Renormalization of an inverse-scattering theory for discontinuous profiles

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Abstract
A renormalized solution that is based upon the exact inversion theories of Gel’fand, Levitan, and Marchenko has been developed by using a multiple-scales analysis. Our previous theory [J. Opt. Soc. Am. A 2, 1916 (1985)] for the inverse-scattering problem for inhomogeneous, continuously varying regions has been extended to include discontinuities in the dielectric permittivity. A singular perturbation method has been used to obtain a uniformly valid expression for the electric field within the dielectric region. The advantage of the multiple-scales analysis of the interior electric field is that it rigorously indicates the dielectric region over which the weak-scattering or high-frequency approximation is valid. Furthermore, it is an effective renormalization technique that is physically motivated by the requirement of energy conservation and that allows a systematic investigation of the various scales associated with the inverse problem. The singular perturbation method for the solution of the inverse problem associated with the electromagnetic reflection data from a discontinuous dielectric region utilizes the high-frequency Born approximation to determine the magnitude of the discontinuity in the neighborhood of the origin. The method used for reconstructing discontinuous profiles is also appropriate for the reconstruction of profiles with turning points. The theory is demonstrated by two examples.

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