Combs, Fast and Slow: Non‐Adiabatic Mean‐Field Theory of Active Cavities

Integrated frequency combs in active cavities are appealing for a broad spectrum of applications. A powerful framework for describing these cavities is mean‐field theory, which captures the averaged effect of internal dynamics over 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 can also arise in slower gain media. An operator‐based mean‐field theory is developed that non‐adiabatically describes the dynamics of bidirectional active cavities, both fast and slow. This first‐principles formalism provides a semi‐exact replacement for the Maxwell–Bloch equations and accommodates non‐trivial gain lineshapes and population dynamics. As an example, this formalism is used to establish an additional constraint on the formation of frequency‐modulated combs. These results are broadly applicable to bidirectional and unidirectional active cavities alike (including both Fabry–Pérot and ring cavities), and they extend naturally to nearly any chip‐scale laser system.