Ewald–Oseen extinction theorem
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In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, the Ewald–Oseen extinction theorem, sometimes referred to as just the 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 Paul Peter Ewald, FRS (January 23, 1888 in Berlin, Germany – August 22, 1985 in Ithaca, New York) was a German crystallographer and physicist, a pioneer of X-ray diffraction methods. Education Ewald received his early education in the classi ...
and
Carl Wilhelm Oseen Carl Wilhelm Oseen (17 April 1879 in Lund – 7 November 1944 in Uppsala) was a theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in Stockholm. Life Oseen was born in Lund, and took a Fil. Kand. degre ...
, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. 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.


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 In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
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 In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
. 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 superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
, 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 In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
.) 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. A proof of the classical theorem can be found in ''
Principles of Optics ''Principles of Optics'', colloquially known as ''Born and Wolf'', is an optics textbook written by Max Born and Emil Wolf that was initially published in 1959 by Pergamon Press. After going through six editions with Pergamon Press, the book wa ...
'', by Born and Wolf., and that of its extension has been presented by
Akhlesh Lakhtakia Akhlesh Lakhtakia is Evan Pugh University Professor and Charles Godfrey Binder Professor of engineering science and mechanics at the Pennsylvania State University. His research focuses on electromagnetic fields in complex materials, such as scul ...
.


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

The following is a derivation based on a work by Ballenegger and Weber. 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 \mathbf_ and the sum of the fields generated by the oscillating electrons in the medium \mathbf_(z, t). The total field at any point z in space is then given by the superposition of the two contributions, \mathbf(z, t)= \mathbf_(z, t) + \mathbf_(z, t). To match what we already observe, \mathbf_ has this form. However, we already know that inside the medium, z>0, we will only observe what we call the transmitted E-field \mathbf_ which travels through the material at speed c/n. Therefore in this formalism, \mathbf_(z, t) = -\mathbf_(z, t) + \mathbf_(z, t) 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 \mathbf_ traveling at speed c in the opposite direction to the incident field. \mathbf_(z, t) = -\mathbf_(z, t) -\mathbf_(z, t) 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 \begin \nabla \cdot \mathbf&=0 \\ \nabla \cdot \mathbf&=0 \\ \nabla \times \mathbf&=-\frac \\ \nabla \times \mathbf&=\boldsymbol_ \mathbf+\epsilon_ \boldsymbol_ \frac \end both the total electric and magnetic fields \mathbf=\mathbf_+\mathbf_, \quad \mathbf=\mathbf_+\mathbf_ the set of Maxwell equations inside the dielectric \begin \\ \\ \\ \end where \mathbf includes the true and polarization current induced in the material by the outside electric field. We assume a linear relationship between the current and the electric field, hence \mathbf= \left(\mathbf_ + \mathbf_\right) The set of Maxwell equations outside the dielectric has no current density term \begin \\ \\ \\ \end 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 \mathbf_(z, t)=\mathbf_ \exp (k z- \omega t) with k=\omega /. Now to solve for \mathbf_, we take the curl of the third equation in the first set of Maxwell equation and combine it with the fourth. \begin \nabla\times\nabla\times \mathbf_ &= -\frac \partial (\nabla\times\mathbf_) \\ ex\nabla\times\nabla \times \mathbf_ &= - \frac \partial \left(\mu_ \mathbf+\epsilon_ \mu_ \frac \right) \end We simplify the double curl in a couple of steps using
