Inverse Diffraction Theory and Computation of Minimum Source Regions of Far Fields

Abstract

A methodology based on the multipole expansion is developed to estimate the minimum source region of a given far field. The support of any source that produces the given far field must contain this minimum source region. The results are derived in the framework of the scalar Helmholtz equation in two-dimensional free space, which is relevant to transverse magnetic electromagnetic waves. The proposed approach consists of two steps. First we address, via an exterior inverse diffraction framework, the estimation of the minimum convex source region, which is the convex hull of the minimum source region. Next we compute, via a complementary interior inverse diffraction approach, nonconvex bounds for the minimum source region. This allows, in theory, the estimation of the minimum source region which can be nonconvex. The derived approach is illustrated with analytical and numerical examples relevant to inverse source and scattering problems.