The problem of finding the dielectric constant of a slab from the reflection data is considered. It is shown that a recently developed method by Trantanella et al. [J. Opt. Soc. Am. A 12, 1469 (1995)], which is based on an interesting extension of the Born approximation, is intimately related to the Schwinger variational solution. This remarkable relation enables one to extend the class of profiles that can be reconstructed, evaluate analytically some of the quantities that were found approximately by Trantanella et al., thus simplifying the numerical computations, and also arrive at a systematic development of a sequence of more accurate extensions of the Born inversion method. To illustrate the proposed inversion procedures, an exactly solvable analytical example is presented.
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