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id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Circular polarization</span></h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Circular_polarization" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Circular_polarization" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled 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menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <p>In <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">electrodynamics</a>, <b>circular polarization</b> of an <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic wave</a> is a <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> state in which, at each point, the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to the direction of the wave. </p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Circular.Polarization.Circularly.Polarized.Light_Left.Hand.Animation.305x190.255Colors.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Circular.Polarization.Circularly.Polarized.Light_Left.Hand.Animation.305x190.255Colors.gif" decoding="async" width="305" height="190" class="mw-file-element" data-file-width="305" data-file-height="190"></a><figcaption>The <a href="/wiki/Electric_field" title="Electric field">electric field</a> 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.</figcaption></figure> <p>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 <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a>, 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 <a href="/wiki/Helix" title="Helix">helix</a> oriented along the direction of propagation. A circularly polarized wave can rotate in one of two possible senses: <i>right-handed circular polarization (RHCP)</i> in which the electric field vector rotates in a <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand</a> sense with respect to the direction of propagation, and <i>left-handed circular polarization (LHCP)</i> in which the vector rotates in a <a href="/wiki/Left_hand_rule" class="mw-redirect" title="Left hand rule">left-hand</a> sense. </p><p><i>Circular polarization</i> is a <a href="/wiki/Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting case</a> of <i><a href="/wiki/Elliptical_polarization" title="Elliptical polarization">elliptical polarization</a></i>. The other <a href="/wiki/Special_case" title="Special case">special case</a> is the easier-to-understand <i><a href="/wiki/Linear_polarization" title="Linear polarization">linear polarization</a></i>. All three terms were coined by <a href="/wiki/Augustin-Jean_Fresnel" title="Augustin-Jean Fresnel">Augustin-Jean Fresnel</a>, in a memoir read to the <a href="/wiki/French_Academy_of_Sciences" title="French Academy of Sciences">French Academy of Sciences</a> on 9 December 1822.<sup id="cite_ref-fresnel-1822z_1-0" class="reference"><a href="#cite_note-fresnel-1822z-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Fresnel had first described the case of circular polarization, without yet naming it, in 1821.<sup id="cite_ref-fresnel-1821a_3-0" class="reference"><a href="#cite_note-fresnel-1821a-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The phenomenon of polarization arises as a consequence of the fact that <a href="/wiki/Light" title="Light">light</a> behaves as a two-dimensional <a href="/wiki/Transverse_wave#Explanation" title="Transverse wave">transverse wave</a>. </p><p>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. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Characteristics"><span class="tocnumber">1</span> <span class="toctext">Characteristics</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Reversal_of_handedness"><span class="tocnumber">1.1</span> <span class="toctext">Reversal of handedness</span></a> <ul> <li class="toclevel-3 tocsection-3"><a href="#Waveplate"><span class="tocnumber">1.1.1</span> <span class="toctext">Waveplate</span></a></li> <li class="toclevel-3 tocsection-4"><a href="#Reflection"><span class="tocnumber">1.1.2</span> <span class="toctext">Reflection</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-5"><a href="#Conversion_to_linear_polarization"><span class="tocnumber">1.2</span> <span class="toctext">Conversion to linear polarization</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Handedness_conventions"><span class="tocnumber">2</span> <span class="toctext">Handedness conventions</span></a> <ul> <li class="toclevel-2 tocsection-7"><a href="#From_the_point_of_view_of_the_source"><span class="tocnumber">2.1</span> <span class="toctext">From the point of view of the source</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#From_the_point_of_view_of_the_receiver"><span class="tocnumber">2.2</span> <span class="toctext">From the point of view of the receiver</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Uses_of_the_two_conventions"><span class="tocnumber">2.3</span> <span class="toctext">Uses of the two conventions</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#FM_radio"><span class="tocnumber">3</span> <span class="toctext">FM radio</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Dichroism"><span class="tocnumber">4</span> <span class="toctext">Dichroism</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Luminescence"><span class="tocnumber">5</span> <span class="toctext">Luminescence</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Mathematical_description"><span class="tocnumber">6</span> <span class="toctext">Mathematical description</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Antennas"><span class="tocnumber">7</span> <span class="toctext">Antennas</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#In_quantum_mechanics"><span class="tocnumber">8</span> <span class="toctext">In quantum mechanics</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#In_nature"><span class="tocnumber">9</span> <span class="toctext">In nature</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#See_also"><span class="tocnumber">10</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#References"><span class="tocnumber">11</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-19"><a href="#Further_reading"><span class="tocnumber">12</span> <span class="toctext">Further reading</span></a></li> <li class="toclevel-1 tocsection-20"><a href="#External_links"><span class="tocnumber">13</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand 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data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg.png" data-alt="" data-width="440" data-height="258" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg/660px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg/880px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Right.Handed.svg.png 2x" data-class="mw-file-element"> </span></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:442px;max-width:442px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg.png" decoding="async" width="440" height="286" class="mw-file-element" data-file-width="329" data-file-height="214"></noscript><span class="lazy-image-placeholder" style="width: 440px;height: 286px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg.png" data-alt="" data-width="440" data-height="286" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg/660px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg/880px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Right.Handed.svg.png 2x" data-class="mw-file-element"> </span></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:442px;max-width:442px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Circular_polarization_cross_section.gif" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circular_polarization_cross_section.gif/440px-Circular_polarization_cross_section.gif" decoding="async" width="440" height="440" class="mw-file-element" data-file-width="2369" data-file-height="2369"></noscript><span class="lazy-image-placeholder" style="width: 440px;height: 440px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circular_polarization_cross_section.gif/440px-Circular_polarization_cross_section.gif" data-alt="" data-width="440" data-height="440" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circular_polarization_cross_section.gif/660px-Circular_polarization_cross_section.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circular_polarization_cross_section.gif/880px-Circular_polarization_cross_section.gif 2x" data-class="mw-file-element"> </span></a></span></div></div></div><div class="trow" style="display:flow-root"><div class="thumbcaption" style="text-align:left">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.