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.

翻译：本文提出了一种快速稳健的数值方法，用于在给定固定频率柯西数据条件下重构时谐麦克斯韦方程中的点状源。这是一个具有广泛应用的电磁逆源问题，例如天线综合与设计、医学成像及污染源追踪。我们引入了新的成像函数与计算算法，以确定点源的数量、位置及其关联的矩向量，即使这些向量具有显著不同的幅值。源的数量与位置通过成像函数的显著峰值进行估计，而矩向量则通过显式简洁的公式计算。本文研究了成像函数的理论分析与稳定性，其主要难点在于分析格林张量虚部列向量与未知矩向量之间点积的行为特性。此外，我们将该方法扩展至通过辐射电场的渐近展开重构小体积源。我们提供了三维数值算例以验证所提方法的性能。