Treatment of nonconvergence of the Fourier modal method and C method arising from field hypersingularities at lossless metal–dielectric arbitrary-angle edges

Abstract

Here, we report perturbative approaches to overcome the recently reported nonconvergence of the Fourier modal method (FMM) and the coordinate transformation method (C method) caused by the field hypersingularities (also called irregular field singularities) at lossless metal-dielectric arbitrary-angle edges. For the example of triangular gratings, we replace the sharp edge with a rounded edge to remove the hypersingularities at the edge. With such profile perturbations, we observe the convergence of the C method. The converged values of the diffraction efficiency are tested by the finite element method. However, with the radius of the rounded edge approaching zero, the converged values of the diffraction efficiency cannot approach a fixed value. For the example of parallelogram gratings, we smooth the sharp lamellar boundaries with a medium having a gradually varied refractive index to remove the hypersingularities. With the decrease of the width of the perturbative medium, the converged values of the diffraction efficiency can approach a fixed value for some numerical examples but cannot for other examples. For parallelogram gratings with a period much smaller than the wavelength, we surprisingly find that the FMM tends to converge despite the existence of hypersingularities, and the converged value consists well with the theoretical value given by the effective medium theory.