Analytical design of flat-top transmission filters composed of several resonant structures

Resonant properties of composite structures consisting of several identical resonant structures (e.g. multilayer thin-film structures or guided-mode resonance gratings) separated by phase-shift layers are investigated theoretically. Using the scattering matrix formalism, we analytically demonstrate that, at properly chosen thicknesses of the phase-shift layers, the composite structures comprising two or four resonant diffractive structures with a Lorentzian transmittance profile optically implement the Butterworth filters of the order two or three, respectively, and enable achieving flat-top transmission spectra with steep slopes and low sidebands. In addition, we show that the composite structures consisting of three or four second-order Butterworth filters can accurately approximate the fourth- or fifth-order Butterworth filters, respectively. The presented theoretical results are confirmed by rigorous numerical simulations of composite structures consisting of the so-called W-structures (simple three-layer resonant structures comprising a high-index core layer and two low-index cladding layers in a high-index dielectric environment). The simulation results confirm the formation of flat-top transmittance peaks, the shape of which fully agrees with the derived theoretical description. Moreover, we demonstrate an exceptionally simple mechanism of controlling the transmittance peak width, which consists in changing the thicknesses of the cladding layers of the initial W-structure and enables generating flat-top transmission peaks with a significantly subnanometer width.