Green's Function and Radiation over a Periodic Surface: Reciprocity and Reversal Green's Function

This paper deals with the scattering of a cylindrical wave by a perfectly conductive periodic surface. This problem is equivalent to finding the Green's function G(x, z|xs, zs), where (x, z) and (xs, zs) are the observation and radiation source positions above the periodic surface, respectively. It is widely known that the Green's function satisfies the reciprocity: G(x, z|xs, zs) = G(xs, zs|x, z), where G(xs, zs|x, z) is named the reversal Green's function in this paper. So far, there is no numerical method to synthesize the Green's function with the reciprocal property in the grating theory. By combining the shadow theory, the reciprocity theorem for scattering factors and the average filter introduced previously, this paper gives a new numerical method to synthesize the Green's function with reciprocal property. The reciprocity means that any properties of the Green's function can be obtained from the reversal Green's function. Taking this fact, this paper obtains several new formulae on the radiation and scattering from the reversal Green's function, such as a spectral representation of the Green's function, an asymptotic expression of the Green's function in the far region, the angular distribution of radiation power, the total power of radiation and the relative error of power balance. These formulae are simple and easy to use. Numerical examples are given for a very rough periodic surface. Several properties of the radiation and scattering are calculated for a transverse magnetic (TM) case and illustrated in figures.