Analytical stability boundaries for quantum cascade lasers subject to optical feedback

We consider nonlinear rate equations appropriate for a quantum cascade laser subject to optical feedback. We analyze the conditions for a Hopf bifurcation in the limit of large values of the delay. We obtain a simple expression for the critical feedback rate that highlights the effects of key parameters such as the linewidth enhancement factor and the pump. All our asymptotic approximations are validated numerically by using a path continuation technique that allows us to follow Hopf bifurcation points in parameter space.