Modelling of energy-dependent nonlinearities in microresonator filtered lasers

Laser cavity-solitons (LCSs) are characteristic states of microresonator-filtered lasers [1] and represent one of the most promising solutions for implementing optical frequency combs in such systems [2]–[4]. Mean-field models are powerful mathematical frameworks that provide an effective and comprehensive description of the key features of cavity solitons [5], [6]. In this work, we present a general approach to effectively incorporate localised losses and energy-dependent, slow-time effects (i.e. thermal detuning and gain saturation), in the mean field model for LCS starting from general principles. Specifically, we employ a multiple-scale approach to integrate gain effects derived from the Maxwell-Bloch equations and thermal decay, linking key parameters such as the amplification gain and the common frequency detuning between the microresonator and the laser cavity modes to reach a set of ordinary differential equations. We demonstrate that the inclusion of these equations effectively implements an energy-based feedback mechanism [4].