Inverse Diffraction and a New Reciprocity Theorem*

Abstract

A solution is obtained to the inverse diffraction problem for a monochromatic scalar wave field propagated into the half space z>0. It is shown how to determine the field distribution throughout the region 0≤z<z1 from the knowledge of the field distribution in the plane z=z1>0. The solution takes a particularly simple form when the spatial-frequency spectrum of the distribution in the plane z=z1 (or in any other plane z=const>0) is bandlimited to a circle whose radius is equal to the wavenumber of the field. In this case, the solution to the inverse diffraction problem may be expressed in a form strictly similar to that for the direct-propagation problem (exterior boundary-value problem), given by Rayleigh’s diffraction formula of the first kind. A comparison of these two solutions leads to the formulation of a new reciprocity theorem, valid for a wide class of wave fields.