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{{Short description|Polarization of electromagnetic radiation}}
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{{more footnotes|date=November 2018}}
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In [[electrodynamics]], '''elliptical polarization''' is the [[Polarization (waves)|polarization]] of [[electromagnetic radiation]] such that the tip of the [[electric field]] [[vector (geometry)|vector]] describes an [[ellipse]] in any fixed plane intersecting, and [[Surface normal|normal]] to, the direction of propagation. An elliptically polarized wave may be resolved into two [[linear polarization|linearly polarized wave]]s in [[Quadrature phase|phase quadrature]], with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit [[chirality (physics)|chirality]].
''[[Circular polarization]]'' and ''[[linear polarization]]'' can be considered to be special cases of ''elliptical polarization''. This terminology was introduced 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 December 1822; printed in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol. 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 access); §§9–10.</ref> before the electromagnetic nature of light waves was known.
[[Image:Elliptical polarization schematic.png|right|Elliptical polarization diagram]]
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The [[Classical physics|classical]] [[sinusoidal]] plane wave solution of the [[electromagnetic wave equation]] for the [[Electric field|electric]] and [[Magnetic field|magnetic]] fields is ([[Gaussian units]])
:<math> \mathbf{E} ( \mathbf{r} , t ) = \
:<math> \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t ) ,</math>
where <math>k</math> is the [[wavenumber]], <math display=inline> \omega = c k</math> is the [[angular frequency]] of the wave propagating in the +z direction, and <math>
▲is the [[angular frequency]] of the wave propagating in the +z direction, and <math> c </math> is the [[speed of light]].
:<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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==Polarization ellipse==
:<math> A=|\mathbf{E}|\sqrt{\frac{1+\sqrt{1-\sin^2(2\theta)\sin^2\beta}}{2}}</math>
and
:<math> B=|\mathbf{E}|\sqrt{\frac{1-\sqrt{1-\sin^2(2\theta)\sin^2\beta}}{2}}</math>,
where <math>\beta =\alpha_y-\alpha_x</math> with the phases <math>\alpha_x</math> and <math>\alpha_y</math>.
The orientation of the ellipse is given by the angle <math>\phi </math> the semi-major axis makes with the x-axis. This angle can be calculated from
:<math> \tan2\phi=\tan2\theta\cos\beta</math>.
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If <math>\beta=\pm\pi/2</math> and <math>\theta=\pi/4</math>, <math> A=B=|\mathbf{E}|/\sqrt{2}</math>, i.e., the wave is [[circular polarization|circularly polarized]]. When <math>\beta=\pi/2</math>, the wave is left-circularly polarized, and when <math>\beta=-\pi/2</math>, the wave is right-circularly polarized.
===Parameterization===
{{main|Polarization (waves)#Parameterization}}
{{anchor|Axial ratio}} Any fixed polarization can be described in terms of the shape and orientation of the polarization ellipse, which is defined by two parameters: axial ratio AR and tilt angle <math>\tau</math>. The axial ratio is the ratio of the lengths of the major and minor axes of the ellipse, and is always greater than or equal to one.
Alternatively, polarization can be represented as a point on the surface of the [[Poincaré sphere (optics)|Poincaré sphere]], with <math>2\times \tau</math> as the [[longitude]] and <math>2\times \epsilon</math> as the [[latitude]], where <math>\epsilon=\arccot(\pm AR)</math>. The sign used in the argument of the <math>\arccot</math> depends on the handedness of the polarization. Positive indicates left hand polarization, while negative indicates right hand polarization, as defined by IEEE.
For the special case of [[circular polarization]], the axial ratio equals 1 (or 0 dB) and the tilt angle is undefined. For the special case of [[linear polarization]], the axial ratio is infinite.
==In nature==
The reflected light from some beetles (e.g. ''[[Cetonia aurata]]'') is elliptical polarized.<ref>
==See also==
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*{{FS1037C MS188}}
{{reflist}}
* [[Henri Poincaré]] (1889) [https://archive.org/details/leonssurlath00poin/page/n8 Théorie Mathématique de la Lumière, volume 1] and [https://archive.org/details/thoriemathma00poin/page/n8 Volume 2] (1892) via [[Internet Archive]].
* H. Poincaré (1901) [https://archive.org/details/lectricitetopti04poingoog/page/n12 Électricité et Optique : La Lumière et les Théories Électrodynamiques], via Internet Archive
==External links==
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