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<title>Fresnelsche Formeln</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Fresnelsche Formeln</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">Die fresnelschen Formeln (nach <a href="Augustin_Jean_Fresnel" class="mw-redirect" title="Augustin Jean Fresnel">Augustin Jean Fresnel</a>) beschreiben quantitativ die <a href="Reflexion_(Physik)" title="Reflexion (Physik)">Reflexion</a> und <a href="Transmission_(Physik)" title="Transmission (Physik)">Transmission</a> einer <a href="Ebene_Welle" title="Ebene Welle">ebenen</a>, <a href="Elektromagnetische_Welle" title="Elektromagnetische Welle">elektromagnetischen Welle</a> an einer ebenen Grenzfläche. Der zunächst berechnete Reflexions- und Transmissionsfaktor ist das Verhältnis der reflektierten bzw. transmittierten Amplitude zu jener der einfallenden Welle. Durch Quadrieren erhält man den Reflexions- bzw. den Transmissionsgrad, welche als <a href="Energiegr%C3%B6%C3%9Fe" class="mw-redirect" title="Energiegröße">Energiegrößen</a> Intensitätsverhältnisse darstellen.

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

Die fresnelschen Formeln können aus den <a href="Maxwellsche_Gleichungen" class="mw-redirect" title="Maxwellsche Gleichungen">maxwellschen Gleichungen</a> hergeleitet werden, dabei nutzt man Sonderfälle der <a href="Grenzbedingungen_(Elektrodynamik)" title="Grenzbedingungen (Elektrodynamik)">Randbedingungen</a> elektromagnetischer Wellen an einer ladungs- und stromfreien Grenzschicht:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}\times ({\vec {E}}_{2}-{\vec {E}}_{1})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/566a48952fc14299f7fabe5a7f8bb377ad41baa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.805ex; height:3.343ex;" alt="{\displaystyle {\vec {n}}\times ({\vec {E}}_{2}-{\vec {E}}_{1})=0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}\times ({\vec {H}}_{2}-{\vec {H}}_{1})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93ddad03fcf73b12169fca978207c675a9953188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.381ex; height:3.343ex;" alt="{\displaystyle {\vec {n}}\times ({\vec {H}}_{2}-{\vec {H}}_{1})=0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}\cdot ({\vec {D}}_{2}-{\vec {D}}_{1})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/272464e3c5b4282393c6690466267d5f18280f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.941ex; height:3.343ex;" alt="{\displaystyle {\vec {n}}\cdot ({\vec {D}}_{2}-{\vec {D}}_{1})=0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}\cdot ({\vec {B}}_{2}-{\vec {B}}_{1})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfa7ebabbf9d8791d7ab87307b1387af9a3c494e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.621ex; height:3.343ex;" alt="{\displaystyle {\vec {n}}\cdot ({\vec {B}}_{2}-{\vec {B}}_{1})=0}" loading="lazy">

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Hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"> die Normale auf die Grenzfläche und die anderen Größen beschreiben Magnetfeld und elektrisches Feld in den beiden Medien. Die Tangentialkomponente der <a href="Elektrische_Feldst%C3%A4rke" title="Elektrische Feldstärke">elektrischen Feldstärke</a> E und der <a href="Magnetische_Feldst%C3%A4rke" title="Magnetische Feldstärke">magnetischen Feldstärke</a> H sind an der Grenzfläche stetig, ebenso wie die Normalkomponente der <a href="Elektrische_Flussdichte" title="Elektrische Flussdichte">elektrischen Flussdichte</a> D und der <a href="Magnetische_Flussdichte" title="Magnetische Flussdichte">magnetischen Flussdichte</a> B (tangential und normal bezieht sich auf die Grenzfläche).

