Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface

Abstract

We study theoretically the angular dependence of the correlation functions of the scattering amplitudes occurring in the interaction of a beam of polarized light with a one-dimensional, perfectly conducting rough surface. For an ensemble of surface-profile functions that are realizations of a stationary stochastic process, a necessary condition for the exact scattering amplitudes to be correlated is established: the projection on the mean surface of the difference between the incident wave vectors must equal that of the scattered wave vectors. The exact expressions for the amplitudes of the scattered plane waves constituting the far field are derived from Green’s second integral theorem. By numerical simulations, the dependence of these amplitude correlation functions on the angle of incidence and on the incident and the scattered polarization states is computed for Gaussian surfaces producing enhanced backscattering. Results are presented for the complex correlation functions of p- and s-polarized scattering amplitudes. However, it is argued that, for an incident field polarized at +45°, the single- and the multiple-scattering contributions to the amplitude correlation functions are clearly separated if the scattered field is resolved into −45° and +45° polarized plane waves. In this case the real part of the correlation function of the −45° scattering amplitudes displays two peaks of large angular width. The maxima occur for angles of incidence at which the correlation between the +45° polarized scattering amplitudes has sharp peaks of angular width approximately equal to λ/a, where the surface correlation length a may be interpreted as an estimate for the mean distance between successive scattering points on the surface. One of these maxima occurs when the correlated amplitudes are identical, and the other occurs when the role of the incoming and the outgoing directions is interchanged between the two scattering amplitudes, which are then related by the reciprocity condition. The coherent addition of the amplitudes arising from multiple-scattering processes and those of their time-reversed partners in directions close to the retroreflection direction is demonstrated in the calculations of the correlation functions.