Physical and geometrical optics for 2-D rough surfaces with power-law height spectra

Random rough surfaces with slowly decaying power spectral density can have infinite slope variance. Such surfaces do not satisfy the classical curvature criterion for validity of the physical optics (PO) approximation, and the infinite frequency geometrical optics limit or specular point scattering model breaks down. We show for two-dimensional surfaces with infinite slope variance that the Gaussian form of the classical geometrical limit generalizes to a stable distribution function. We also show that the PO integral is insensitive to surface components with spatial frequency above a cutoff wavenumber, which explains past observations that PO can be accurate for surfaces with power law spectra. This result leads to a general validity condition for the PO approximation in the backscattering direction for power-law surfaces, which in the case of a k^-4 spectrum requires that the significant slope of the surface be less than 0.03.