Transient scattering of a beam from a periodic surface

Transient scattering from a periodic surface has been examined for an incident plane wave. For this plane wave case the authors used a number of techniques to determine the scattered fields. One such technique, known as the Rayleigh hypothesis, is only valid for sufficiently smooth surfaces. A more encompassing integral equation theory was also developed to provide solutions for rougher surfaces. These two methods can be modified to a more general nonuniform (beam) incidence wave, by properly choosing a number of plane waves, with which the resulting sum of plane waves can be formed into a beam. There are a number of ways in which the plane waves to be used can be determined. One method is to Fourier synthesize the plane waves needed to create a predetermined aperture field, this method is similar to antenna array beam synthesis. Another method, although less general, is to simply create a square wave using Fourier techniques which could then be space windowed to create the effect of a single beam. Both methods would ideally involve an infinite number a plane waves to exactly reconstruct the intended pattern, therefore a feasible solution would involve a finite number of plane waves, with the smallest number of plane waves needed to accurately create the beam as an optimal choice. By setting this limitation a slightly imperfect beam is expected, in fact the aperture field method will create a periodic beam (square-like), which will also need to be space windowed, and the two methods somewhat degrade into one.