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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Kerr_electro-optic_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kerr_electro-optic_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Kerr electro-optic effect</span> </div> </a> <ul id="toc-Kerr_electro-optic_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optical_Kerr_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Optical_Kerr_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Optical Kerr effect</span> </div> </a> <ul id="toc-Optical_Kerr_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magneto-optic_Kerr_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Magneto-optic_Kerr_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Magneto-optic Kerr effect</span> </div> </a> <ul id="toc-Magneto-optic_Kerr_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Theory</span> </div> </a> <button aria-controls="toc-Theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Theory subsection</span> </button> <ul id="toc-Theory-sublist" class="vector-toc-list"> <li id="toc-DC_Kerr_effect" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#DC_Kerr_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>DC Kerr effect</span> </div> </a> <ul id="toc-DC_Kerr_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-AC_Kerr_effect" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#AC_Kerr_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>AC Kerr effect</span> </div> </a> <ul id="toc-AC_Kerr_effect-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div 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class="mw-page-title-main">Kerr effect</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 29 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B8%D8%A7%D9%87%D8%B1%D8%A9_%D9%83%D9%8A%D8%B1" title="ظاهرة كير – Arabic" lang="ar" hreflang="ar" data-title="ظاهرة كير" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Efeutu_Kerr" title="Efeutu Kerr – Asturian" lang="ast" hreflang="ast" data-title="Efeutu Kerr" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D1%84%D0%B5%D0%BA%D1%82_%D0%9A%D0%B5%D1%80%D0%B0" title="Эфект Кера – Belarusian" lang="be" hreflang="be" data-title="Эфект Кера" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Efecte_Kerr" title="Efecte Kerr – Catalan" lang="ca" hreflang="ca" data-title="Efecte Kerr" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kerr-effekt" title="Kerr-effekt – Danish" lang="da" hreflang="da" data-title="Kerr-effekt" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kerr-Effekt" title="Kerr-Effekt – German" lang="de" hreflang="de" data-title="Kerr-Effekt" 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/Efecto_Kerr" title="Efecto Kerr – Spanish" lang="es" hreflang="es" data-title="Efecto Kerr" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kerr_efektua" title="Kerr efektua – Basque" lang="eu" hreflang="eu" data-title="Kerr efektua" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%B1_%DA%A9%D8%B1" title="اثر کر – Persian" lang="fa" hreflang="fa" data-title="اثر کر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Effet_Kerr" title="Effet Kerr – French" lang="fr" hreflang="fr" data-title="Effet Kerr" 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%BB%A4_%ED%9A%A8%EA%B3%BC" title="커 효과 – Korean" lang="ko" hreflang="ko" data-title="커 효과" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%A5%D6%80%D5%AB_%D5%A5%D6%80%D6%87%D5%B8%D6%82%D5%B5%D5%A9" title="Քերի երևույթ – Armenian" lang="hy" hreflang="hy" data-title="Քերի երևույթ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Effetto_Kerr" title="Effetto Kerr – Italian" lang="it" hreflang="it" data-title="Effetto Kerr" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%86%E0%B2%B0%E0%B3%8D_%E0%B2%AA%E0%B2%B0%E0%B2%BF%E0%B2%A3%E0%B2%BE%E0%B2%AE" title="ಕೆರ್ ಪರಿಣಾಮ – Kannada" lang="kn" hreflang="kn" data-title="ಕೆರ್ ಪರಿಣಾಮ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kerreffect" title="Kerreffect – Dutch" lang="nl" hreflang="nl" data-title="Kerreffect" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%83%BC%E5%8A%B9%E6%9E%9C" title="カー効果 – Japanese" lang="ja" hreflang="ja" data-title="カー効果" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kerreffekt" title="Kerreffekt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kerreffekt" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kerra_effekti" title="Kerra effekti – Uzbek" lang="uz" hreflang="uz" data-title="Kerra effekti" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zjawiska_Kerra" title="Zjawiska Kerra – Polish" lang="pl" hreflang="pl" data-title="Zjawiska Kerra" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Efeito_Kerr" title="Efeito Kerr – Portuguese" lang="pt" hreflang="pt" data-title="Efeito Kerr" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Efectul_Kerr" title="Efectul Kerr – Romanian" lang="ro" hreflang="ro" data-title="Efectul