Instantaneous modes in dispersive laser cavities

We develop a unified instantaneous-mode description for lasers with dispersive cavities, exploiting the separation of timescales between fast cavity fields and slow carrier dynamics. The resulting reduced rate equations retain the essential effects of frequency-dependent mirrors through a dynamic modal gain and an effective confinement factor determined directly by the mirror reflectivity. Applied to a Fano laser, the reduced description captures the essential dynamics, closely reproduces the full-model behavior and clarifies the physical origin of dispersive instabilities. More generally, the approach provides a transparent framework for deriving reduced models and simplifying the stability analysis of dispersive laser cavities.