Abhängig von der Polarisation der einfallenden Welle ergeben sich unterschiedliche Randbedingungen für das Auftreffen einer elektromagnetischen Welle auf eine optische Grenzfläche. Jede beliebig <a href="Polarisation#Polarisation_elektromagnetischer_Wellen" title="Polarisation">polarisierte elektromagnetische Welle</a> lässt sich als Superposition zweier linear polarisierter Wellen, die senkrecht zueinander schwingen, darstellen. Als Bezugsebene dient die <a href="Einfallsebene" title="Einfallsebene">Einfallsebene</a>, die vom <a href="Wellenvektor" title="Wellenvektor">Wellenvektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k_{e}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {k_{e}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ad06bd6aa5215efbc681288bfe33a98f0ab55f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:4.009ex;" alt="{\displaystyle {\vec {k_{e}}}}" loading="lazy"> der einfallenden Welle und der <a href="Fl%C3%A4chennormale" class="mw-redirect" title="Flächennormale">Flächennormalen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}=\left[(E_{0e})_{s}\ {\vec {e}}_{s}\ e^{i\delta _{s}}+(E_{0e})_{p}\ {\vec {e}}_{p}\ e^{i\delta _{p}}\right]\ e^{i({\vec {k}}_{e}\cdot {\vec {r}}-\omega t)}=(E_{0e})_{s}\ {\vec {e}}_{s}\ e^{i({\vec {k}}_{e}\cdot {\vec {r}}-\omega t+\delta _{s})}+(E_{0e})_{p}\ {\vec {e}}_{p}\ e^{i({\vec {k}}_{e}\cdot {\vec {r}}-\omega t+\delta _{p})}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f534963b5e272a51a599d420ff7f0af67de8db19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:95.402ex; height:4.009ex;" alt="{\displaystyle {\vec {E}}=\left[(E_{0e})_{s}\ {\vec {e}}_{s}\ e^{i\delta _{s}}+(E_{0e})_{p}\ {\vec {e}}_{p}\ e^{i\delta _{p}}\right]\ e^{i({\vec {k}}_{e}\cdot {\vec {r}}-\omega t)}=(E_{0e})_{s}\ {\vec {e}}_{s}\ e^{i({\vec {k}}_{e}\cdot {\vec {r}}-\omega t+\delta _{s})}+(E_{0e})_{p}\ {\vec {e}}_{p}\ e^{i({\vec {k}}_{e}\cdot {\vec {r}}-\omega t+\delta _{p})}}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}" loading="lazy"> der Feldvektor des elektrischen Feldes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a86dbaf7593cb1a89a4d90d740b2c8b010551cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.023ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{i}}" loading="lazy"> sind die Einheitsvektoren für s- und p-Polarisation, und die Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0c5e905acee9cc0cf8bc01c08a4e876f43d5c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.832ex; height:2.676ex;" alt="{\displaystyle \delta _{i}}" loading="lazy"> entsprechen beliebigen Phasenverschiebungen.

Wegen des Superpositionsprinzips reicht es aus, die Amplitudenverhältnisse für parallel und senkrecht zur Einfallsebene linear polarisierte Wellen zu berechnen.

Die Polarisationsrichtung (senkrecht bzw. parallel zur Einfallsebene) bleibt nach der Reflexion unverändert.

<div class="mw-heading mw-heading2"><h2 id="Allgemeiner_Fall">Allgemeiner Fall</h2></div>

Im allgemeinen Fall haben beide Medien eine unterschiedliche <a href="Permittivit%C3%A4t" title="Permittivität">Permittivität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{r}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/287bce0849b56661bbaebca9942dfe3a6cadc21c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.057ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{r}}" loading="lazy"> und <a href="Permeabilit%C3%A4t_(Magnetismus)" class="mw-redirect" title="Permeabilität (Magnetismus)">Permeabilität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{r}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu _{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70e2dc5a760017c379bd66f0043d213db98bf77b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.375ex; height:2.176ex;" alt="{\displaystyle \mu _{r}}" loading="lazy"> sowie einen komplexen <a href="Brechungsindex" title="Brechungsindex">Brechungsindex</a>

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<annotation encoding="application/x-tex">{\displaystyle N=n+\mathrm {i} K\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdd3fefa402e2b6ed6875b83258fbb1532a9ca94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.497ex; height:2.343ex;" alt="{\displaystyle N=n+\mathrm {i} K\,}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Vorbetrachtung_für_Gleichungen_mit_eliminiertem_Brechungswinkel">Vorbetrachtung für Gleichungen mit eliminiertem Brechungswinkel</h3></div>

Im Allgemeinen sind für die Berechnung der Reflexions- bzw. Transmissionsgrade mit den fresnelschen Formeln sowohl der Brechungsindex der beteiligten Medien als auch der Einfalls- und Brechungswinkel notwendig.

