A band-limited uniform diffuser in transmission

In this work we consider a structure consisting of vacuum in the region x(3) > zeta (x(1)); a dielectric film characterized by a real, positive, dielectric constant epsilon in the region -D < x(3) < zeta (x(1)); and vacuum in the region x(3) < -D. The surface profile function <zeta>(x(1)) is assumed to be a single-valued function of x(1), that is differentiable, and constitutes a random process. This structure is illuminated from the region x(3) > zeta (x(1)) by s-polarized light whose plane of incidence is the x(1)x(3)-plane. By the use of the geometrical optics limit of phase perturbation theory we show how to design the surface profile function zeta (x(1)) in such a way that the mean differential transmission coefficient has a prescribed form within a specified range of the angle of transmission, and vanishes outside this range. In particular, we consider the case in which the transmitted intensity is constant within a specified range of the angle of transmission, and vanishes outside it. Rigorous numerical simulation calculations show that the transmitted intensity indeed has this property.