In this paper we present an approach to the design of a two-dimensional circularly symmetric randomly rough Dirichlet surface that, when illuminated at normal incidence by a scalar plane wave, scatters it with a specified dependence of its intensity along the normal to the surface in the absence of the roughness. This approach is illustrated by applying it to the design of a surface that produces a constant scattered intensity within a finite region of this normal, and no scattered intensity outside this region. It is validated by the results of rigorous computer simulation calculations of scattering from the surface designed in this way. It is then shown that a random surface of this type can be used to produce a pseudo-nondiffracting beam in reflection, namely a scattered beam with a finite beam aperture, a finite propagation range, a variation in transverse beam profiles, and an intensity peak in the direction of propagation.
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