Uniqueness for a linearized, inverse obstacle problem using backscattering data

For the scattering of time-harmonic, acoustic plane waves by a sound-soft obstacle D in R-2 we prove injectivity for the Frechet derivative of the mapping F which maps the boundary of the obstacle to the backscattered far field for all incident directions and an interval of wave numbers. We reduce the injectivity of the Frechet derivative to the question whether the squares of the normal derivatives of the total fields are complete on the boundary deltaD. With the help of a conformal mapping we simplify the geometry and give a positive answer to this question.