Walk-Off-Induced Modulation Instability, Temporal Pattern Formation, and Frequency Comb Generation in Cavity-Enhanced Second-Harmonic Generation

Abstract

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 ...

Dynamics of microresonator frequency comb generation: models and stability

“…Wideband comb generation from such integrated sources is currently not possible due to high pump power requirements, but the use of novel materials with large Kerr nonlinearities may be exploited in order to help reduce the threshold power. Another promising avenue of exploration is the use of quadratic nonlinearities [63,64], which, according to preliminary estimates, may increase the efficiency and reduce the pump power requirement for frequency comb generation even further.…”

Frequency comb generation in a continuously pumped optical parametric oscillator

“…The symbol ⊗ denotes convolution and I(τ) is the same delayed response already derived for the comb dynamics in singly resonant cavity SHG [4,5]. Stability analysis of Eq.…”

“…1(a) and (b) display a resolved comb structure made of lines separated by about 20 GHz (RBW= 3 GHz). Starting from the coupled mode equations that describe the propagation of the amplitude envelopes of the pump and parametric fields, along with the boundary condition for the fields at the end of each round trip [4], we derived the following mean field equation for the parametric field amplitude A…”

Nonlinear dynamics of optical frequency combs

“…Figure 3 provides a schematic illustration of optical frequency comb generation around the FF and SH field in a singly resonant quadratic microresonator. 16,18 As can be seen, the FF driving field A in is coupled into the microring and recirculates in the cavity, whereas the SH field that is generated during propagation through the ring is fully out-coupled at each round-trip. This means that θ 2 = 1, hence B m+1 (0, τ ) = 0 in Eq.…”

“…In this case, the nonlinear response function of Eq. (8) reads as 16,18 The use of a single cavity averaged Eq. (8) strongly facilitates the study of the CW solutions for the intracavity fundamental field intensity, and its MI gain profile.…”