Lines of Quasi-BICs and Butterworth Line Shape in Stacked Resonant Gratings: Analytical Description

Abstract

We propose analytical approximations of the reflection and transmission spectra of a stacked dielectric diffraction grating consisting of two identical resonant guided-mode gratings with a Lorentzian line shape. These approximations, derived using the scattering matrix formalism, are functions of both angular frequency ω and the tangential wave vector component kx of the incident wave. We analytically demonstrate and, using full-wave simulations with rigorous coupled-wave analysis technique, numerically confirm that by a proper choice of the thickness of the dielectric layer separating the gratings, one can tailor the resonant optical properties of the stacked structure. In particular, it is possible to obtain lines of quasi-bound states in the continuum in the ω–kx parameter space with the quality factor decaying proportionally to kx−4 or kx−6. In addition, the stacked structure can be used as a spectral or spatial Butterworth filter operating in reflection. The presented results may find application in the design of optical filters and sensors based on stacked resonant gratings.