Study of the Resonance Diffraction Phenomena on Gratings by the Rigorous Coupled Wave Method with Modified Equation System

Abstract

Results of a numerical analysis of the diffraction of optical waves of transverse magnetic (TM) polarization on gratings under the electromagnetic field resonance conditions are presented. Rigorous coupled wave analysis (RCWA) with modified equation system has been used. Modification is in change of the Toeplitz matrices in differential equation systems. These matrices are formed by complex Fourier series of a periodic function of the dielectric permittivity and its inversed function. Rectangular function of the dielectric permittivity per one period is changed by trapezoidal function; however, grating relief remains rectangular. Such replacement of the dielectric permittivity function more precisely corresponds to actual gratings. Oscillations of the reflectance and transmittance depending on the number of used coupled waves were reduced using modified equation system. Moreover, there is better convergence, especially in case of metallic gratings on the dielectric substrates, in which the localized plasmon resonance or the electromagnetic field resonance can occur in the grating slit. Any changes of the convergence with the use of modified systems of equations are not present in the case of dielectric gratings on the metallic substrates in comparison to the traditional expansion of the dielectric constant into a Fourier series. Surface plasmon-polariton resonance of a metal grating on metal substrate in case of which the reflectance from the grating is equal to zero was researched. There is a particularly complicated dependence of the reflectance on the number of coupled waves used in the analysis for both modifications of the equation system. Strong oscillations of the reflection coefficient from the grating are observed at certain values of coupled waves, which decrease with their growth.

A theoretical design of evanescent wave biosensors based on gate-controlled graphene surface plasmon resonance

“…Through the conventional mode-matching method, the scattering characteristics of the graphene ribbons array can be determined; however, the Gibbs phenomenon in Floquet–Fourier series due to the discontinuity of tangential electric field causes poor numerical convergence in scattering analysis. Specifically, as was reported 46 , 47 , traditional approach of rigorous coupled-wave analysis experienced a poor convergence in particular under the resonance of surface plasmon-polariton wave. Therefore, local basis functions inherently satisfying the field nature in graphene ribbon and slit regions were employed to modify the input–output relation of the periodic graphene ribbons array while the whole formulation can still fit into the standard procedure of Fourier modal method 12 .…”