From coupled plane waves to the coupled-mode theory of guided-mode resonant gratings

We establish the connection between two theoretical approaches widely used for describing resonant optical properties of guided-mode resonant gratings (GMRGs): the coupled plane waves model and the coupled-mode theory (CMT). In the coupled plane waves model (also known as the multiple interference model), the field inside and outside the structure is represented as a small number of propagating plane waves; these waves are coupled at one or several interfaces, which leads to a system of linear equations. In the CMT, the optical properties are described by several coupled differential equations, each corresponding to one of the modes supported by the structure. The approach proposed in the present work provides explicit expressions for the coupling coefficients of the CMT through "local" reflection, transmission, and diffraction coefficients of the coupled plane waves model, which is important for the design and analysis of optical resonators, filters, and sensors based on GMRGs.