Subharmonic transition in an optically injected semiconductor laser: theory and experiments

The equations for a semiconductor laser subject to detuned optical injection are analysed using asymptotic methods. We derive a third-order equation for the phase of the laser field which is then investigated for small injection but arbitrary frequency detuning. The long-time solution is a small amplitude time-periodic solution with a frequency close to the detuning except if the detuning is close to a multiple of the free-running laser relaxation frequency (resonance). We examine the case of subharmonic resonance, injecting at twice the relaxation resonance frequency, in detail. In addition to period-doubling bifurcations, we show the coexistence of bifurcation and isolated branches of solutions. Our approximate results are in good agreement with the numerical bifurcation diagram obtained from the original laser equations. Our analysis motivated a series of new experiments on laser diodes operating in the weak injection but large detuning regime. The experimental spectra show clearly the period-doubling bifurcation as well as the shifting of the slave-laser frequency predicted by our analysis.