Abstract
A brief historical introduction to the visual and wave-theoretic consequences of high frequency electromagnetic scattering by large spheres is given, with special emphasis on backscattering. Exact electromagnetic solutions for radially inhomogeneous dielectric lenses are unavailable for many functional dependences of the refractive index on the radial distance, so the high-frequency behavior based on an asymptotic analysis of the exact solution has been obtained in very few cases. In this chapter existing results for the asymptotic behavior of backscattered radiation are extended to a broader class of refractive index profiles. Additionally, by exploiting some known results from quantum mechanics, asymptotic solutions for two scalar problems (decoupled from the electromagnetic cases) are derived for the case of small variations in the refractive index across the scattering sphere. By using a Liouville transformation the electromagnetic wavenumber-dependent scattering potential is converted to a wavenumber-independent form, and the resulting inverse problem is solved for several refractive index profiles.
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