Einstein summation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
. \begin (\nabla\times\nabla \times \mathbf)_ &= \epsilon_\epsilon_ \partial_ \partial_ E_ \\&=(\delta_ \delta_-\delta_ \delta_)\partial_ \partial_ E_ \\&=\partial_(\partial_E_)-\partial_\partial_ E_ \end Hence we obtain, \nabla\times\nabla \times \mathbf_=\nabla(\nabla\cdot\mathbf_)-\nabla^2\mathbf_ Then substituting \mathbf by \left(\mathbf_ + \mathbf_\right) , using the fact that \nabla\cdot\mathbf_ = 0 we obtain, \nabla^2\mathbf_\mathrm = \frac \partial (\mu_\mathbf_ + \mu_0 \mathbf_ + \epsilon_ \mu_ \partial \mathbf_ / \partial t) Realizing that all the fields have the same time dependence \exp (-i \omega t) , the time derivatives are straightforward and we obtain the following inhomogeneous wave equation \nabla^ \mathbf_+\mu_ \omega^\left(\epsilon_+i \sigma / \omega\right) \mathbf_= -i \mu_ \omega \sigma \mathbf_(z) with particular solution \mathbf_^ = -\mathbf_(z) 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 \left(\mathbf_^\right)_ = \int g_(\boldsymbol, \boldsymbol) \exp \left(i \mathbf' \cdot \mathbf\right) d \Omega where k' is found from the homogeneous equation to be k^=\mu_ \epsilon_ \omega^2 \left(1+i \frac\right) Note that we have taken the solution as a coherent superposition of plane waves. Because of symmetry, we expect the fields to be the same in a plane perpendicular to the z axis. Hence \mathbf' \cdot \mathbf = 0, where \mathbf a is a displacement perpendicular to z. Since there are no boundaries in the region z>0, we expect a wave traveling to the right. The solution to the homogeneous equation becomes, \mathbf_^ = \mathbf_ \exp \left(i k' z\right) Adding this to the particular solution, we get the radiated wave inside the medium (z > 0) \mathbf_ = -\mathbf_(z) + \mathbf_ \exp \left(i k' z\right) The total field at any position z is the sum of the incident and radiated fields at that position. Adding the two components inside the medium, we get the total field \mathrm(z) = \mathrm_ \exp \left(i k' z\right), \qquad z>0 This wave travels inside the dielectric at speed c/n, n = c k' / \omega = \sqrt We can simplify the above n to a familiar form of the index of refraction of a linear isotropic dielectric. To do so, we remember that in a linear dielectric an applied electric field \mathbf E induces a polarization \mathbf P proportional to the electric field \mathbf P = \epsilon_ \chi_ \mathbf. When the electric field changes, the induced charges move and produces a current density given by \partial \mathbf / \partial t. Since the time dependence of the electric field is \exp(-i\omega t), we get \mathbf=-i \epsilon_ \omega \chi_ \mathbf, which implies that the conductivity \sigma=-i \epsilon_ \omega \chi_. Then substituting the conductivity in the equation of n, gives n=\sqrt which is a more familiar form. For the region z<0, one imposes the condition of a wave traveling to the left. By setting the conductivity in this region \sigma=0, we obtain the reflected wave \mathrm(z) = \mathrm_ \exp \left(-i k z\right), traveling at the speed of light. Note that the coefficients nomenclature, \mathbf E_ and \mathbf E_, are only adopted to match what we already expect.


