A Robust Multi-Scale Field-Only Formulation of Electromagnetic Scattering

TL;DR

This paper introduces a stable, multi-scale boundary integral method for electromagnetic scattering that directly solves for electric and magnetic fields, avoiding common numerical issues and applicable to complex geometries.

Contribution

It presents a novel scalar Helmholtz-based formulation that is free of zero-frequency instability and suitable for multiscale problems, generalizing classical Fresnel and Snell laws.

Findings

Numerical results converge to static solutions in the long wavelength limit.

Method handles multiscale problems with large and small characteristic lengths.

Accurate near and far field calculations for arbitrary surface curvatures.

Abstract

We present a boundary integral formulation of electromagnetic scattering by homogeneous bodies that are characterized by linear constitutive equations in the frequency domain. By working with the Cartesian components of the electric, E and magnetic, H fields and with the scalar functions (r*E) and (r*H), the problem is cast as solving a set of scalar Helmholtz equations for the field components that are coupled by the usual electromagnetic boundary conditions at material boundaries. This facilitates a direct solution for E and H rather than working with surface currents as intermediate quantities in existing methods. Consequently, our formulation is free of the well-known numerical instability that occurs in the zero frequency or long wavelength limit in traditional surface integral solutions of Maxwell's equations and our numerical results converge uniformly to the static results in the long wavelength limit. Furthermore, we use a formulation of the scalar Helmholtz equation that is expressed as classically convergent integrals and does not require the evaluation of principal value integrals or any knowledge of the solid angle. Therefore, standard quadrature and higher order surface elements can readily be used to improve numerical precision. In addition, near and far field values can be calculated with equal precision and multiscale problems in which the scatterers possess characteristic length scales that are both large and small relative to the wavelength can be easily accommodated. From this we obtain results for the scattering and transmission of electromagnetic waves at dielectric boundaries that are valid for any ratio of the local surface curvature to the wave number. This is a generalization of the familiar Fresnel formula and Snell's law, valid at planar dielectric boundaries, for the scattering and transmission of electromagnetic waves at surfaces of arbitrary curvature.