Unified theory for frequency combs in ring and Fabry–Perot quantum cascade lasers: An order-parameter equation approach

We present a unified model to describe the dynamics of optical frequency combs in quantum cascade lasers (QCLs), incorporating both ring and Fabry–Pérot (FP) cavity configurations. The model derives a modified complex Ginzburg–Landau equation (CGLE), leveraging an order parameter approach, and is capable of capturing the dynamics of both configurations, thus enabling a comparative analysis. This result demonstrates that FP QCLs, in addition to ring QCLs, belong to the same universality class of physical systems described by the CGLE, which includes, among others, systems in the fields of superconductivity and hydrodynamics. In the modified CGLE, a nonlinear integral term appears that is associated with the coupling between counterpropagating fields in the FP cavity and whose suppression yields the ring model, which is known to be properly described by a conventional CGLE. We show that this crucial term holds a key role in inhibiting the formation of harmonic frequency combs (HFCs), associated with multi-peaked localized structures, due to its anti-patterning effect. We provide support for a comprehensive campaign of numerical simulations in which we observe a higher occurrence of HFCs in the ring configuration compared to the FP case. Furthermore, the simulations demonstrate the model’s capability to reproduce experimental observations, including the coexistence of amplitude and frequency modulation, linear chirp, and typical dynamic scenarios observed in QCLs. Finally, we perform a linear stability analysis of the single-mode solution for the ring case, confirming its consistency with numerical simulations and highlighting its predictive power regarding the formation of harmonic combs.