Editing Linear polarization

{{Short description|Electromagnetic radiation special case}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
{{More footnotes|date=May 2020}}
[[File:Linear polarization schematic.png|162px|thumb|right|Diagram of the electric field of a light wave (blue), linear-polarized along a plane (purple line), and consisting of two orthogonal, in-phase components (red and green waves)]]

In [[electrodynamics]], '''linear polarization''' or '''plane polarization''' of [[electromagnetic radiation]] is a confinement of the [[electric field]] vector or [[magnetic field]] vector to a given plane along the direction of propagation. The term ''linear polarization'' (French: ''polarisation rectiligne'') was coined by [[Augustin-Jean Fresnel]] in 1822.<ref name=fresnel-1822z>A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe", read 9&nbsp;December 1822; printed in H.&nbsp;de Senarmont, E.&nbsp;Verdet, and L.&nbsp;Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol.&nbsp;1 (1866), pp.{{nnbsp}}731–51; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis", {{Zenodo|4745976}}, 2021 (open&nbsp;access); §9.</ref> See ''[[Polarization (waves)|polarization]]'' and ''[[plane of polarization]]'' for more information.

The orientation of a linearly polarized electromagnetic wave is defined by the direction of the [[electric field]] vector.<ref name="Shapira,">{{cite book   
  | last = Shapira
  | first = Joseph 
  |author2=Shmuel Y. Miller
   | title = CDMA radio with repeaters
  | publisher = Springer
  | date = 2007
  | pages = 73
  | url = https://books.google.com/books?id=Yd56YZY1RpAC&q=%5Bpolarization+of+radio+waves%5D&pg=PA73
  | isbn = 978-0-387-26329-8}}</ref>  For example, if the electric field vector is vertical (alternately up and down as the wave travels) the radiation is said to be vertically polarized.

==Mathematical description==
The [[Classical physics|classical]] [[sinusoidal]] plane wave solution of the [[electromagnetic wave equation]] for the [[Electric field|electric]] and [[Magnetic field|magnetic]] fields is (cgs units)
 
:<math> \mathbf{E} ( \mathbf{r} , t ) = |\mathbf{E}|  \mathrm{Re} \left \{  |\psi\rangle  \exp \left [ i \left  ( kz-\omega t  \right ) \right ] \right \}  </math>

:<math> \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )/c   </math>

for the magnetic field, where k is the [[wavenumber]],

:<math> \omega_{ }^{ } = c k</math>

is the [[angular frequency]] of the wave, and <math> c </math> is the [[speed of light]].

Here <math>  \mid\mathbf{E}\mid    </math> is the [[amplitude]] of the field and

:<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>

is the [[Jones vector]] in the x-y plane.

The wave is linearly polarized when the phase angles <math> \alpha_x^{ } , \alpha_y </math> are equal,

:<math>    \alpha_x =  \alpha_y \ \stackrel{\mathrm{def}}{=}\   \alpha    </math>.

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

:<math>   |\psi\rangle  =   \begin{pmatrix} \cos\theta    \\ \sin\theta   \end{pmatrix} \exp \left ( i \alpha \right )   </math>.

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that

:<math>   |x\rangle  \ \stackrel{\mathrm{def}}{=}\    \begin{pmatrix} 1    \\ 0  \end{pmatrix}    </math>

and

:<math>   |y\rangle  \ \stackrel{\mathrm{def}}{=}\    \begin{pmatrix} 0    \\ 1  \end{pmatrix}    </math>

then the polarization state can be written in the "x-y basis" as

:<math>   |\psi\rangle  =  \cos\theta \exp \left ( i \alpha \right ) |x\rangle + \sin\theta \exp \left ( i \alpha \right ) |y\rangle = \psi_x |x\rangle + \psi_y |y\rangle </math>.

== See also ==
*[[Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
*[[Polarization (waves)|Polarization]]
**[[Circular polarization]]
**[[Elliptical polarization]]
**[[Plane of polarization]]
*[[Photon polarization]]

==References==
*{{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|date=1998|isbn=0-471-30932-X}}

{{Reflist}}

==External links==
*[https://www.youtube.com/watch?v=oDwqUgDFe94 Animation of Linear Polarization (on YouTube) ]
*[https://www.youtube.com/watch?v=Q0qrU4nprB0 Comparison of Linear Polarization with Circular and Elliptical Polarizations (YouTube Animation)]

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[[Category:Polarization (waves)]]

[[ja:直線偏光]]
[[pl:Polaryzacja_fali#Polaryzacja_liniowa]]