number of frequency components: no models have been put forward that would allow for the full frequency comb dynamics to be examined. For quadratically nonlinear, spatially diffractive cavities, full models do exist, and their study has revealed numerous spatial phenomena whose temporal analogs would be associated with frequency combs, including pattern formation [17][18][19] and quadratic cavity solitons [20]. Unfortunately, insights obtained ...

A study is made of frequency-comb generation described by the driven and damped nonlinear Schrödinger equation on a finite interval. It is shown that frequency-comb generation can be interpreted as a modulational instability of the continuous-wave pump mode, and a linear stability analysis, taking into account the cavity boundary conditions, is performed. Further, a truncated three-wave model is derived, which allows one to gain additional insight into the dynamical behavior of the comb generation. This formalism describes the pump mode and the most unstable sideband and is found to connect the coupled mode theory with the conventional theory of modulational instability. An in-depth analysis is done of the nonlinear three-wave model. It is demonstrated that stable frequency-comb states can be interpreted as attractive fixed points of a dynamical system. The possibility of soft and hard excitation states in both the normal and the anomalous dispersion regime is discussed. Investigations are made of bistable comb states and the dependence of the final state on the way the comb has been generated. The analytical predictions are verified by means of direct comparison with numerical simulations of the full equation and the agreement is discussed.

Continuously pumped passive nonlinear cavities can be harnessed for the creation of novel optical frequency combs. While most research has focused on third-order "Kerr" nonlinear interactions, recent studies have shown that frequency comb formation can also occur via second-order nonlinear effects. Here, we report on the formation of quadratic combs in optical parametric oscillator (OPO) configurations. Specifically, we demonstrate that optical frequency combs can be generated in the parametric region around half of the pump frequency in a continuously driven OPO. We also model the OPO dynamics through a single time-domain mean-field equation, identifying previously unknown dynamical regimes, induced by modulation instabilities, which lead to comb formation. Numerical simulation results are in good agreement with experimentally observed spectra. Moreover, the analysis of the coherence properties of the simulated spectra shows the existence of correlated and phase-locked combs. Our results reveal previously unnoticed dynamics of an apparently well assessed optical system, and can lead to a new class of frequency comb sources that may stimulate novel applications by enabling straightforward access to elusive spectral regions, such as the midinfrared.

A study is made of the nonlinear dynamics of bichromatically pumped microresonator Kerr frequency combs described by a driven and damped nonlinear Schrödinger equation, with an additional degree of freedom in the form of the modulation frequency. A truncated four-wave model is derived for the pump modes and the dominant sideband pair, which is found to be able to describe much of the essential dynamical behavior of the full equation. The stability of stationary states within the four-wave model is investigated, and numerical simulations are made to demonstrate that a large range of solutions, including cavity solitons, are possible beyond previously considered low-intensity patterns.

The generation of optical frequency combs in microresonators is considered without resorting to the mean-field approximation. New dynamical regimes are found to appear for high intracavity power that cannot be modeled using the Lugiato-Lefever equation. Using the Ikeda map we show the existence of multi-valued stationary states and analyse their stability. Period doubled patterns are considered and a novel type of super cavity soliton associated with the multi-stable states is predicted.