A direct reconstruction method for radiating sources in Maxwell's equations with single-frequency data

TL;DR

This paper introduces a fast, stable numerical method for reconstructing point-like electromagnetic sources from single-frequency Maxwell data, with applications in antenna design, medical imaging, and pollution tracing.

Contribution

The paper presents a novel imaging function and algorithm for source reconstruction that accurately estimates the number, locations, and moments of sources, even with varying magnitudes.

Findings

Successfully reconstructs multiple sources in 3D simulations.

Accurately estimates source locations and moments.

Extends method to small-volume sources using asymptotic expansion.

Abstract

This paper presents a fast and robust numerical method for reconstructing point-like sources in the time-harmonic Maxwell's equations given Cauchy data at a fixed frequency. This is an electromagnetic inverse source problem with broad applications, such as antenna synthesis and design, medical imaging, and pollution source tracing. We introduce new imaging functions and a computational algorithm to determine the number of point sources, their locations, and associated moment vectors, even when these vectors have notably different magnitudes. The number of sources and locations are estimated using significant peaks of the imaging functions, and the moment vectors are computed via explicitly simple formulas. The theoretical analysis and stability of the imaging functions are investigated, where the main challenge lies in analyzing the behavior of the dot products between the columns of the imaginary part of the Green's tensor and the unknown moment vectors. Additionally, we extend our method to reconstruct small-volume sources using an asymptotic expansion of their radiated electric field. We provide numerical examples in three dimensions to demonstrate the performance of our method.