We theoretically modeled and experimentally realized the generation of quadratic frequency combs in a cw-pumped, nearly degenerate, doubly resonant optical parametric oscillator (OPO) [Mos18].

We derived a time-domain mean-field equation that governs the evolution of the slowly varying field amplitude at the parametric frequency. The mean field equation is a generalization of the dissipative nonlinear Schrödinger equation, with a parametric driving term and a noninstantaneous interaction term, admitting a trivial zero solution and a nontrivial time independent solution. By means of a linear stability analysis we found that both solutions can exhibit modulation instability (MI) gain, which reinforces random fluctuations, eventually leading to the formation of frequency combs. Indeed, while the instability of the constant solution does depend on the temporal walk-off, the instability of the trivial zero solution is rather induced by the group-velocity dispersion. Figure 1 shows the calculated MI gain for the constant finite (left) and zero (right) solutions, as a function of the offset frequency with respect to the degeneracy frequency and the laser-to-cavity detuning. For positive detunings, both solutions display MI gain, with frequency symmetric pairs of gain maxima. For zero and negative detunings, the zero solution exhibits MI gain, with peak gain at Ω = 0.

Fig.1: (left) MI gain of the nonzero constant solution and (right) trivial zero solution, as a function of the offset frequency Ω/2π and the detuning Δ.

The degenerate OPO is based on a lithium niobate nonlinear crystal placed in a bow-tie optical cavity, pumped by a frequency doubled cw Nd:YAG laser. Comb emission was experimentally realized both around the visible pump frequency of the OPO and the parametric region around the degeneracy frequency, as we confirmed by recording the beat notes at the cavity free spectral range (FSR), both in the visible and infrared region. We also investigated the effect of the cavity detuning on the comb spectra. Figure 2(a)-(c) show experimental comb spectra recorded for 300 mW of pump power and three different detunings. Experimental spectra for negative and zero detunings are very similar, displaying a 1 FSR line spacing, while, for the positive detuning, the spectrum consists of two pairs of symmetric lines. Corresponding spectra obtained from numerical simulations obtained from the mean field equation are plotted in Fig. 2(d)-(f), showing a very good with experimental observations.

Finally, in our study, we also performed a numerical analysis of intracavity fields temporal profiles, showing that frequency combs can exhibit a high degree of coherence for a wide range of pump powers and detunings [Mos18]. In particular, we simulated the comb generated in the correspondence of an input pump power of 300 mW and detuning Δ = –2. The results of our simulations revealed a nonlinear compensation of the walk-off induced linear spectral phase, an almost perfect correlation over a 100 μs time scale for all the comb teeth emerging from the noise floor, and the antisymmetrization of the phase profiles around the degeneracy frequency.

Fig.2: Normalized infrared OPO spectra for cavity detunings Δ = −0.3, 0, +0.3, respectively. Comparison between experimental data (a), (b), (c) and corresponding numerically calculated spectra (d), (e), (f).

[Mos18] S. Mosca, M. Parisi, I. Ricciardi, F. Leo, T. Hansson, M. Erkintalo, P. Maddaloni, P. De Natale, S. Wabnitz and M. De Rosa, Phys. Rev. Lett. 121, 093903 (2018).