Deterministic Chaos Maps in External-Cavity Semiconductor Lasers with Short-Delay Optical Feedback

In this work, we investigate deterministic chaos in external-cavity semiconductor lasers with delayed optical self-feedback. A noise-free quadrature-based delay differential model is used to isolate the intrinsic nonlinear dynamics produced by phase-sensitive delayed reinjection and carrier–photon interactions. Sensitivity to initial conditions is quantified by computing the leading Lyapunov exponents through a variational approach that integrates the base delay differential equations together with their delayed variational equations using a fourth-order Runge–Kutta method of steps and periodic QR orthonormalization. High-resolution Lyapunov maps are constructed in the (log10C,ϕf) parameter space for different pump ratios and selected short-feedback delays. The delay values are interpreted through the reference-normalized ratio τf/TR,ref, where TR,ref≈131.9ps is a fixed reference timescale derived from a reference solitary-laser operating point. The results show that both the spatial organization of positive-λ1 regions and the mean positive Lyapunov exponent are strongly affected by feedback delay, feedback phase, feedback strength, and pump ratio. Within the selected short-delay set, delayed self-feedback produces broader, more connected, and more strongly unstable chaotic regions as the external-cavity memory time increases toward the fixed reference timescale, particularly at larger pump ratios. These findings show that short external-cavity self-feedback can support robust deterministic chaotic regimes relevant to compact and integrated photonic implementations. The proposed framework provides a reproducible deterministic reference for identifying and interpreting feedback-induced chaos in short-delay external-cavity semiconductor lasers, while stochastic effects such as spontaneous-emission and Langevin noise are left for future robustness studies.