Design of one-dimensional Lambertian diffusers of light

We describe a method for designing a one-dimensional random surface that acts as a Lambertian diffuser. The method is tested using rigorous computer simulations and is shown to yield the desired scattering pattern. Optical devices that give rise to a scattered intensity that is proportional to the cosine of the scattering angle are frequently used in the optical industry, e.g. for calibrating scatterometers [1]. Such diffusers have the property that their radiance or luminance is the same in all scattering directions. Due to this angular dependence such devices are often referred to as Lambertian diffusers. In the visible region of the optical spectrum volume disordered media, e.g. compacted powdered barium sulfate, and freshly smoked magnesium oxide [2] are used as Lambertian diffusers. However, this type of diffuser is inapplicable in the infrared region due to its strong absorption and the presence of a specular component in the scattered light, in this frequency range. The design of a random surface that acts as a Lambertian diffuser, especially in the infrared region of the optical spectrum, is therefore a desirable goal, and one that has been regarded as difficult to achieve [3]. In this paper we present a solution to this problem that is based on an approach used in several recent papers to design one-dimensional random surfaces with specified scattering properties [4–6], and to fabricate them in the laboratory [5,7]. The design of a two-dimensional random surface that acts as a Lambertian diffuser will be described elsewhere [8]. To motivate the calculations that follow we begin by considering the scattering of spolarized light of frequency ω from a one-dimensional, randomly rough, perfectly conducting surface defined by x3 = ζ(x1). The region x3 > ζ(x1) is vacuum, the region x3 < ζ(x1) is the perfect conductor (figure 1). The plane of incidence is the x1x3-plane. The surface profile function ζ(x1) is assumed to be a single-valued function of x1 that is differentiable, and to constitute a random process. 0959-7174/01/040529+05$30.00 © 2001 IOP Publishing Ltd Printed in the UK 529 530 A A Maradudin et al Figure 1. The scattering geometry assumed in this paper. The mean differential reflection coefficient 〈∂R/∂θs〉, where the angle brackets denote an average over the ensemble of realizations of the surface profile function, is defined such that 〈∂R/∂θs〉 dθs is the fraction of the total time-averaged flux incident on the surface that is scattered into the angular interval (θs, θs + dθs) in the limit as dθs → 0. In the geometrical optics limit of the Kirchhoff approximation it is given by [5] 〈 ∂R