Pulse instabilities in harmonic active mode-locking: a time-delayed approach

We propose a time-delayed model for the study of active mode-locking that is devoid of restriction regarding the values of the round-trip gain and losses. It allows us to access the typical regimes encountered in semiconductor lasers and to perform an extended bifurcation analysis. Close to the harmonic resonances and to the lasing threshold, we recover the Hermite-Gauss solutions. However, the presence of the linewidth enhancement factor induces complex regimes in which even the fundamental solution becomes unstable. Finally, we discover a global bifurcation scenario in which a single pulse can jump, over a slow time scale, between the different minima of the modulation potential.