Um neben diesen allgemeinen Gleichungen auch eine vom Brechungswinkel unabhängige Form anzugeben, muss der Brechungswinkel aus der allgemeinen Form eliminiert werden. Da beide Winkel (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">) über das <a href="Snelliussches_Brechungsgesetz" title="Snelliussches Brechungsgesetz">snelliussche Brechungsgesetz</a> verknüpft sind, kann dies wie folgt (mit Hilfe einer Falleingrenzung) erreicht werden:

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<annotation encoding="application/x-tex">{\displaystyle N_{1}\sin \alpha =N_{2}\sin \beta \,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/698d4b4e15dc2449a08eb84e02f7cd50dee16d98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.406ex; height:2.509ex;" alt="{\displaystyle N_{1}\sin \alpha =N_{2}\sin \beta \,}" loading="lazy"> (Brechungsgesetz)</dd></dl>

Quadrieren liefert (unter Nutzung einer <a href="Formelsammlung_Trigonometrie#Gegenseitige_Darstellung" title="Formelsammlung Trigonometrie">trigonometrischen Umrechnung</a>) folgenden Zusammenhang:

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<annotation encoding="application/x-tex">{\displaystyle N_{1}^{2}\sin ^{2}\alpha =N_{2}^{2}\sin ^{2}\beta =N_{2}^{2}\left(1-\cos ^{2}\beta \right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08f33ef74016e522714babc1056a7fe7df569fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.32ex; height:3.343ex;" alt="{\displaystyle N_{1}^{2}\sin ^{2}\alpha =N_{2}^{2}\sin ^{2}\beta =N_{2}^{2}\left(1-\cos ^{2}\beta \right)}" loading="lazy"></dd></dl>

Umgestellt ergibt sich daraus:

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<annotation encoding="application/x-tex">{\displaystyle \cos \beta =\pm {\frac {\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}{N_{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daf4bfe203c0cbc031bb5fe238a0b07555c92420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.263ex; height:8.009ex;" alt="{\displaystyle \cos \beta =\pm {\frac {\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}{N_{2}}}}" loading="lazy"></dd></dl>

Als Lösung wird der Fall mit dem positiven <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> genutzt, damit später der Reflexionsfaktor r ≤ 1 ist.

<div class="mw-heading mw-heading3"><h3 id="Senkrechte_Polarisation">Senkrechte Polarisation</h3></div>

Als erstes betrachtet man die Komponente, die linear senkrecht (Index: s) zur Einfallsebene polarisiert ist. Sie wird in der Literatur auch als <a href="Transversalelektromagnetische_Welle" title="Transversalelektromagnetische Welle">transversalelektrische</a> (TE) Komponente bezeichnet.

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2N_{1}\cos \alpha }{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}N_{2}\cos \beta }}={\frac {2N_{1}\cos \alpha }{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c32e0a74e6cb1530675684672f2a04e70e39bdd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:76.224ex; height:8.343ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2N_{1}\cos \alpha }{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}N_{2}\cos \beta }}={\frac {2N_{1}\cos \alpha }{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {N_{1}\cos \alpha -{\frac {\mu _{r1}}{\mu _{r2}}}N_{2}\cos \beta }{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}N_{2}\cos \beta }}={\frac {N_{1}\cos \alpha -{\frac {\mu _{r1}}{\mu _{r2}}}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d42a79a85751486b4153f0a68324843c095a84da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:76.433ex; height:10.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {N_{1}\cos \alpha -{\frac {\mu _{r1}}{\mu _{r2}}}N_{2}\cos \beta }{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}N_{2}\cos \beta }}={\frac {N_{1}\cos \alpha -{\frac {\mu _{r1}}{\mu _{r2}}}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}{N_{1}\cos \alpha +{\frac {\mu _{r1}}{\mu _{r2}}}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}" loading="lazy"></dd></dl>

Mit dem Transmissionsfaktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{s}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c13991f34d5ab3f8117ef25b53dad2dc328972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.197ex; height:2.176ex;" alt="{\displaystyle \mu _{r1}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{r2}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08761dd39effce246cbdac7e329faca2d23d0d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.197ex; height:2.176ex;" alt="{\displaystyle \mu _{r2}}" loading="lazy">. Hierbei beziehen sich die Koeffizienten auf das elektrische Feld.