</div></div></div></div> <p><br> In a circularly polarized electromagnetic wave, the individual electric field vectors, as well as their combined vector, have a constant <a href="/wiki/Magnitude_(vector)" class="mw-redirect" title="Magnitude (vector)">magnitude</a>, and with changing phase angle. Given that this is a <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a>, 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 <a href="/wiki/Plane_wave#Polarized_electromagnetic_plane_waves" title="Plane wave">circularly polarized plane wave</a>, these vectors indicate that the electric field, from plane to plane, has a constant strength while its direction steadily rotates. Refer to <a href="/wiki/Plane_wave#Polarized_electromagnetic_plane_waves" title="Plane wave">these two images</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged January 2021">dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup> 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 <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic wave</a>, each <a href="/wiki/Electric_field" title="Electric field">electric field</a> vector has a corresponding, but not illustrated, <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> vector that is at a <a href="/wiki/Right_angle" title="Right angle">right angle</a> to the electric field vector and <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> in magnitude to it. As a result, the magnetic field vectors would trace out a second helix if displayed. </p><p>Circular polarization is often encountered in the field of optics and, in this section, the electromagnetic wave will be simply referred to as <a href="/wiki/Light" title="Light">light</a>. </p><p>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 <a href="/wiki/Euclidean_vector" title="Euclidean vector">components</a> 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 <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>, a 90° phase difference. It is this <a href="/wiki/In-phase_and_quadrature_components" title="In-phase and quadrature components">quadrature phase</a> relationship that creates the <a href="/wiki/Helix" title="Helix">helix</a> 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. </p><p>To appreciate how this quadrature <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> 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 <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacements</a> of the dot, relative to the center of the circle, vary <a href="/wiki/Sine_wave" title="Sine wave">sinusoidally</a> 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. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:444px;max-width:444px"><div class="trow"><div class="tsingle" style="width:442px;max-width:442px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg.png" decoding="async" width="440" height="258" class="mw-file-element" data-file-width="329" data-file-height="193"></noscript><span class="lazy-image-placeholder" style="width: 440px;height: 258px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg.png" data-alt="" data-width="440" data-height="258" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg/660px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg/880px-Circular.Polarization.Circularly.Polarized.Light_Without.Components_Left.Handed.svg.png 2x" data-class="mw-file-element"> </span></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:442px;max-width:442px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg.png" decoding="async" width="440" height="286" class="mw-file-element" data-file-width="329" data-file-height="214"></noscript><span class="lazy-image-placeholder" style="width: 440px;height: 286px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg/440px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg.png" data-alt="" data-width="440" data-height="286" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg/660px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg/880px-Circular.Polarization.Circularly.Polarized.Light_With.Components_Left.Handed.svg.png 2x" data-class="mw-file-element"> </span></a></span></div></div></div><div class="trow" style="display:flow-root"><div class="thumbcaption" style="text-align:left">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.</div></div></div></div> <p>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 <i>lagging</i> the vertical component by one quarter of a wavelength, rather than leading it. </p> <div class="mw-heading mw-heading3"><h3 id="Reversal_of_handedness">Reversal of handedness</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=2" title="Edit section: Reversal of handedness" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <div class="mw-heading mw-heading4"><h4 id="Waveplate">Waveplate</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=3" title="Edit section: Waveplate" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>To convert circularly polarized light to the other handedness, one can use a half-<a href="/wiki/Waveplate" title="Waveplate">waveplate</a>. A half-waveplate shifts a given linear component of light one half of a wavelength relative to its orthogonal linear component. </p> <div class="mw-heading mw-heading4"><h4 id="Reflection">Reflection</h4><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=4" title="Edit section: Reflection" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The handedness of polarized light is reversed reflected off a surface at normal incidence. Upon such reflection, the rotation of the <a href="/wiki/Plane_of_polarization" title="Plane of polarization">plane of polarization</a> of the reflected light is identical to that of the incident field. However, with propagation now in the <i>opposite</i> 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 <a href="/wiki/Birefringence" title="Birefringence">birefringent</a> surface). </p><p>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 <a href="/wiki/Brewster%27s_angle" title="Brewster's angle">Brewster angle</a>) 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 <a href="/wiki/Plane_of_incidence" title="Plane of incidence">plane of incidence</a>, commonly denoted <i>p</i> and <i>s</i> respectively. The reflected components in the <i>p</i> and <i>s</i> linear polarizations are found by applying the <a href="/wiki/Fresnel_equations" title="Fresnel equations">Fresnel coefficients</a> 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 <i>p</i> and <i>s</i>, are the Fresnel coefficients for the two components identical, leading to the above property. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif/220px-Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif" decoding="async" width="220" height="137" class="mw-file-element" data-file-width="620" data-file-height="385"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 137px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif/220px-Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif" data-width="220" data-height="137" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif/330px-Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif/440px-Reversal_of_handedness_of_circularly_polarized_light_reflected_by_mirror_2s.gif 2x" data-class="mw-file-element"> </span></a><figcaption>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.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Conversion_to_linear_polarization">Conversion to linear polarization</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=5" title="Edit section: Conversion to linear polarization" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Circularly polarized light can be converted into linearly polarized light by passing it through a quarter-<a href="/wiki/Waveplate" title="Waveplate">waveplate</a>. 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 <i>other</i> than 45° will generally produce elliptical polarization. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Handedness_conventions">Handedness conventions <span class="anchor" id="Left/Right"></span></h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=6" title="Edit section: Handedness conventions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Circular.Polarization.Circularly.Polarized.Light_Left.Hand.Animation.305x190.255Colors.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Circular.Polarization.Circularly.Polarized.Light_Left.Hand.Animation.305x190.255Colors.gif" decoding="async" width="305" height="190" class="mw-file-element" data-file-width="305" data-file-height="190"></noscript><span class="lazy-image-placeholder" style="width: 305px;height: 190px;" data-src="//upload.wikimedia.org/wikipedia/commons/d/d1/Circular.Polarization.Circularly.Polarized.Light_Left.Hand.Animation.305x190.255Colors.gif" data-width="305" data-height="190" data-class="mw-file-element"> </span></a><figcaption>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.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Circular.Polarization.Circularly.Polarized.Light_Right.Handed.Animation.305x190.255Colors.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/8/81/Circular.Polarization.Circularly.Polarized.Light_Right.Handed.Animation.305x190.255Colors.gif" decoding="async" width="305" height="190" class="mw-file-element" data-file-width="305" data-file-height="190"></noscript><span class="lazy-image-placeholder" style="width: 305px;height: 190px;" data-src="//upload.wikimedia.org/wikipedia/commons/8/81/Circular.Polarization.Circularly.Polarized.Light_Right.Handed.Animation.305x190.255Colors.gif" data-width="305" data-height="190" data-class="mw-file-element"> </span></a><figcaption>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.