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

Abstract

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 $\mathbb{R}^d,$ $d\geq 2$. In this region, we consider a hyperplane $X$ with sufficiently large distance $s$ from the origin in ${\mathbb R}^d.$ 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 $s \to +\infty.$ A numerical implementation is also presented.