Kerr" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D1%84%D1%84%D0%B5%D0%BA%D1%82_%D0%9A%D0%B5%D1%80%D1%80%D0%B0" title="Эффект Керра – Russian" lang="ru" hreflang="ru" data-title="Эффект Керра" 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/Kerrov_pojav" title="Kerrov pojav – Slovenian" lang="sl" hreflang="sl" data-title="Kerrov pojav" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B5%D1%80%D0%BE%D0%B2_%D0%B5%D1%84%D0%B5%D0%BA%D0%B0%D1%82" title="Керов ефекат – Serbian" lang="sr" hreflang="sr" data-title="Керов ефекат" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kerreffekt" title="Kerreffekt – Swedish" lang="sv" hreflang="sv" data-title="Kerreffekt" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D1%84%D0%B5%D0%BA%D1%82_%D0%9A%D0%B5%D1%80%D1%80%D0%B0" title="Ефект Керра – Ukrainian" lang="uk" hreflang="uk" data-title="Ефект Керра" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Hi%E1%BB%87u_%E1%BB%A9ng_Kerr" title="Hiệu ứng Kerr – Vietnamese" lang="vi" hreflang="vi" data-title="Hiệu ứng Kerr" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Change in refractive index of a material in response to an applied electric field</div> <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">This article is about the Kerr nonlinear optical effect. For the magneto-optic phenomenon of the same name, see <a href="/wiki/Magneto-optic_Kerr_effect" title="Magneto-optic Kerr effect">magneto-optic Kerr effect</a>.</div> <p> The <b>Kerr effect</b>, also called the <b>quadratic electro-optic</b> (<b>QEO</b>) <b>effect</b>, is a change in the <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a> of a material in response to an applied <a href="/wiki/Electric_field" title="Electric field">electric field</a>. The Kerr effect is distinct from the <a href="/wiki/Pockels_effect" title="Pockels effect">Pockels effect</a> in that the induced index change for the Kerr effect is <a href="/wiki/Directly_proportional" class="mw-redirect" title="Directly proportional">directly proportional</a> to the <i>square</i> of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by Scottish physicist <a href="/wiki/John_Kerr_(physicist)" title="John Kerr (physicist)">John Kerr</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Two special cases of the Kerr effect are normally considered, these being the Kerr electro-optic effect, or DC Kerr effect, and the optical Kerr effect, or AC Kerr effect. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Kerr_electro-optic_effect">Kerr electro-optic effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=1" title="Edit section: Kerr electro-optic effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Kerr electro-optic effect, or DC Kerr effect, is the special case in which a slowly varying external electric field is applied by, for instance, a <a href="/wiki/Voltage" title="Voltage">voltage</a> on electrodes across the sample material. Under this influence, the sample becomes <a href="/wiki/Birefringent" class="mw-redirect" title="Birefringent">birefringent</a>, with different indices of refraction for light <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarized</a> parallel to or perpendicular to the applied field. The difference in index of refraction, <i>Δn</i>, 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 \Delta n=\lambda KE^{2},\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>n</mi> <mo>=</mo> <mi>λ</mi> <mi>K</mi> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta n=\lambda KE^{2},\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43603ab87803a2a1dddfbf52264b54b6cf79f2be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.313ex; height:3.009ex;" alt="{\displaystyle \Delta n=\lambda KE^{2},\ }"></span></dd></dl> <p>where <i>λ</i> is the wavelength of the light, <i>K</i> is the <i>Kerr constant</i>, and <i>E</i> is the strength of the electric field. This difference in index of refraction causes the material to act like a <a href="/wiki/Waveplate" title="Waveplate">waveplate</a> when light is incident on it in a direction perpendicular to the electric field. If the material is placed between two "crossed" (perpendicular) linear <a href="/wiki/Polarizer" title="Polarizer">polarizers</a>, no light will be transmitted when the electric field is turned off, while nearly all of the light will be transmitted for some optimum value of the electric field. Higher values of the Kerr constant allow complete transmission to be achieved with a smaller applied electric field. </p><p>Some <a href="/wiki/Polar_molecule" class="mw-redirect" title="Polar molecule">polar</a> liquids, such as <a href="/wiki/Nitrotoluene" class="mw-redirect" title="Nitrotoluene">nitrotoluene</a> (C<sub>7</sub>H<sub>7</sub>NO<sub>2</sub>) and <a href="/wiki/Nitrobenzene" title="Nitrobenzene">nitrobenzene</a> (C<sub>6</sub>H<sub>5</sub>NO<sub>2</sub>) exhibit very large Kerr