Singly resonant second-harmonic-generation frequency combs

We consider frequency comb generation in dispersive singly resonant second-harmonic-generation cavity systems. Using a single temporal mean-field equation for the fundamental field that features a noninstantaneous nonlinear response function, we model the temporal and spectral dynamics and analyze comb generation, continuous wave bistability, and modulational instability. It is found that, owing to the significant temporal walk-off between the fundamental and second-harmonic fields, modulational instability can occur even in the complete absence of group-velocity dispersion. We further consider the relation of our model to a previously proposed modal expansion approach, and present a derivation of a general system of coupled mode equations. We show that the two models provide very similar predictions and become exactly equivalent in the limit that absorption losses and group-velocity dispersion at the fundamental frequency are neglected. Finally, we perform numerical simulations that show examples of the variety of comb states that are possible in phase-matched quadratic resonators, and discuss the dynamics of the comb formation process.