Transition from traveling to motionless pulses in semiconductor lasers with saturable absorber

Semiconductor lasers show interesting pulse dynamics. We investigate, analytically and numerically, the transition from traveling to motionless pulses in vertical-cavity semiconductor lasers with a saturable absorber. Based on two different approaches, (i) the adiabatic elimination of the charge carriers corresponding to a class A laser and (ii) considering the laser dynamics close to lasing instability (class B lasers), we figure out the relationship between these pulses. The pulses exhibit a continuous transition between traveling to motionless ones. Starting from the asymptotic behavior of the pulse, we are able to set a formula for the pulse speed. In the limit of a small electric field envelope, we elucidate that the observed transition corresponds to the spontaneous breaking instability of reflection symmetry. Employing a reduction method, we set equations for the position, asymmetrical amplitude, and frequency of the pulse. This reduced pulse model shows quite fair agreement with numerical simulations for the different approaches considered for semiconductor lasers with a saturable absorber.