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<title>Stokes-Parameter</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Stokes-Parameter</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Stokes-Parameter</b> sind ein Satz von vier Werten, meist als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}\ldots S_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{0}\ldots S_{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e5ae79b662a7ebebe2f00c6a79d64a58159e660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.456ex; height:2.509ex;" alt="{\displaystyle S_{0}\ldots S_{3}}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><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,Q,U}">

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<annotation encoding="application/x-tex">{\displaystyle I,Q,U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22cb8899190f1b7b6bc14003bbd883fb77ad7440.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.861ex; height:2.509ex;" alt="{\displaystyle I,Q,U}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> bezeichnet, die 1852 von <a href="George_Gabriel_Stokes" title="George Gabriel Stokes">George Gabriel Stokes</a> zur Beschreibung des <a href="Polarisation" title="Polarisation">Polarisations</a>zustandes <a href="Elektromagnetische_Wellen" class="mw-redirect" title="Elektromagnetische Wellen">elektromagnetischer Wellen</a> (meist Licht) eingeführt wurden. Das Besondere an diesen Werten ist, dass sie durch einfache Messungen der <a href="Strahlungsleistung" title="Strahlungsleistung">Strahlungsleistung</a> nach Durchgang durch verschiedene <a href="Polarisator" title="Polarisator">Polarisatoren</a> berechnet werden können und so der Polarisationszustand recht einfach bestimmt werden kann.

</p><p>Die Stokes-Parameter können zum <b>Stokes-Vektor</b> zusammengefasst werden. Analog zum <a href="Jones-Vektor" class="mw-redirect" title="Jones-Vektor">Jones-Vektor</a> und der <a href="Jones-Matrix" class="mw-redirect" title="Jones-Matrix">Jones-Matrix</a> – auch <a href="Jones-Formalismus" title="Jones-Formalismus">Jones-Formalismus</a> genannt – kann die Wirkung <a href="Optisches_System" class="mw-redirect" title="Optisches System">optischer Systeme</a> auf Stokes-Vektoren im Müller-Formalismus durch Anwendung entsprechender Matrizen (<a href="M%C3%BCller-Matrix" title="Müller-Matrix">Müller-Matrix</a>) behandelt werden. Im Unterschied zum Jones-Formalismus kann zwar die <a href="Bestrahlungsst%C3%A4rke" title="Bestrahlungsstärke">Bestrahlungsstärke</a> beschrieben werden, jedoch nur von <a href="Koh%C3%A4renz_(Physik)" title="Kohärenz (Physik)">inkohärentem</a> Licht. Das heißt, es sind keine Phaseninformationen enthalten, und erlaubt damit nicht die Berechnung von <a href="Interferenz_(Physik)" title="Interferenz (Physik)">Interferenzeffekten</a>.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<caption>Beispiele

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<th>Polarisation

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<th>Polarisationszustand

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<th>Stokes-Vektor

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\1\\0\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4adaccdff7596db26afad0bb57c55f40315070e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:5.981ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\1\\0\\0\end{pmatrix}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\-1\\0\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2df8a5c966f71af54ab05b00f95111386bd7bfd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:7.789ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\-1\\0\\0\end{pmatrix}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/776a4dd6624081b648544d58fdb9b956537e261a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:5.981ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\\0\\-1\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f244aaab64d798961474d294823df5f34cfebb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:7.789ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\\0\\-1\end{pmatrix}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3a9eab96de1e0db94a61873f5cbf217eb158295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:5.981ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}}" loading="lazy"></span>

