Theory of modulation instability in Kerr Fabry-Perot resonators beyond the mean-field limit

We analyze the nonlinear dynamics of Fabry-Perot cavities of arbitrary finesse filled by a dispersive Kerr medium, pumped by a continuous-wave laser or a synchronous train of flat-top pulses. The combined action of feedback, group velocity dispersion, and Kerr nonlinearity leads to temporal instability with respect to perturbations at specified frequencies. We characterize the generation of new spectral bands by deriving the exact dispersion relation, and we find approximate analytical expressions for the instability threshold and gain spectrum of modulation instability (MI). We show that, in contrast to ring resonators, both the stationary solutions and the gain spectrum are significantly affected by the duration of the pump pulse. We derive the extended Lugiato-Lefever equation for the Fabry-Perot resonator (FP-LLE) starting from coupled nonlinear Schr\"odinger equations, and we compare the outcome of the stability analysis of the two models. While FP-LLE gives overall good results, we show regimes that are not captured by the mean-field limit, namely, the period-two modulation instability, which may appear in highly detuned or nonlinear regimes. We report numerical simulations of the generation of MI-induced Kerr combs by solving FP-LLE and the coupled Schr\"odinger equations.