A High-Order Perturbation of Envelopes (HOPE) Method for Vector Electromagnetic Scattering by Periodic Inhomogeneous Media

TL;DR

This paper introduces a high-order spectral perturbation method for simulating electromagnetic wave scattering by three-dimensional periodic inhomogeneous media, extending previous work to full vector Maxwell equations with rigorous analyticity guarantees.

Contribution

It develops a novel perturbative spectral algorithm for vector Maxwell equations in periodic media, with proven analyticity in deformation size and spatial variables.

Findings

Provides a rigorous analyticity theory for the method

Extends previous scalar Helmholtz results to vector Maxwell equations

Enables accurate and efficient simulations of complex periodic structures

Abstract

The scattering of electromagnetic waves by three--dimensional periodic structures is important for many problems of crucial scientific and engineering interest. Due to the complexity and three-dimensional nature of these waves, the fast, accurate, and reliable numerical simulations of these are indispensable for engineers and scientists alike. For this, High Order Spectral methods are frequently employed and here we describe an algorithm in this class. Our approach is perturbative in nature where we view the deviation of the permittivity from a constant value as the deformation and we pursue regular perturbation theory. This work extends our previous contribution regarding the Helmholtz equation to the full vector Maxwell equations, by providing a rigorous analyticity theory, both in deformation size and spatial variable (provided that the permittivity is, itself, analytic).