Pulsating laser oscillations depend on extremely-small-amplitude noise

Pulsating laser oscillations consist of short and intense pulses separated by long regimes with nearly zero intensities. These oscillations appear if the dimensionless relaxation rate of the inversion of population γ is a small quantity (class-B lasers). During the low-intensity regimes, the laser is particularly sensitive to small-amplitude noise. We investigate the effect of noise by studying the limit-cycle oscillations of the laser with a saturable absorber. Specifically, we analyze how the size of the limit cycle is modified as the amplitude of the noise d is progressively increased from zero. We show that if γ≪1 and d>${\mathit{d}}_{\mathit{c}}$=O(exp(-1/γ)), the size of the limit cycle decreases by an O(1) quantity and depends on d. The mathematical problem is a singular perturbation problem that involves two small parameters (d and γ). We first show that the slow evolution of the limit-cycle solution can be studied as a slow passage through a steady bifurcation point. We then analyze the case of a constant imperfection (small injected signal) and the case of additive noise (small-amplitude Gaussian white noise). Both cases lead to the same conclusions.