Hertz vector approach

The following is a derivation based on a work by Wangsness and a similar derivation found in chapter 20 of Zangwill's text, Modern Electrodynamics. The setup is as follows, let the infinite half-space z<0 be vacuum and the infinite half-space z>0 be a uniform, isotropic, dielectric material with
electric susceptibility In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applie ...
, \chi. The
inhomogeneous electromagnetic wave equation In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero sou ...
for the electric field can be written in terms of the electric Hertz Potential, \boldsymbol_ , in the Lorenz gauge as \nabla^ \boldsymbol_-\frac \frac=-\frac. The electric field in terms of the Hertz vectors is given as \mathbf=\nabla \times \nabla \times \boldsymbol\pi_-\frac- \frac \left(\nabla \times \boldsymbol_ \right), but the magnetic Hertz vector \boldsymbol_ is 0 since the material is assumed to be non-magnetizable and there is no external magnetic field. Therefore the electric field simplifies to \mathbf = \nabla \times \nabla \times \boldsymbol\pi_ - \frac. In order to calculate the electric field we must first solve the inhomogeneous wave equation for \boldsymbol_ . To do this, split \boldsymbol_ in the homogeneous and particular solutions \boldsymbol_(\mathbf, t) = \boldsymbol_(\mathbf, t) + \boldsymbol_(\mathbf, t). Linearity then allows us to write \mathbf(\mathbf, t)=\mathbf_(\mathbf, t)+\mathbf_(\mathbf, t). The homogeneous solution, \mathbf_(\mathbf, t) , is the initial plane wave traveling with wave vector k_0 = \omega/c in the positive z direction \mathbf_(\mathbf, t) = \mathbf_ e^ . We do not need to explicitly find \boldsymbol_(\mathbf, t) since we are only interested in finding the field. The particular solution, \boldsymbol_(\mathbf, t) and therefore, \mathbf_(\mathbf, t) , is found using a time dependent
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
method on the inhomogeneous wave equation for \boldsymbol_ which produces the retarded integral \boldsymbol_(\mathbf, t) = \frac \int d^3 r' \frac. Since the initial electric field is polarizing the material, the polarization vector must have the same space and time dependence \mathbf(\mathbf, t) = \mathbf_ e^. More detail about this assumption is discussed by Wangsness. Plugging this into the integral and expressing in terms of Cartesian coordinates produces \boldsymbol_(\mathbf, t)= \frac \int_^ dz' e^ \int_^ dx' \int_^ dy' \frac. First, consider only the integration over x' and y' and convert this to
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
(x,y,z)\rightarrow(\rho,\varphi,z) and call \left, \mathbf-\mathbf'\ = R I:=\int_^ d x^ \int_^ d y^ \frac = \int_^ d \varphi' \int_^ d \rho' \frac = 2\pi \int_^ d \rho' \frac. Then using the substitution R^=\rho^+\left, z'-z\^ \Rightarrow \rho^ = R^-\left, z'-z\^ \Rightarrow \rho \, d \rho = R \, dR and \rho=\sqrt so the limits become \rho=0=\sqrt \Rightarrow R=\left, z'-z\ and \rho = \infty=\sqrt \Rightarrow R=\infty. Then introduce a convergence factor e^ with \epsilon \in \R into the integrand since it does not change the value of the integral, \begin I &= 2\pi \int_^ d R e^ \\&= 2\pi \lim_\int_^ d R e^ \\&= \left. 2\pi \lim_ \left frac\right\_^\infty \\&= 2\pi \lim_ \left frac-\frac\right \end Then \epsilon \in \R implies \lim_ e^=0 , hence \lim_ e^ = \lim_e^ e^=0 . Therefore, \begin I &= 2\pi \lim_ \left -\frac\right\\&= -2\pi \frac \\&= 2\pi i \frac. \end Now, plugging this result back into the z-integral yields \boldsymbol_(z, t)= \frac \int_^ d z' e^ e^ Notice that \boldsymbol_ is now only a function of z and not \mathbf, which was expected for the given symmetry. This integration must be split into two due to the absolute value \left, z-z' \ inside the integrand. The regions are z<0 and z>0. Again, a convergence factor must be introduced to evaluate both integrals and the result is \boldsymbol_(z, t)=-\frac \begin & \\ & \end Instead of plugging \boldsymbol_ directly into the expression for the electric field, several simplifications can be made. Begin with the curl of the curl vector identity, \nabla \times(\nabla \times \mathbf)=\nabla(\nabla \cdot \mathbf)-\nabla^ \mathbf, therefore, \begin \mathbf_ &= \nabla \times \nabla \times \boldsymbol\pi_-\frac \\&= \nabla(\nabla \cdot \boldsymbol_)-\nabla^ \boldsymbol_-\frac. \end Notice that \nabla \cdot \boldsymbol_ = 0 because \mathbf has no dependence and is always perpendicular to \hat. Also, notice that the second and third terms are equivalent to the inhomogeneous wave equation, therefore, \begin \mathbf_ &= -\frac \frac \\&= -\frac (-i\omega)^2 \boldsymbol_ \\&= k_^2 \boldsymbol_ \end Therefore, the total field is \mathbf(z, t)=\mathbf_ e^ +k_^ \boldsymbol\pi_(z, t) which becomes, \mathbf(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(k z-\omega t) & {z>0.} \end{array}\right. Now focus on the field inside the dielectric. Using the fact that \mathbf{E}(z, t) is complex, we may immediately write \mathbf{E}(z>0, t) = \mathbf{E} e^{i\left(k_{0} z-\omega t\right)} recall also that inside the dielectric we have \mathbf{P} =\epsilon_{0} \chi \mathbf{E}. Then by coefficient matching we find, e^{i\left(k z-\omega t\right)} \Rightarrow 1=-\chi \frac{k_{0}^{2{k_{0}^{2}-k^{2 and 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 \begin{align} k &=\sqrt{1+\chi} k_{0} \\&=n k_{0}. \end{align} Using this result and the definition of \mathbf{P} in the second expression yields the polarization vector in terms of the incident electric field as \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 \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(n k_{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.


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. In interstellar space, the extinction length for light is 2 light years. 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.
Special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
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. 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 The de Sitter effect was described by Willem de Sitter in 1913 (as well as by Daniel Frost Comstock in 1910) and used to support the special theory of relativity against a competing 1908 emission theory by Walther Ritz that postulated a variable ...
.) 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. However, experiments using X-rays emitted by binary pulsars, with much longer extinction length, have been successful.


References

{{DEFAULTSORT:Ewald-Oseen extinction theorem Physics theorems Scattering, absorption and radiative transfer (optics)