This paper derives rigorous results pertaining to the validity of the far-field approximation for scattering from randomly rough, perfectly conducting surfaces having arbitrary statistics. The methodology employs the stochastic Fourier transform of the current induced on the infinite surface by either a bounded or unbounded incident plane wave. The results are general in that no approximate simplifying forms for the current are employed. Exact expressions are obtained for the mean and variance of the scattered field for unbounded illumination, and they are compared to the far-field approximations to illustrate how the latter simplifications fail in this limit. Some of the pitfalls of the far-field approximation in the case of beam illumination are discussed. When the incident plane wave is bounded, the conventional far-field form for the mean scattered field can be rigorously derived for arbitrary surfaces provided the cross-sectional area of the incident beam is large in comparison to the square of the electromagnetic wavelength. The conventional far-field result for the variance of the scattered field is shown to require the additional stipulation that the cross-sectional area of the incident beam contain many decorrelation intervals of the surface roughness. The results obtained herein are important because they hold for arbitrary surface statistics. Whereas they appear to duplicate previous results, it must be remembered that the earlier results were only valid for a special class of surface statistics, i.e., surfaces for which single scattering theory holds.
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