Fast and accurate modelling of Kerr-Brillouin combs in Fabry-Perot resonators

We introduce a new mean-field equation for modeling Fabry-Perot resonators filled with a dispersive medium exhibiting both Brillouin and Kerr nonlinearities, e.g. an optical fiber. This model is derived from a unified framework that accounts for Brillouin scattering and four-wave mixing. It involves two coupled nonlinear Schr\"odinger equations for the forward and backward propagating fields, alongside a single equation governing the acoustic oscillation. Under the standard assumptions for the mean-field approach (high finesse, weak nonlinearity, and weak dispersion) we demonstrate that our model closely matches the original system. The simplified and elegant mathematical structure of our equation provides valuable physical insights. As a key example, we derive an expression for the growth rate of harmonic perturbations to the steady states. Additionally, our model facilitates fast and accurate numerical simulations using standard Fourier split-step methods. We highlight the effectiveness of this approach by simulating frequency comb generation in state-of-the-art high-Q fiber Fabry-Perot resonators.