Rate equation analysis for deterministic and unidirectional lasing in ring resonators with an S-shaped coupler

Ring resonators are traditionally popular optical devices that apply to various components in photonic integrated circuits. They also play an important role in the on-chip generation of many novel optical states in topological systems and non-Hermitian systems. Unidirectional lasing of ring resonators is used in many such systems to create exotic states of light including optical vortexes and optical skyrmions, but the unidirectional behavior has not been fully understood. Previous research has constructed a simplified model to explain the steady state behaviors of unidirectional ring resonators, but the carrier dynamics and spontaneous emission were omitted. In this work, we give a numerical analysis of unidirectional ring resonators with an S-shaped coupler. We identified the importance of the gain saturation to robustness against backscattering and high unidirectionality by comparing to the linear model without saturation. We also discuss the effect of amount of asymmetrical coupling to the realization of unidirectionality.