Ewald–Oseen extinction theorem

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.

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.[2]

The following is a derivation based on a work by Wangsness [5] and a similar derivation found in chapter 20 of Zangwill's text, Modern Electrodynamics.[6] The setup is as follows, let the infinite half-space ${\displaystyle z<0}$ be vacuum and the infinite half-space ${\displaystyle z>0}$ be a uniform, isotropic, dielectric material with electric susceptibility, ${\displaystyle \chi .}$

The inhomogeneous electromagnetic wave equation for the electric field can be written in terms of the electric Hertz Potential, ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e} }}$, in the Lorenz gauge as

$${\displaystyle \nabla ^{2}{\boldsymbol {\pi }}_{\mathrm {e} }-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {\pi }}_{\mathrm {e} }}{\partial t^{2}}}=-{\frac {\mathbf {P} }{\epsilon _{0}}}.}$$

The electric field in terms of the Hertz vectors is given as

$${\displaystyle \mathbf {E} =\nabla \times \nabla \times {\boldsymbol {\pi }}_{\mathrm {e} }-{\frac {\mathbf {P} }{\epsilon _{0}}}-{\frac {\partial }{\partial t}}\left(\nabla \times {\boldsymbol {\pi }}_{\mathrm {m} }\right),}$$

but the magnetic Hertz vector ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {m} }}$ is 0 since the material is assumed to be non-magnetizable and there is no external magnetic field. Therefore the electric field simplifies to

$${\displaystyle \mathbf {E} =\nabla \times \nabla \times {\boldsymbol {\pi }}_{\mathrm {e} }-{\frac {\mathbf {P} }{\epsilon _{0}}}.}$$

In order to calculate the electric field we must first solve the inhomogeneous wave equation for ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e} }}$. To do this, split ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e} }}$ in the homogeneous and particular solutions

$${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e} }(\mathbf {r} ,t)={\boldsymbol {\pi }}_{e,h}(\mathbf {r} ,t)+{\boldsymbol {\pi }}_{\mathrm {e,p} }(\mathbf {r} ,t).}$$

Linearity then allows us to write

$${\displaystyle \mathbf {E} (\mathbf {r} ,t)=\mathbf {E} _{h}(\mathbf {r} ,t)+\mathbf {E} _{\mathrm {p} }(\mathbf {r} ,t).}$$

The homogeneous solution, ${\displaystyle \mathbf {E} _{h}(\mathbf {r} ,t)}$, is the initial plane wave traveling with wave vector ${\displaystyle k_{0}=\omega /c}$ in the positive ${\displaystyle z}$ direction

$${\displaystyle \mathbf {E} _{h}(\mathbf {r} ,t)=\mathbf {E} _{0}e^{i\left(k_{0}z-\omega t\right)}.}$$

We do not need to explicitly find ${\displaystyle {\boldsymbol {\pi }}_{e,h}(\mathbf {r} ,t)}$since we are only interested in finding the field.

The particular solution, ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }(\mathbf {r} ,t)}$ and therefore, ${\displaystyle \mathbf {E} _{\mathrm {p} }(\mathbf {r} ,t)}$, is found using a time dependent Green's function method on the inhomogeneous wave equation for ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }}$ which produces the retarded integral

$${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }(\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int d^{3}r'{\frac {\mathbf {P} \left(\mathbf {r} ',t-\left|\mathbf {r} -\mathbf {r} '\right|/c\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}.}$$

Since the initial electric field is polarizing the material, the polarization vector must have the same space and time dependence ${\displaystyle \mathbf {P} (\mathbf {r} ,t)=\mathbf {P} _{0}e^{i(kz-\omega t)}.}$ More detail about this assumption is discussed by Wangsness. Plugging this into the integral and expressing in terms of Cartesian coordinates produces

$${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }(\mathbf {r} ,t)={\frac {\mathbf {P} _{0}e^{i(kz-\omega t)}}{4\pi \epsilon _{0}}}\int _{0}^{\infty }dz'e^{ik\left(z'-z\right)}\int _{-\infty }^{\infty }dx'\int _{-\infty }^{\infty }dy'{\frac {e^{ik_{0}\left|\mathbf {r} -\mathbf {r} '\right|}}{\left|\mathbf {r} -\mathbf {r} '\right|}}.}$$

First, consider only the integration over ${\displaystyle x'}$ and ${\displaystyle y'}$ and convert this to cylindrical coordinates ${\displaystyle (x,y,z)\rightarrow (\rho ,\varphi ,z)}$ and call ${\displaystyle \left|\mathbf {r} -\mathbf {r} '\right|=R}$

$${\displaystyle I:=\int _{-\infty }^{\infty }dx^{\prime }\int _{-\infty }^{\infty }dy^{\prime }{\frac {e^{ik_{0}\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}}{\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}}=\int _{0}^{2\pi }d\varphi '\int _{0}^{\infty }d\rho '{\frac {e^{ik_{0}R}}{R}}=2\pi \int _{0}^{\infty }d\rho '{\frac {e^{ik_{0}R}}{R}}.}$$

