A theoretical study of electromagnetic wave scattering from deep perfectly conducting one-dimensional random rough surfaces and reflection gratings is performed by means of the extinction theorem. The scattering equations are solved numerically (instead of being solved by the usual analytical procedures, which are valid only for slight corrugations). This permits us to obtain an exhaustive collection of results for the mean scattered intensity as a function of polarization and surface parameters. In particular, Lambertian scattering and enhanced backscattering are predicted for random surfaces. Also, the range of validity of the Kirchhoff approximation is established for random surfaces whose correlation length is comparable with or smaller than the wavelength. Concerning gratings, generalizations of the blaze for large angles of incidence, large periods, and arbitrary shapes are obtained. Finally, it is shown that the blaze of the antispecular order for gratings is at the root of the enhanced backscattering for random surfaces.
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