Truncation of the Feigenbaum sequence in directly modulated semiconductor lasers

In this paper, we analyze the dynamic behavior of directly modulated semiconductor lasers as the modulation index is varied, with an emphasis on the influence of noise at two different biasing levels. We studied the route to chaos followed by the deterministic system when the modulation frequency is two times the resonance frequency. We found that the behavior is more complex than that of a Feigenbaum sequence. In addition, a period tripling stable solution, which appears due to a fold bifurcation, coexists with the Feigenbaum sequence for certain values of the modulation index. Their coexistence gives rise to hysteresis loops and chaotic bifurcations, namely boundary crisis. Due to the coexistence of solutions in the deterministic system, the role of noise can be expected to be of great importance. When noise fluctuations are introduced in the model, the behavior evolves from the single periodic response through period doubling, period quadrupling, and period tripling in accordance with recent experimental studies. We have also found agreement in the behavior at different conditions of the analysis by varying the biasing point and the modulation frequency. Our results show that in the route to chaos, the period-doubling sequence is effectively truncated due to random noise. The reason for the truncation is found in the nearby coexisting period-three solution.