<div class="mw-heading mw-heading3"><h3 id="Parallele_Polarisation">Parallele Polarisation</h3></div>

Im anderen Fall wird die Amplitude einer in der Einfallsebene linear parallel (Index: p) polarisierten Welle betrachtet. Sie wird in der Literatur auch als transversalmagnetische (TM) Komponente bezeichnet. Hierbei beziehen sich die Koeffizienten auf das magnetische Feld.

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2N_{1}\cos \alpha }{N_{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}\cos \beta }}={\frac {2N_{1}N_{2}\cos \alpha }{N_{2}^{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f200020685f68b9436f438dd0e86cb39a7a083ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:79.513ex; height:8.343ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2N_{1}\cos \alpha }{N_{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}\cos \beta }}={\frac {2N_{1}N_{2}\cos \alpha }{N_{2}^{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {N_{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha -N_{1}\cos \beta }{N_{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}\cos \beta }}={\frac {N_{2}^{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha -N_{1}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}{N_{2}^{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9abd9b8bd38e04f0a8d6e0881523a53505723fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:79.722ex; height:10.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {N_{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha -N_{1}\cos \beta }{N_{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}\cos \beta }}={\frac {N_{2}^{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha -N_{1}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}{N_{2}^{2}{\frac {\mu _{r1}}{\mu _{r2}}}\cos \alpha +N_{1}{\sqrt {N_{2}^{2}-N_{1}^{2}\sin ^{2}\alpha }}}}}" loading="lazy"></dd></dl>

Die Richtungen der elektrischen Feldvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}_{r}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/539be8d919ce9516c4aa6d8059168e825d97ac10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.749ex; height:3.176ex;" alt="{\displaystyle {\vec {E}}_{r}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}_{t}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7304aefe271e7f51da303972091ece2fd5fc103b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.602ex; height:3.176ex;" alt="{\displaystyle {\vec {E}}_{t}}" loading="lazy"> entsprechen den Richtungen der Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}_{e}\times {\vec {k}}_{r}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/049259010002f501a89fbf03f865463640540c15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.419ex; height:3.176ex;" alt="{\displaystyle {\vec {n}}_{e}\times {\vec {k}}_{r}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}_{e}\times {\vec {k}}_{t}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65c1764e3501f9aa8e6b2f92012ebab5e15c50bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.271ex; height:3.176ex;" alt="{\displaystyle {\vec {n}}_{e}\times {\vec {k}}_{t}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}_{e}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a81bcadc97b89b6638e9dd96ea106079b27ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle {\vec {n}}_{e}}" loading="lazy"> der Normalenvektor der Einfallsebene ist.

<div class="mw-heading mw-heading2"><h2 id="Spezialfall:_gleiche_magnetische_Permeabilität">Spezialfall: gleiche magnetische Permeabilität</h2></div>

Für den in der Praxis häufigen Spezialfall, dass die beteiligten Materialien näherungsweise die gleiche <a href="Permeabilit%C3%A4t_(Magnetismus)" class="mw-redirect" title="Permeabilität (Magnetismus)">magnetische Permeabilität</a> besitzen (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{r1}=\mu _{r2}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69bbc3785a89bd838394168681e5c16049febd0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.493ex; height:2.176ex;" alt="{\displaystyle \mu _{r1}=\mu _{r2}}" loading="lazy">), z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{r}=1}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/667125088cfe966ae7d351b6a9cf9c76838c381a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.636ex; height:2.676ex;" alt="{\displaystyle \mu _{r}=1}" loading="lazy"> für nicht-magnetische Materialien, vereinfachen sich die fresnelschen Formeln wie folgt:

<dl><dt>Senkrechte Polarisation (TE)</dt></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2N_{1}\cos {\alpha }}{N_{1}\cos {\alpha }+N_{2}\cos {\beta }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c91ad9b8166f7015c3709042e0d6cffff73605ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.944ex; height:6.176ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2N_{1}\cos {\alpha }}{N_{1}\cos {\alpha }+N_{2}\cos {\beta }}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {N_{1}\cos {\alpha }-N_{2}\cos {\beta }}{N_{1}\cos {\alpha }+N_{2}\cos {\beta }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7671a831b3ad7ed9b790380a3db0c96877c177dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.153ex; height:6.176ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {N_{1}\cos {\alpha }-N_{2}\cos {\beta }}{N_{1}\cos {\alpha }+N_{2}\cos {\beta }}}}" loading="lazy"></dd></dl>

