Reflection and transmission of light by a square non-polar lattice

Abstract

The reflectance R and transmittance T of an infinite square lattice of polarizable nonpolar particles with retarded dipole-dipole interactions are rigorously calculated. The incident light has wavelength λ ⪢ lattice constant a and impinges upon the surface with an angle θ. First the amplitudes of the dipole moments of the particles are evaluated as functions of the electric field strength of the incident wave. Then the secondary fields emitted by all dipoles are summed and the reflected and transmitted waves are calculated. The general results for R and T as functions of θ, λ, a and the polarizability α of the particles obtained in the lattice model, are applied to the special case of a square lattice of metallic spheres of equal radius r ⪡ λ. The resulting formulae are compared with those found by Maxwell Garnett in a much more elementary, but very successful, approach for very thin films, consisting of a very large number of these metallic particles. The qualitative agreement between the expressions for R and T in both theories is consistent with recent optical experiments on very thin metallic island films which confirm the applicability of Maxwell Garnett's theory for calculations on the optical properties of these films.