Analytical stability boundaries for a semiconductor laser subject to optical injection

We determine analytically the stability boundaries of a semiconductor laser subject to external optical injection, Specifically, we derive the exact conditions for a steady state limit point and a Hopf bifurcation point in terms of the injection strength and the frequency detuning. These conditions are formulated in parametric form and are appropriate for asymptotic approximations, We investigate the limit of a large ratio of the carrier and photon lifetimes and we discuss the resulting expressions in terms of the linewidth enhancement factor. For negative detuning, the unstable domain is bounded by two Hopf bifurcation points which coincide at a critical negative value of the detuning. For positive detunings, all steady state solutions are unstable, Useful scaling laws characterizing different parts of the stability diagram are derived.