Bifurcation analysis of lasers with delay

Lasers, especially semiconductor lasers, are known to exhibit an amazing range of dynamical behaviour when subjected to external influences. In practically all investigations of laser dynamics the question is: how does the dynamics of a given laser system depend on the control parameters? The approach in a typical experiment is to change one such parameter and record the dynamics, sweeping back and forth to catch possible hysteresis loops. A particular problem for semiconductor lasers is that their dynamics can generally only be recorded in the form of spectra. This is due to the fast time scales in such lasers. When a mathematical model is available, for example in the form of suitable rate equations, then it is possible to perform numerical simulations. The approach is essentially the same as in an experiment: one or more parameters are changed and the dynamics is recorded. The big advantage is that more information on the solution, and not just its spectrum, is now available, which can be used to interpret and explain the experimental results. Numerical simulation is relatively straightforward, but may be quite expensive in terms of run-time and data storage. A complimentary approach is that of bifurcation analysis. The idea is to find a parameter value where the laser’s dynamics changes qualitatively, for example, from stable output to periodic intensity variations. Such a qualitative change is called a bifurcation. It is now possible to follow or continue this bifurcation in several parameters with specialised software, called continuation software. In this way, one traces out the boundary between two regions of qualitatively different laser dynamics. The collection of bifurcation curves bounding regions of different dynamical behaviour is called a bifurcation diagram. Bifurcation analysis is an established tool that has been used with great success in many fields of application; see Refs. [29, 44, 55, 57] as general references and entry points