Layer-by-Layer Methods in the Study of Photonic Crystals and Related Problems

Abstract

The optical properties of a two-dimensional array of metallic particles on a dielectric slab have been investigated by a number of authors, mainly because of the possible applications of such systems in coating technology. In most cases the metallic particles are randomly distributed on the substrate surface and vary both in volume and shape [1], although in some instances a periodic arrangement in space of nearly identical particles has been achieved [2]. Usually, the average diameter of the metallic particles is of the order of 100Å and together the non-overlapping particles cover 30–70% of the substrate surface. The traditional analysis of such experiments is based on the Maxwell-Garnett theory and various extensions of it [1]. Essentially, the particles are replaced by interacting dipoles; an effective (local) field is evaluated by the use of the Clausius-Mossotti equation, and an effective dielectric function for the composite medium is obtained. This analysis (we refer to as the dipolar or electrostatic approximation) breaks down when the wavelength of the incident radiation is relatively small, of the same order of magnitude as the size of the particles and/or the interparticle distance, or when the volume occupied by the particles is about half or more of the total volume. We shall deal with this problem in section 2. We shall represent the particles by spheres characterised by a certain dielectric function and we shall assume they are arranged periodically on a plane.