View source for Circular polarization

{{short description|Polarization state}}
[[Image:Circular.Polarization.Circularly.Polarized.Light Left.Hand.Animation.305x190.255Colors.gif|thumb|305px|The [[electric field]] vectors of a traveling circularly polarized electromagnetic wave. This wave is right-handed/clockwise circularly polarized as defined from the point of view of the source, or left-handed/anti-clockwise circularly polarized if defined from the point of view of the receiver.]]

In [[Classical electromagnetism|electrodynamics]], '''circular polarization''' of an [[Electromagnetic radiation|electromagnetic wave]] is a [[Polarization (waves)|polarization]] state in which, at each point, the [[electromagnetic field]] of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to the direction of the wave.

In electrodynamics, the strength and direction of an electric field is defined by its electric field vector. In the case of a circularly polarized wave, the tip of the electric field [[Euclidean vector|vector]], at a given point in space, relates to the phase of the light as it travels through time and space. At any instant of time, the electric field vector of the wave indicates a point on a [[helix]] oriented along the direction of propagation. A circularly polarized wave can rotate in one of two possible senses: ''right-handed circular polarization (RHCP)'' in which the electric field vector rotates in a [[Right-hand rule|right-hand]] sense with respect to the direction of propagation, and ''left-handed circular polarization (LHCP)'' in which the vector rotates in a [[left hand rule|left-hand]] sense.

''Circular polarization'' is a [[limiting case (mathematics)|limiting case]] of ''[[elliptical polarization]]''. The other [[special case]] is the easier-to-understand ''[[linear polarization]]''. All three terms were coined by [[Augustin-Jean Fresnel]], in a memoir read to the [[French Academy of Sciences]] on 9&nbsp;December 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); §§9–10.</ref><ref>Académie des Sciences, ''Procès-verbaux des séances de l'Académie tenues depuis la fondation de l'Institut jusqu'au mois d'août 1835'', vol.&nbsp;7 (for 1820–23), Hendaye, Basses Pyrénées: Imprimerie de l'Observatoire d'Abbadia, 1916, p.&nbsp;401.</ref> Fresnel had first described the case of circular polarization, without yet naming it, in 1821.<ref name=fresnel-1821a>A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et&nbsp;seq., ''Annales de Chimie et de Physique'', Ser.&nbsp;2, vol.&nbsp;17, pp.&nbsp;102–11 (May&nbsp;1821), 167–96 (June&nbsp;1821), 312–15 ("Postscript", July&nbsp;1821); reprinted (with added section nos.) in H.&nbsp;de Senarmont, E.&nbsp;Verdet, and L.&nbsp;Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol.&nbsp;1 (1866), pp.&nbsp;609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, &amp;&nbsp;postscript", {{Zenodo|4058004}} (Creative Commons), 2021;&nbsp; author's footnote to §16.</ref>

The phenomenon of polarization arises as a consequence of the fact that [[light]] behaves as a two-dimensional [[Transverse wave#Explanation|transverse wave]].

Circular polarization occurs when the two orthogonal electric field component vectors are of equal magnitude and are out of phase by exactly 90°, or one-quarter wavelength.

== Characteristics ==
{{multiple image
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 | footer  = Right-handed/counterclockwise circularly polarized light displayed with and without the use of components. This would be considered clockwise circularly polarized if defined from the point of view of the source rather than the receiver. Handedness is independent of the perspective of the source or receiver.
 | footer_align = left
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 | image1  = Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg
 | image2  = Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg
 | image3 =  Circular polarization cross section.gif
}}

In a circularly polarized electromagnetic wave, the individual electric field vectors, as well as their combined vector, have a constant [[Magnitude (vector)|magnitude]], and with changing phase angle. Given that this is a [[plane wave]], each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the optical axis. Specifically, given that this is a [[Plane wave#Polarized electromagnetic plane waves|circularly polarized plane wave]], these vectors indicate that the electric field, from plane to plane, has a constant strength while its direction steadily rotates. Refer to [[Sinusoidal_plane_wave#Polarized electromagnetic plane waves|these two images]] in the plane wave article to better appreciate this dynamic. This light is considered to be right-hand, clockwise circularly polarized if viewed by the receiver. Since this is an [[Electromagnetic radiation|electromagnetic wave]], each [[electric field]] vector has a corresponding, but not illustrated, [[magnetic field]] vector that is at a [[right angle]] to the electric field vector and [[Proportionality (mathematics)|proportional]] in magnitude to it. As a result, the magnetic field vectors would trace out a second helix if displayed.

Circular polarization is often encountered in the field of optics and, in this section, the electromagnetic wave will be simply referred to as [[light]].

