Editing Elliptical polarization

{{Short description|Polarization of electromagnetic radiation}}
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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&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 access); §§9–10.</ref> before the electromagnetic nature of light waves was known.

[[Image:Elliptical polarization schematic.png|right|Elliptical polarization diagram]]

==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 ([[Gaussian units]])

:<math> \mathbf{E} ( \mathbf{r} , t ) = \left| \mathbf{E} \right|  \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 )   </math>

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

:<math> \omega = c k</math>

is the [[angular frequency]] of the wave propagating in the +z direction, and <math> c </math> is the [[speed of light]].

Here <math>| \mathbf{E} |</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 normalized [[Jones vector]]. This is the most complete representation of polarized electromagnetic radiation and corresponds in general to elliptical polarization.

==Polarization ellipse==
At a fixed point in space (or for fixed z), the electric vector <math> \mathbf{E} </math> traces out an ellipse in the x-y plane. The semi-major and semi-minor axes of the ellipse have lengths A and B, respectively, that are given by
:<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>.
If <math>\beta= 0</math>, the wave is [[linear polarization|linearly polarized]]. The ellipse collapses to a straight line <math>(A=|\mathbf{E}|, B=0</math>) oriented at an angle <math>\phi=\theta</math>. This is the case of superposition of two simple harmonic motions (in phase), one in the x direction with an amplitude <math>|\mathbf{E}| \cos\theta</math>, and the other in the y direction with an amplitude <math>|\mathbf{E}| \sin\theta </math>. When <math>\beta</math> increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking in the direction of the propagating wave); this then corresponds to ''left-handed elliptical polarization''; the semi-major axis is now oriented at an angle <math>\phi\neq\theta </math>. Similarly, if <math>\beta</math> becomes negative from zero, the line evolves into an ellipse that is being traced out in the clockwise direction; this corresponds to ''right-handed elliptical polarization''.

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>{{Cite journal|title=Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson|first1=Hans|last1=Arwin|first2=Roger|last2=Magnusson|first3=Jan|last3=Landin|first4=Kenneth|last4=Järrendahl|date=April 21, 2012|journal=Philosophical Magazine|volume=92|issue=12|pages=1583–1599|doi=10.1080/14786435.2011.648228|bibcode = 2012PMag...92.1583A|s2cid=13988658 |url = http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-77876}}</ref>

==See also==
*[[Ellipsometry]]
*[[Fresnel rhomb]]
*[[Photon polarization]]
*[[Sinusoidal plane-wave solutions of the electromagnetic wave equation]]

==References==
*{{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==
*[https://www.youtube.com/watch?v=KZz25bmTWXo Animation of Elliptical Polarization (on YouTube) ]
*[https://www.youtube.com/watch?v=Q0qrU4nprB0 Comparison of Elliptical Polarization with Linear and Circular Polarizations (YouTube Animation)]

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[[Category:Polarization (waves)]]

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