Integral representation of time-harmonic solutions to Maxwell's equations with fast numerical convergence

TL;DR

This paper develops integral representations for time-harmonic Maxwell solutions that enable fast numerical approximation and can be applied to various wave phenomena, including acoustic and elastic waves.

Contribution

It introduces a broad integral form for Maxwell's equations solutions, allowing exponential convergence in numerical approximation and applications to symmetric structures.

Findings

Solutions can be approximated with exponentially fast convergence using trapezoidal rules.

Finite sources of plane waves can effectively approximate broad classes of solutions.

The integral representation extends to Helmholtz-type equations for different physical phenomena.

Abstract

The robustness of XRD methods for the determination of the lattice parameters of crystals is well established. These methods have been extended to helical atomic structures using twisted x-rays \cite{friesecke_twisted_2016}. Building on an integral form used in \cite{friesecke_twisted_2016}, we construct integral representations of a broad class of time-harmonic solutions to Maxwell's equations in a vacuum or, more generally, in a homogeneous medium without source terms. The representation includes assignable generalized functions (distributions) that can be tailored to specific boundary or far-field conditions. When the assignable functions satisfy mild periodicity and smoothness conditions, the solutions can be approximated using multi-dimensional trapezoidal rules with exponentially fast convergence. This approximation can be physically interpreted as utilizing finite sources of plane waves to approximate the broad class of time-harmonic solutions to Maxwell's equations. Using these solutions, we show that radiation from suitably placed and oriented sources can serve as incoming radiation for structures with icosahedral symmetry to achieve constructive interference after interacting with the icosahedral structure. The finite source approximations are sufficiently general to satisfy the general Dirichlet conditions at an arbitrarily large number of assigned locations in a source-free domain. The integral representation also extends to a broad class of physical phenomena governed by Helmholtz-type equations. Examples include the scalar wave equation for acoustic waves and elastic wave propagation in linear isotropic solids, which involve both scalar and vector wave equations.