The nature of circular polarization and its relationship to other polarizations is often understood by thinking of the electric field as being divided into two [[Euclidean vector|components]] that are perpendicular to each other. The vertical component and its corresponding plane are illustrated in blue, while the horizontal component and its corresponding plane are illustrated in green. Notice that the rightward (relative to the direction of travel) horizontal component leads the vertical component by one quarter of a [[wavelength]], a 90° phase difference. It is this [[In-phase and quadrature components|quadrature phase]] relationship that creates the [[helix]] and causes the points of maximum magnitude of the vertical component to correspond with the points of zero magnitude of the horizontal component, and vice versa. The result of this alignment are select vectors, corresponding to the helix, which exactly match the maxima of the vertical and horizontal components.

To appreciate how this quadrature [[phase (waves)|phase]] shift corresponds to an electric field that rotates while maintaining a constant magnitude, imagine a dot traveling clockwise in a circle. Consider how the vertical and horizontal [[Displacement (vector)|displacements]] of the dot, relative to the center of the circle, vary [[Sine wave|sinusoidally]] in time and are out of phase by one quarter of a cycle. The displacements are said to be out of phase by one quarter of a cycle because the horizontal maximum displacement (toward the left) is reached one quarter of a cycle before the vertical maximum displacement is reached. Now referring again to the illustration, imagine the center of the circle just described, traveling along the axis from the front to the back. The circling dot will trace out a helix with the displacement toward our viewing left, leading the vertical displacement. Just as the horizontal and vertical displacements of the rotating dot are out of phase by one quarter of a cycle in time, the magnitude of the horizontal and vertical components of the electric field are out of phase by one quarter of a wavelength.

{{multiple image
| align = right
| direction = vertical
| footer = Left-handed/counterclockwise circularly polarized light displayed with and without the use of components. This would be considered right-handed/clockwise circularly polarized if defined from the point of view of the source rather than the receiver.
| footer_align = left
| width = 440
| image1 = Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg
| caption1 = 
| image2 = Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg
| caption2 = 
}}
The next pair of illustrations is that of left-handed, counterclockwise circularly polarized light when viewed by the receiver. Because it is left-handed, the rightward (relative to the direction of travel) horizontal component is now ''lagging'' the vertical component by one quarter of a wavelength, rather than leading it.

===Reversal of handedness===

====Waveplate====
To convert circularly polarized light to the other handedness, one can use a half-[[waveplate]]. A half-waveplate shifts a given linear component of light one half of a wavelength relative to its orthogonal linear component.

====Reflection====
The handedness of polarized light is reversed reflected off a surface at normal incidence. Upon such reflection, the rotation of the [[plane of polarization]] of the reflected light is identical to that of the incident field. However, with propagation now in the ''opposite'' direction, the same rotation direction that would be described as "right-handed" for the incident beam, is "left-handed" for propagation in the reverse direction, and vice versa. Aside from the reversal of handedness, the ellipticity of polarization is also preserved (except in cases of reflection by a [[Birefringence|birefringent]] surface).

Note that this principle only holds strictly for light reflected at normal incidence. For instance, right circularly polarized light reflected from a dielectric surface at grazing incidence (an angle beyond the [[Brewster's angle|Brewster angle]]) will still emerge as right-handed, but elliptically polarized. Light reflected by a metal at non-normal incidence will generally have its ellipticity changed as well. Such situations may be solved by decomposing the incident circular (or other) polarization into components of linear polarization parallel and perpendicular to the [[plane of incidence]], commonly denoted ''p'' and ''s'' respectively. The reflected components in the ''p'' and ''s'' linear polarizations are found by applying the [[Fresnel equations|Fresnel coefficients]] of reflection, which are generally different for those two linear polarizations. Only in the special case of normal incidence, where there is no distinction between ''p'' and ''s'', are the Fresnel coefficients for the two components identical, leading to the above property.

[[File:Reversal of handedness of circularly polarized light reflected by mirror 2s.gif|thumbnail|A 3-slide series of pictures taken with and without a pair of MasterImage 3D circularly polarized movie glasses of some dead European rose chafers (Cetonia aurata) whose shiny green color comes from left-polarized light. Note that, without glasses, both the beetles and their images have shiny color. The right-polarizer removes the color of the beetles but leaves the color of the images. The left-polarizer does the opposite, showing reversal of handedness of the reflected light.]]

===Conversion to linear polarization===
Circularly polarized light can be converted into linearly polarized light by passing it through a quarter-[[waveplate]]. Passing linearly polarized light through a quarter-waveplate with its axes at 45° to its polarization axis will convert it to circular polarization. In fact, this is the most common way of producing circular polarization in practice. Note that passing linearly polarized light through a quarter-waveplate at an angle ''other'' than 45° will generally produce elliptical polarization.

