Coexistence of stable ECM solutions in the Lang–Kobayashi system

Abstract

The Lang–Kobayashi system of delay differential equations describes the behavior of the complex electric field $\mathcal{ℰ}$ and inversion $N$ inside an external cavity semiconductor laser. This system has a family of special periodic solutions known as external cavity modes (ECMs). It is well known that these ECM solutions appear through saddle-node bifurcations, then lose stability through a Hopf bifurcation before new ECM solutions are born through a secondary saddle-node bifurcation. Employing analytical and numerical techniques, we show that for certain short external cavity lasers the loss of stability happens only after the secondary saddle-node bifurcations, which means that stable ECM solutions can coexist in these systems. We also investigate the basins of these ECM attractors.