Linear polarization: Difference between revisions

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{{Short description|Electromagnetic radiation special case}}

[[Image: Linear_polarization_schematic.png|right|Linear polarization diagram]]

{{Use American English|date=March 2021}}

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. See [[polarization]] for more information.

{{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.

Historically, the orientation of a polarized electromagnetic wave has been defined in the optical regime by the orientation of the electric vector, and in the [[radio]] regime, by the orientation of the magnetic vector.

The orientation of a linearly polarized electromagnetic wave is defined by the direction of the [[electric field]] vector.<ref name="Shapira,">{{cite book

==Mathematical description of linear polarization==

| last = Shapira

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)

| 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 of linear polarization==

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 ) = \mid |\mathbf{E} \mid| \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

Here

:<math> |\psi\rangle \equiv \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>

:<math> \mid \mathbf{E} \mid </math>

is the [[amplitudeJones vector]] ofin the fieldx-y andplane.

:<math> |\psi\rangle \equiv \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 \equiv \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>.

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If unit vectors are defined such that

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

and

:<math> |y\rangle \equiv \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]]

*[[Category: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|yeardate=1998|idisbn=ISBN 047130932X0-471-30932-X}}

{{Reflist}}

== See also ==

==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)]

*[[Polarization of classical electromagnetic waves]]

* [[Circular polarization]]

* [[Elliptical polarization]]

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[[Category:Polarization]]

[[Category:Polarization (waves)]]

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