On the Derivation of the Integral Equation for the Propagation of Light in Dielectric Crystals

Abstract

A new derivation is given of the integral equation for the macroscopic polarization set up in a dielectric crystal by an incident light wave. The molecules are assumed to lie on the points of a perfect, simple lattice and to interact only via the retarded dipole–dipole interaction. The derivation is based on a direct averaging of the microscopic equations for the dipole moments induced by the incident wave and the dipole fields. The averaging is a space averaging with a weight function with a width Δ satisfying [Formula: see text], where a is the lattice constant and λ the wavelength of the incident field. The derivation is an application of Nijboer and De Wette's method for the evaluation of lattice sums.In contrast to the more indirect derivation given earlier by Hoek, (i) the present derivation is not based on an expansion in powers of the molecular polarizability, α, so that, e.g., there is no limitation to frequencies outside the resonance regions, and (ii) the integral equation is shown to be valid everywhere in the crystal except in a macroscopically negligible boundary layer of thickness Δ rather than λ. The latter improvement is crucial in the presence of superradiance.The integral equation is shown to be equivalent to the usual wave equation derived from Maxwell's phenomenological theory supplemented by the appropriate constitutive equations. An explicit expression is derived for the frequency dependent dielectric tensor in terms of rapidly convergent lattice sums.