</figcaption></figure> <p>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. </p> <div class="mw-heading mw-heading3"><h3 id="From_the_point_of_view_of_the_source">From the point of view of the source</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=7" title="Edit section: From the point of view of the source" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>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</em> from the source, in the <em>same</em> 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</em> from the source and in the <em>same</em> direction of the wave's propagation, one observes the direction of the field's temporal rotation. </p><p>Using this convention, the electric field vector of a left-handed circularly polarized wave is as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(E_{x},\,E_{y},\,E_{z}\right)\propto \left(\cos {\frac {2\pi }{\lambda }}\left(ct-z\right),\,-\sin {\frac {2\pi }{\lambda }}\left(ct-z\right),\,0\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∝</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(E_{x},\,E_{y},\,E_{z}\right)\propto \left(\cos {\frac {2\pi }{\lambda }}\left(ct-z\right),\,-\sin {\frac {2\pi }{\lambda }}\left(ct-z\right),\,0\right).}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b155d7d7e16896125033dfa35b7694e4827625f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.893ex; height:6.176ex;" alt="{\displaystyle \left(E_{x},\,E_{y},\,E_{z}\right)\propto \left(\cos {\frac {2\pi }{\lambda }}\left(ct-z\right),\,-\sin {\frac {2\pi }{\lambda }}\left(ct-z\right),\,0\right).}"></noscript><span class="lazy-image-placeholder" style="width: 56.893ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b155d7d7e16896125033dfa35b7694e4827625f" data-alt="{\displaystyle \left(E_{x},\,E_{y},\,E_{z}\right)\propto \left(\cos {\frac {2\pi }{\lambda }}\left(ct-z\right),\,-\sin {\frac {2\pi }{\lambda }}\left(ct-z\right),\,0\right).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </p><p>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. </p><p>This convention is in conformity with the <a href="/wiki/Institute_of_Electrical_and_Electronics_Engineers" title="Institute of Electrical and Electronics Engineers">Institute of Electrical and Electronics Engineers</a> (IEEE) standard and, as a result, it is generally used in the engineering community.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Orfanidis_5-0" class="reference"><a href="#cite_note-Orfanidis-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Quantum physicists also use this convention of handedness because it is consistent with their convention of handedness for a particle's spin.<sup id="cite_ref-Lectures_on_Physics(Vol_1_ch_33-1)_7-0" class="reference"><a href="#cite_note-Lectures_on_Physics(Vol_1_ch_33-1)-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Radio astronomers also use this convention in accordance with an <a href="/wiki/International_Astronomical_Union" title="International Astronomical Union">International Astronomical Union (IAU)</a> resolution made in 1973.<sup id="cite_ref-IAU_8-0" class="reference"><a href="#cite_note-IAU-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="From_the_point_of_view_of_the_receiver">From the point of view of the receiver</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=8" title="Edit section: From the point of view of the receiver" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>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</em> the source, <em>against</em> the direction of propagation, and then matching the curling of one's fingers to the temporal rotation of the field. </p><p>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. </p><p>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</em> the source, <em>against</em> the direction of propagation, one observes the direction of the field's temporal rotation. </p><p>Just as in the other convention, right-handedness corresponds to a clockwise rotation, and left-handedness corresponds to an anti-clockwise rotation. </p><p>Many optics textbooks use this second convention.<sup id="cite_ref-Polarization_in_Spectral_Lines_Section_1.2_9-0" class="reference"><a href="#cite_note-Polarization_in_Spectral_Lines_Section_1.2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> It is also used by <a href="/wiki/SPIE" title="SPIE">SPIE</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> as well as the <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" title="International Union of Pure and Applied Chemistry">International Union of Pure and Applied Chemistry</a> (IUPAC).<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Uses_of_the_two_conventions">Uses of the two conventions</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=9" title="Edit section: Uses of the two conventions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>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.<sup id="cite_ref-Orfanidis_5-1" class="reference"><a href="#cite_note-Orfanidis-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lectures_on_Physics(Vol_1_ch_33-1)_7-1" class="reference"><a href="#cite_note-Lectures_on_Physics(Vol_1_ch_33-1)-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-IAU_8-1" class="reference"><a href="#cite_note-IAU-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-Lectures_on_Physics(Vol_1_ch_33-1)_7-2" class="reference"><a href="#cite_note-Lectures_on_Physics(Vol_1_ch_33-1)-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Polarization_in_Spectral_Lines_Section_1.2_9-1" class="reference"><a href="#cite_note-Polarization_in_Spectral_Lines_Section_1.2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>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. </p><p>The archive of the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090822015912/http://www.its.bldrdoc.gov/fs-1037/">US Federal Standard 1037C</a> proposes two contradictory conventions of handedness.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>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. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="FM_radio">FM radio</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=10" title="Edit section: FM radio" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:KHTB-FM_broadcasting_antennas_LakeMountain.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/KHTB-FM_broadcasting_antennas_LakeMountain.jpg/130px-KHTB-FM_broadcasting_antennas_LakeMountain.jpg" decoding="async" width="130" height="495" class="mw-file-element" data-file-width="174" data-file-height="663"></noscript><span class="lazy-image-placeholder" style="width: 130px;height: 495px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/KHTB-FM_broadcasting_antennas_LakeMountain.jpg/130px-KHTB-FM_broadcasting_antennas_LakeMountain.jpg" data-width="130" data-height="495" data-srcset="//upload.wikimedia.org/wikipedia/commons/f/fb/KHTB-FM_broadcasting_antennas_LakeMountain.jpg 1.5x" data-class="mw-file-element"> </span></a><figcaption>Crossed-dipole antenna array of station <a href="/wiki/KENZ_(FM)" title="KENZ (FM)">KENZ</a>'s <span class="nowrap">94.9 MHz</span>, <span class="nowrap">48 kW</span> transmitter on Lake Mountain, Utah. It radiates <a class="mw-selflink selflink">circularly polarized</a> radio waves.</figcaption></figure> <p><a href="/wiki/FM_broadcasting" title="FM broadcasting">FM broadcast</a> radio stations sometimes employ circular polarization to improve signal penetration into buildings and vehicles. It is one example of what the <a href="/wiki/International_Telecommunication_Union" title="International Telecommunication Union">International Telecommunication Union</a> refers to as "mixed polarization", i.e. radio emissions that include both horizontally- and vertically-polarized components.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In the United States, <a href="/wiki/Federal_Communications_Commission" title="Federal Communications Commission">Federal Communications Commission</a> regulations state that horizontal polarization is the standard for FM broadcasting, but that "circular or elliptical polarization may be employed if desired".