Ewald–Oseen extinction theorem
It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915.
[1] Originally, the theorem applied to scattering by an isotropic dielectric objects in free space.
The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media.
In particular, there is a derivation of how the refractive index works and where it comes from, starting from microscopic physics.
A more complete description can be found in Classical Optics and its Applications, by Masud Mansuripur.
[3] A proof of the classical theorem can be found in Principles of Optics, by Born and Wolf.,[1] and that of its extension has been presented by Akhlesh Lakhtakia.
This result is, however, counterintuitive to the practical wave one observes in the dielectric moving at a speed of c/n, where n is the medium index of refraction.
[4] Let's consider a simplified situation in which a monochromatic electromagnetic wave is normally incident on a medium filling half the space in the region z>0 as shown in Figure 1.
In our case, we separate the fields in two categories based on their generating sources.
assume that the wavelength is much larger than the average separation of atoms so that the medium can be considered continuous.
We use the usual macroscopic E and B fields and take the medium to be nonmagnetic and neutral so that Maxwell's equations read
includes the true and polarization current induced in the material by the outside electric field.
The set of Maxwell equations outside the dielectric has no current density term
The two sets of Maxwell equations are coupled since the vacuum electric field appears in the current density term.
For a monochromatic wave at normal incidence, the vacuum electric field has the form
We simplify the double curl in a couple of steps using Einstein summation.
, the time derivatives are straightforward and we obtain the following inhomogeneous wave equation
to a familiar form of the index of refraction of a linear isotropic dielectric.
When the electric field changes, the induced charges move and produces a current density given by
In order to calculate the electric field we must first solve the inhomogeneous wave equation for
, is found using a time dependent Green's function method on the inhomogeneous wave equation for
Plugging this into the integral and expressing in terms of Cartesian coordinates produces
in the second expression yields the polarization vector in terms of the incident electric field as
The characteristic "extinction length" of a medium is the distance after which the original wave can be said to have been completely replaced.
For visible light, traveling in air at sea level, this distance is approximately 1 mm.
[8] At very high frequencies, the electrons in the medium can't "follow" the original wave into oscillation, which lets that wave travel much further: for 0.5 MeV gamma rays, the length is 19 cm of air and 0.3 mm of Lucite, and for 4.4 GeV, 1.7 m in air, and 1.4 mm in carbon.
[9] Special relativity predicts that the speed of light in vacuum is independent of the velocity of the source emitting it.
This widely believed prediction has been occasionally tested using astronomical observations.
Unfortunately, the extinction length of light in space nullifies the results of any such experiments using visible light, especially when taking account of the thick cloud of stationary gas surrounding such stars.
[7] However, experiments using X-rays emitted by binary pulsars, with much longer extinction length, have been successful.
