A Unified Theory of Multidimensional Electromagnetic Vector Inverse Scattering Within the Kirchhoff or Born Approximation

Abstract

Multidimensional inverse scattering has very important applications in radar, medical diagnostics, geophysical exploration and nondestructive testing. As such, acoustic, electromagnetic and elastic waves are involved. The mathematics of wave propagation, scattering and inverse scattering differs considerably in complexity for these various types of waves. Acoustic waves can be considered as strictly scalar, whereas electromagnetic waves require field vectors, and field quantities to describe elastic waves are vectors as well as tensors. In terms of Green’s functions: a scalar Green function is sufficient for acoustic waves, a dyadic one is most appropriate for electromagnetic waves, and for elastic wave propagation, a dyadic and a triadic Green function is needed. Green’s functions determine the scattering of waves, the “inverse” of Green’s functions determines inverse scattering of waves; therefore, acoustic inverse scattering has found the earliest solutions (compare [4.1, 2] for a summary, as far as multidimensional linearized inverse scattering is considered). On the other hand, the utilization of polarization information for electromagnetic imaging has been considerably stimulated in two subsequent workshops [4.3, 4]. This can be achieved finding polarization-dependent target descriptors and signatures, or, more quantitatively, solving the electromagnetic vector inverse scattering problem. Starting from some ideas developed for the scalar case, a number of authors [4.5–9] have already tried to extend them to the vector case. Here, we want to evaluate a unified theory of multidimensional acoustic inverse scattering [4.1, 2] for electromagnetic waves; our underlying model of the direct scattering process is linear and relies on the weak scattering (Born) approximation for the penetrable scatterer, and the physical optics (Kirchhoff) approximation for the perfect scatterer. It is essentially based on the definition of the generalized holographic field, the derivation of the Porter-Bojarski integral equation, and its solution in terms of a multifrequency (transient) or multi-look-angle experiment; we call those experimental modes of operation frequency diversity, and angular diversity. The theory can be formulated in a coordinate-free version for arbitrarily located and arbitrarily shaped measurement surfaces, or in a diffraction tomographic version using Cartesian coordinates; then, the measurement surface is considered to be planar. Various data processing schemes can be established, for instance, multidimensional Fourier inversion and observation space backpropagation techniques; in the frequency diversity mode, time domain backpropagation schemes are available. The algorithms can either be derived for multistatic or monostatic arrangements; for the latter case, frequency diversity is mandatory.