Abstract

We consider the backscattering coefficient of a perfectly conducting, one-dimensional random rough surface in the physical-optics approximation. The high-frequency limit of physical optics yields the geometrical-optics scattering coefficient with Gaussian dependence on incidence angle. We demonstrate that for finite frequencies and surfaces with infinite slope variance the Gaussian form of the geometrical-optics limit generalizes to an α-stable distribution function. The proof of this result employs an asymptotic method that can be interpreted as a refinement of the central-limit theorem of probability theory for infinite-variance random variables. The theory leads to an effective cutoff of the surface-height power spectral density. The backscatter is not sensitive to surface components with wave number above this spectral cutoff, thus eliminating the nonphysical dependence of geometrical optics on surface features much smaller than the wavelength of the incident field. The composite or two-scale surface model is also derived as a term in a series expansion of the stable distribution. Comparison with numerical results shows that the approximation, although asymptotic, remains accurate for relatively low values of the surface roughness parameter.

© 1998 Optical Society of America

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