WIAS Preprint No. 1725, (2012)

Reflection of plane waves by rough surfaces in the sense of Born approximation


  • Arnold, Thomas
  • Rathsfeld, Andreas

2010 Mathematics Subject Classification

  • 78A45 35J25 35J05 35B40


  • Electromagnetic scattering, rough surfaces, Fourier analysis, Born approximation


The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, i.e., by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns' formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions if the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far-field terms like those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces.

Appeared in

  • Math. Methods Appl. Sci., 37 (2014) pp. 2091--2111.

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