  Resonant properties of composite structures consisting of several identical
resonant diffractive 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 size.
