Combs, fast and slow: non-adiabatic mean field theory of active cavities

Integrated frequency combs based on active cavities are of interest for a wide range of applications. An elegant description of these cavities is based on mean-field theory, which averages the effect of internal dynamics occurring within a round trip. Lasers based on media with slow gain dynamics can be described by solving the population over many round trips, while lasers based on fast gain media can be described by adiabatic elimination. However, most gain media actually have both fast and slow components, and effects often ascribed to fast gain media are known to arise even in slower gain media. Here, we develop an operator-based mean-field theory that non-adiabatically describes the dynamics of bidirectional active cavities, both fast and slow. It is based on first principles and semi-exactly replaces the Maxwell-Bloch equations, but is flexible enough to accomodate non-trivial lineshapes and population dynamics. As an example, we use this formalism to establish an additional constraint on the formation of frequency-modulated combs. Our results are general and apply to any bidirectional or unidirectional active cavity, and as a result, generalize to essentially any chip-scale laser.