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*[[{{Short description|Polarization of classical electromagnetic waves]]radiation}}

In [[electrodynamics]], '''elliptical polarization''' is the [[polarization]] of [[electromagnetic radiation]] such that the tip of the [[electric field]] [[Vector (spatial)|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.

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{{more footnotes|date=November 2018}}

{{Technical|date=February 2010}}

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In [[electrodynamics]], '''elliptical polarization''' is the [[Polarization (waves)|polarization]] of [[electromagnetic radiation]] such that the tip of the [[electric field]] [[Vectorvector (spatialgeometry)|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_phaseQuadrature 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 Elliptical_polarization_schematicpolarization schematic.png|right|Elliptical polarization diagram]]

==Example==

If an electromagnetic wave is created by multiple dipole antennas with 90° [[phase shift]]s relative to one another, the electric field vector will rotate, with the direction (clockwise or counterclockwise) depending on the sign of the phase shift.

==Mathematical description of linear polarization==

The angle between the electric field vector and the '''x''' axis is given by

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[[Gaussian units]])

:<math>\Phi(\mathbf{E})=\tan^{-1}{E_y\over E_x},</math>

where <math>E_x</math> and <math>E_y</math> are perpendicular [[vector component|components]] of the electric field vector. If the x and y components have a 90° phase shift between them, these components are given by

:<math>E_y=E_{yo} \sin(wt+\phi),\ \mathrm{and}\,</math>

:<math>E_x=E_{xo} \sin(wt+\phi+{\pi \over 2}),</math>

where <math>\phi\,</math> is an arbitrary phase, and <math>E_xo</math> and <math>E_yo</math> are the [[amplitude]]s of the x and y components of the field. It can then be shown that

:<math>\Phi(\mathbf{E}) = \omega t,</math>

which indicates that the electric field vector rotates with an [[angular velocity]] <math>\omega</math>. Since the phase shift on <math>E_x</math> is positive, the rotation is counterclockwise. If <math>E_{yo} = E_{xo}</math>, the polarization is circular. If they are different, it is elliptical.

:<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>

==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 ) ,</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

:<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>

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

is the normalized [[Jones vector]]. This is the most complete representation of polarized electromagnetic radiation and corresponds in general to elliptical polarization.

Here

==Polarization ellipse==

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

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.

is the [[amplitude]] of the field and

===Parameterization===

:<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>

{{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.

is the [[Jones vector]] in the x-y plane. Here <math> \theta </math> is an angle that determines the tilt of the ellipse and <math> \alpha_x - \alpha_y </math> determines the aspect ratio of the ellipse.

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]]

*[[Polarization of classical electromagnetic waves]]

*[[Fresnel rhomb]]

{{optics-stub}}

*[[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)]

[[Category{{DEFAULTSORT:Elliptical Polarization]]}}

[[Category:Polarization (waves)]]

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