Bifurcation analysis of a multi-transverse-mode VCSEL

We study the behaviour of a multi-transverse-mode vertical-cavity surface-emitting laser subject to optical feedback in which the optical modes are coupled through the external round-trip. Starting from a delayed partial differential equation description of the spatial optical mode profiles and the carrier diffusion, we first use eigenfunction expansion techniques to resolve the spatial dependence. The resulting system of delay differential equations is then amenable to a full nonlinear bifurcation analysis by means of numerical continuation techniques. As illustration, we present bifurcation diagrams of a two-mode VCSEL in the plane of feedback strength versus feedback phase. In this way, we identify a number of changes in the structure and bifurcations of the VCSEL's dynamics. In particular, we find coexisting stable steady state solutions, which bifurcate to stable in-phase and anti-phase periodic solutions with vastly differing frequencies. We show how these periodic solutions give rise to quasiperiodic and chaotic laser dynamics.