  Coherent multimode instabilities are responsible for several phenomena of
recent interest in semiconductor lasers, such as the generation of frequency
combs and ultrashort pulses. These techonologies have proven disruptive in
optical telecommunications and spectroscopy applications. While the standard
Maxwell-Bloch equations encompass such complex lasing phenomena, their
integration is computationally expensive and offers limited analytical insight.
In this paper, we demonstrate an efficient spectral approach to the simulation
of multimode instabilities via a quantitative analysis of the instability of
single-frequency lasing in ring lasers, referred to as the Lorenz-Haken (LH)
instability or the Risken-Nummedal-Graham-Haken (RNGH) instability in distinct
parameter regimes. Our approach, referred to as CFTD, uses generally
non-Hermitian Constant Flux modes to obtain projected Time Domain equations.
CFTD provides excellent agreement with finite-difference integration of the
Maxwell-Bloch equations across a wide range of parameters in regimes of
non-stationary inversion, including frequency comb formation and spatiotemporal
chaos. We also develop a modal linear stability analysis using CFTD to
efficiently predict multimode instabilities in lasers. The combination of
numerical accuracy, speedup, and semi-analytic insight across a variety of
dynamical regimes make the CFTD approach ideal to analyze multimode
instabilities in lasers, especially in more complex geometries or coupled laser
arrays.
