On the Geometrical Optics (Hagfors´ Law) and Physical Optics Approximations for Scattering From Exponentially Correlated Surfaces

High-frequency approximations to the physical optics (PO) theory of scattering from exponentially correlated rough surfaces are examined and used to interpret the expected accuracy of the PO theory. As an introduction, a review of the PO theory for Gaussian-correlated surfaces is provided, and in this process an analytical summation of the PO series for specular scattering from Gaussian-correlated surfaces is obtained. A similar form is then derived for specular scattering from exponentially correlated surfaces and contrasted to the Gaussian case. These series allow the accuracy of the leading order term (i.e., the geometrical optics limit) in the high-frequency approximation of PO scattering for Gaussian or exponentially correlated surfaces to be investigated analytically. The leading order term in the high-frequency expansion for general PO scattering from exponentially correlated surfaces (Hagfors´ Law) is then reviewed and interpreted in terms of a recently published theory of PO for surfaces with infinite rms slopes. The approximate ldquocutoffrdquo wavenumber from Hagfors´ Law at which the high-frequency portion of the spectrum of an exponentially correlated surface can be truncated without producing large errors in PO predicted scattering is also discussed. Using this cutoff wavenumber, an approximate region of validity of the complete PO theory for exponentially correlated surfaces is obtained. The validity condition indicates that, for fixed surface statistics, the PO method produces accurate predictions of true surface scattering only up to a specific frequency, and that PO is inaccurate in the high-frequency limit. Comparisons of PO predictions with those of a Monte Carlo numerical simulation are used to show that the validity condition derived appears to provide a reasonable indication of PO accuracy. These results have important implications for current investigations of scattering from exponentially correlated surfaces and for the use of Hagfors´ La- w, as it is traditional to accept PO as the appropriate high-frequency limit in most existing approximate models of surface scattering.