constants. A glass cell filled with one of these liquids is called a <i>Kerr cell</i>. These are frequently used to <a href="/wiki/Modulation" title="Modulation">modulate</a> light, since the Kerr effect responds very quickly to changes in electric field. Light can be modulated with these devices at frequencies as high as 10 <a href="/wiki/Gigahertz" class="mw-redirect" title="Gigahertz">GHz</a>. Because the Kerr effect is relatively weak, a typical Kerr cell may require voltages as high as 30 <a href="/wiki/Kilovolt" class="mw-redirect" title="Kilovolt">kV</a> to achieve complete transparency. This is in contrast to <a href="/wiki/Pockels_cell" class="mw-redirect" title="Pockels cell">Pockels cells</a>, which can operate at much lower voltages. Another disadvantage of Kerr cells is that the best available material, <a href="/wiki/Nitrobenzene" title="Nitrobenzene">nitrobenzene</a>, is poisonous. Some transparent crystals have also been used for Kerr modulation, although they have smaller Kerr constants. </p><p>In media that lack <a href="/wiki/Inversion_symmetry" class="mw-redirect" title="Inversion symmetry">inversion symmetry</a>, the Kerr effect is generally masked by the much stronger <a href="/wiki/Pockels_effect" title="Pockels effect">Pockels effect</a>. The Kerr effect is still present, however, and in many cases can be detected independently of Pockels effect contributions.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Optical_Kerr_effect">Optical Kerr effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=2" title="Edit section: Optical Kerr effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The optical Kerr effect, or AC Kerr effect is the case in which the electric field is due to the light itself. This causes a variation in index of refraction which is proportional to the local <a href="/wiki/Irradiance" title="Irradiance">irradiance</a> of the light.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This refractive index variation is responsible for the <a href="/wiki/Nonlinear_optics" title="Nonlinear optics">nonlinear optical</a> effects of <a href="/wiki/Self-focusing" title="Self-focusing">self-focusing</a>, <a href="/wiki/Self-phase_modulation" title="Self-phase modulation">self-phase modulation</a> and <a href="/wiki/Modulational_instability" title="Modulational instability">modulational instability</a>, and is the basis for <a href="/wiki/Kerr-lens_modelocking" title="Kerr-lens modelocking">Kerr-lens modelocking</a>. This effect only becomes significant with very intense beams such as those from <a href="/wiki/Laser" title="Laser">lasers</a>. The optical Kerr effect has also been observed to dynamically alter the mode-coupling properties in <a href="/wiki/Multi-mode_optical_fiber" title="Multi-mode optical fiber">multimode fiber</a>, a technique that has potential applications for all-optical switching mechanisms, nanophotonic systems and low-dimensional photo-sensors devices.<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><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Magneto-optic_Kerr_effect">Magneto-optic Kerr effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=3" title="Edit section: Magneto-optic Kerr effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magneto-optic_Kerr_effect" title="Magneto-optic Kerr effect">Magneto-optic Kerr effect</a></div> <p>The magneto-optic Kerr effect (MOKE) is the phenomenon that the light reflected from a magnetized material has a slightly rotated plane of polarization. It is similar to the <a href="/wiki/Faraday_effect" title="Faraday effect">Faraday effect</a> where the plane of polarization of the transmitted light is rotated. </p> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=4" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="DC_Kerr_effect">DC Kerr effect</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=5" title="Edit section: DC Kerr effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a nonlinear material, the <a href="/wiki/Polarization_(electrostatics)" class="mw-redirect" title="Polarization (electrostatics)">electric polarization</a> <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 {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }"></span> will depend on the electric field <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 {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span>:<sup id="cite_ref-New,_Intro_to_Nonlinear_Optics_8-0" class="reference"><a href="#cite_note-New,_Intro_to_Nonlinear_Optics-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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 \mathbf {P} =\varepsilon _{0}\chi ^{(1)}\mathbf {E} +\varepsilon _{0}\chi ^{(2)}\mathbf {EE} +\varepsilon _{0}\chi ^{(3)}\mathbf {EEE} +\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\varepsilon _{0}\chi ^{(1)}\mathbf {E} +\varepsilon _{0}\chi ^{(2)}\mathbf {EE} +\varepsilon _{0}\chi ^{(3)}\mathbf {EEE} +\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef211f54250e5df634ec64486f5f0dcc593f4c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:44.491ex; height:3.176ex;" alt="{\displaystyle \mathbf {P} =\varepsilon _{0}\chi ^{(1)}\mathbf {E} +\varepsilon _{0}\chi ^{(2)}\mathbf {EE} +\varepsilon _{0}\chi ^{(3)}\mathbf {EEE} +\cdots }"></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 \varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}"></span> is the vacuum <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> 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 \chi ^{(n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{(n)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691c8d2b9f17d893358489a84c39962ef4bfd4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.953ex; height:3.176ex;" alt="{\displaystyle \chi ^{(n)}}"></span> is the <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-th order component of the <a href="/wiki/Electric_susceptibility" title="Electric susceptibility">electric susceptibility</a> of the medium. We can write that relationship explicitly; the <i>i-</i>th component for the vector <i>P</i> can be expressed as:<sup id="cite_ref-Moreno,_Kerr_Effect_9-0" class="reference"><a href="#cite_note-Moreno,_Kerr_Effect-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </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 P_{i}=\varepsilon _{0}\sum _{j=1}^{3}\chi _{ij}^{(1)}E_{j}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\chi _{ijk}^{(2)}E_{j}E_{k}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\sum _{l=1}^{3}\chi _{ijkl}^{(3)}E_{j}E_{k}E_{l}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}=\varepsilon _{0}\sum _{j=1}^{3}\chi _{ij}^{(1)}E_{j}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\chi _{ijk}^{(2)}E_{j}E_{k}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\sum _{l=1}^{3}\chi _{ijkl}^{(3)}E_{j}E_{k}E_{l}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424323c707d684abe7a4b93093aa30c4cd0572f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.439ex; height:7.509ex;" alt="{\displaystyle P_{i}=\varepsilon _{0}\sum _{j=1}^{3}\chi _{ij}^{(1)}E_{j}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\chi _{ijk}^{(2)}E_{j}E_{k}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\sum _{l=1}^{3}\chi _{ijkl}^{(3)}E_{j}E_{k}E_{l}+\cdots }"></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 i=1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a87b7e5d81c62b040a35078417fb9294b1ec7c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.456ex; height:2.509ex;" alt="{\displaystyle i=1,2,3}"></span>. It is often assumed that <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 P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> ∥ <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 P_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00798226b1d789f7fd779244d1afc7174b8199b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.665ex; height:2.509ex;" alt="{\displaystyle P_{x}}"></span>, i.e., the component parallel to <i>x</i> of the polarization field; <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 E_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6ee346e54f38302f47b5cf3016d8718f2040c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{2}}"></span> ∥ <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 E_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff71d8c6b39b429ae113e95c77c0016bd4aa62ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.765ex; height:2.843ex;" alt="{\displaystyle E_{y}}"></span> and so on. </p><p>For a linear medium, only the first term of this equation is significant and the polarization varies linearly with the electric field. </p><p>For materials exhibiting a non-negligible Kerr effect, the third, χ<sup>(3)</sup> term is significant, with the even-order terms typically dropping out due to inversion symmetry of the Kerr medium. Consider the net electric field <b>E</b> produced by a light wave of frequency ω together with an external electric field <b>E</b><sub>0</sub>: </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 \mathbf {E} =\mathbf {E} _{0}+\mathbf {E} _{\omega }\cos(\omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =\mathbf {E} _{0}+\mathbf {E} _{\omega }\cos(\omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f93fcf5e1c769141d0e6e7da2b37eb55b11bcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.759ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} =\mathbf {E} _{0}+\mathbf {E} _{\omega }\cos(\omega t),}"></span></dd></dl> <p>where <b>E</b><sub>ω</sub> is the vector amplitude of the wave. </p><p>Combining these two equations produces a complex expression for <b>P</b>. For the DC Kerr effect, we can neglect all except the linear terms and those in <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 \chi ^{(3)}|\mathbf {E} _{0}|^{2}\mathbf {E} _{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{(3)}|\mathbf {E} _{0}|^{2}\mathbf {E} _{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec63e9a83446d185e608ca4a7425f09d375c6390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.96ex; height:3.343ex;" alt="{\displaystyle \chi ^{(3)}|\mathbf {E} _{0}|^{2}\mathbf {E} _{\omega }}"></span>: </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 \mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+3\chi ^{(3)}|\mathbf {E} _{0}|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>≃</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+3\chi ^{(3)}|\mathbf {E} _{0}|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da1b990a4eeffde44ca0e506fbed2ce47b7ae31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.603ex; height:4.843ex;" alt="{\displaystyle \mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+3\chi ^{(3)}|\mathbf {E} _{0}|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t),}"></span></dd></dl> <p>which is similar to the linear relationship between polarization and an electric field of a wave, with an additional non-linear susceptibility term proportional to the square of the amplitude of the external field. </p><p>For non-symmetric media (e.g. liquids), this induced change of susceptibility produces a change in refractive index in the direction of the electric field: </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 n=\lambda _{0}K|\mathbf {E} _{0}|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>n</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta n=\lambda _{0}K|\mathbf {E} _{0}|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbacef4bd4331e539a1df3dc9fbfa9c836c73d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.711ex; height:3.343ex;" alt="{\displaystyle \Delta n=\lambda _{0}K|\mathbf {E} _{0}|^{2},}"></span></dd></dl> <p>where λ<sub>0</sub> is the vacuum <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> and <i>K</i> is the <i>Kerr constant</i> for the medium. The applied field induces <a href="/wiki/Birefringence" title="Birefringence">birefringence</a> in the medium in the direction of the field. A Kerr cell with a transverse field can thus act as a switchable <a href="/wiki/Wave_plate" class="mw-redirect" title="Wave plate">wave plate</a>, rotating the plane of polarization of a wave travelling through it. In combination with polarizers, it can be used as a shutter or modulator. </p><p>The values of <i>K</i> depend on the medium and are about 9.4×10<sup>−14</sup> m·<a href="/wiki/Volt" title="Volt">V</a><sup>−2</sup> for <a href="/wiki/Water" title="Water">water</a>,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2013)">citation needed</span></a></i>]</sup> and 4.4×10<sup>−12</sup> m·V<sup>−2</sup> for <a href="/wiki/Nitrobenzene" title="Nitrobenzene">nitrobenzene</a>.<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> </p><p>For <a href="/wiki/Crystal" title="Crystal">crystals</a>, the susceptibility of the medium will in general be a <a href="/wiki/Tensor" title="Tensor">tensor</a>, and the Kerr effect produces a modification of this tensor. </p> <div class="mw-heading mw-heading3"><h3 id="AC_Kerr_effect">AC Kerr effect</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=6" title="Edit section: AC Kerr effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the optical or AC Kerr effect, an intense beam of light in a medium can itself provide the modulating electric field, without the need for an external field to be applied. In this case, the electric field 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 \mathbf {E} =\mathbf {E} _{\omega }\cos(\omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =\mathbf {E} _{\omega }\cos(\omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28449bf85358359236812f79903a682248248b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.107ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} =\mathbf {E} _{\omega }\cos(\omega t),}"></span></dd></dl> <p>where <b>E</b><sub>ω</sub> is the amplitude of the wave as before. </p><p>Combining this with the equation for the polarization, and taking only linear terms and those in χ<sup>(3)</sup>|<b>E</b><sub>ω</sub>|<sup>3</sup>:<sup id="cite_ref-New,_Intro_to_Nonlinear_Optics_8-1" class="reference"><a href="#cite_note-New,_Intro_to_Nonlinear_Optics-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 81–82">: 81–82 </span></sup> </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 \mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+{\frac {3}{4}}\chi ^{(3)}|\mathbf {E} _{\omega }|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>≃</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+{\frac {3}{4}}\chi ^{(3)}|\mathbf {E} _{\omega }|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b4ef0f5064d2408bc064a0383f412144b051b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.286ex; height:6.176ex;" alt="{\displaystyle \mathbf {P} \simeq \varepsilon _{0}\left(\chi ^{(1)}+{\frac {3}{4}}\chi ^{(3)}|\mathbf {E} _{\omega }|^{2}\right)\mathbf {E} _{\omega }\cos(\omega t).