Time- and frequency-domain approaches to modeling complex semiconductor laser systems

Recent advances in semiconductor lasers have greatly expanded their scope of applications. Initially limited to optical communications, these lasers now compete effectively with fiber and solid-state lasers across diverse advanced fields, including sensing, microwave photonics, LiDAR, frequency comb generation, and quantum computing. These expanded capabilities are driven by continuous improvement in their stability, output power, reliability, and efficiency. Moreover, semiconductor lasers serve as a versatile platform for investigating emerging physical phenomena in fields like non-Hermitian and topological physics. In this review, we discuss the key theoretical approaches used to model complex semiconductor laser systems, focusing on both time- and frequency-domain approaches. We first introduce the fundamental equations underlying each approach and highlight essential terms accounting for cavity dispersion and nonlinear effects. We then extend these models to address complex and coupled cavities and detail their numerical implementations. Finally, we illustrate the practical applications of these models by exploring narrow-linewidth operation in self-injection locked lasers as well as passive and active mode-locking dynamics.