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<td>unpolarisiert

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b66674354a25035b42cb97912d29aca7158ab1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:5.981ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle S_{0}=I=P_{0^{\circ }}+P_{90^{\circ }}=\langle E_{x}^{2}+E_{y}^{2}\rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/faa428edf4a12186f6094a5aee57a03451b16a7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.75ex; height:3.176ex;" alt="{\displaystyle S_{0}=I=P_{0^{\circ }}+P_{90^{\circ }}=\langle E_{x}^{2}+E_{y}^{2}\rangle }" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle S_{1}=Q=P_{0^{\circ }}-P_{90^{\circ }}=\langle E_{x}^{2}-E_{y}^{2}\rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b02df424b4442f237afc857c7c30848349d9ad20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.417ex; height:3.176ex;" alt="{\displaystyle S_{1}=Q=P_{0^{\circ }}-P_{90^{\circ }}=\langle E_{x}^{2}-E_{y}^{2}\rangle }" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle S_{2}=U=P_{45^{\circ }}-P_{135^{\circ }}=\langle 2E_{x}E_{y}\cos \delta \rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5240bdd31a9a464eda6c36b472dc925f1da6f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.178ex; height:3.009ex;" alt="{\displaystyle S_{2}=U=P_{45^{\circ }}-P_{135^{\circ }}=\langle 2E_{x}E_{y}\cos \delta \rangle }" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle S_{3}=V=P_{\rm {RZ}}-P_{\rm {LZ}}\,=\langle 2E_{x}E_{y}\sin \delta \rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19cfc225463a78c6796e031e16b44f4ffce47429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.786ex; height:3.009ex;" alt="{\displaystyle S_{3}=V=P_{\rm {RZ}}-P_{\rm {LZ}}\,=\langle 2E_{x}E_{y}\sin \delta \rangle }" loading="lazy"></span></dd></dl>

<p>Die Leistungen sind dabei die gemessene Leistung nach Durchgang durch einen horizontal (0°), vertikal (90°), 45° und 135° orientierten, idealen Polarisator sowie der rechts- und links-zirkular polarisierte Anteil des Lichts.

</p><p>Alternativ lassen sie sich über die zeitgemittelten Amplituden <span class="mwe-math-element mwe-math-element-inline"><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_{\mathrm {x} }}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59a5724f78251575d1dff3a411a009f6bf861968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.815ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {x} }}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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_{\mathrm {y} }}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/287d833159caf0049af62cb7a660e79204d4e1a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.815ex; height:2.843ex;" alt="{\displaystyle E_{\mathrm {y} }}" loading="lazy"></span> der elektrischen Wellenvektoren in einem orthogonalen Koordinatensystem, sowie deren relativer Phase <span class="mwe-math-element mwe-math-element-inline"><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 }">

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<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"></span> definieren.

</p><p>Üblicherweise werden die Stokes-Parameter auf die einfallende Leistung normiert, indem alle vier Werte durch <i>S</i><sub>0</sub> dividiert werden, man spricht in diesem Zusammenhang vom <b>normierten Stokes-Vektor</b>.

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<annotation encoding="application/x-tex">{\displaystyle {\vec {S}}_{\mathrm {N} }={\frac {1}{S_{0}}}{\begin{pmatrix}S_{0}\\S_{1}\\S_{2}\\S_{3}\end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1419c31999fdb21ac8472ab2463fccd75d6a1142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:16.714ex; height:12.509ex;" alt="{\displaystyle {\vec {S}}_{\mathrm {N} }={\frac {1}{S_{0}}}{\begin{pmatrix}S_{0}\\S_{1}\\S_{2}\\S_{3}\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<caption>Definition der Stokes-Parameter in weiteren Bezugssystemen<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>

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<annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebe0ac45a38c4437bd2689a14ec434cd499e7e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70e15f3e200aaa247f69c43110cc5a09ecc91b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}" loading="lazy"></span>

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<td>Lichtintensität</td>

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<annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {x} }+I_{\mathrm {y} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5c0df783cac32c9a87c8319fa14d098b689b30e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.087ex; height:2.843ex;" alt="{\displaystyle I_{\mathrm {x} }+I_{\mathrm {y} }}" loading="lazy"></span></td>

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<annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {x} }-I_{\mathrm {y} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5211b0f1c98242a9aaccc3bce438230820866aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.087ex; height:2.843ex;" alt="{\displaystyle I_{\mathrm {x} }-I_{\mathrm {y} }}" loading="lazy"></span></td>

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<annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {+45^{\circ }} }-I_{\mathrm {-45^{\circ }} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c1aa634b5a979b13bb62670c5114959e4c4f9e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.859ex; height:2.676ex;" alt="{\displaystyle I_{\mathrm {+45^{\circ }} }-I_{\mathrm {-45^{\circ }} }}" loading="lazy"></span></td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {R} }-I_{\mathrm {L} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06d6c3a477e4c03dbab59e09eab77b99a43fd6f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.588ex; height:2.509ex;" alt="{\displaystyle I_{\mathrm {R} }-I_{\mathrm {L} }}" loading="lazy"></span>