Then using the substitution

$${\displaystyle R^{2}=\rho ^{2}+\left|z'-z\right|^{2}\Rightarrow \rho ^{2}=R^{2}-\left|z'-z\right|^{2}\Rightarrow \rho \,d\rho =R\,dR}$$

and

$${\displaystyle \rho ={\sqrt {R^{2}-\left|z'-z\right|^{2}}}}$$

so the limits become

$${\displaystyle \rho =0={\sqrt {R^{2}-\left|z'-z\right|^{2}}}\Rightarrow R=\left|z'-z\right|}$$

and

$${\displaystyle \rho =\infty ={\sqrt {R^{2}-\left|z'-z\right|^{2}}}\Rightarrow R=\infty .}$$

Then introduce a convergence factor ${\displaystyle e^{-\epsilon R}}$ with ${\displaystyle \epsilon \in \mathbb {R} }$ into the integrand since it does not change the value of the integral,

$${\displaystyle {\begin{aligned}I&=2\pi \int _{\left|z'-z\right|}^{\infty }dRe^{ik_{0}R}\\&=2\pi \lim _{\epsilon \to 0}\int _{\left|z'-z\right|}^{\infty }dRe^{(ik_{0}-\epsilon )R}\\&=\left.2\pi \lim _{\epsilon \to 0}\left[{\frac {e^{(ik_{0}-\epsilon )R}}{ik_{0}-\epsilon }}\right]\right|_{\left|z'-z\right|}^{\infty }\\&=2\pi \lim _{\epsilon \to 0}\left[{\frac {e^{(ik_{0}-\epsilon )\infty }}{ik_{0}-\epsilon }}-{\frac {e^{(ik_{0}-\epsilon ){\left|z'-z\right|}}}{ik_{0}-\epsilon }}\right].\end{aligned}}}$$

Then ${\displaystyle \epsilon \in \mathbb {R} }$ implies ${\displaystyle \lim _{\epsilon \to 0}e^{-\epsilon \infty }=0}$, hence ${\displaystyle \lim _{\epsilon \to 0}e^{(ik_{0}-\epsilon )\infty }=\lim _{\epsilon \to 0}e^{ik_{0}\infty }e^{-\epsilon \infty }=0}$. Therefore,

$${\displaystyle {\begin{aligned}I&=2\pi \lim _{\epsilon \to 0}\left[0-{\frac {e^{(ik_{0}-\epsilon ){\left|z^{\prime }-z\right|}}}{ik_{0}-\epsilon }}\right]\\&=-2\pi {\frac {e^{ik_{0}{\left|z'-z\right|}}}{ik_{0}}}\\&=2\pi i{\frac {e^{ik_{0}{\left|z'-z\right|}}}{k_{0}}}.\end{aligned}}}$$

Now, plugging this result back into the z-integral yields

$${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }(z,t)={\frac {i\mathbf {P} _{0}e^{i(kz-\omega t)}}{2k_{0}\epsilon _{0}}}\int _{0}^{\infty }dz'e^{ik\left(z'-z\right)}e^{ik_{0}\left|z-z'\right|}}$$

Notice that ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }}$ is now only a function of ${\displaystyle z}$ and not ${\displaystyle \mathbf {r} }$, which was expected for the given symmetry.

This integration must be split into two due to the absolute value ${\displaystyle \left|z-z'\right|}$ inside the integrand. The regions are ${\displaystyle z<0}$ and ${\displaystyle z>0}$. Again, a convergence factor must be introduced to evaluate both integrals and the result is

$${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }(z,t)=-{\frac {\mathbf {P} e^{-i\omega t}}{2\epsilon _{0}k_{0}}}{\begin{cases}{{\frac {1}{k+k_{0}}}e^{-ik_{0}z}}&{z<0}\\{{\frac {2k_{0}}{k_{0}^{2}-k^{2}}}e^{ikz}+{\frac {1}{k-k_{0}}}e^{ik_{0}z}}&{z>0.}\end{cases}}}$$

Instead of plugging ${\displaystyle {\boldsymbol {\pi }}_{\mathrm {e,p} }}$ directly into the expression for the electric field, several simplifications can be made. Begin with the curl of the curl vector identity,

$${\displaystyle \nabla \times (\nabla \times \mathbf {{\boldsymbol {\pi }}_{\mathrm {e,p} }} )=\nabla (\nabla \cdot \mathbf {{\boldsymbol {\pi }}_{\mathrm {e,p} }} )-\nabla ^{2}\mathbf {{\boldsymbol {\pi }}_{\mathrm {e,p} }} ,}$$

therefore,

$${\displaystyle {\begin{aligned}\mathbf {E} _{\mathrm {p} }&=\nabla \times \nabla \times {\boldsymbol {\pi }}_{\mathrm {e} }-{\frac {\mathbf {P} }{\epsilon _{0}}}\\&=\nabla (\nabla \cdot {\boldsymbol {\pi }}_{\mathrm {e,p} })-\nabla ^{2}{\boldsymbol {\pi }}_{\mathrm {e,p} }-{\frac {\mathbf {P} }{\epsilon _{0}}}.\end{aligned}}}$$