}"></span></dd></dl> <p>As before, this looks like a linear susceptibility with an additional non-linear term: </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 \chi =\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }=\chi ^{(1)}+{\frac {3\chi ^{(3)}}{4}}|\mathbf {E} _{\omega }|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi =\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }=\chi ^{(1)}+{\frac {3\chi ^{(3)}}{4}}|\mathbf {E} _{\omega }|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6c5422557c804caed42e303a8c8ddf38027fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.403ex; height:5.843ex;" alt="{\displaystyle \chi =\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }=\chi ^{(1)}+{\frac {3\chi ^{(3)}}{4}}|\mathbf {E} _{\omega }|^{2},}"></span></dd></dl> <p>and since: </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 n=(1+\chi )^{1/2}=\left(1+\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }\right)^{1/2}\simeq n_{0}\left(1+{\frac {1}{2{n_{0}}^{2}}}\chi _{\mathrm {NL} }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>χ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>≃</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=(1+\chi )^{1/2}=\left(1+\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }\right)^{1/2}\simeq n_{0}\left(1+{\frac {1}{2{n_{0}}^{2}}}\chi _{\mathrm {NL} }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d6aa44b225fdd4696d89356ccbc541fdafe304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.203ex; height:6.176ex;" alt="{\displaystyle n=(1+\chi )^{1/2}=\left(1+\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }\right)^{1/2}\simeq n_{0}\left(1+{\frac {1}{2{n_{0}}^{2}}}\chi _{\mathrm {NL} }\right)}"></span></dd></dl> <p>where <i>n</i><sub>0</sub>=(1+χ<sub>LIN</sub>)<sup>1/2</sup> is the linear refractive index. Using a <a href="/wiki/Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> since χ<sub>NL</sub> ≪ <i>n</i><sub>0</sub><sup>2</sup>, this gives an <i>intensity dependent refractive index</i> (IDRI) of: </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 n=n_{0}+{\frac {3\chi ^{(3)}}{8n_{0}}}|\mathbf {E} _{\omega }|^{2}=n_{0}+n_{2}I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=n_{0}+{\frac {3\chi ^{(3)}}{8n_{0}}}|\mathbf {E} _{\omega }|^{2}=n_{0}+n_{2}I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e30b670ec1193ce28f94fc65f8348419bd105ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.938ex; height:6.176ex;" alt="{\displaystyle n=n_{0}+{\frac {3\chi ^{(3)}}{8n_{0}}}|\mathbf {E} _{\omega }|^{2}=n_{0}+n_{2}I}"></span></dd></dl> <p>where <i>n</i><sub>2</sub> is the second-order nonlinear refractive index, and <i>I</i> is the intensity of the wave. The refractive index change is thus proportional to the intensity of the light travelling through the medium. </p><p>The values of <i>n</i><sub>2</sub> are relatively small for most materials, on the order of 10<sup>−20</sup> m<sup>2</sup> W<sup>−1</sup> for typical glasses. Therefore, beam intensities (<a href="/wiki/Irradiance" title="Irradiance">irradiances</a>) on the order of 1 GW cm<sup>−2</sup> (such as those produced by lasers) are necessary to produce significant variations in refractive index via the AC Kerr effect. </p><p>The optical Kerr effect manifests itself temporally as self-phase modulation, a self-induced phase- and frequency-shift of a pulse of light as it travels through a medium. This process, along with <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a>, can produce optical <a href="/wiki/Soliton" title="Soliton">solitons</a>. </p><p>Spatially, an intense beam of light in a medium will produce a change in the medium's refractive index that mimics the transverse intensity pattern of the beam. For example, a <a href="/wiki/Gaussian_beam" title="Gaussian beam">Gaussian beam</a> results in a Gaussian refractive index profile, similar to that of a <a href="/wiki/Gradient-index_lens" class="mw-redirect" title="Gradient-index lens">gradient-index lens</a>. This causes the beam to focus itself, a phenomenon known as <a href="/wiki/Self-focusing" title="Self-focusing">self-focusing</a>. </p><p>As the beam self-focuses, the peak intensity increases which, in turn, causes more self-focusing to occur. The beam is prevented from self-focusing indefinitely by nonlinear effects such as <a href="/wiki/Multiphoton_ionization" class="mw-redirect" title="Multiphoton ionization">multiphoton ionization</a>, which become important when the intensity becomes very high. As the intensity of the self-focused spot increases beyond a certain value, the medium is ionized by the high local optical field. This lowers the refractive index, defocusing the propagating light beam. Propagation then proceeds in a series of repeated focusing and defocusing steps.