<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Dichroism">Dichroism</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=11" title="Edit section: Dichroism" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Circular_dichroism" title="Circular dichroism">Circular dichroism</a></div> <p><b>Circular dichroism</b> (<b>CD</b>) is the differential absorption of left- and right-handed circularly polarized <a href="/wiki/Light" title="Light">light</a>. Circular dichroism is the basis of a form of <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a> that can be used to determine the <a href="/wiki/Chirality_(chemistry)" title="Chirality (chemistry)">optical isomerism</a> and secondary structure of <a href="/wiki/Molecule" title="Molecule">molecules</a>. </p><p>In general, this phenomenon will be exhibited in absorption bands of any <a href="/wiki/Optical_activity" class="mw-redirect" title="Optical activity">optically active</a> molecule. As a consequence, circular dichroism is exhibited by most biological molecules, because of the <a href="/wiki/Dextrorotation_and_levorotation" class="mw-redirect" title="Dextrorotation and levorotation">dextrorotary</a> (e.g., some <a href="/wiki/Sugar" title="Sugar">sugars</a>) and <a href="/wiki/Dextrorotation_and_levorotation" class="mw-redirect" title="Dextrorotation and levorotation">levorotary</a> (e.g., some <a href="/wiki/Amino_acid" title="Amino acid">amino acids</a>) molecules they contain. Noteworthy as well is that a <a href="/wiki/Secondary_structure" class="mw-redirect" title="Secondary structure">secondary structure</a> will also impart a distinct CD to its respective molecules. Therefore, the <a href="/wiki/Alpha_helix" title="Alpha helix">alpha helix</a>, <a href="/wiki/Beta_sheet" title="Beta sheet">beta sheet</a> and <a href="/wiki/Random_coil" title="Random coil">random coil</a> regions of proteins and the <a href="/wiki/Nucleic_acid_double_helix" title="Nucleic acid double helix">double helix</a> of <a href="/wiki/Nucleic_acid" title="Nucleic acid">nucleic acids</a> have CD spectral signatures representative of their structures. </p><p>Also, under the right conditions, even non-chiral molecules will exhibit <a href="/wiki/Magnetic_circular_dichroism" title="Magnetic circular dichroism">magnetic circular dichroism</a> — that is, circular dichroism induced by a magnetic field. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Luminescence">Luminescence</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=12" title="Edit section: Luminescence" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <p><i>Circularly polarized luminescence</i> (CPL) can occur when either a <a href="/wiki/Luminophore" title="Luminophore">luminophore</a> or an ensemble of luminophores is <a href="/wiki/Chirality_(chemistry)" title="Chirality (chemistry)">chiral</a>. The extent to which emissions are polarized is quantified in the same way it is for <a href="/wiki/Circular_dichroism" title="Circular dichroism">circular dichroism</a>, in terms of the <i>dissymmetry factor</i>, also sometimes referred to as the <i><a href="/wiki/Anisotropy" title="Anisotropy">anisotropy</a> factor</i>. This value is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{em}\ =\ 2\left({\theta _{\mathrm {left} }-\theta _{\mathrm {right} } \over \theta _{\mathrm {left} }+\theta _{\mathrm {right} }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>m</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>2</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{em}\ =\ 2\left({\theta _{\mathrm {left} }-\theta _{\mathrm {right} } \over \theta _{\mathrm {left} }+\theta _{\mathrm {right} }}\right)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a7e42a7d56bfd56af6b95978fac25628b9be8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.019ex; height:6.509ex;" alt="{\displaystyle g_{em}\ =\ 2\left({\theta _{\mathrm {left} }-\theta _{\mathrm {right} } \over \theta _{\mathrm {left} }+\theta _{\mathrm {right} }}\right)}"></noscript><span class="lazy-image-placeholder" style="width: 25.019ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a7e42a7d56bfd56af6b95978fac25628b9be8d" data-alt="{\displaystyle g_{em}\ =\ 2\left({\theta _{\mathrm {left} }-\theta _{\mathrm {right} } \over \theta _{\mathrm {left} }+\theta _{\mathrm {right} }}\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\mathrm {left} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\mathrm {left} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326ce98500df15d4a3a5050528051ccfa1136a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.762ex; height:2.509ex;" alt="{\displaystyle \theta _{\mathrm {left} }}"></noscript><span class="lazy-image-placeholder" style="width: 3.762ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326ce98500df15d4a3a5050528051ccfa1136a13" data-alt="{\displaystyle \theta _{\mathrm {left} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> corresponds to the quantum yield of left-handed circularly polarized light, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\mathrm {right} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\mathrm {right} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30bbfe6cd47757e128b064c7098bd39995504d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.8ex; height:2.843ex;" alt="{\displaystyle \theta _{\mathrm {right} }}"></noscript><span class="lazy-image-placeholder" style="width: 4.8ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30bbfe6cd47757e128b064c7098bd39995504d6" data-alt="{\displaystyle \theta _{\mathrm {right} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to that of right-handed light. The maximum absolute value of <i>g</i><sub>em</sub>, corresponding to purely left- or right-handed circular polarization, is therefore 2. Meanwhile, the smallest absolute value that <i>g</i><sub>em</sub> can achieve, corresponding to linearly polarized or unpolarized light, is zero. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Mathematical_description">Mathematical description</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=13" title="Edit section: Mathematical description" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <p>The <a href="/wiki/Classical_physics" title="Classical physics">classical</a> <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> plane wave solution of the <a href="/wiki/Electromagnetic_wave_equation" title="Electromagnetic wave equation">electromagnetic wave equation</a> for the <a href="/wiki/Electric_field" title="Electric field">electric</a> and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic</a> fields is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow> <mo>|</mo> <mi>ψ</mi> <mo>⟩</mo> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>z</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36f9076326b8ada76bd59eeec2b31b4472d0c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.36ex; margin-bottom: -0.311ex; width:41.32ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 41.32ex;height: 8.509ex;vertical-align: -3.36ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36f9076326b8ada76bd59eeec2b31b4472d0c1a" data-alt="{\displaystyle {\begin{aligned}\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{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where k is the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a>; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =ck}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>c</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =ck}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a121709f0c11324e6983899f0606e33ce4e7072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.762ex; height:2.176ex;" alt="{\displaystyle \omega =ck}"></noscript><span class="lazy-image-placeholder" style="width: 6.762ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a121709f0c11324e6983899f0606e33ce4e7072" data-alt="{\displaystyle \omega =ck}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the wave; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} =\left[{\hat {\mathbf {x} }},{\hat {\mathbf {y} }}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} =\left[{\hat {\mathbf {x} }},{\hat {\mathbf {y} }}\right]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6f209c9644cf3ab69fd75e6d696bb9278c0c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.256ex; height:2.843ex;" alt="{\displaystyle \mathbf {Q} =\left[{\hat {\mathbf {x} }},{\hat {\mathbf {y} }}\right]}"></noscript><span class="lazy-image-placeholder" style="width: 10.256ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6f209c9644cf3ab69fd75e6d696bb9278c0c12" data-alt="{\displaystyle \mathbf {Q} =\left[{\hat {\mathbf {x} }},{\hat {\mathbf {y} }}\right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is an orthogonal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></noscript><span class="lazy-image-placeholder" style="width: 5.165ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" data-alt="{\displaystyle 2\times 2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> matrix whose columns span the transverse x-y plane; and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></noscript><span class="lazy-image-placeholder" style="width: 1.007ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" data-alt="{\displaystyle c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. </p><p>Here, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\,\mathbf {E} \,\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\,\mathbf {E} \,\right|}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18cfd0d294a9c556e2337ac6a8dac4698fa9c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.825ex; height:2.843ex;" alt="{\displaystyle \left|\,\mathbf {E} \,\right|}"></noscript><span class="lazy-image-placeholder" style="width: 3.825ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18cfd0d294a9c556e2337ac6a8dac4698fa9c23" data-alt="{\displaystyle \left|\,\mathbf {E} \,\right|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>is the <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> of the field, and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\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}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6649b4fc98156e55841b592a8b7a7eec5bac44f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.411ex; height:6.509ex;" alt="{\displaystyle |\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}}}"></noscript><span class="lazy-image-placeholder" style="width: 34.411ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6649b4fc98156e55841b592a8b7a7eec5bac44f4" data-alt="{\displaystyle |\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}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>is the normalized <a href="/wiki/Jones_calculus" title="Jones calculus">Jones vector</a> in the x-y plane. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{y}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9236645a8f56e17b2d56fa7aa8d76852df77dfdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.537ex; height:2.343ex;" alt="{\displaystyle \alpha _{y}}"></noscript><span class="lazy-image-placeholder" style="width: 2.537ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9236645a8f56e17b2d56fa7aa8d76852df77dfdb" data-alt="{\displaystyle \alpha _{y}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is rotated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}"></noscript><span class="lazy-image-placeholder" style="width: 3.657ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67" data-alt="{\displaystyle \pi /2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> radians with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{x}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6aea4f482cc815dc367ff2686b84188beb9ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.66ex; height:2.009ex;" alt="{\displaystyle \alpha _{x}}"></noscript><span class="lazy-image-placeholder" style="width: 2.66ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6aea4f482cc815dc367ff2686b84188beb9ea0" data-alt="{\displaystyle \alpha _{x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and the x amplitude equals the y amplitude, the wave is circularly polarized. The Jones vector is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle ={1 \over {\sqrt {2}}}{\begin{pmatrix}1\\\pm i\end{pmatrix}}\exp \left(i\alpha _{x}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>±</mo> <mi>i</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle ={1 \over {\sqrt {2}}}{\begin{pmatrix}1\\\pm i\end{pmatrix}}\exp \left(i\alpha _{x}\right)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8483339e771af995e8152674700d180068a479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.092ex; height:6.509ex;" alt="{\displaystyle |\psi \rangle ={1 \over {\sqrt {2}}}{\begin{pmatrix}1\\\pm i\end{pmatrix}}\exp \left(i\alpha _{x}\right)}"></noscript><span class="lazy-image-placeholder" style="width: 26.092ex;height: 6.509ex;vertical-align: -2.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8483339e771af995e8152674700d180068a479" data-alt="{\displaystyle |\psi \rangle ={1 \over {\sqrt {2}}}{\begin{pmatrix}1\\\pm i\end{pmatrix}}\exp \left(i\alpha _{x}\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>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 <i>x</i>-<i>y</i> plane. </p><p>If basis vectors are defined such that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathrm {R} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\-i\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathrm {R} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\-i\end{pmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1c97a9835a1014dd9fcf59b1037e5f56f560e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.397ex; height:6.509ex;" alt="{\displaystyle |\mathrm {R} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\-i\end{pmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.397ex;height: 6.509ex;vertical-align: -2.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1c97a9835a1014dd9fcf59b1037e5f56f560e3" data-alt="{\displaystyle |\mathrm {R} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\-i\end{pmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>and: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathrm {L} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\i\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathrm {L} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\i\end{pmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2600343c7f728c8e3436c429bd14cd4991213d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.691ex; height:6.509ex;" alt="{\displaystyle |\mathrm {L} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\i\end{pmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 15.691ex;height: 6.509ex;vertical-align: -2.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2600343c7f728c8e3436c429bd14cd4991213d" data-alt="{\displaystyle |\mathrm {L} \rangle \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {2}}}{\begin{pmatrix}1\\i\end{pmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>then the polarization state can be written in the "R-L basis" as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle =\psi _{\mathrm {R} }|\mathrm {R} \rangle +\psi _{\mathrm {L} }|\mathrm {L} \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\psi _{\mathrm {R} }|\mathrm {R} \rangle +\psi _{\mathrm {L} }|\mathrm {L} \rangle }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d9ea96690b39ffbb459efb0f2668746a268ce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.998ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle =\psi _{\mathrm {R} }|\mathrm {R} \rangle +\psi _{\mathrm {L} }|\mathrm {L} \rangle }"></noscript><span class="lazy-image-placeholder" style="width: 20.998ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d9ea96690b39ffbb459efb0f2668746a268ce9" data-alt="{\displaystyle |\psi \rangle =\psi _{\mathrm {R} }|\mathrm {R} \rangle +\psi _{\mathrm {L} }|\mathrm {L} \rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6711d7f20055547156b404337ade7a956630835c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.017ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 43.017ex;height: 12.509ex;vertical-align: -5.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6711d7f20055547156b404337ade7a956630835c" data-alt="{\displaystyle {\begin{aligned}\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{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>and: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =\alpha _{y}-\alpha _{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =\alpha _{y}-\alpha _{x}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4924cda3d0fc6d4ac121308d8f961d407ba3e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.831ex; height:3.009ex;" alt="{\displaystyle \delta =\alpha _{y}-\alpha _{x}.}"></noscript><span class="lazy-image-placeholder" style="width: 12.831ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4924cda3d0fc6d4ac121308d8f961d407ba3e98" data-alt="{\displaystyle \delta =\alpha _{y}-\alpha _{x}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Antennas">Antennas</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=14" title="Edit section: Antennas" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Over-quotation plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section <b>contains <a href="/wiki/Wikipedia:Manual_of_Style#Quotations" title="Wikipedia:Manual of Style">too many or overly lengthy quotations</a></b>.<span class="hide-when-compact"> Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Circular_polarization&action=edit">summarize the quotations</a>. Consider transferring direct quotations to <a href="https://en.wikiquote.org/wiki/Special:Search/Circular_polarization" class="extiw" title="q:Special:Search/Circular polarization">Wikiquote</a> or excerpts to <a href="https://en.wikisource.org/wiki/Special:Search/Circular_polarization" class="extiw" title="s:Special:Search/Circular polarization">Wikisource</a>.</span> <span class="date-container"><i>(<span class="date">April 2018</span>)</i></span></div></td></tr></tbody></table> <p>A number of different types of antenna elements can be used to produce circularly polarized (or nearly so) radiation; following <a href="/wiki/Constantine_A._Balanis" title="Constantine A. Balanis">Balanis</a>,<sup id="cite_ref-Balanis_16-0" class="reference"><a href="#cite_note-Balanis-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> one can use <a href="/wiki/Dipole_antenna" title="Dipole antenna"><i>dipole elements</i></a>: </p> <blockquote><p>"... 