Ewald–Oseen extinction theorem
Ewald–Oseen extinction theorem
In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just "extinction theorem", is a theorem that underlies the common understanding of scattering (as well as refraction, reflection, and diffraction). 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.[2]
Overview
An important part of optical physics theory is starting with microscopic physics—the behavior of atoms and electrons—and using it to derive the familiar, macroscopic, laws of optics. In particular, there is a derivation of how the refractive index works and where it comes from, starting from microscopic physics. The Ewald–Oseen extinction theorem is one part of that derivation (as is the Lorentz–Lorenz equation etc.).
When light traveling in vacuum enters a transparent medium like glass, the light slows down, as described by the index of refraction. Although this fact is famous and familiar, it is actually quite strange and surprising when you think about it microscopically. After all, according to the superposition principle, the light in the glass is a superposition of:
The original light wave, and
The light waves emitted by oscillating electrons in the glass.
(Light is an oscillating electromagnetic field that pushes electrons back and forth, emitting dipole radiation.)
Individually, each of these waves travels at the speed of light in vacuum, not at the (slower) speed of light in glass. Yet when the waves are added up, they surprisingly create only a wave that travels at the slower speed.
The Ewald–Oseen extinction theorem says that the light emitted by the atoms has a component traveling at the speed of light in vacuum, which exactly cancels out ("extinguishes") the original light wave. Additionally, the light emitted by the atoms has a component which looks like a wave traveling at the slower speed of light in glass. Altogether, the only wave in the glass is the slow wave, consistent with what we expect from basic optics.
Derivation from Maxwell's equations
Introduction
When an electromagnetic wave enters a dielectric medium, it excites (resonates) the material’s electrons whether they are free or bound, setting them into a vibratory state with the same frequency as the wave. These electrons will in turn radiate their own electromagnetic fields as a result of their oscillation (EM fields of oscillating charges). Due to the linearity of Maxwell equations, one expects the total field at any point in space to be the sum of the original field and the field produced by oscillating electrons. 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. The Ewald-Oseen extinction theorem seek to address the disconnect by demonstrating how the superposition of these two waves reproduces the familiar result of a wave that moves at a speed of c/n.
Derivation
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.
The electric field at a point in space is the sum of the electric fields due to all the various sources. In our case, we separate the fields in two categories based on their generating sources. We denote the incident field
and the sum of the fields generated by the oscillating electrons in the medium
The total field at any point z in space is then given by the superposition of the two contributions,
Therefore in this formalism,
This to say that the radiated field cancels out the incident field and creates a transmitted field traveling within the medium at speed c/n. Using the same logic, outside the medium the radiated field produces the effect of a reflected field Er traveling at speed c in the opposite direction to the incident field.
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
both the total electric and magnetic fields
the set of Maxwell equations inside the dielectric
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
![{\displaystyle \mathbf {E} _{\mathrm {vac} }(z,t)=\mathbf {E} _{\mathrm {vac} }\exp[i(kz-\omega t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3052adb623ffecfdf157d03288b9c91110573f7)
We simplify the double curl in a couple of steps using Einstein summation Einstein summation.
Hence we obtain,
![{\displaystyle \exp[-i\omega t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f51789162acfb9afb17f7900a1f6b84f2403d13)
with particular solution
For the complete solution, we add to the particular solution the general solution of the homogeneous equation which is a superposition of plane waves traveling in arbitrary directions[13]
Which implies that the conductivity
traveling at the speed of light.
Hertz vector approach
The electric field in terms of the Hertz vectors is given as
Linearity then allows us to write
Then using the substitution
and
so the limits become
and
![{\displaystyle {\begin{aligned}I&=2\pi \int _{\left|z^{\prime }-z\right|}^{\infty }dRe^{ik_{0}R}\\&=2\pi \lim _{\epsilon \to 0}\int _{\left|z^{\prime }-z\right|}^{\infty }dRe^{(ik_{0}-\epsilon )R}\\&=\left.2\pi \lim _{\epsilon \to 0}\left[{\frac {e^{(ik_{0}-\epsilon )R}}{ik_{0}-\epsilon }}\right]\right|_{\left|z^{\prime }-z\right|}^{\infty }\\&=2\pi \lim _{\epsilon \to 0}\left[{\frac {e^{(ik_{0}-\epsilon )\infty }}{ik_{0}-\epsilon }}-{\frac {e^{(ik_{0}-\epsilon ){\left|z^{\prime }-z\right|}}}{ik_{0}-\epsilon }}\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b93af10b1e33d65ddc0b0b0cc8dee527227e73ef)
![{\displaystyle {\begin{aligned}I&=2\pi \lim _{\epsilon \to 0}\left[0-{\frac {e^{(ik_{0}-\epsilon ){\left|z^{\prime }-z\right|}}}{ik_{0}-\epsilon }}\right]\\&=-2\pi {\frac {e^{ik_{0}{\left|z^{\prime }-z\right|}}}{ik_{0}}}\\&=2\pi i{\frac {e^{ik_{0}{\left|z^{\prime }-z\right|}}}{k_{0}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1d3d804252b33f36d630af72daa60749857d20)
Now, plugging this result back into the z-integral yields
therefore,
Therefore, the total field is
which becomes,
Then by coefficient matching we find,
and
The first relation quickly yields the wave vector in the dielectric in terms of the incident wave as
Both of these results can be substituted into the expression for the electric field to obtain the final expression
This is exactly the result as expected. There is only one wave inside the medium and it has wave speed reduced by n. The expected reflection and transmission coefficients are also recovered.
Extinction lengths and tests of special relativity
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.[6] In interstellar space, the extinction length for light is 2 light years[7] 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.[8]
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.[6][7] For example, in a binary star system, the two stars are moving in opposite directions, and one might test the prediction by analyzing their light. (See, for instance, the De Sitter double star experiment.) 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.[6] However, experiments using X-rays emitted by binary pulsars, with much longer extinction length, have been successful.[7]