Modeling quasi-dark states with temporal coupled-mode theory

Coupled resonators are commonly used to achieve tailored spectral responses and allow novel functionalities in a broad range of applications, from optical modulation and filtering in integrated photonic circuits to the study of nonlinear dynamics in arrays of resonators.The Temporal Coupled-Mode Theory (TCMT) provides a simple and general tool that is widely used to model these devices and has proved to yield very good results in many different systems of low-loss, weakly coupled resonators.Relying on TCMT to model coupled resonators might however be misleading in some circumstances due to the lumped-element nature of the model.In this article, we report an important limitation of TCMT related to the prediction of dark states.Studying a coupled system composed of three microring resonators, we demonstrate that TCMT predicts the existence of a dark state that is in disagreement with experimental observations and with the more general results obtained with the Transfer Matrix Method (TMM) and the Finite-Difference Time-Domain (FDTD) simulations.We identify the limitation in the TCMT model to be related to the mechanism of excitation/decay of the supermodes and we propose a correction that effectively reconciles the model with expected results.A comparison with TMM and FDTD allows to verify both steady-state and transient solutions of the modified-TCMT model.The proposed correction is derived from general considerations, energy conservation and the non-resonant power circulating in the system, therefore it provides good insight on how the TCMT model should be modified to eventually account for the same limitation in a different coupled-resonator design.Moreover, our discussion based on coupled microring resonators can be useful for other electromagnetic resonant systems due to the generality and far-reach of the TCMT formalism.