== Handedness conventions {{Anchor|Left/Right}} ==

[[Image:Circular.Polarization.Circularly.Polarized.Light Left.Hand.Animation.305x190.255Colors.gif|thumb|305px|A right-handed/clockwise circularly polarized wave as defined from the point of view of the source. It would be considered left-handed/anti-clockwise circularly polarized if defined from the point of view of the receiver.]]
 [[Image:Circular.Polarization.Circularly.Polarized.Light Right.Handed.Animation.305x190.255Colors.gif|thumb|305px|A left-handed/anti-clockwise circularly polarized wave as defined from the point of view of the source. It would be considered right-handed/clockwise circularly polarized if defined from the point of view of the receiver.]]

Circular polarization may be referred to as right-handed or left-handed, and clockwise or anti-clockwise, depending on the direction in which the electric field vector rotates. Unfortunately, two opposing historical conventions exist.

=== From the point of view of the source ===

Using this convention, polarization is defined from the point of view of the source. When using this convention, left- or right-handedness is determined by pointing one's left or right thumb {{em|away}} from the source, in the {{em|same}} direction that the wave is propagating, and matching the curling of one's fingers to the direction of the temporal rotation of the field at a given point in space. When determining if the wave is clockwise or anti-clockwise circularly polarized, one again takes the point of view of the source, and while looking {{em|away}} from the source and in the {{em|same}} direction of the wave's propagation, one observes the direction of the field's temporal rotation.

Using this convention, the electric field vector of a left-handed circularly polarized wave is as follows:
<math> \left( E_x ,\, E_y ,\, E_z \right) \propto \left(\cos \frac{2 \pi}{\lambda} \left(c t - z \right),\, -\sin \frac{2 \pi}{\lambda} \left(c t - z \right),\, 0 \right) . </math>

As a specific example, refer to the circularly polarized wave in the first animation. Using this convention, that wave is defined as right-handed because when one points one's right thumb in the same direction of the wave's propagation, the fingers of that hand curl in the same direction of the field's temporal rotation. It is considered clockwise circularly polarized because, from the point of view of the source, looking in the same direction of the wave's propagation, the field rotates in the clockwise direction. The second animation is that of left-handed or anti-clockwise light, using this same convention.

This convention is in conformity with the [[Institute of Electrical and Electronics Engineers]] (IEEE) standard and, as a result, it is generally used in the engineering community.<ref>IEEE Std 149-1979 (R2008), "IEEE Standard Test Procedures for Antennas". Reaffirmed December 10, 2008, Approved December 15, 1977, IEEE-SA Standards Board. Approved October 9, 2003, American National Standards Institute. {{ISBN|0-471-08032-2}}. {{doi|10.1109/IEEESTD.1979.120310}}, sec. 11.1, p.&nbsp;61."the sense of polarization, or handedness ... is called right handed (left handed) if the direction of rotation is clockwise (anti-clockwise) for an observer looking in the direction of propagation"</ref><ref name=Orfanidis>Electromagnetic Waves & Antennas – S. J. Orfanidis: Footnote p.45, "most engineering texts use the IEEE convention and most physics texts, the opposite convention."</ref><ref>Electromagnetic Waves & Antennas – S. J. Orfanidis Pg 44 "Curl the fingers of your left and right hands into a fist and point both thumbs ''towards'' the direction of propagation"</ref>

Quantum physicists also use this convention of handedness because it is consistent with their convention of handedness for a particle's spin.<ref name="Lectures_on_Physics(Vol_1_ch_33-1)" >Lectures on Physics Feynman (Vol. 1, ch.33-1) "If the end of the electric vector, when we look at it as the light comes straight toward us, goes around in an anti-clockwise direction, we call it right-hand circular polarization. ... Our convention for labeling left-hand and right-hand circular polarization is consistent with that which is used today for all the other particles in physics which exhibit polarization (e.g., electrons). However, in some books on optics the opposite conventions are used, so one must be careful."</ref>

Radio astronomers also use this convention in accordance with an [[International Astronomical Union|International Astronomical Union (IAU)]] resolution made in 1973.<ref name=IAU>IAU General Assembly Meeting, 1973, Commission 40 (Radio Astronomy/Radioastronomie), 8. POLARIZATION DEFINITIONS -- "A working Group chaired by Westerhout was convened to discuss the definition of polarization brightness temperatures used in the description of polarized extended objects and the galactic
background. The following resolution was adopted by Commissions 25 and 40: 'RESOLVED, that the frame of reference for the Stokes parameters is that of Right Ascension and Declination with the position angle of electric-vector maximum, q, starting from North and increasing through East. Elliptical polarization is defined in conformity with the definitions of the Institute of Electrical and Electronics Engineers (IEEE Standard 211, 1969). This means that the polarization of incoming radiation, for which the position angle, q, of the electric vector, measured at a fixed point in space, increases with time, is described as right-handed and positive.'"</ref>

