Permutation-Entropy Mapping of Dynamical Complexity in Modulated Optically Injected Semiconductor Lasers

Semiconductor lasers subject to external optical injection and current modulation exhibit a wide range of nonlinear behaviors, including phase locking, quasiperiodicity, and fully developed chaos. Ιdentifying the boundaries between these regimes is essential for understanding the onset of broadband chaotic emission and for applications such as secure communications, sensing, and high-rate chaos-based signal generation. In this work, we investigate the dynamical complexity of a directly modulated, optically injected semiconductor laser using permutation entropy (PE) applied to both the full rate-equation model and a reduced one-dimensional Poincaré phase map. PE provides a model-free quantifier of temporal complexity that readily distinguishes regular, quasiperiodic, and chaotic dynamics. By constructing PE-based resonance diagrams across a broad range of modulation amplitudes and detuning fractions, we show that the reduced phase map accurately reproduces the global organization of complexity observed in the full system, including the high-entropy regions emerging at the intersections of Arnol’d tongues. Representative cases demonstrate close qualitative agreement between the two descriptions, while also revealing systematic quantitative offsets associated with the higher intra-cycle variability present in the full intensity waveform. These results demonstrate that permutation-entropy analysis, combined with phase-map reduction, offers an efficient framework for mapping and interpreting the complexity landscape of modulated optically injected semiconductor lasers, enabling rapid exploration of parameter spaces relevant to chaos-based photonic applications.