The Ewald-Oseen Extinction Theorem

Abstract

When a beam of light enters a material medium, it sets in motion the resident electrons, whether these electrons are free or bound. The electronic oscillations in turn give rise to electromagnetic radiation which, in the case of linear media, possess the frequency of the exciting beam. Because Maxwell's equations are linear, one expects the total field at any point in space to be the sum of the original (exciting) field and the radiation produced by all the oscillating electrons. However, in practice the original beam appears to be absent within the medium, as though it had been replaced by a different beam, one having a shorter wavelength and propagating in a different direction. The Ewald-Oseen theorem resolves this paradox by showing how the oscillating electrons conspire to produce a field that exactly cancels out the original beam everywhere inside the medium. The net field is indeed the sum of the incident beam and the radiated field of the oscillating electrons, but the latter field completely masks the former. Although the proof of the Ewald-Oseen theorem is fairly straightforward, it involves complicated integrations over dipolar fields in three-dimensional space, making it a brute-force drill in calculus and devoid of physical insight. It is possible, however, to prove the theorem using plane-waves interacting with thin slabs of material, while invoking no physics beyond Fresnel's reflection coefficients. The thin slabs represent sheets of electric dipoles, and the use of Fresnel's coefficients allows one to derive exact expressions for the electromagnetic field radiated by these dipolar sheets. The goal of the present article is to outline a proof of the Ewald-Oseen theorem using arguments that are based primarily on thin-film optics.