Bifurcation analysis of a spatially extended laser with optical feedback

Vertical cavity surface-emitting lasers (VCSELs) are a new type of semiconductor laser characterized by the spatial extent of their disk-shaped output apertures. As a result, a VCSEL supports several optical modes (patterns of light) transverse to the direction of light propagation. When any laser is coupled to other optical elements, there is unavoidable optical feedback via reflecting surfaces, which influences the stability of the laser output. For a VCSEL, the question is how the transverse optical modes interact dynamically in the presence of optical feedback and how this affects stability of the system. In this paper, we start from a PDE description of the VCSEL. We proceed by using an expansion in suitable eigenfunctions to resolve the spatial dependence. In the presence of optical feedback we obtain a model in the form of a system of delay differential equations (DDEs). As we show with the example of a VCSEL that supports two transverse modes, the spatially expanded DDE model is small enough to allow for a multiparameter bifurcation analysis with numerical continuation tools. Specifically, we present stability regions of steady states and periodic solutions in dependence on the feedback strength and a homotopy parameter that models the amount of self- versus crossfeedback between the two modes. Bifurcations of more complicated spatiotemporal mode dynamics are then discussed. © 2009 Society for Industrial and Applied Mathematics