A Method for the Calculation of the Distribution of Energy Reflected from a Periodic Surface

Abstract

The problem consists of finding the amplitudes of the plane waves reflected from a periodic surface due to a plane wave incident upon the surface. Consider first the function formed by the value of the incident wave on the surface and likewise the functions formed from the values of the diffracted waves (consisting of both homogeneous and inhomogeneous plane waves) on the surface. A typical diffraction problem consists of expressing the incident function as a linear combination of the diffracted functions. Unfortunately the diffracted functions are not in general mutually orthogonal so that the problem remaining consists of inverting an infinite system of linear equations. In order to make progress on this inversion, an orthonormal system of functions has been formed from the non-orthogonal diffracted functions. One can then write out the successive approximations to the reflection coefficients of the periodic surface using any given number of diffracted waves. The method has the advantage that it secures the least square fit to the boundary condition using the given set of diffracted waves. The problem has been programmed for MIDAC (the University of Michigan digital computer) in the above form. Numerical results will be presented and compared with the results of experiment. This work has been supported in part by the Office of Naval Research.