Accumulating regions of winding periodic orbits in optically driven lasers

We investigate the route to locking in class B lasers subject to optically injected light for injection strengths and detunings near a codimension-two saddle-node Hopf point. This is the parameter region where the Adler approximation is not valid and where Yeung and Strogatz recently reported a self-similar cascade of periodic orbits in the case of a solid-state laser. We explain this cascade as an accumulation of large regions bounded by saddle-node bifurcations of periodic orbits, but also containing further bifurcations, such as period doubling, torus bifurcations and small pockets of chaos. In the vicinity of the simultaneous saddle-node and Hopf bifurcations, successive periodic orbits wind more and more near the point in phase space where the saddle-node bifurcation is about to occur. This leads to a self-similar period-adding cascade. By varying the linewidth enhancement parameter alpha from zero, the case of a solid state or CO2 laser, to values larger than one, the case of semiconductor lasers, we show how the accumulating regions of winding periodic orbits change rapidly. This explains why the period-adding cascade has not been found in injected semiconductor lasers. Moreover, we are able to identify certain regions with complex dynamics in injected semiconductor lasers as `remains' of the accumulating regions.