Theory of a multimode semiconductor laser with optical feedback

We discuss the derivation of multimode rate equations for the description of a semiconductor laser with external cavity. We adopt a formulation where the complex field amplitudes are coupled to the nonlinear gains. For N lasing modes, this leads to 2N equations that display in-phased and antiphased time-dependent solutions. A simplified reference model is obtained by assuming that the key parameters are frequency independent. A general linear stability analysis leads to the prediction of two types of Hopf bifurcations. A nondegenerate Hopf bifurcation occurs with the relaxation oscillation frequency as the characteristic bifurcation frequency. A (N-1)-degenerate Hopf bifurcation occurs with a lower characteristic frequency. To assess the nature and stability of the solutions emerging from the Hopf bifurcations, we perform a nonlinear stability analysis on a reduced model obtained in the limit of large linewidth enhancement factor. In this asymptotic limit, the steady state is always destabilized in favor of a stable periodic inphased or antiphased state. A numerical analysis yields a bifurcation diagram of the multimode equations, which confirms the analytic results and reveals further complex regimes (quasiperiodic and chaotic, in-phased and antiphased) as the amplitude of the field fed back into the laser is increased.