</td></tr>

<tr>

<td>ε-θ-System</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span></td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {2\varepsilon }\cdot \cos {2\theta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/648ea8778b5a1c7ae35523de63d1632a923b155f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.174ex; height:2.176ex;" alt="{\displaystyle \cos {2\varepsilon }\cdot \cos {2\theta }}" loading="lazy"></span></td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {2\varepsilon }\cdot \sin {2\theta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7b291a6f7984ccf797c1bd0d60ef2218a62a1e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.919ex; height:2.176ex;" alt="{\displaystyle \cos {2\varepsilon }\cdot \sin {2\theta }}" loading="lazy"></span></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin {2\varepsilon }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a04962d023fb44afaa1af0de421ee5c78cef83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.489ex; height:2.176ex;" alt="{\displaystyle \sin {2\varepsilon }}" loading="lazy"></span>

</td></tr>

<tr>

<td>ψ-Δ-System</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\cos {2\psi }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3daea6a1dd2c5eceaf7be7283d71f2b4d84d230.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.369ex; height:2.509ex;" alt="{\displaystyle -\cos {2\psi }}" loading="lazy"></span></td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin {2\psi }\cdot \cos {\Delta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fb87ca299e3d346cc99b56f6f9bfc7b757c6a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.032ex; height:2.509ex;" alt="{\displaystyle \sin {2\psi }\cdot \cos {\Delta }}" loading="lazy"></span></td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\sin {2\psi }\cdot \sin {\Delta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2d2a8ffbc7e87b375988c8e2a2a01d768c3aa26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.971ex; height:2.509ex;" alt="{\displaystyle -\sin {2\psi }\cdot \sin {\Delta }}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Definition_in_Kugelkoordinaten">Definition in Kugelkoordinaten</h2></div>

<p>Eine andere Formulierung findet sich in Kugelkoordinaten:

</p>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}S_{0}&=&I\\S_{1}&=&I_{\mathrm {p} }\cos 2\psi \cos 2\chi \\S_{2}&=&I_{\mathrm {p} }\sin 2\psi \cos 2\chi \\S_{3}&=&I_{\mathrm {p} }\sin 2\chi \end{matrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e1128daacb1181b8450b1448b1fe6a3b0be6a18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:24.917ex; height:13.176ex;" alt="{\displaystyle {\begin{matrix}S_{0}&=&I\\S_{1}&=&I_{\mathrm {p} }\cos 2\psi \cos 2\chi \\S_{2}&=&I_{\mathrm {p} }\sin 2\psi \cos 2\chi \\S_{3}&=&I_{\mathrm {p} }\sin 2\chi \end{matrix}}}" loading="lazy"></span></dd></dl>

<p>Wobei <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> die Gesamtintensität, <span class="mwe-math-element mwe-math-element-inline"><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_{\mathrm {p} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\mathrm {p} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d45f4d256d75e8fa81578821d4e83f5ac749e3ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.169ex; height:2.843ex;" alt="{\displaystyle I_{\mathrm {p} }}" loading="lazy"></span> der polarisierte Anteil der Intensität, <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> die Verkippung der Polarisationsellipse und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \chi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef94ee7fead714dc5709291d674791284d80d509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.202ex; height:2.343ex;" alt="{\displaystyle \tan \chi }" loading="lazy"></span> das Verhältnis der beiden Hauptachsen der Polarisationsellipse ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Polarisationsgrad">Polarisationsgrad</h2></div>

<p>Der <b>Polarisationsgrad</b> Π gibt an, wie groß der geordnete Anteil der Welle ist. Er ist definiert durch:

</p>

<mfrac>

<msqrt>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ={\frac {\sqrt {S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3b86a7729e28efffb9d4c53a499590d9ad4dab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.41ex; height:8.176ex;" alt="{\displaystyle \Pi ={\frac {\sqrt {S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}}}}" loading="lazy"></span></dd></dl>

<p>Beziehungsweise für nur linear polarisiertes Licht:

</p>

<mfrac>

<msqrt>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ={\frac {\sqrt {S_{1}^{2}+S_{2}^{2}}}{S_{0}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c45cf5fea93a3907488776f723982991165103f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.994ex; height:8.176ex;" alt="{\displaystyle \Pi ={\frac {\sqrt {S_{1}^{2}+S_{2}^{2}}}{S_{0}}}}" loading="lazy"></span></dd></dl>

<p>Für vollständig polarisiertes Licht gilt:

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{0}^{2}=S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f5fe47608f018cb033b417906c6013f99a47b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.083ex; height:3.176ex;" alt="{\displaystyle S_{0}^{2}=S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}" loading="lazy"></span></dd></dl>

<p>Für unpolarisiertes Licht hingegen gilt:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}=S_{2}=S_{3}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1bd34f27441af682788f71384d5bf7dd62e081c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.895ex; height:2.509ex;" alt="{\displaystyle S_{1}=S_{2}=S_{3}=0}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Winkel_der_maximalen_Polarisation">Winkel der maximalen Polarisation</h2></div>

<p>Der Winkel der maximalen Polarisation ist definiert durch:

</p>

<mfrac>

</mfrac>

</mrow>

<mi>arctan</mi>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta ={\frac {1}{2}}\arctan {\frac {S_{2}}{S_{1}}}+n{\frac {\pi }{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b226b6b0c298fb4dd28965ce09465300c2245575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.864ex; height:5.843ex;" alt="{\displaystyle \Theta ={\frac {1}{2}}\arctan {\frac {S_{2}}{S_{1}}}+n{\frac {\pi }{2}}}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}<0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}<0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f217e0ed9d2ba5ba1cdaf2c2ba5f1230e1f0aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.74ex; height:2.509ex;" alt="{\displaystyle S_{1}<0}" loading="lazy"></span>, ansonsten ist <span class="mwe-math-element mwe-math-element-inline"><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=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26819344e55f5e671c76c07c18eb4291fcec85ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}" loading="lazy"></span>. Anders ausgedrückt bedeutet das, dass man 90° zum Winkel <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }" loading="lazy"></span> hinzu zählen muss, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"></span> kleiner als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>William A. Shurcliff: <cite style="font-style:italic">Polarized Light: Production and Use</cite>. Harvard University Press, Cambridge, Mass. 1962, ISBN 0-674-68250-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Stokes-Parameter&rft.au=William+A.+Shurcliff&rft.btitle=Polarized+Light%3A+Production+and+Use&rft.date=1962&rft.genre=book&rft.isbn=0674682505&rft.place=Cambridge%2C+Mass.&rft.pub=Harvard+University+Press" style="display:none"> </span></li>

<li>Craig F. Bohren, Donald R. Huffman: <cite style="font-style:italic">Absorption and scattering of light by small particles</cite>. Wiley, New York 1998, ISBN 0-471-29340-7.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Stokes-Parameter&rft.au=Craig+F.+Bohren%2C+Donald+R.+Huffman&rft.btitle=Absorption+and+scattering+of+light+by+small+particles&rft.date=1998&rft.genre=book&rft.isbn=0471293407&rft.place=New+York&rft.pub=Wiley" style="display:none"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><span class="cite">Jürgen Weiprecht: <a rel="nofollow" class="external text" href="https://www.astro.uni-jena.de/Teaching/Praktikum/pra2002/node235.html"><i>Polarisation und Stokes-Parameter.</i></a> In: <i>Kompendium für das Astronomische Praktikum.</i> Hans-Georg Reimann, Olaf Fischer, Christian Friedemann, Reinhard E. Schielicke, 29. Oktober 2002,<span class="Abrufdatum"> abgerufen am 2. Februar 2010</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AStokes-Parameter&rft.title=Polarisation+und+Stokes-Parameter&rft.description=Polarisation+und+Stokes-Parameter&rft.identifier=https%3A%2F%2Fwww.astro.uni-jena.de%2FTeaching%2FPraktikum%2Fpra2002%2Fnode235.html&rft.creator=J%C3%BCrgen+Weiprecht&rft.publisher=Hans-Georg+Reimann%2C+Olaf+Fischer%2C+Christian+Friedemann%2C+Reinhard+E.+Schielicke&rft.date=2002-10-29"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Hiroyuki Fujiwara: <cite style="font-style:italic">Spectroscopic Ellipsometry: Principles and Applications</cite>. Wiley-Interscience, 2007, ISBN 978-0-470-01608-4, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>75</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Stokes-Parameter&rft.au=Hiroyuki+Fujiwara&rft.btitle=Spectroscopic+Ellipsometry%3A+Principles+and+Applications&rft.date=2007&rft.genre=book&rft.isbn=9780470016084&rft.pages=75&rft.pub=Wiley-Interscience" style="display:none"> </span></span>

</li>

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