<dl><dt>Parallele Polarisation (TM)</dt></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2N_{1}\cos {\alpha }}{N_{2}\cos {\alpha }+N_{1}\cos {\beta }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ffc919bd5568e5a7c1503fa506fde66f77a8473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.055ex; height:6.509ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2N_{1}\cos {\alpha }}{N_{2}\cos {\alpha }+N_{1}\cos {\beta }}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {N_{2}\cos {\alpha }-N_{1}\cos {\beta }}{N_{2}\cos {\alpha }+N_{1}\cos {\beta }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/877adda54c44f382d7ec1f2451b6129c86cf5b38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.264ex; height:6.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {N_{2}\cos {\alpha }-N_{1}\cos {\beta }}{N_{2}\cos {\alpha }+N_{1}\cos {\beta }}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Spezialfall:_dielektrische_Materialien">Spezialfall: dielektrische Materialien</h2></div>

Ein weiterer Spezialfall ergibt sich für ideale <a href="Dielektrika" class="mw-redirect" title="Dielektrika">Dielektrika</a>, bei denen der <a href="Absorptionskoeffizient" title="Absorptionskoeffizient">Absorptionskoeffizient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> des komplexen Brechungsindex gleich null ist. Das heißt, das Material auf beiden Seiten der Grenzfläche absorbiert die entsprechende elektromagnetische Strahlung nicht (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}=k_{2}=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00ca4301cb7ed43ecf2fd7b15e899fbe6b32b59d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.89ex; height:2.509ex;" alt="{\displaystyle k_{1}=k_{2}=0}" loading="lazy">). Es gilt:

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<annotation encoding="application/x-tex">{\displaystyle N_{i}=n_{i}(1+\mathrm {i} \kappa _{i})=n_{i}+\mathrm {i} k_{i}\quad {\xrightarrow[{}]{k_{i}=0}}\quad N_{i}=n_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76ea4c5807682bd685b13184b300c7746a7092b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.654ex; margin-top: -0.351ex; margin-bottom: -0.518ex; width:44.857ex; height:5.676ex;" alt="{\displaystyle N_{i}=n_{i}(1+\mathrm {i} \kappa _{i})=n_{i}+\mathrm {i} k_{i}\quad {\xrightarrow[{}]{k_{i}=0}}\quad N_{i}=n_{i}}" loading="lazy"></dd></dl>

Durch den Wegfall des Imaginärteils vereinfachen sich die fresnelschen Formeln wie folgt:<a href="#cite_note-Bass-1">[1]</a>

<dl><dt>Senkrechte Polarisation (TE)</dt></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2n_{1}\cos {\alpha }}{n_{1}\cos {\alpha }+n_{2}\cos {\beta }}}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/488cd2301af1fe390eb7ebb454b32e4a0c53fc9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.433ex; height:6.343ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2n_{1}\cos {\alpha }}{n_{1}\cos {\alpha }+n_{2}\cos {\beta }}}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {n_{1}\cos {\alpha }-n_{2}\cos {\beta }}{n_{1}\cos {\alpha }+n_{2}\cos {\beta }}}=-{\frac {\sin {(\alpha -\beta )}}{\sin {(\alpha +\beta )}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0072d8e2e32090c75358440daad750ab019717f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.665ex; height:6.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {n_{1}\cos {\alpha }-n_{2}\cos {\beta }}{n_{1}\cos {\alpha }+n_{2}\cos {\beta }}}=-{\frac {\sin {(\alpha -\beta )}}{\sin {(\alpha +\beta )}}}}" loading="lazy"></dd></dl>

<dl><dt>Parallele Polarisation (TM)</dt></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2n_{1}\cos {\alpha }}{n_{2}\cos {\alpha }+n_{1}\cos {\beta }}}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}\cos {(\alpha -\beta )}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/504c11142920c3076e28792843bf6bfd8dc44c63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:62.114ex; height:6.509ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2n_{1}\cos {\alpha }}{n_{2}\cos {\alpha }+n_{1}\cos {\beta }}}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}\cos {(\alpha -\beta )}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {n_{2}\cos {\alpha }-n_{1}\cos {\beta }}{n_{2}\cos {\alpha }+n_{1}\cos {\beta }}}={\frac {\tan {(\alpha -\beta )}}{\tan {(\alpha +\beta )}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/796e1ca8fa4363b881efa068e04e39ca8a639268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:51.472ex; height:6.676ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {n_{2}\cos {\alpha }-n_{1}\cos {\beta }}{n_{2}\cos {\alpha }+n_{1}\cos {\beta }}}={\frac {\tan {(\alpha -\beta )}}{\tan {(\alpha +\beta )}}}}" loading="lazy"></dd></dl>

