Hopf bifurcation cascade in small-$\alpha$ laser diodes subject to optical feedback

We analyze theoretically the dynamics of a semiconductor laser subject to optical feedback, on the basis of the well-known Lang-Kobayashi equations. Previous investigations on this laser system suggest that a small linewidth enhancement factor $(\alpha$ factor) stabilizes the laser dynamics. By contrast, we unveil here optical feedback induced instabilities which are present for a small value of $\alpha$ but which disappear when $\alpha$ increases above $\alpha \sim 1.$ By combining numerical simulations and modern continuation methods for delay-differential equations, we unveil cascades of subcritical and supercritical Hopf bifurcations on the first external-cavity mode (ECM). We unveil for the first time, to our knowledge, the occurrence of subcritical Hopf bifurcation points for intermediate values of the EC length, i.e. close to the boundary between the short and the long EC regimes. They lead to severe laser instabilities such as large intensity and possibly chaotic pulsations. Moreover, these Hopf bifurcation cascades for small values of $\alpha$ are shown to be responsible for different bifurcation scenarios leading to restabilization of the first ECM and to ECM bistability.