A holographic uniqueness theorem for the two-dimensional Helmholtz equation

We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\mathbb R^2$. We consider a straight line in this region, such that the direction of propagation of the plane wave is not parallel to this line. We show that the radiation solution in the exterior region is uniquely determined by the intensity of the total solution on an interval of this line. In particular, this result solves one of the old mathematical questions of holography in its two-dimensional setting. Our proofs also contribute to the theory of the Karp expansion of radiation solutions in two dimensions.