Hinweis: Das jeweils dritte Gleichheitszeichen ergibt sich durch Anwenden des <a href="Brechungsgesetz" class="mw-redirect" title="Brechungsgesetz">Brechungsgesetzes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n_{1}}{n_{2}}}={\frac {\sin \beta }{\sin \alpha }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {n_{1}}{n_{2}}}={\frac {\sin \beta }{\sin \alpha }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/702ec884129c2faddcc590bb840cca0bdb5e68cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.95ex; height:5.676ex;" alt="{\displaystyle {\frac {n_{1}}{n_{2}}}={\frac {\sin \beta }{\sin \alpha }}}" loading="lazy"> und <a href="Formelsammlung_Trigonometrie#Additionstheoreme" title="Formelsammlung Trigonometrie">Additionstheoremen</a>.<a href="#cite_note-2">[2]</a> Die dabei getroffenen Annahmen sind für Einfallswinkel von 0° und 90° nicht gültig und die Formeln können daher nicht genutzt werden. Hierfür muss die ursprüngliche Form aus reinen Kosinustermen verwendet werden

<div class="mw-heading mw-heading3"><h3 id="Senkrechter_Einfall">Senkrechter Einfall</h3></div>

Eine weitere Vereinfachung ergibt sich für den Fall, dass der Einfallswinkel α gleich 0 ist (senkrechter Einfall):<a href="#cite_note-Bass-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}=-r_{p}=-\left({\frac {E_{0r}}{E_{0e}}}\right)_{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/225e603a27515c2baa55654a826f5919da292b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.393ex; height:6.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}={\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}=-r_{p}=-\left({\frac {E_{0r}}{E_{0e}}}\right)_{p}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2n_{1}}{n_{1}+n_{2}}}=t_{p}=\left({\frac {E_{0t}}{E_{0e}}}\right)_{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d3a30fcb2f853625168dae40383b2ae882df5fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.358ex; height:6.509ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2n_{1}}{n_{1}+n_{2}}}=t_{p}=\left({\frac {E_{0t}}{E_{0e}}}\right)_{p}}" loading="lazy"></dd></dl>

Fällt beispielsweise sichtbares Licht senkrecht auf die Grenzfläche Luft/<a href="Quarzglas" title="Quarzglas">Quarzglas</a>, dann wird der Anteil

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<annotation encoding="application/x-tex">{\displaystyle R=r_{s}^{2}=(-r_{p})^{2}=\left({\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right)^{2}=\left({\frac {1-1{,}46}{1+1{,}46}}\right)^{2}=0{,}035=3{,}5\,\%}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0be81e29576c493a22322872c6104ae02aa7ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.327ex; height:6.509ex;" alt="{\displaystyle R=r_{s}^{2}=(-r_{p})^{2}=\left({\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right)^{2}=\left({\frac {1-1{,}46}{1+1{,}46}}\right)^{2}=0{,}035=3{,}5\,\%}" loading="lazy"></dd></dl>

der einfallenden Intensität unabhängig von der Polarisation reflektiert (vgl. Abschnitt <a href="#Zusammenhang_mit_Reflexions-_und_Transmissionsgrad">Zusammenhang mit Reflexions- und Transmissionsgrad</a>).

<div class="mw-heading mw-heading3"><h3 id="Diskussion_der_Amplitudenverhältnisse">Diskussion der Amplitudenverhältnisse</h3></div>

Dort, wo die Amplitudenkoeffizienten reell und negativ sind, tritt ein <a href="Phasensprung" title="Phasensprung">Phasensprung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 180^{\circ }=\pi }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9408974059bef819b53b8265d1c43c4e861c0385.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.972ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }=\pi }" loading="lazy"> auf (bei reell und positiv keine Phasenänderung):

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<annotation encoding="application/x-tex">{\displaystyle r=-|r|=|r|\cdot e^{i\pi }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e38481bbd647576de57c09204fc63c7b0469ae9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.243ex; height:3.176ex;" alt="{\displaystyle r=-|r|=|r|\cdot e^{i\pi }}" loading="lazy"></dd></dl>