<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> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Jeffree_cell" title="Jeffree cell">Jeffree cell</a>, an early acousto-optic modulator</li> <li><a href="/wiki/Filament_propagation" title="Filament propagation">Filament propagation</a></li> <li><a href="/wiki/Rapatronic_camera" title="Rapatronic camera">Rapatronic camera</a>, which used a Kerr cell to take sub-millisecond photographs of nuclear explosions</li> <li><a href="/wiki/Optical_heterodyne_detection" title="Optical heterodyne detection">Optical heterodyne detection</a></li> <li><a href="/wiki/Zeeman_effect" title="Zeeman effect">Zeeman effect</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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><cite id="CITEREFWeinberger2008" class="citation journal cs1">Weinberger, P. 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Retrieved <span class="nowrap">2023-11-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Kerr+Effect&rft.date=2018-06-14&rft.aulast=Moreno&rft.aufirst=Michelle&rft_id=https%3A%2F%2Fwww.ifsc.usp.br%2F~strontium%2FTeaching%2FMaterial2018-1%2520SFI5708%2520Eletromagnetismo%2FMonografia%2520-%2520Michelle%2520-%2520Kerr.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+effect" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoelho2012" class="citation book cs1">Coelho, Roland (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oT4wH0E5wbUC&q=kerr+constant+table+dielectrics&pg=PA51"><i>Physics of Dielectrics for the Engineer</i></a>. <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>. p. 52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-60180-3" title="Special:BookSources/978-0-444-60180-3"><bdi>978-0-444-60180-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+of+Dielectrics+for+the+Engineer&rft.pages=52&rft.pub=Elsevier&rft.date=2012&rft.isbn=978-0-444-60180-3&rft.aulast=Coelho&rft.aufirst=Roland&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoT4wH0E5wbUC%26q%3Dkerr%2Bconstant%2Btable%2Bdielectrics%26pg%3DPA51&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+effect" class="Z3988"></span></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 id="CITEREFDharmadhikariDharmadhikariMathur2008" class="citation journal cs1">Dharmadhikari, A. K.; Dharmadhikari, J. A.; Mathur, D. (2008). "Visualization of focusing–refocusing cycles during filamentation in BaF<sub>2</sub>". <i>Applied Physics B</i>. <b>94</b> (2): 259. <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/2009ApPhB..94..259D">2009ApPhB..94..259D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00340-008-3317-7">10.1007/s00340-008-3317-7</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:122865446">122865446</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Physics+B&rft.atitle=Visualization+of+focusing%E2%80%93refocusing+cycles+during+filamentation+in+BaF%3Csub%3E2%3C%2Fsub%3E&rft.volume=94&rft.issue=2&rft.pages=259&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122865446%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00340-008-3317-7&rft_id=info%3Abibcode%2F2009ApPhB..94..259D&rft.aulast=Dharmadhikari&rft.aufirst=A.+K.&rft.au=Dharmadhikari%2C+J.+A.&rft.au=Mathur%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+effect" class="Z3988"></span></span> </li> </ol></div></div> <p><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><span class="citation 1037C"><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/18px-PD-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/24px-PD-icon.svg.png 2x" data-file-width="196" data-file-height="196" /></span></span> This article incorporates <a href="/wiki/Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220122224547/https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm"><i>Federal Standard 1037C</i></a>. <a href="/wiki/General_Services_Administration" title="General Services Administration">General Services Administration</a>. Archived from <a rel="nofollow" class="external text" href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">the original</a> on 2022-01-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Federal+Standard+1037C&rft.pub=General+Services+Administration&rft_id=https%3A%2F%2Fwww.its.bldrdoc.gov%2Ffs-1037%2Ffs-1037c.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKerr+effect" class="Z3988"></span></span> </p> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kerr_effect&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.tvhistory.tv/1935%20TV%20Today%20Part%202.htm">Kerr cells in early television</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161001033146/http://www.tvhistory.tv/1935%20TV%20Today%20Part%202.htm">Archived</a> 2016-10-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (Scroll down the page for several early articles on Kerr cells.)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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