Unique determination of a sound-soft ball by the modulus of a single far field datum

In this paper, we prove uniqueness in determining a sound-soft ball in inverse acoustic scattering by the modulus of the far field patterns. Since the modulus of the far field pattern is invariant under translation (Kress and Rundell, 1997 [6], Kwon and Seo, 2000 [7]), we can only reconstruct the shape of the obstacle but not the location. Assume that the ball is centered at the origin. Under the condition that the ball is small, it is uniquely determined by the modulus of a single far field datum measured at a fixed spot corresponding to a single incident plane wave.