Frequency-comb formation in doubly resonant second-harmonic generation

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.