=== From the point of view of the receiver ===

In this alternative convention, polarization is defined from the point of view of the receiver. Using this convention, left- or right-handedness is determined by pointing one's left or right thumb {{em|toward}} the source, {{em|against}} the direction of propagation, and then matching the curling of one's fingers to the temporal rotation of the field.

When using this convention, in contrast to the other convention, the defined handedness of the wave matches the handedness of the screw type nature of the field in space. Specifically, if one freezes a right-handed wave in time, when one curls the fingers of one's right hand around the helix, the thumb will point in the direction of progression for the helix, given the sense of rotation. Note that, in the context of the nature of all screws and helices, it does not matter in which direction you point your thumb when determining its handedness.

When determining if the wave is clockwise or anti-clockwise circularly polarized, one again takes the point of view of the receiver and, while looking {{em|toward}} the source, {{em|against}} the direction of propagation, one observes the direction of the field's temporal rotation.

Just as in the other convention, right-handedness corresponds to a clockwise rotation, and left-handedness corresponds to an anti-clockwise rotation.

Many optics textbooks use this second convention.<ref name="Polarization_in_Spectral_Lines_Section_1.2">Polarization in Spectral Lines. 2004 E. Landi Degl'innocenti, M Landolfi Section 1.2 "When ... the tip of the electric field vector rotates clockwise for an observer facing the radiation source, ... (it will be considered)... positive (or righthanded) circular polarization, Our convention ... agrees with those proposed in the classical textbooks on polarized light by Shurcliff (1952) and by Clarke and Grainger (1971). The same convention is also used, although with some few exceptions, by optical astronomers working in the field of polarimetry. Many radio astronomers, on the other hand, use the opposite convention. [https://books.google.com/books?id=8sl2CkmZNWIC&dq=circular+polarization+conventions&pg=PA5]</ref><ref>HANDBOOK OPTICS Volume I, Devices, Measurements and Properties, Michael Bass Page 272 Footnote: "Right-circularly polarized light is defined as a clockwise rotation of the electric vector when the observer is looking ''against'' the direction the wave is traveling."</ref> It is also used by [[SPIE]]<ref>{{cite web|title=The Polarization Ellipse|url=https://spie.org/publications/fg05_p07-09_polarization_ellipse|website=spie.org|access-date=13 April 2018}}</ref> as well as the  [[International Union of Pure and Applied Chemistry]] (IUPAC).<ref>{{cite journal |title=Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006) |author=S. E. Braslavsky |url=https://www.degruyter.com/downloadpdf/journals/pac/79/3/article-p293.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.degruyter.com/downloadpdf/journals/pac/79/3/article-p293.pdf |archive-date=2022-10-09 |url-status=live |journal=Pure and Applied Chemistry |volume=79 |issue=3 |pages=293–465 |date=1 January 2009 |doi=10.1351/pac200779030293|s2cid=96601716 }}</ref>

=== Uses of the two conventions ===

As stated earlier, there is significant confusion with regards to these two conventions. As a general rule, the engineering, quantum physics, and radio astronomy communities use the first convention, in which the wave is observed from the point of view of the source.<ref name=Orfanidis /><ref name="Lectures_on_Physics(Vol_1_ch_33-1)" /><ref name=IAU /> In many physics textbooks dealing with optics, the second convention is used, in which the light is observed from the point of view of the receiver.<ref name="Lectures_on_Physics(Vol_1_ch_33-1)" /><ref name="Polarization_in_Spectral_Lines_Section_1.2" />

To avoid confusion, it is good practice to specify "as defined from the point of view of the source" or "as defined from the point of view of the receiver" when discussing polarization matters.

