An extinction theorem for electromagnetic waves in a point dipole model

Abstract

A transfer matrix method is used to show that an electromagnetic wave in a cubic lattice of point dipoles propagates as if its wave speed has been reduced from c in vacuo to c/n, where n is the index of refraction, while in the interstices the wave speed is c. Because the lattice is not a continuum, there are evanescent waves between lattice planes. These waves ultimately give rise to the Lorentz–Lorenz result for n and produce rapid variations of the polarization in the first few lattice planes. These rapid variations indicate that there is an extinction length over which the external wave is converted to the internal wave propagating at the reduced speed. This length is the distance over which the local field assumes its bulk value. The possibility of an extinction length in recent work on the Ewald–Oseen theorem was missed because only continuum models were considered.