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

TL;DR

This paper investigates the nonlinear dynamics and modulation instability in Kerr Fabry-Perot resonators, deriving new analytical expressions and extending the Lugiato-Lefever equation to better understand spectral band generation and regimes beyond the mean field limit.

Contribution

It introduces an extended FP-LLE derived from coupled nonlinear Schrödinger equations, capturing regimes beyond the mean-field approximation, including period-two modulation instability.

Findings

The pump pulse duration significantly influences stationary solutions and gain spectrum.

The extended FP-LLE accurately predicts certain regimes not captured by the mean-field limit.

Numerical simulations confirm the presence of MI-induced Kerr combs and novel instability regimes.

Abstract

We analyse 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 characterise the generation of new spectral bands by deriving the exact dispersion relation and we find approximate analytical expressions for the instabilities 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 dramatically 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 (rather than Maxwell-Bloch 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.