Quadrature by Parity Asymptotic eXpansions (QPAX) for scattering by high aspect ratio particles

TL;DR

This paper introduces QPAX, a novel method for accurately solving high aspect ratio scattering problems by accounting for parity effects and nearly singular behaviors in boundary integral equations.

Contribution

The paper presents QPAX, a new approach that effectively handles the challenges of high aspect ratio particles by incorporating parity asymptotic expansions in boundary integral methods.

Findings

QPAX accurately solves scattering problems with high aspect ratio particles.

The method captures parity-dependent asymptotic behaviors.

Numerical examples demonstrate QPAX's efficiency and precision.

Abstract

We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provide valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called Quadrature by Parity Asymptotic eXpansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples.