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.
Highlights
We consider in the framework of the scalar Helmholtz equation the inverse problem of estimating the minimum source region of a given far field
We derive new closed form formulas for the minimum source region that are based on the multipole expansion [3] and that complement the counterpart plane-wave-theorybased, closed form method of a previous paper [2]
The derived formulas facilitate analytical demonstration of the computation of the minimum source region for far fields that are known in closed form, which is of interest in antenna synthesis and as a theoretical tool to handle certain inverse scattering problems analytically
Summary
We consider in the framework of the scalar Helmholtz equation the inverse problem of estimating the minimum source region of a given far field. This region is sufficiently big so that there exists a source supported in it that radiates the given far field This region is optimal in the sense that it is the smallest support of any source that can produce the given far field. We do not specify in the following the class of sources for which the methods of this paper (the nonconvex bounds, in particular) are useful in practice This challenging question is related to theoretical issues raised in [4] which go beyond the scope of our presentation and is left for the future
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