The inverse electromagnetic scattering problem for a perfectly conducting cylinder

The problem of determining the shape of the cross section of a simply connected perfectly conducting infinite cylinder from a knowledge of the far-field pattern for all angles of observation and small values of the wavenumber is considered. The method proposed relies heavily on conformal mapping techniques. In particular it is shown that if the transfinite diameter is known each Fourier coefficient of the far-field pattern of the electric field determines a Laurent coefficient of the conformal mapping taking the exterior of the unit disk onto the exterior of the unknown cross section. The transfinite diameter is determined by changing the polarization of the incoming wave and measuring the far-field pattern of the resulting magnetic field. Of particular interest is the case when only a finite number of the Fourier coefficients of the far-field pattern are known. In this situation error estimates are obtained by using results on coefficient estimates for univalent functions.