A multiple time scale approach to the stability of external cavity modes in the Lang-Kobayashi system using the limit of large delay

The Lang-Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so-called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In th is case, both the number of ECMs and the number of critical eigenvalues is large. Applying a new ly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.