Rigorous Derivation of Temporal Coupled Mode Theory Expressions for Travelling and Standing Wave Resonators Coupled to Optical Waveguides

Temporal coupled mode theory (CMT) has so far been applied phenomenologically in the analysis of optical cavity-waveguide structures, and relies on a priori knowledge of the to-be-excited resonator mode. Thus a rigorous derivation from Maxwell’s equations, and without any prior knowledge of the resonator type is needed. In this paper we derive temporal CMT of optical cavities coupled to waveguides. Starting from Maxwell’s equations and considering a proper expansion of the modes of the waveguide and resonator, and using mode orthogonality, the temporal CMT for this structure is obtained. We show that this formulation is general and can be applied to both traveling wave and standing wave type resonators. The results are validated against full-wave simulations.