Inverse Source Problems for Maxwell's Equations and the Windowed Fourier Transform

Abstract

The inverse source problem for time-harmonic Maxwell's equations in three-dimensional free space is considered. Given far field observations of a wave radiated by a compactly supported volume current density at a single frequency, we generalize a qualitative reconstruction algorithm, recently developed in [R. Griesmaier, M. Hanke, and T. Raasch, SIAM J. Sci. Comput., 34 (2012), A1544--A1562], to this three-dimensional vector-valued setting. The method is based on the fact that a windowed Fourier transform of the far field data approximates an exponential transverse ray transform with a purely imaginary exponent of a smoothed version of the unknown source. Applying a filtered backprojection for the standard transverse ray transform to a suitably generalized squared absolute value of the windowed Fourier transform of the far field data, we recover information on the support of the volume current density. We analyze this algorithm, discuss its numerical implementation, and supplement our theoretical findings...