The archive of the [https://web.archive.org/web/20090822015912/http://www.its.bldrdoc.gov/fs-1037/ US Federal Standard 1037C] proposes two contradictory conventions of handedness.<ref>In one location it is stated..."Note 1. ... In general, the figure, i.e., polarization, is elliptical and is traced in a clockwise or anti-clockwise sense, as viewed in the direction of propagation. ... Rotation of the electric vector in a clockwise sense is designated right-hand polarization, and rotation in an anti-clockwise sense is designated left-hand polarization. "[http://www.its.bldrdoc.gov/fs-1037//dir-028/_4059.htm] {{Webarchive|url=https://web.archive.org/web/20110514080812/http://www.its.bldrdoc.gov/fs-1037/dir-028/_4059.htm|date=2011-05-14}} In another location it is stated... "Note 4: Circular polarization may be referred to as "right-hand" or "left-hand", depending on whether the helix describes the thread of a right-hand or left-hand screw, respectively". [http://www.its.bldrdoc.gov/fs-1037/dir-007/_0972.htm] {{Webarchive|url=https://web.archive.org/web/20110606113103/http://www.its.bldrdoc.gov/fs-1037/dir-007/_0972.htm|date=2011-06-06}}</ref>

Note that the IEEE defines RHCP and LHCP the opposite as those used by physicists.  The IEEE 1979 Antenna Standard will show RHCP on the South Pole of the Poincare Sphere. The IEEE defines RHCP using the right hand with thumb pointing in the direction of transmit, and the fingers showing the direction of rotation of the E field with time.  The rationale for the opposite conventions used by Physicists and Engineers is that Astronomical Observations are always done with the incoming wave traveling toward the observer, where as for most engineers, they are assumed to be standing behind the transmitter watching the wave traveling away from them.  This article is not using the IEEE 1979 Antenna Standard and is not using the +t convention typically used in IEEE work.

== FM radio ==
[[File:KHTB-FM broadcasting antennas LakeMountain.jpg|thumb|upright=0.6|Crossed-dipole antenna array of station [[KENZ (FM)|KENZ]]'s {{nowrap|94.9 MHz}}, {{nowrap|48 kW}} transmitter on Lake Mountain, Utah. It radiates circularly polarized radio waves.]]
[[FM broadcasting|FM broadcast]] radio stations sometimes employ circular polarization to improve signal penetration into buildings and vehicles. It is one example of what the [[International Telecommunication Union]] refers to as "mixed polarization", i.e. radio emissions that include both horizontally- and vertically-polarized components.<ref>{{cite report |title=Report 464-5, "Polarization of Emissions in Frequency-Modulation Broadcasting in Band 8 (VHF)" |year=1990 |url=https://www.itu.int/dms_pub/itu-r/opb/rep/r-rep-bs.464-5-1990-pdf-e.pdf |publisher=International Telecommunications Union}}</ref> In the United States, [[Federal Communications Commission]] regulations state that horizontal polarization is the standard for FM broadcasting, but that "circular or elliptical polarization may be employed if desired".<ref>{{CodeFedReg |47|73|316}}</ref>

==Dichroism==
{{Main article|Circular dichroism}}
'''Circular dichroism''' ('''CD''') is the differential absorption of left- and right-handed circularly polarized [[light]]. Circular dichroism is the basis of a form of [[spectroscopy]] that can be used to determine the [[Chirality (chemistry)|optical isomerism]] and secondary structure of [[molecule]]s.

In general, this phenomenon will be exhibited in absorption bands of any [[optical activity|optically active]] molecule. As a consequence, circular dichroism is exhibited by most biological molecules, because of the [[Dextrorotation and levorotation|dextrorotary]] (e.g., some [[sugar]]s) and [[Dextrorotation and levorotation|levorotary]] (e.g., some [[amino acid]]s) molecules they contain. Noteworthy as well is that a [[secondary structure]] will also impart a distinct CD to its respective molecules. Therefore, the [[alpha helix]], [[beta sheet]] and [[random coil]] regions of proteins and the [[Nucleic acid double helix|double helix]] of [[nucleic acid]]s have CD spectral signatures representative of their structures.

Also, under the right conditions, even non-chiral molecules will exhibit [[magnetic circular dichroism]] — that is, circular dichroism induced by a magnetic field.

==Luminescence==
''Circularly polarized luminescence'' (CPL) can occur when either a [[luminophore]] or an ensemble of luminophores is [[Chirality (chemistry)|chiral]]. The extent to which emissions are polarized is quantified in the same way it is for [[circular dichroism]], in terms of the ''dissymmetry factor'', also sometimes referred to as the ''[[anisotropy]] factor''. This value is given by:

:<math> g_{em} \ =\  2\left ( {\theta_\mathrm{left} - \theta_\mathrm{right} \over \theta_\mathrm{left} + \theta_\mathrm{right} } \right )   </math>

where <math> \theta_\mathrm{left} </math> corresponds to the quantum yield of left-handed circularly polarized light, and <math> \theta_\mathrm{right} </math> to that of right-handed light. The maximum absolute value of ''g''<sub>em</sub>, corresponding to purely left- or right-handed circular polarization, is therefore 2. Meanwhile, the smallest absolute value that ''g''<sub>em</sub> can achieve, corresponding to linearly polarized or unpolarized light, is zero.

