Relaxation Oscillations and Pulse Stability in Harmonically Mode-Locked Semiconductor Lasers

In this paper, we discuss pulse dynamics in harmonically mode-locked semiconductor lasers and present the conditions necessary for stability. In a laser mode-locked at the
N
th harmonic, the pulse energy fluctuations have (
N
+1) different modes of relaxation oscillations. Different modes correspond to different patterns for the energy fluctuations in the
N
different pulses inside the laser cavity. In the higher order relaxation oscillation modes, the energy fluctuations are negatively correlated in different pulses inside the laser cavity, and these modes can cause instability. Gain saturation on time scales of the order of the pulse width (dynamic gain saturation) stabilizes pulse energy fluctuations with respect to relaxation oscillations. The precise limits on the stable operating regime depend on the gain dynamics at both slow and fast time scales. We also discuss harmonic mode-locking in the presence of a slow saturable absorber. Dynamic loss saturation in a saturable absorber can work against dynamic gain saturation and limit the stability range for harmonic mode-locking.