The Ewald–Oseen extinction theorem and extinction lengths

Abstract

We give an elementary demonstration of the extinction theorem for electromagnetic waves at normal incidence on a plane surface of a medium. We stress that the extinction of the incident wave and its replacement with a wave of index of refraction n takes place throughout the medium rather than in the surface layers. Although the extinction theorem is usually thought to apply only to dielectrics, we extend the theorem to include conductors. We use the macroscopic fields in which the contributions of the oscillating dipoles in a dielectric or the induced currents in a conductor are already summed. Our elementary derivation of the extinction theorem should be readily accessible to advanced undergraduates since it depends only on the superposition principle and the solution of the wave equation. An analogy is made between this extinction and the cancellation of the electric field inside a conductor placed in a static electric field. We also study the more advanced case of propagation of radiation in a dilute random medium in which the wavelength is small relative to the interparticle distance and find that an analogous extinction of the incident wave takes place. Furthermore, for the dilute random medium, we estimate the length into the medium for which a large fraction, (1−1/e), of the incident radiation has interacted with the particles making up the medium. This length is much larger than any lengths associated with the extinction theorem itself.