==Mathematical description==
The [[Classical physics|classical]] [[Sine wave|sinusoidal]] plane wave solution of the [[electromagnetic wave equation]] for the [[Electric field|electric]] and [[Magnetic field|magnetic]] fields is:
:<math>\begin{align}
 \mathbf{E} ( \mathbf{r}, t ) &= \left| \,\mathbf{E}\, \right| \mathrm{Re} \left\{ \mathbf{Q} \left|\psi\right\rangle \exp \left[ i \left( kz - \omega t \right) \right] \right\} \\
 \mathbf{B} ( \mathbf{r}, t ) &= \dfrac{1}{c} \hat{ \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )
\end{align}</math>

where k is the [[wavenumber]];
:<math> \omega = c k</math>

is the [[angular frequency]] of the wave; <math> \mathbf{Q} = \left [ \hat{ \mathbf{x}}, \hat{\mathbf{y}} \right ] </math> is an orthogonal <math> 2 \times 2</math> matrix whose columns span the transverse x-y plane; and <math> c </math> is the [[speed of light]].

Here,
:<math> \left| \,\mathbf{E}\, \right| </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 calculus|Jones vector]] in the x-y plane.

If <math>  \alpha_y  </math> is rotated by <math>  \pi / 2  </math> radians with respect to <math>  \alpha_x  </math> and the x amplitude equals the y amplitude, the wave is circularly polarized. The Jones vector is:
:<math>  |\psi\rangle = {1\over \sqrt{2}} \begin{pmatrix} 1  \\ \pm i \end{pmatrix} \exp \left ( i \alpha_x \right )  </math>

where the plus sign indicates left circular polarization, and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the ''x''-''y'' plane.

If basis vectors are defined such that:
:<math>  |\mathrm{R}\rangle \ \stackrel{\mathrm{def}}{=}\  {1\over \sqrt{2}}\begin{pmatrix} 1  \\ -i \end{pmatrix}  </math>

and:
:<math>  |\mathrm{L}\rangle \ \stackrel{\mathrm{def}}{=}\  {1\over \sqrt{2}}\begin{pmatrix} 1  \\ i \end{pmatrix}  </math>

then the polarization state can be written in the "R-L basis" as:
:<math>  |\psi\rangle = \psi_\mathrm{R} |\mathrm{R}\rangle + \psi_\mathrm{L} |\mathrm{L}\rangle </math>

where:
:<math>\begin{align}
 \psi_\mathrm{R} ~&\stackrel{\mathrm{def}}{=}~ \frac{1}{\sqrt{2}} \left( \cos\theta + i\sin\theta \exp \left( i \delta \right) \right) \exp \left( i \alpha_x \right) \\
 \psi_\mathrm{L} ~&\stackrel{\mathrm{def}}{=}~ \frac{1}{\sqrt{2}} \left( \cos\theta - i\sin\theta \exp \left( i \delta \right) \right) \exp \left( i \alpha_x \right)
\end{align}</math>

and:

:<math> \delta= \alpha_y - \alpha_x. </math>

==Antennas==
{{Over-quotation|section|date=April 2018}}
A number of different types of antenna elements can be used to produce circularly polarized (or nearly so) radiation; following [[Constantine A. Balanis|Balanis]],<ref name=Balanis>Balanis, Constantine A. "Antenna Theory: Analysis and Design", 2016, 4th Edition, John Wiley & Sons.</ref> one can use [[Dipole antenna|''dipole elements'']]:
<blockquote>"... two crossed dipoles provide the two orthogonal field components.... If the two dipoles are identical, the field intensity of each along zenith ... would be of the same intensity. Also, if the two dipoles were fed with a 90° degree time-phase difference (phase quadrature), the polarization along zenith would be circular.... One way to obtain the 90° time-phase difference between the two orthogonal field components, radiated respectively by the two dipoles, is by feeding one of the two dipoles with a transmission line which is 1/4 wavelength longer or shorter than that of the other," p.80;</blockquote>
or [[Helical antenna|''helical elements'']]:
<blockquote>"To achieve circular polarization [in axial or end-fire mode] ... the circumference ''C'' of the helix must be ... with ''C''/wavelength = 1 near optimum, and the spacing about ''S'' = wavelength/4," p.571;</blockquote>
or [[Patch antenna#Circular polarization|''patch elements'']]:
<blockquote>"... circular and elliptical polarizations can be obtained using various feed arrangements or slight modifications made to the elements.... Circular polarization can be obtained if two orthogonal modes are excited with a 90° time-phase difference between them. This can be accomplished by adjusting the physical dimensions of the patch.... For a square patch element, the easiest way to excite ideally circular polarization is to feed the element at two adjacent edges.... The quadrature phase difference is obtained by feeding the element with a 90° power divider," p.859.</blockquote>