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;</p></blockquote> <p>or <a href="/wiki/Helical_antenna" title="Helical antenna"><i>helical elements</i></a>: </p> <blockquote><p>"To achieve circular polarization [in axial or end-fire mode] ... the circumference <i>C</i> of the helix must be ... with <i>C</i>/wavelength = 1 near optimum, and the spacing about <i>S</i> = wavelength/4," p.571;</p></blockquote> <p>or <a href="/wiki/Patch_antenna#Circular_polarization" title="Patch antenna"><i>patch elements</i></a>: </p> <blockquote><p>"... 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.</p></blockquote> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="In_quantum_mechanics">In quantum mechanics</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=15" title="Edit section: In quantum mechanics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Photon_polarization" title="Photon polarization">Photon polarization</a></div> <p>In the <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> view, light is composed of <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a>. Polarization is a manifestation of the <a href="/wiki/Spin_angular_momentum_of_light" title="Spin angular momentum of light">spin angular momentum of light</a>. 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.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="In_nature">In nature</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=16" title="Edit section: In nature" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cetonia-aurata.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Cetonia-aurata.jpg/220px-Cetonia-aurata.jpg" decoding="async" width="220" height="183" class="mw-file-element" data-file-width="1800" data-file-height="1500"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 183px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Cetonia-aurata.jpg/220px-Cetonia-aurata.jpg" data-width="220" data-height="183" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Cetonia-aurata.jpg/330px-Cetonia-aurata.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Cetonia-aurata.jpg/440px-Cetonia-aurata.jpg 2x" data-class="mw-file-element"> </span></a><figcaption>The <a href="/wiki/Cetonia_aurata" title="Cetonia aurata">rose chafer</a>'s external surface reflects almost exclusively left-circularly polarized light.</figcaption></figure> <p>Only a few mechanisms in nature are known to systematically produce circularly polarized <a href="/wiki/Light" title="Light">light</a>. In 1911, <a href="/wiki/Albert_A._Michelson" title="Albert A. Michelson">Albert Abraham Michelson</a> discovered that light reflected from the golden scarab beetle <i><a href="/wiki/Chrysina_resplendens" title="Chrysina resplendens">Chrysina resplendens</a></i> is preferentially left-polarized. Since then, circular polarization has been measured in several other <a href="/wiki/Scarabaeidae" title="Scarabaeidae">scarab beetles</a> such as <i><a href="/wiki/Chrysina_gloriosa" title="Chrysina gloriosa">Chrysina gloriosa</a></i>,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> as well as some <a href="/wiki/Crustacean" title="Crustacean">crustaceans</a> such as the <a href="/wiki/Mantis_shrimp" title="Mantis shrimp">mantis shrimp</a>. In these cases, the underlying mechanism is the molecular-level helicity of the <a href="/wiki/Chitin" title="Chitin">chitinous</a> <a href="/wiki/Cuticle" title="Cuticle">cuticle</a>.<sup id="cite_ref-Hegedüs_19-0" class="reference"><a href="#cite_note-Heged%C3%BCs-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Bioluminescence" title="Bioluminescence">bioluminescence</a> of the <a href="/wiki/Larva" title="Larva">larvae</a> of <a href="/wiki/Firefly" title="Firefly">fireflies</a> is also circularly polarized, as reported in 1980 for the species <i><a href="/wiki/Photuris" title="Photuris">Photuris lucicrescens</a></i> and <i><a href="/wiki/Photuris_versicolor" title="Photuris versicolor">Photuris versicolor</a></i>. 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 <a href="/wiki/Linear_polarization" title="Linear polarization">linear polarization</a> due to inhomogeneities inside aligned <a href="/wiki/Photocyte" title="Photocyte">photocytes</a>, and it picks up circular polarization while passing through linearly <a href="/wiki/Birefringence" title="Birefringence">birefringent</a> tissue.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Circular polarization has been detected in light reflected from leaves and photosynthetic microbes.<sup id="cite_ref-b966_21-0" class="reference"><a href="#cite_note-b966-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>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 <a href="/wiki/Total_internal_reflection" title="Total internal reflection">totally internally reflected</a> back down, its vertical component undergoes a phase shift. To an underwater observer looking up, the faint light outside <a href="/wiki/Snell%27s_window" title="Snell's window">Snell's window</a> therefore is (partially) circularly polarized.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Weaker sources of circular polarization in nature include multiple scattering by linear polarizers<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (June 2021)">dubious</span></a> – <a href="/wiki/Talk:Circular_polarization#Dubious" title="Talk:Circular polarization">discuss</a></i>]</sup>, as in the circular polarization of starlight, and selective absorption by <a href="/wiki/Circular_dichroism" title="Circular dichroism">circularly dichroic</a> media. </p><p>Radio emission from pulsars can be strongly circularly polarized.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Two species of <a href="/wiki/Mantis_shrimp" title="Mantis shrimp">mantis shrimp</a> have been reported to be able to detect circular polarized light.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kleinlogel_et_al._25-0" class="reference"><a href="#cite_note-Kleinlogel_et_al.-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=17" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <ul><li><a href="/wiki/Polarizer" title="Polarizer">Polarizer</a></li> <li><a href="/wiki/3D_film" title="3D film">3D film</a></li> <li><a href="/wiki/Chirality" title="Chirality">Chirality</a></li> <li><a href="/wiki/Sinusoidal_plane-wave_solutions_of_the_electromagnetic_wave_equation" title="Sinusoidal plane-wave solutions of the electromagnetic wave equation">Sinusoidal plane-wave solutions of the electromagnetic wave equation</a></li> <li><a href="/wiki/Starlight_polarization" class="mw-redirect" title="Starlight polarization">Starlight polarization</a></li> <li><a href="/wiki/Waveplate" title="Waveplate">Waveplate</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=18" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-11 collapsible-block" id="mf-section-11"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-fresnel-1822z-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-fresnel-1822z_1-0">^</a></b></span> <span class="reference-text">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.), <i>Oeuvres complètes d'Augustin Fresnel</i>, vol. 1 (1866), pp. 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", <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/Zenodo" title="Zenodo">Zenodo</a>: <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://zenodo.org/record/4745976">4745976</a></span>, 2021 (open access); §§9–10.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Académie des Sciences, <i>Procès-verbaux des séances de l'Académie tenues depuis la fondation de l'Institut jusqu'au mois d'août 1835</i>, vol. 7 (for 1820–23), Hendaye, Basses Pyrénées: Imprimerie de l'Observatoire d'Abbadia, 1916, p. 401.</span> </li> <li id="cite_note-fresnel-1821a-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-fresnel-1821a_3-0">^</a></b></span> <span class="reference-text">A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et seq., <i>Annales de Chimie et de Physique</i>, Ser. 2, vol. 17, pp. 102–11 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821); reprinted (with added section nos.) in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), <i>Oeuvres complètes d'Augustin Fresnel</i>, vol. 1 (1866), pp. 609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, & postscript", <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Zenodo" title="Zenodo">Zenodo</a>: <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://zenodo.org/record/4058004">4058004</a></span> (Creative Commons), 2021; author's footnote to §16.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">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. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-08032-2" title="Special:BookSources/0-471-08032-2">0-471-08032-2</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEEESTD.1979.120310">10.1109/IEEESTD.1979.120310</a>, sec. 11.1, p. 