Das Amplitudenverhältnis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{p}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d02ec163fac8837ac757005d783b375e52808b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.108ex; height:2.343ex;" alt="{\displaystyle r_{p}}" loading="lazy"> besitzt einen Nulldurchgang am <a href="Brewster-Winkel" title="Brewster-Winkel">Brewster-Winkel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{B}}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{p}={\frac {\tan(\alpha -\beta )}{\tan(\alpha +\beta )}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0929f0ba3cd92a5cd8e9cc32a41bfde5c302c931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.132ex; height:6.509ex;" alt="{\displaystyle r_{p}={\frac {\tan(\alpha -\beta )}{\tan(\alpha +\beta )}}=0}" loading="lazy">   genau bei   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta =90{}^{\circ }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha +\beta =90{}^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/111e44c61efbb6d5d316b05458f8c2e9bf207ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.138ex; height:2.676ex;" alt="{\displaystyle \alpha +\beta =90{}^{\circ }}" loading="lazy"></dd></dl>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {\sin \alpha }{\sin \beta }}={\frac {\sin \alpha }{\sin(90{}^{\circ }-\alpha )}}={\frac {\sin \alpha }{\cos \alpha }}=\tan \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c71954f000eea65ee04568ed8924ff994f2130a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.51ex; height:6.009ex;" alt="{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {\sin \alpha }{\sin \beta }}={\frac {\sin \alpha }{\sin(90{}^{\circ }-\alpha )}}={\frac {\sin \alpha }{\cos \alpha }}=\tan \alpha }" loading="lazy">   also   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{B}}=\arctan {\frac {n_{2}}{n_{1}}}}">

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{B}}=\arctan {\frac {n_{2}}{n_{1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceaea32556ecbc2b68f25207bf2037644865d898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.12ex; height:5.009ex;" alt="{\displaystyle \alpha _{\text{B}}=\arctan {\frac {n_{2}}{n_{1}}}}" loading="lazy"></dd></dl>

Beispiele: Brewster-Winkel für Luft-Glas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1{,}5}{1}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1{,}5}{1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57e4adee1d5fe3798c7d5e385952b227f5c73d92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.69ex; height:4.009ex;" alt="{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1{,}5}{1}}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{B}}=56{,}3{}^{\circ }}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{B}}=56{,}3{}^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3552741ef680286444b4114650d60f826c42b49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.676ex;" alt="{\displaystyle \alpha _{\text{B}}=56{,}3{}^{\circ }}" loading="lazy"> und für Glas-Luft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1}{1{,}5}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1}{1{,}5}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f52cb44a845dfaffa7a70a3a3d0fd2a9f09fc9bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.69ex; height:3.843ex;" alt="{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1}{1{,}5}}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{B}}=33{,}7{}^{\circ }}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{B}}=33{,}7{}^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf8ff0c176153c68f7c69974df8ca67e658a4c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.17ex; height:2.676ex;" alt="{\displaystyle \alpha _{\text{B}}=33{,}7{}^{\circ }}" loading="lazy">.

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}<n_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7250d6bd98b516c22bc6716496e17319fc133e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.996ex; height:2.176ex;" alt="{\displaystyle n_{2}<n_{1}}" loading="lazy"> werden ab einem bestimmten Winkel die Amplitudenverhältnisse komplex. Ab diesem <a href="Kritischer_Winkel" title="Kritischer Winkel">kritischen Winkel</a> oder Grenzwinkel tritt <a href="Totalreflexion" title="Totalreflexion">Totalreflexion</a> auf. Der Grenzwinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{c}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6b9e8e6bb9100c6453325f8f93094bd2619a211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.45ex; height:2.009ex;" alt="{\displaystyle \alpha _{\text{c}}}" loading="lazy"> entspricht dem Brechungswinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =90{}^{\circ }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =90{}^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31b5305febbe460e6aef5b239090999d95ac425b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.81ex; height:2.676ex;" alt="{\displaystyle \beta =90{}^{\circ }}" loading="lazy"> also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \beta =1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0129f28e2c19ffb2be44de264f3b194505314864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.836ex; height:2.509ex;" alt="{\displaystyle \sin \beta =1}" loading="lazy">, d. h., die Welle läuft an der Grenzfläche entlang.

