Period locking due to delayed feedback in a laser with saturable absorber

We consider laser with saturable absorber operating in the pulsating regime that is subject to delayed feedback. Alone, both the saturable absorber and delayed feedback cause the clockwise output to become unstable to periodic output via Hopf bifurcations. The delay feedback causes the laser pulse period to lock to an integer fraction of the feedback time. We derive a map from the original model to describe the periodic pulsations of the laser. Equations for the period of the laser predict the occurrence of the different locking states as well as the value of the pump when there is a switch between the locked states.