Validity of the Kirchhoff approximation for electromagnetic wave scattering from fractal surfaces

Valid application of the Kirchhoff approximation (KA) for scattering from rough surfaces requires that the surface radius of curvature exceed approximately the electromagnetic wavelength λ. Fractal surface models have characteristic features on arbitrarily small scales, thereby posing problems in application of the electromagnetic boundary conditions in general as well as in the evaluation of surface radius of curvature pertinent to KA. Experiments and numerical simulations show variations in scattering behavior that are consistent with scattering from progressively smoother surfaces with increasing wavelength, demonstrating surface smoothing effects in the wave-surface interaction. We hypothesize control of KA scattering from fractal surfaces by an effective average radius of curvature c> as a function of the smallest lateral scale Δx contributing to scattering at λ. Solution of c(Δx[λ])>=λ for λ is one possible method for approximating the limit of KA validity, assuming that Δx[λ] is known. Investigation of the validity of KA for the calculation of scattering from perfectly conducting Weierstrass-Mandelbrot and fractional Brownian process fractal surface models shows that for both models the region of applicability of KA grows with increases in λ and the Hurst exponent H controlling large-scale roughness. Numerical simulations using the method of moments demonstrate the dependence of Δx on λ and the surface parameters.