Modeling dispersive and active ring resonators with discrete optical components: from injected loops to mode-locked lasers

This article introduces a general framework for modeling the dynamics of active ring resonators with both continuous and discrete optical elements under the traveling-wave approximation. The framework is built upon a rigorous derivation of the two-time-variable formulation, which serves as the basis for several models, including the Haus master equations for lasers and the Lugiato-Lefever equation for microresonators. Starting from the slowly varying envelope approximation, we demonstrate how to integrate discrete optical components into the model, such as modulators, optical couplers, and saturable absorbers, and address gain dynamics using auxiliary differential equations. By explicitly incorporating these discrete elements rather than relying on mean-field approximations, we enhance simulation accuracy without increasing computational complexity. Additionally, we emphasize the crucial role of selecting an appropriate round-trip time, highlighting its importance in avoiding non-physical solutions. Special attention is given to the subtleties of the model, focusing on its effective applications and limitations. The consistency of the model and its improved accuracy compared to mean-field models are supported by numerical simulations at the end of the article, with additional tutorial-like examples provided in the associated supplementary materials.