==In quantum mechanics==
{{further|Photon polarization}}
In the [[quantum mechanical]] view, light is composed of [[photons]]. Polarization is a manifestation of the [[spin angular momentum of light]]. More specifically, in quantum mechanics, the direction of spin of a photon is tied to the handedness of the circularly polarized light, and the spin of a beam of photons is similar to the spin of a beam of particles, such as electrons.<ref>Introduction to Quantum Theory 2ED David Park Sec 2.2 Pg32 "... the polarization of a beam of light is exactly the same kind of thing as the spin of a beam of electrons, the differences of terminology reflecting only the accidents of the historical order of discovery."</ref>

==In nature==
[[File:Cetonia-aurata.jpg|thumb|right|The [[Cetonia aurata|rose chafer]]'s external surface reflects almost exclusively left-circularly polarized light.]]

Only a few mechanisms in nature are known to systematically produce circularly polarized [[light]]. In 1911, [[Albert A. Michelson|Albert Abraham Michelson]] discovered that light reflected from the golden scarab beetle ''[[Chrysina resplendens]]'' is preferentially left-polarized. Since then, circular polarization has been measured in several other [[Scarabaeidae|scarab beetles]] such as ''[[Chrysina gloriosa]]'',<ref>{{Cite journal|url=https://www.science.org/doi/10.1126/science.1172051|title=Structural Origin of Circularly Polarized Iridescence in Jeweled Beetles|first1=Mohan|last1=Srinivasarao|first2=Jung Ok|last2=Park|first3=Matija|last3=Crne|first4=Vivek|last4=Sharma|date=July 24, 2009|journal=Science|volume=325|issue=5939|pages=449–451|via=science.sciencemag.org|doi=10.1126/science.1172051|pmid=19628862|bibcode=2009Sci...325..449S|s2cid=206519071}}</ref> as well as some [[crustacean]]s such as the [[mantis shrimp]]. In these cases, the underlying mechanism is the molecular-level helicity of the [[chitin]]ous [[cuticle]].<ref name="Hegedüs">{{cite journal |title=Imaging polarimetry of the circularly polarizing cuticle of scarab beetles (Coleoptera: Rutelidae, Cetoniidae) |author1=Hegedüs, Ramón |author2=Győző Szélb |author3=Gábor Horváth |doi=10.1016/j.visres.2006.02.007 |journal=Vision Research |volume=46 |issue=17 |date=September 2006 |pages=2786–2797 |pmid=16564066 |s2cid=14974820 |doi-access=free }}</ref>

The [[bioluminescence]] of the [[larva]]e of [[firefly|fireflie]]s is also circularly polarized, as reported in 1980 for the species ''[[Photuris|Photuris lucicrescens]]'' and ''[[Photuris versicolor]]''. For fireflies, it is more difficult to find a microscopic explanation for the polarization, because the left and right lanterns of the larvae were found to emit polarized light of opposite senses. The authors suggest that the light begins with a [[linear polarization]] due to inhomogeneities inside aligned [[photocyte]]s, and it picks up circular polarization while passing through linearly [[Birefringence|birefringent]] tissue.<ref>{{cite journal |title=Circular polarization observed in bioluminescence |author1=Wynberg, Hans |author2=Meijer, E.W. |author3=Hummelen, J.C. |author4=Dekkers, H.P.J.M. |author5=Schippers, P.H. |author6=Carlson, A.D. |url=http://keur.eldoc.ub.rug.nl/FILES/wetenschappers/10/29/29.pdf |journal=Nature |volume=286 |issue=5773 |pages=641–642 |date=7 August 1980 |doi=10.1038/286641a0 |bibcode=1980Natur.286..641W |s2cid=4324467 |url-status=dead |archive-url=https://web.archive.org/web/20110724164914/http://keur.eldoc.ub.rug.nl/FILES/wetenschappers/10/29/29.pdf |archive-date=24 July 2011 }}</ref>

Circular polarization has been detected in light reflected from leaves and photosynthetic microbes.<ref name="b966">{{cite journal | last1=Sparks | first1=William B. | last2=Hough | first2=James | last3=Germer | first3=Thomas A. | last4=Chen | first4=Feng | last5=DasSarma | first5=Shiladitya | last6=DasSarma | first6=Priya | last7=Robb | first7=Frank T. | last8=Manset | first8=Nadine | last9=Kolokolova | first9=Ludmilla | last10=Reid | first10=Neill | last11=Macchetto | first11=F. Duccio | last12=Martin | first12=William | title=Detection of circular polarization in light scattered from photosynthetic microbes | journal=Proceedings of the National Academy of Sciences | volume=106 | issue=19 | date=2009-05-12 | issn=0027-8424 | pmid=19416893 | pmc=2674403 | doi=10.1073/pnas.0810215106 | pages=7816–7821| doi-access=free | arxiv=0904.4646 | bibcode=2009PNAS..106.7816S }}</ref>

