Contoured patterns from reflector systems - A spherical wave expansion solution

Any electromagnetic field - e.g., a contoured antenna pattern may be expanded into a sum of spherical waves, similar to a one-dimensional Fourier expansion. This expanded wave could then be scattered from a source reflector to determine a resulting required feed field in the vicinity of the focal region. This is of questionable value, and the more significant information to produce would be the required feed pattern itself. Hence the required contour pattern is instead scattered from the back of the reflector, and the result is the required feed pattern with origin at the focus of the chosen reflector.