Notice that ${\displaystyle \nabla \cdot {\boldsymbol {\pi }}_{\mathrm {e,p} }=0}$ because ${\displaystyle \mathbf {P} }$ has no ${\displaystyle {\mathbf {z} }}$ dependence and is always perpendicular to ${\displaystyle {\hat {\mathbf {z} }}}$. Also, notice that the second and third terms are equivalent to the inhomogeneous wave equation, therefore,

$${\displaystyle {\begin{aligned}\mathbf {E} _{\mathrm {p} }&=-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\boldsymbol {\pi }}_{\mathrm {e,p} }}{\partial t^{2}}}\\&=-{\frac {1}{c^{2}}}(-i\omega )^{2}{\boldsymbol {\pi }}_{\mathrm {e,p} }\\&=k_{0}^{2}{\boldsymbol {\pi }}_{\mathrm {e,p} }\end{aligned}}}$$

Therefore, the total field is

$${\displaystyle \mathbf {E} (z,t)=\mathbf {E} _{0}e^{i\left(k_{0}z-\omega t\right)}+k_{0}^{2}{\boldsymbol {\pi }}_{\mathrm {e,p} }(z,t)}$$

which becomes,

$${\displaystyle \mathbf {E} (z,t)=\left\{{\begin{array}{l}{\mathbf {E} _{0}e^{i\left(k_{0}z-\omega t\right)}-{\frac {\mathbf {P} }{2\epsilon _{0}}}{\frac {k_{0}}{k+k_{0}}}e^{-i\left(k_{0}z+\omega t\right)}}&{z<0}\\{\mathbf {E} _{0}e^{i\left(k_{0}z-\omega t\right)}-{\frac {\mathbf {P} }{2\epsilon _{0}}}{\frac {k_{0}}{k-k_{0}}}e^{i\left(k_{0}z-\omega t\right)}-{\frac {\mathbf {P} }{\epsilon _{0}}}{\frac {k_{0}^{2}}{k_{0}^{2}-k^{2}}}e^{i(kz-\omega t)}}&{z>0.}\end{array}}\right.}$$

Now focus on the field inside the dielectric. Using the fact that ${\displaystyle \mathbf {E} (z,t)}$ is complex, we may immediately write

$${\displaystyle \mathbf {E} (z>0,t)=\mathbf {E} e^{i\left(k_{0}z-\omega t\right)}}$$

recall also that inside the dielectric we have ${\displaystyle \mathbf {P} =\epsilon _{0}\chi \mathbf {E} }$.

Then by coefficient matching we find,

$${\displaystyle e^{i\left(kz-\omega t\right)}\Rightarrow 1=-\chi {\frac {k_{0}^{2}}{k_{0}^{2}-k^{2}}}}$$

and

$${\displaystyle e^{i\left(k_{0}z-\omega t\right)}\Rightarrow 0=\mathbf {E} _{0}-{\frac {\chi }{2}}{\frac {k_{0}}{k-k_{0}}}\mathbf {E} .}$$

The first relation quickly yields the wave vector in the dielectric in terms of the incident wave as

$${\displaystyle {\begin{aligned}k&={\sqrt {1+\chi }}k_{0}\\&=nk_{0}.\end{aligned}}}$$

Using this result and the definition of ${\displaystyle \mathbf {P} }$ in the second expression yields the polarization vector in terms of the incident electric field as

$${\displaystyle \mathbf {P} =2\epsilon _{0}(n-1)\mathbf {E} _{0}.}$$

Both of these results can be substituted into the expression for the electric field to obtain the final expression

$${\displaystyle \mathbf {E} (z,t)=\left\{{\begin{array}{l}{\mathbf {E} _{0}e^{i\left(k_{0}z-\omega t\right)}-\left({\frac {n-1}{n+1}}\right)\mathbf {E} _{0}e^{-i\left(k_{0}z+\omega t\right)}}&{z<0}\\{\left({\frac {2}{n+1}}\right)\mathbf {E} _{0}e^{i\left(nk_{0}z-\omega t\right)}}&{z>0.}\end{array}}\right.}$$

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.

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.[7] In interstellar space, the extinction length for light is 2 light years.[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.[7][8] 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.[7] However, experiments using X-rays emitted by binary pulsars, with much longer extinction length, have been successful.[8]