A two-point phase recovering from holographic data on a single plane

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 ℝ^d, d ⩾ 2. In this region, we consider a hyperplane X with sufficiently large distance s from the origin in ℝ^d. We give two-point local formulas for approximate recovering the radiation solution restricted to the plane X from the intensity of the total solution at X, that is, from holographic data. The recovering is given in terms of the far-field pattern of the radiation solution with a decaying error term as s → + ∞. A numerical implementation is also presented.