Low pump limit of the bifurcation to periodic intensities in a semiconductor laser subject to external optical feedback

The conditions for a bifurcation to periodic intensities (Hopf bifurcation) for low values of the pump current are examined using singular perturbation methods. We show that the frequency of the oscillations at the bifurcation point remains close to the relaxation oscillation frequency of the solitary laser until the pump parameter is close to its effective threshold value. This part of our analysis adds substance to previous estimates of the frequency of the oscillations, which were guided by experiments. In the second part of our analysis, we show that, very close to threshold, the frequency exhibits a sharp transition from the relaxation oscillation frequency to a frequency proportional to the external cavity frequency (i.e., 2\ensuremath{\pi}/\ensuremath{\tau}, where \ensuremath{\tau} is the external round-trip time).