Dynamics of multimode Fabry-Perot lasers: A nonlinear analysis

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We apply the multiple time scale method to perform a nonlinear analysis of the Tang, Statz, and deMars rate equations, which describe an $N$-mode Fabry-Perot laser in which all modes have an equal gain and a large loss rate. In addition to the two relaxation oscillation frequencies ${\ensuremath{\Omega}}_{L}$ and ${\ensuremath{\Omega}}_{R}$ known from the linearized analysis, we find the four frequencies $2{\ensuremath{\Omega}}_{L},$ ${\ensuremath{\Omega}}_{R}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{L},$ and $2{\ensuremath{\Omega}}_{R}.$ The signature of antiphased dynamics for the new frequencies is that there are no relaxation oscillations in the total intensity at ${\ensuremath{\Omega}}_{R}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{L}.$ The laser steady state is shown to be stable, being characterized by damping rates derived explicitly. Relations among these damping rates are obtained. We also study the role played by the initial condition in governing the manifestation of the antiphase dynamics and the relative magnitude of the modal intensity power spectrum peak heights at the two main frequencies ${\ensuremath{\Omega}}_{L}$ and ${\ensuremath{\Omega}}_{R}.$ Finally, we deal with the resonant case ${\ensuremath{\Omega}}_{R}=2{\ensuremath{\Omega}}_{L}.$ In this case, inphased dynamics is shown to appear at $2{\ensuremath{\Omega}}_{R},$ instead of at ${\ensuremath{\Omega}}_{R}.$