Math 2.0

Optics, electromagnetic theory Scattering Theorem

Ewald–Oseen extinction theorem

Builds on Polarization densityElectric dipole momentPlane waveRefractive indexSnell's law

Statement

Let a homogeneous, isotropic, linear, non-magnetic dielectric fill a half-space, with vacuum in the other half-space. Let a monochromatic plane wave of angular frequency ω\omega come from the vacuum side. Regard the medium as vacuum filled with oscillating electric dipoles with polarization density PP. Let each dipole be driven by the total field at its location. Then, inside the medium, the field radiated by all the dipoles is the sum of two waves:

  1. a wave that travels with speed cc and exactly cancels the incident wave;
  2. a wave that travels with speed c/nc / n, where nn is the refractive index of the medium.

So the total field inside the medium is a single wave of speed c/nc / n.

Remarks

The proof writes the field at a point as the incident field plus the integral of the dipole fields over the half-space. One looks for a polarization of the form of a plane wave with unknown wave vector inside the medium. The integral over the half-space then splits into a contribution from the bulk and a contribution from the boundary surface. The bulk part must reproduce the assumed wave, which fixes its speed c/nc / n; the surface part has the vacuum wave vector and must cancel the incident wave. The cancellation condition gives the amplitude of the transmitted wave, and the field radiated backward by the dipoles gives the reflected wave.

In this way the theorem derives Snell's law and the Fresnel equations from a microscopic picture.

Going deeper

For a student

The theorem justifies the macroscopic description of a dielectric by a refractive index. It shows how the boundary conditions of macroscopic electromagnetism arise from the radiation of the dipoles near the surface. Together with a local-field model such as the Clausius–Mossotti relation, it links the index nn to the polarizability of the molecules.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Calculus of variations, concept, no evidence levelCalculus of variationsconceptElectric dipole moment, concept, no evidence levelElectric dipole momentconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptNormal vector, concept, no evidence levelNormal vectorconceptPlane wave, concept, no evidence levelPlane waveconceptPolarization density, concept, no evidence levelPolarization densityconceptRefractive index, concept, no evidence levelRefractive indexconceptFermat's principle, statement, no evidence levelFermat's principlestatementSnell's law, statement, no evidence levelSnell's lawstatementEwald–Oseen extinction theorem, statement, no evidence levelEwald–Oseen extinctio…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Calculus of variations concept
  2. Electric dipole moment concept
  3. Euler–Lagrange equation concept
  4. Geodesic concept
  5. Normal vector concept
  6. Plane wave concept
  7. Polarization density concept
  8. Refractive index concept
  9. Fermat's principle statement
  10. Snell's law statement
  11. Ewald–Oseen extinction theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Ewald%E2%80%93Oseen_extinction_theorem
About this page
TagN04C2CC2JJ
URIm2://N04C2CC2JJ
Kindstatement, theorem
WritingW0 stub
MSC78A45, 78A40, 78A10
StatusThis copy of the site has no build data. Run m2 status N04C2CC2JJ.
AboutRefractive index
UsesPolarization density, Electric dipole moment, Plane wave, Refractive index, Snell's law
Same as
Changed3 November 2026

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