Influence of noise in the route to chaos of directly modulated semiconductor lasers

In this article we carry out a careful investigation of the route to chaos exhibited by directly modulated semiconductor lasers as the modulation current amplitude is varied. The role played by the noise fluctuations in the route to chaos is analyzed. The laser is described by the rate equation model in which the noise is introduced through appropriate Langevin sources. For this analysis, we have first studied the dynamics of the underlying deterministic system. Using time integration, the bifurcation diagrams, when the modulation index is swept forwards and backwards, were obtained. These have provided only the stable solutions of the system. From these diagrams, the well known period doubling route to chaos is shown to coexist at certain intervals of the modulation index with a periodic three solution, showing a hysteresis phenomenon. Thorough explanation of the deterministic dynamics requires knowledge of unstable solutions, which we have found using a continuation method. This method reveals that the periodic three solution appears by a tangent bifurcation and that the hysteresis cycle involves chaotic bifurcations. At this point, we have introduced noise fluctuations, which allow the solutions to probe the surrounding dynamics. The route to chaos is found to be reversible and via period doubling, period quadrupling, and period tripling in agreement with experimental data. From this study, a consistent interpretation of the route to chaos was found in agreement with different experimental results.