Theoretical analysis of semiconductor ring lasers with short and long time-delayed optoelectronic and incoherent feedback

We report results on the asymptotic, bifurcation and numerical analysis of semiconductor ring lasers with negative optoelectronic feedback or incoherent optical feedback. We find that the dynamical behavior of both systems can be adequately described by two differential equations and one map with time delay on time-scales longer than the relaxation oscillations. On these timescales, the dynamics for both types of feedback is identical. We study the stationary solutions of this model and are able to obtain the analytical expressions of all local bifurcations. As we vary the feedback strength for small delay times comparable to the period of relaxation oscillations, the devices under consideration in the paper display both continuous wave operation and a period-doubling route to chaos. The intensities of two counter propagating modes of both systems exhibit in-phase chaotic behavior similar to single mode semiconductor lasers. However, for long delay times, the counter propagating modes show anti-phase chaotic oscillations. This anti-phase chaotic regime does not involve carrier dynamics and is a result of the bistable character of the semiconductor ring laser.