The study presents a sophisticated traveling‑wave (TW) model for a semiconductor laser (SL) with delayed optical feedback (DOF) from an external cavity (EC). By solving a set of partial differential equations that describe the forward and backward propagating optical fields, the carrier density, and the polarization, the authors obtain steady‑state solutions and instantaneous optical modes that capture the full spatial and temporal dynamics of the system. The model incorporates Lorentzian‑shaped material gain dispersion, carrier‑density‑dependent gain compression, and linewidth‑enhancement effects, and it includes Langevin noise sources to account for spontaneous emission. Boundary conditions at the front and rear facets of the SL are treated explicitly, allowing the model to handle arbitrary front‑facet reflectivities, including the vanishing‑reflectivity limit that is problematic for the conventional Lang‑Kobayashi (LK) delay‑differential equation approach.

A key innovation is the use of instantaneous optical modes to decompose the chaotic field into a set of modal components that evolve in time. By synthesizing these modes with their corresponding steady‑state amplitudes, the authors construct time‑frequency representations of the chaotic trajectories. This approach reveals how large‑amplitude mode clusters form and interact, producing the characteristic mode‑beating oscillations observed in the radio‑frequency (RF) spectrum. The model reproduces the coherence‑collapse (CC) regime, where the laser abruptly transitions from single‑frequency operation to broadband chaos, and it captures low‑frequency fluctuations (LFF) that modulate the chaotic state. The RF bandwidth of the simulated chaotic output reaches several to tens of gigahertz, matching the requirements of secure‑communication, random‑number generation, and reservoir‑computing applications. Because the TW model naturally includes multiple longitudinal modes and spatial field distributions, it remains valid at high feedback levels where the LK model fails, and it can describe systems with small or zero front‑facet reflectivity.

The analysis demonstrates that the external‑cavity modes (ECMs) of the SL, which are the steady‑state solutions of the TW equations, play a decisive role in shaping the chaotic attractor. By examining the stable and unstable manifolds of these ECMs, the authors identify the mechanisms that drive the CC and the subsequent mode‑beating dynamics. The field expansion into modal components provides a clear physical interpretation of the optical and RF spectra, linking specific spectral features to underlying mode interactions and feedback pathways.

The work is a joint effort between the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) in Berlin and the Research School of Physics at the Australian National University (ANU) in Canberra. Lead author Mindaugas Radziunas from WIAS developed the TW model and performed the numerical simulations, while co‑author Deborah M. Kane from ANU contributed to the theoretical framework and interpretation of the results. The preprint was submitted on 19 July 2024, indicating that the research was carried out over the preceding years as part of a broader collaboration on semiconductor laser dynamics. While the report does not specify a particular funding source, it is part of the WIAS preprint series, suggesting institutional support from the German research community and the ANU research infrastructure. The collaboration combines expertise in applied mathematics, laser physics, and numerical analysis to advance the understanding of chaotic laser dynamics in the coherence‑collapse regime.