Spatiotemporal coupled-mode theory for Fabry-Pérot resonators and its application to linear variable filters

Coupled-mode theory (CMT) is a widely used approach for describing resonances and eigenmodes in various photonic structures. Here, we propose a formulation of the CMT describing resonant multilayer structures. In particular, we revisit the conventional Fabry-P\'erot resonator and describe its optical properties from the point of view of the spatiotemporal formulation of the CMT. This formulation provides partial differential equations describing both temporal and spatial evolution of the field distribution, thus generalizing the conventional temporal and spatial versions of the CMT. The developed CMT takes into account the symmetry of the considered structure, energy conservation law, reciprocity, and causality. By considering the parameters of the developed CMT to be spatially dependent, we apply it to describe the optical properties of linear variable filters (LVFs) comprising two Bragg mirrors separated by a wedge-shaped (tapered) layer. In good agreement with the results of the rigorous numerical solution of Maxwell's equations, the proposed CMT accurately reproduces the broadening of the resonant peak and the appearance of Fizeau fringes when increasing the wedge angle of the LVF.