Pole-Expansion of the T-Matrix Based on a Matrix-Valued AAA-Algorithm

TL;DR

This paper introduces a pole-expansion method for the T-matrix using a matrix-valued AAA-algorithm, enabling efficient and physically interpretable spectral analysis of scattering responses across frequencies.

Contribution

It presents a novel pole-expansion technique for the T-matrix based on a matrix-valued AAA-algorithm, reducing computational cost and improving spectral representation.

Findings

Efficient pole-expansion of T-matrix demonstrated on various scatterers.

Reduces computational cost compared to naive frequency sampling.

Provides open-source tools for the community.

Abstract

The transition matrix (T-matrix) is a complete description of an object's linear scattering response. As such, it has found wide adoption for the theoretical and computational description of multiple-scattering phenomena. In its original form, the T-matrix describes the interaction of a scatterer with a monochromatic source. In practice, however, information about the T-matrix is usually needed in an extended spectral domain. To access the frequency-dispersion, one might naively sample T-matrices over a finely resolved set of discrete frequencies and store one T-matrix per frequency. This approach has multiple drawbacks: it is computationally expensive, requires excessive memory, and it disregards the physical origin of the spectral features, weakening physical interpretability. To overcome these major limitations, we leverage a pole-expansion technique to represent the T-matrix with arbitrary frequency resolution within a selected frequency domain via a set of resonant contributions. A matrix-valued variant of the recently established adaptive Antoulas-Anderson (AAA) algorithm for rational approximation enables us to compute the pole-expansion at minimal computational cost using only a small number of direct evaluations. We demonstrate the benefits of such a representation with examples ranging from semi-analytically accessible scatterers to quasi-dual bound states in the continuum. To allow the wider community to capitalize on these findings, we provide open-source tools to perform the presented pole-expansion of the T-matrix.