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| '''Sinusoidal plane-wave solutions''' are particular solutions to the [[electromagnetic wave equation]].
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| The general solution of the electromagnetic [[wave equation]] in homogeneous, linear, time-independent media can be written as a [[Superposition principle|linear superposition]] of plane-waves of different frequencies and [[Polarization (waves)|polarizations]].
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| The treatment in this article is [[Classical physics|classical]] but, because of the generality of [[Maxwell's equations]] for electrodynamics, the treatment can be converted into the [[Quantum mechanics|quantum mechanical]] treatment with only a reinterpretation of classical quantities (aside from the quantum mechanical treatment needed for charge and current densities).
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| The reinterpretation is based on the theories of [[Max Planck]] and the interpretations by [[Albert Einstein]] of those theories and of other experiments. The quantum generalization of the classical treatment can be found in the articles on [[Photon polarization]] and Photon dynamics in the double-slit experiment.
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| == Explanation ==
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| Experimentally, every light signal can be decomposed into a [[Electromagnetic spectrum|spectrum]] of frequencies and wavelengths associated with sinusoidal solutions of the wave equation. Polarizing filters can be used to decompose light into its various polarization components. The polarization components can be [[linear polarization|linear]], [[circular polarization|circular]] or [[elliptical polarization|elliptical]].
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| ==Plane waves==
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| The plane [[Sine wave|sinusoidal]] solution for an [[Electromagnetic wave equation|electromagnetic wave]] traveling in the z direction is (cgs units and SI units)
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| :<math> \mathbf{E} ( \mathbf{r} , t ) = \begin{pmatrix} E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \\ E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \\ 0 \end{pmatrix} = E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \hat {\mathbf{x}} \; + \; E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \hat {\mathbf{y}} </math>
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| for the electric field and
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| :<math> c \, \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t ) = \begin{pmatrix} -E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \\ E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \\ 0 \end{pmatrix} = -E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \hat {\mathbf{x}} \; + \; E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \hat {\mathbf{y}} </math>
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| for the magnetic field, where k is the [[wavenumber]],
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| :<math> \omega_{ }^{ } = c k</math>
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| is the [[angular frequency]] of the wave, and <math> c </math> is the [[speed of light]]. The hats on the [[vector (geometric)|vectors]] indicate [[unit vectors]] in the x, y, and z directions. | |
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| The plane wave is parameterized by the [[amplitude]]s
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| [[Image:Light-wave.svg|thumb|right|350px|Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarized wave propagating from right to left. The magnetic field (labeled M) is in a horizontal plane, and the electric field (labeled E) is in a vertical plane.]]
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| :<math> E_x^0 = \mid \mathbf{E} \mid \cos \theta </math>
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| :<math> E_y^0 = \mid \mathbf{E} \mid \sin \theta </math>
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| and [[Phase (waves)|phases]]
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| :<math> \alpha_x^{ } , \alpha_y </math>
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| where
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| :<math> \theta \ \stackrel{\mathrm{def}}{=}\ \tan^{-1} \left ( { E_y^0 \over E_x^0 } \right ) </math>.
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| and
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| :<math> \mid \mathbf{E} \mid^2 \ \stackrel{\mathrm{def}}{=}\ \left ( E_x^0 \right )^2 + \left ( E_y^0 \right )^2 </math>.
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| ==Polarization state vector==
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| {{main|Jones calculus}}
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| [[Image:Linear polarization schematic.png|thumb|right|350px|Linear polarization.]]
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| ===Jones vector===
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| All the polarization information can be reduced to a single vector, called the [[Jones vector]], in the x-y plane. This vector, while arising from a purely classical treatment of polarization, can be interpreted as a [[quantum state]] vector. The connection with quantum mechanics is made in the article on [[photon polarization]].
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| The vector emerges from the plane-wave solution. The electric field solution can be rewritten in [[Complex number|complex]] notation as
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| :<math> \mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \} </math>
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| where
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| :<math> |\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix} </math>
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| is the Jones vector in the x-y plane. The notation for this vector is the [[bra-ket notation]] of [[Paul Dirac|Dirac]], which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector.
