Abstract
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.
Highlights
Optical filters that selectively reflect or transmit input light are ubiquitously used in spectroscopic measurements, laser systems, optical communications, displays, hyperspectral imaging and many other practical applications
Using the scattering matrix formalism, we analytically demonstrate that composite structures comprising two or four resonant diffractive structures (RDSs) optically implement the Butterworth filters of the order two or three, respectively, and enable achieving flat-top transmittance spectra with steep slopes and no sidelobes
In the present work, we theoretically investigated resonant properties of composite structures consisting of several identical resonant diffractive structures (RDSs) with a Lorentzian transmittance spectrum
Summary
Optical filters that selectively reflect or transmit input light are ubiquitously used in spectroscopic measurements, laser systems, optical communications, displays, hyperspectral imaging and many other practical applications. Let us suppose that the lossless RDSs constituting the composite structure have a Lorentzian transmittance spectrum In this case, the complex transmission and reflection coefficients of the RDS considered as functions of the angular frequency of the incident light can be approximated by the following expressions: r1. Let us consider a composite structure consisting of two identical RDSs described by the scattering matrix of Eq (2) and separated by a phase-shift layer with the thickness l1. In this case, the scattering matrix of the composite structure can be expressed through the matrix S1 ( ) in the form [17, 21]. The reflection coefficient r2 ( ) has two zeros corresponding to the roots of the polynomial being the numerator of the corresponding expression in Eq (9)
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