On the Efficient Evaluation of the Azimuthal Fourier Components of the Green's Function for Helmholtz's Equation in Cylindrical Coordinates

TL;DR

This paper introduces a fast, scalable algorithm for computing azimuthal Fourier components of the Green's function in cylindrical coordinates, crucial for electromagnetic scattering problems, with performance unaffected by proximity or wavenumber.

Contribution

We develop a novel algorithm that efficiently evaluates the modal Green's function with linear complexity in Fourier mode and is highly parallelizable, overcoming limitations of existing methods.

Findings

Performance independent of source-target proximity

Performance independent of wavenumber

Algorithm complexity grows as O(m)

Abstract

In this manuscript, we develop an efficient algorithm to evaluate the azimuthal Fourier components of the Green's function for the Helmholtz equation in cylindrical coordinates. A computationally efficient algorithm for this modal Green's function is essential for solvers for electromagnetic scattering from bodies of revolution (e.g., radar cross sections, antennas). Current algorithms to evaluate this modal Green's function become computationally intractable when the source and target are close or when the wavenumber is large. Furthermore, most state of the art methods cannot be easily parallelized. In this manuscript, we present an algorithm for evaluating the modal Green's function that has performance independent of both source-to-target proximity and wavenumber, and whose cost grows as $O(m)$, where $m$ is the Fourier mode. Furthermore, our algorithm is embarrassingly parallelizable.