Water-air interfaces provide another source of circular polarization. Sunlight that gets scattered back up towards the surface is linearly polarized. If this light is then [[total internal reflection|totally internally reflected]] back down, its vertical component undergoes a phase shift. To an underwater observer looking up, the faint light outside [[Snell's window]] therefore is (partially) circularly polarized.<ref>{{cite book |title=Polarized Light in Animal Vision: Polarization Patterns in Nature |author1=Horváth, Gábor |author2=Dezsö Varjú |year=2003 |publisher=Springer |isbn=978-3-540-40457-6 |pages=100–103}}</ref>

Weaker sources of circular polarization in nature include multiple scattering by linear polarizers{{dubious|date=June 2021}}, as in the circular polarization of starlight, and selective absorption by [[circular dichroism|circularly dichroic]] media.

Radio emission from pulsars can be strongly circularly polarized.<ref>{{cite journal|doi=10.1111/j.1365-2966.2005.09681.x |title=On the origin of the circular polarization in radio pulsars |date=2005 |last1=Gogoberidze |first1=G. |last2=Machabeli |first2=G. Z. |journal=Monthly Notices of the Royal Astronomical Society |volume=364 |issue=4 |pages=1363–1366 |doi-access=free |arxiv=astro-ph/0510116 |bibcode=2005MNRAS.364.1363G }}</ref>

Two species of [[mantis shrimp]] have been reported to be able to detect circular polarized light.<ref>{{cite journal|author1=Tsyr-Huei Chiou |author2=Sonja Kleinlogel |author3=Tom Cronin |author4=Roy Caldwell |author5=Birte Loeffler |author6=Afsheen Siddiqi |author7=Alan Goldizen |author8=Justin Marshall |title=Circular polarization vision in a stomatopod crustacean |journal=[[Current Biology]] |year=2008 |volume=18 |issue=6 |pages=429–34 |doi=10.1016/j.cub.2008.02.066 |pmid=18356053|s2cid=6925705 |doi-access=free |bibcode=2008CBio...18..429C }}</ref><ref name="Kleinlogel et al.">{{cite journal |author1=Sonja Kleinlogel |author2=Andrew White |title=The secret world of shrimps: polarisation vision at its best |journal=[[PLoS ONE]] |year=2008 |doi=10.1371/journal.pone.0002190 |volume=3 |issue=5 |pages=e2190 |pmid=18478095 |pmc=2377063 |bibcode=2008PLoSO...3.2190K|arxiv = 0804.2162 |doi-access=free }}</ref>

==See also==
*[[Polarizer]]
*[[3D film]]
*[[Chirality]]
*[[Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
*[[Starlight polarization]]
*[[Waveplate]]

==References==
{{Reflist|30em}}

==Further reading==
*{{Cite book |last=Jackson |first=John D. |title=Classical Electrodynamics |edition=3rd |publisher=Wiley |location=New York |year=1999 |isbn=978-0-471-30932-1}}
*{{Cite book |last1=Born |first1=M. |name-list-style=amp |last2=Wolf |first2=E. |title=[[Principles of Optics]]: Electromagnetic Theory of Propagation, Interference and Diffraction of Light |edition=7th |publisher=Cambridge University Press |location=Cambridge |year=1999 |isbn=978-0-521-64222-4}}

==External links==
* [http://www.polarization.com/beetle/beetle.html Circularly polarized light: beetles and displays]
* [http://optics.org/cws/article/research/34196 Article on the mantis shrimp and circular polarization]
*[https://www.youtube.com/watch?v=jY9hnDzA6Ps Animation of Circular Polarization (on YouTube)]
*[https://www.youtube.com/watch?v=Q0qrU4nprB0 Comparison of Circular Polarization with Linear and Elliptical Polarizations (YouTube Animation)]
*[http://nagyelte.blogspot.hu/2015/01/reversal-of-handedness-of-circularly.html Reversal of handedness of circularly polarized light by mirror. A demonstration – simple, cheap & instructive]

{{DEFAULTSORT:Circular Polarization}}
[[Category:Concepts in astrophysics]]
[[Category:Polarization (waves)]]
[[Category:Stellar astronomy]]