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"</span> </li> <li id="cite_note-Orfanidis-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Orfanidis_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Orfanidis_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Electromagnetic Waves & Antennas – S. J. Orfanidis: Footnote p.45, "most engineering texts use the IEEE convention and most physics texts, the opposite convention."</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Electromagnetic Waves & Antennas – S. J. Orfanidis Pg 44 "Curl the fingers of your left and right hands into a fist and point both thumbs <i>towards</i> the direction of propagation"</span> </li> <li id="cite_note-Lectures_on_Physics(Vol_1_ch_33-1)-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lectures_on_Physics(Vol_1_ch_33-1)_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lectures_on_Physics(Vol_1_ch_33-1)_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lectures_on_Physics(Vol_1_ch_33-1)_7-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">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."</span> </li> <li id="cite_note-IAU-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-IAU_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IAU_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">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.'"</span> </li> <li id="cite_note-Polarization_in_Spectral_Lines_Section_1.2-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Polarization_in_Spectral_Lines_Section_1.2_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Polarization_in_Spectral_Lines_Section_1.2_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">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. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=8sl2CkmZNWIC&dq=circular+polarization+conventions&pg=PA5">[1]</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">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 <i>against</i> the direction the wave is traveling."</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://spie.org/publications/fg05_p07-09_polarization_ellipse">"The Polarization Ellipse"</a>. <i>spie.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">13 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=spie.org&rft.atitle=The+Polarization+Ellipse&rft_id=https%3A%2F%2Fspie.org%2Fpublications%2Ffg05_p07-09_polarization_ellipse&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._E._Braslavsky2009" class="citation journal cs1">S. E. Braslavsky (1 January 2009). <a rel="nofollow" class="external text" href="https://www.degruyter.com/downloadpdf/journals/pac/79/3/article-p293.pdf">"Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)"</a> <span class="cs1-format">(PDF)</span>. <i>Pure and Applied Chemistry</i>. <b>79</b> (3): 293–465. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fpac200779030293">10.1351/pac200779030293</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:96601716">96601716</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/https://www.degruyter.com/downloadpdf/journals/pac/79/3/article-p293.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pure+and+Applied+Chemistry&rft.atitle=Glossary+of+terms+used+in+photochemistry%2C+3rd+edition+%28IUPAC+Recommendations+2006%29&rft.volume=79&rft.issue=3&rft.pages=293-465&rft.date=2009-01-01&rft_id=info%3Adoi%2F10.1351%2Fpac200779030293&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A96601716%23id-name%3DS2CID&rft.au=S.+E.+Braslavsky&rft_id=https%3A%2F%2Fwww.degruyter.com%2Fdownloadpdf%2Fjournals%2Fpac%2F79%2F3%2Farticle-p293.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">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. "<a rel="nofollow" class="external autonumber" href="http://www.its.bldrdoc.gov/fs-1037//dir-028/_4059.htm">[2]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110514080812/http://www.its.bldrdoc.gov/fs-1037/dir-028/_4059.htm">Archived</a> 2011-05-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> 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". <a rel="nofollow" class="external autonumber" href="http://www.its.bldrdoc.gov/fs-1037/dir-007/_0972.htm">[3]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110606113103/http://www.its.bldrdoc.gov/fs-1037/dir-007/_0972.htm">Archived</a> 2011-06-06 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation report cs1"><a rel="nofollow" class="external text" href="https://www.itu.int/dms_pub/itu-r/opb/rep/r-rep-bs.464-5-1990-pdf-e.pdf">Report 464-5, "Polarization of Emissions in Frequency-Modulation Broadcasting in Band 8 (VHF)"</a> <span class="cs1-format">(PDF)</span> (Report). International Telecommunications Union. 1990.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Report+464-5%2C+%22Polarization+of+Emissions+in+Frequency-Modulation+Broadcasting+in+Band+8+%28VHF%29%22&rft.pub=International+Telecommunications+Union&rft.date=1990&rft_id=https%3A%2F%2Fwww.itu.int%2Fdms_pub%2Fitu-r%2Fopb%2Frep%2Fr-rep-bs.464-5-1990-pdf-e.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><span><a href="/wiki/Title_47_of_the_Code_of_Federal_Regulations" title="Title 47 of the Code of Federal Regulations">47 CFR</a> <a rel="nofollow" class="external text" href="https://www.ecfr.gov/current/title-47/section-73.316">73.316</a></span></span> </li> <li id="cite_note-Balanis-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Balanis_16-0">^</a></b></span> <span class="reference-text">Balanis, Constantine A. 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Archived from <a rel="nofollow" class="external text" href="http://keur.eldoc.ub.rug.nl/FILES/wetenschappers/10/29/29.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 24 July 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Circular+polarization+observed+in+bioluminescence&rft.volume=286&rft.issue=5773&rft.pages=641-642&rft.date=1980-08-07&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4324467%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F286641a0&rft_id=info%3Abibcode%2F1980Natur.286..641W&rft.au=Wynberg%2C+Hans&rft.au=Meijer%2C+E.W.&rft.au=Hummelen%2C+J.C.&rft.au=Dekkers%2C+H.P.J.M.&rft.au=Schippers%2C+P.H.&rft.au=Carlson%2C+A.D.&rft_id=http%3A%2F%2Fkeur.eldoc.ub.rug.nl%2FFILES%2Fwetenschappers%2F10%2F29%2F29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-b966-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-b966_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSparksHoughGermerChen2009" class="citation journal cs1">Sparks, William B.; Hough, James; Germer, Thomas A.; Chen, Feng; DasSarma, Shiladitya; DasSarma, Priya; Robb, Frank T.; Manset, Nadine; Kolokolova, Ludmilla; Reid, Neill; Macchetto, F. Duccio; Martin, William (2009-05-12). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2674403">"Detection of circular polarization in light scattered from photosynthetic microbes"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>106</b> (19): 7816–7821. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0904.4646">0904.4646</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009PNAS..106.7816S">2009PNAS..106.7816S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.0810215106">10.1073/pnas.0810215106</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2674403">2674403</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19416893">19416893</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Detection+of+circular+polarization+in+light+scattered+from+photosynthetic+microbes&rft.volume=106&rft.issue=19&rft.pages=7816-7821&rft.date=2009-05-12&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2674403%23id-name%3DPMC&rft_id=info%3Abibcode%2F2009PNAS..106.7816S&rft_id=info%3Aarxiv%2F0904.4646&rft.issn=0027-8424&rft_id=info%3Adoi%2F10.1073%2Fpnas.0810215106&rft_id=info%3Apmid%2F19416893&rft.aulast=Sparks&rft.aufirst=William+B.&rft.au=Hough%2C+James&rft.au=Germer%2C+Thomas+A.&rft.au=Chen%2C+Feng&rft.au=DasSarma%2C+Shiladitya&rft.au=DasSarma%2C+Priya&rft.au=Robb%2C+Frank+T.&rft.au=Manset%2C+Nadine&rft.au=Kolokolova%2C+Ludmilla&rft.au=Reid%2C+Neill&rft.au=Macchetto%2C+F.+Duccio&rft.au=Martin%2C+William&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2674403&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHorváth,_GáborDezsö_Varjú2003" class="citation book cs1">Horváth, Gábor; Dezsö Varjú (2003). <i>Polarized Light in Animal Vision: Polarization Patterns in Nature</i>. Springer. pp. 100–103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-40457-6" title="Special:BookSources/978-3-540-40457-6"><bdi>978-3-540-40457-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polarized+Light+in+Animal+Vision%3A+Polarization+Patterns+in+Nature&rft.pages=100-103&rft.pub=Springer&rft.date=2003&rft.isbn=978-3-540-40457-6&rft.au=Horv%C3%A1th%2C+G%C3%A1bor&rft.au=Dezs%C3%B6+Varj%C3%BA&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGogoberidzeMachabeli2005" class="citation journal cs1">Gogoberidze, G.; Machabeli, G. Z. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-2966.2005.09681.x">"On the origin of the circular polarization in radio pulsars"</a>. <i>Monthly Notices of the Royal Astronomical Society</i>. <b>364</b> (4): 1363–1366. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0510116">astro-ph/0510116</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005MNRAS.364.1363G">2005MNRAS.364.1363G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-2966.2005.09681.x">10.1111/j.1365-2966.2005.09681.x</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Notices+of+the+Royal+Astronomical+Society&rft.atitle=On+the+origin+of+the+circular+polarization+in+radio+pulsars&rft.volume=364&rft.issue=4&rft.pages=1363-1366&rft.date=2005&rft_id=info%3Aarxiv%2Fastro-ph%2F0510116&rft_id=info%3Adoi%2F10.1111%2Fj.1365-2966.2005.09681.x&rft_id=info%3Abibcode%2F2005MNRAS.364.1363G&rft.aulast=Gogoberidze&rft.aufirst=G.&rft.au=Machabeli%2C+G.+Z.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1365-2966.2005.09681.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTsyr-Huei_ChiouSonja_KleinlogelTom_CroninRoy_Caldwell2008" class="citation journal cs1">Tsyr-Huei Chiou; Sonja Kleinlogel; Tom Cronin; Roy Caldwell; Birte Loeffler; Afsheen Siddiqi; Alan Goldizen; Justin Marshall (2008). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cub.2008.02.066">"Circular polarization vision in a stomatopod crustacean"</a>. <i><a href="/wiki/Current_Biology" title="Current Biology">Current Biology</a></i>. <b>18</b> (6): 429–34. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008CBio...18..429C">2008CBio...18..429C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cub.2008.02.066">10.1016/j.cub.2008.02.066</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18356053">18356053</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6925705">6925705</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Current+Biology&rft.atitle=Circular+polarization+vision+in+a+stomatopod+crustacean&rft.volume=18&rft.issue=6&rft.pages=429-34&rft.date=2008&rft_id=info%3Adoi%2F10.1016%2Fj.cub.2008.02.066&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6925705%23id-name%3DS2CID&rft_id=info%3Apmid%2F18356053&rft_id=info%3Abibcode%2F2008CBio...18..429C&rft.au=Tsyr-Huei+Chiou&rft.au=Sonja+Kleinlogel&rft.au=Tom+Cronin&rft.au=Roy+Caldwell&rft.au=Birte+Loeffler&rft.au=Afsheen+Siddiqi&rft.au=Alan+Goldizen&rft.au=Justin+Marshall&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.cub.2008.02.066&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> <li id="cite_note-Kleinlogel_et_al.-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kleinlogel_et_al._25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSonja_KleinlogelAndrew_White2008" class="citation journal cs1">Sonja Kleinlogel; Andrew White (2008). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2377063">"The secret world of shrimps: polarisation vision at its best"</a>. <i><a href="/wiki/PLoS_ONE" class="mw-redirect" title="PLoS ONE">PLoS ONE</a></i>. <b>3</b> (5): e2190. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0804.2162">0804.2162</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PLoSO...3.2190K">2008PLoSO...3.2190K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0002190">10.1371/journal.pone.0002190</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2377063">2377063</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18478095">18478095</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLoS+ONE&rft.atitle=The+secret+world+of+shrimps%3A+polarisation+vision+at+its+best&rft.volume=3&rft.issue=5&rft.pages=e2190&rft.date=2008&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2377063%23id-name%3DPMC&rft_id=info%3Abibcode%2F2008PLoSO...3.2190K&rft_id=info%3Aarxiv%2F0804.2162&rft_id=info%3Apmid%2F18478095&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0002190&rft.au=Sonja+Kleinlogel&rft.au=Andrew+White&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2377063&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></span> </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=19" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-12 collapsible-block" id="mf-section-12"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1999" class="citation book cs1">Jackson, John D. (1999). <i>Classical Electrodynamics</i> (3rd ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-30932-1" title="Special:BookSources/978-0-471-30932-1"><bdi>978-0-471-30932-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.place=New+York&rft.edition=3rd&rft.pub=Wiley&rft.date=1999&rft.isbn=978-0-471-30932-1&rft.aulast=Jackson&rft.aufirst=John+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBornWolf1999" class="citation book cs1">Born, M. & Wolf, E. (1999). <i><a href="/wiki/Principles_of_Optics" title="Principles of Optics">Principles of Optics</a>: Electromagnetic Theory of Propagation, Interference and Diffraction of Light</i> (7th ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-64222-4" title="Special:BookSources/978-0-521-64222-4"><bdi>978-0-521-64222-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Optics%3A+Electromagnetic+Theory+of+Propagation%2C+Interference+and+Diffraction+of+Light&rft.place=Cambridge&rft.edition=7th&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-64222-4&rft.aulast=Born&rft.aufirst=M.&rft.au=Wolf%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACircular+polarization" class="Z3988"></span></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(13)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Circular_polarization&action=edit&section=20" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-13 collapsible-block" id="mf-section-13"> <ul><li><a rel="nofollow" class="external text" href="http://www.polarization.com/beetle/beetle.html">Circularly polarized light: beetles and displays</a></li> <li><a rel="nofollow" class="external text" href="http://optics.org/cws/article/research/34196">Article on the mantis shrimp and circular polarization</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=jY9hnDzA6Ps">Animation of Circular Polarization (on YouTube)</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Q0qrU4nprB0">Comparison of Circular Polarization with Linear and Elliptical Polarizations (YouTube Animation)</a></li> <li><a rel="nofollow" class="external text" href="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</a></li></ul>    </section></div> <noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=mobile" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Circular_polarization&oldid=1258935893">https://en.wikipedia.org/w/index.php?title=Circular_polarization&oldid=1258935893</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Circular_polarization&action=history"> <div class="post-content last-modified-bar__content"> <span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Tallsmallstone" data-user-gender="unknown" data-timestamp="1732280466"> <span>Last edited on 22 November 2024, at 13:01</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D9%82%D8%B7%D8%A7%D8%A8_%D8%AF%D8%A7%D8%A6%D8%B1%D9%8A" title="استقطاب دائري – Arabic" lang="ar" hreflang="ar" data-title="استقطاب دائري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Polaritzaci%C3%B3_circular" title="Polarització circular – Catalan" lang="ca" hreflang="ca" data-title="Polarització circular" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Zirkularpolarisation" title="Zirkularpolarisation – German" lang="de" hreflang="de" data-title="Zirkularpolarisation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Polarizaci%C3%B3n_circular" title="Polarización circular – Spanish" lang="es" hreflang="es" data-title="Polarización circular" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polarisation_circulaire" title="Polarisation circulaire – French" lang="fr" hreflang="fr" data-title="Polarisation circulaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%ED%8E%B8%ED%8C%8C" title="원편파 – Korean" lang="ko" hreflang="ko" data-title="원편파" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%99%D7%98%D7%95%D7%91_%D7%9E%D7%A2%D7%92%D7%9C%D7%99" title="קיטוב מעגלי – Hebrew" lang="he" hreflang="he" data-title="קיטוב מעגלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BF%D0%BE%D0%BB%D1%8F%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F" title="Круговая поляризация – Russian" lang="ru" hreflang="ru" data-title="Круговая поляризация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kro%C5%BEna_polarizacija" title="Krožna polarizacija – Slovenian" lang="sl" hreflang="sl" data-title="Krožna polarizacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%9E%E0%B8%A5%E0%B8%B2%E0%B9%84%E0%B8%A3%E0%B9%80%E0%B8%8B%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%A7%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1" title="โพลาไรเซชันแบบวงกลม – Thai" lang="th" hreflang="th" data-title="โพลาไรเซชันแบบวงกลม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3%D0%BE%D0%B2%D0%B0_%D0%BF%D0%BE%D0%BB%D1%8F%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D1%96%D1%8F_%D1%81%D0%B2%D1%96%D1%82%D0%BB%D0%B0" title="Кругова поляризація світла – Ukrainian" lang="uk" hreflang="uk" data-title="Кругова поляризація світла" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img 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