Improving accuracy of the numerical solution of Maxwell's equations by processing edge singularities of the electromagnetic field

TL;DR

This paper introduces a methodology to improve the accuracy and convergence speed of numerical solutions to Maxwell's equations by addressing electromagnetic field singularities near geometric edges, demonstrated on diffraction gratings.

Contribution

It presents algorithms that incorporate edge singularity treatment into spectral and modal methods, significantly enhancing convergence and accuracy in electromagnetic simulations.

Findings

Achieved exponential convergence in spectral methods.

Demonstrated improved diffraction efficiency calculations.

Enhanced accuracy in permittivity cases where conventional methods fail.

Abstract

In this paper we present a methodology for increasing the accuracy and accelerating the convergence of numerical methods for solution of Maxwell's equations in the frequency domain by taking into account the be-havior of the electromagnetic field near the geometric edges of wedge-shaped structures. Several algorithms for incorporating treatment of singularities into methods for solving Maxwell's equations in two-dimensional structures by the examples of the analytical modal method and the spectral element method are discussed. In test calculations, for which we use diffraction gratings, the significant accuracy improvement and convergence ac-celeration were demonstrated. In the considered cases of spectral methods an enhancement of convergence from algebraic to exponential or close to exponential is observed. Diffraction efficiencies of the gratings, for which the conventional methods fail to converge due to the special values of permittivities, were calculated.