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| ===Dual Jones vector===
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| The Jones vector has a [[Dual space|dual]] given by
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| :<math> \langle \psi | \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x^* & \psi_y^* \end{pmatrix} = \begin{pmatrix} \quad \cos\theta \exp \left ( -i \alpha_x \right ) & \sin\theta \exp \left ( -i \alpha_y \right ) \quad \end{pmatrix} </math>.
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| ===Normalization of the Jones vector===
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| The Jones vector is [[Normalisable wavefunction|normalized]]. The [[inner product]] of the vector with itself is
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| :<math> \langle \psi | \psi\rangle = \begin{pmatrix} \psi_x^* & \psi_y^* \end{pmatrix} \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = 1 </math>.
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| [[Image:Circular polarization schematic.png|thumb|right|350px|Circular polarization.]]
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| ==Polarization states==
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| {{main|Polarization (waves)}}
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| ===Linear polarization===
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| {{main|Linear polarization}}
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| In general, the wave is linearly polarized when the phase angles <math> \alpha_x^{ } , \alpha_y </math> are equal,
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| :<math> \alpha_x = \alpha_y \ \stackrel{\mathrm{def}}{=}\ \alpha </math>.
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| This represents a wave polarized at an angle <math> \theta </math> with respect to the x axis. In that case the Jones vector can be written
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| :<math> |\psi\rangle = \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix} \exp \left ( i \alpha \right ) </math>.
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| ===Circular polarization===
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| {{main|Circular polarization}}
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| If <math> \alpha_y </math> is rotated by <math> \pi / 2 </math> radians with respect to <math> \alpha_x </math> the wave is [[Circular polarization|circularly polarized]]. The Jones vector is
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| :<math> |\psi\rangle = \begin{pmatrix} \cos\theta \\ \pm i\sin\theta \end{pmatrix} \exp \left ( i \alpha_x \right ) </math>
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| where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.
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| If unit vectors are defined such that
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| :<math> |R\rangle \ \stackrel{\mathrm{def}}{=}\ {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix} </math> | |
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| and
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| :<math> |L\rangle \ \stackrel{\mathrm{def}}{=}\ {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix} </math>
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| [[Image:Elliptical polarization schematic.png|thumb|right|350px|Elliptical polarization.]]
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| then a circular polarization state can written in the "R-L basis" as
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| :<math> |c\rangle = \psi_R |R\rangle + \psi_L |L\rangle </math>
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| where
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| :<math> \psi_R \ \stackrel{\mathrm{def}}{=}\ \left ( {\cos\theta -i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) = \left ( {\exp(-i\theta) \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) </math>
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| and
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| :<math> \psi_L \ \stackrel{\mathrm{def}}{=}\ \left ( {\cos\theta +i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) = \left ( {\exp(i\theta) \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) </math>.
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| Any arbitrary state can be written in the R-L basis
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| :<math> |\psi\rangle = a_R \exp \left ( i \alpha_x -i \theta \right ) |R\rangle + a_L \exp \left ( i \alpha_x + i \theta \right ) |L\rangle </math>
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| where
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| :<math> 1 = \mid a_R \mid^2 + \mid a_L \mid^2 </math>. | |
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| ===Elliptical polarization===
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| {{main|Elliptical polarization}}
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| The general case in which the electric field rotates in the x-y plane and has variable magnitude is called [[elliptical polarization]]. The state vector is given by
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| :<math> |\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix} </math>.
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| ==References==
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| *{{cite book |last=Jackson|first= John D.|title=Classical Electrodynamics |edition=3rd|publisher=Wiley|year=1998|isbn=0-471-30932-X}}
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| ==See also==
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| *[[Fourier series]]
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| *[[Transverse mode]]
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| *[[Theoretical and experimental justification for the Schrödinger equation]]
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| *[[Maxwell's equations]]
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| *[[Electromagnetic wave equation]]
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| *[[Mathematical descriptions of the electromagnetic field]]
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| * [http://www.hydrogenlab.de/elektronium/HTML/einleitung_hauptseite_uk.html Polarization from an atomic transition: linear and circular]
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| [[Category:Polarization (waves)|Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
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| [[Category:Electromagnetic radiation|Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
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| [[Category:Antennas (radio)|Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
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Hi there. My title is Garland although it is not the name on my beginning certification. To play badminton is some thing he really enjoys doing. I've usually loved residing in Idaho. My occupation is a messenger.
Take a look at my web site; http://brazil.amor-amore.com