Elliptical polarization: Difference between revisions

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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&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.

Other forms of polarization, such as [[circular polarization|circular]] and [[linear polarization]], can be considered to be special cases of elliptical polarization.

[[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 ) = \midleft| \mathbf{E} \midright| \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>

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> c </math> is the [[speed of light]].

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

:Here <math>| \omega_mathbf{ E}^{ } = c k|</math> is the [[amplitude]] of the field and

is the [[angular frequency]] of the wave propagating in the +z direction, 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>

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==Polarization ellipse==

[[File:Polarisation ellipse.svg|250px|right]]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>.

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==In nature==

The reflected light from some beetles (e.g. ''[[Cetonia aurata]]'') is elliptical polarized.<ref>[http://www.tandfonline.com/doi/full/10.1080/14786435.2011.648228{{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==

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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==