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {\sin \alpha }{\sin 90{}^{\circ }}}=\sin \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e03dc1e7a72e9885d94e4db30deba9311f53856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.67ex; height:5.509ex;" alt="{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {\sin \alpha }{\sin 90{}^{\circ }}}=\sin \alpha }" loading="lazy">   also   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{c}}=\arcsin {\frac {n_{2}}{n_{1}}}}">

<msub>

</mrow>

</msub>

<mi>arcsin</mi>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{c}}=\arcsin {\frac {n_{2}}{n_{1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f0d782f7fd4f1879bb0228e1d1bb1e9df8c9eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.183ex; height:5.009ex;" alt="{\displaystyle \alpha _{\text{c}}=\arcsin {\frac {n_{2}}{n_{1}}}}" loading="lazy"></dd></dl>

Beispiel: Grenzwinkel für Glas-Luft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1}{1{,}5}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n_{2}}{n_{1}}}={\tfrac {1}{1{,}5}}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{c}}=41{,}8{}^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f053da265a6fc1aefdafc46310c4576951d7e8db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.737ex; height:2.676ex;" alt="{\displaystyle \alpha _{\text{c}}=41{,}8{}^{\circ }}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Diskussion_der_Amplitudenverhältnisse_bei_Röntgenstrahlung_unter_streifendem_Einfall">Diskussion der Amplitudenverhältnisse bei Röntgenstrahlung unter streifendem Einfall</h3></div>

Für Röntgenstrahlung mit einer Energie weit weg von den Absoptionskanten des Mediums ist die Absorption vernachlässigbar und der <a href="Brechungsindex" title="Brechungsindex">Brechungsindex</a> ist reell (nicht komplex). Wenn die Strahlung in einem solchen Medium mit Brechungsindex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/840e456e3058bc0be28e5cf653b170cdbfcc3be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}" loading="lazy"> auf die Grenzfläche zu Vakuum oder Luft (mit Brechungsindex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}\approx 1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d737849072ed575329f670cfe7fa82927c54f6a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.509ex;" alt="{\displaystyle n_{1}\approx 1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}>n_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d29344ecad10202589b16130710c7885bc390f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.996ex; height:2.176ex;" alt="{\displaystyle n_{1}>n_{2}}" loading="lazy">) trifft, lautet das <a href="Snelliussches_Brechungsgesetz" title="Snelliussches Brechungsgesetz">snelliussche Brechungsgesetz</a> unter streifendem Einfall (<a href="Einfallswinkel" title="Einfallswinkel">Einfallswinkel</a>:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\tfrac {\pi }{2}}-\psi _{1},\psi _{1}\ll {\tfrac {\pi }{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ={\tfrac {\pi }{2}}-\psi _{1},\psi _{1}\ll {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae11313a3e8b603ea3fc5c312195dfa7f72d45ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.765ex; height:3.176ex;" alt="{\displaystyle \alpha ={\tfrac {\pi }{2}}-\psi _{1},\psi _{1}\ll {\tfrac {\pi }{2}}}" loading="lazy">) und streifendem Ausfall (Ausfallwinkel:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ={\tfrac {\pi }{2}}-\psi _{2},\psi _{2}\ll {\tfrac {\pi }{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta ={\tfrac {\pi }{2}}-\psi _{2},\psi _{2}\ll {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e03e7728f51ae016b5ed276d1d4ab281f1953aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.61ex; height:3.176ex;" alt="{\displaystyle \beta ={\tfrac {\pi }{2}}-\psi _{2},\psi _{2}\ll {\tfrac {\pi }{2}}}" loading="lazy">) und der Definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n_{2}}{n_{1}}}=1-\delta ,\delta >0}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n_{2}}{n_{1}}}=1-\delta ,\delta >0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8ce496fb7ee30b984c1bc8d73c6ff13a0904e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.147ex; height:3.676ex;" alt="{\displaystyle {\tfrac {n_{2}}{n_{1}}}=1-\delta ,\delta >0}" loading="lazy">.

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mtext>, sowie unter Ausnutzung von </mtext>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msqrt>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcrl}n_{1}\sin \alpha =n_{2}\sin \beta &\Rightarrow &\sin({\tfrac {\pi }{2}}-\psi _{1})&=(1-\delta )\sin({\tfrac {\pi }{2}}-\psi _{2})\quad \\&\Rightarrow &\cos \psi _{1}&=(1-\delta )\cos \psi _{2}&{\text{, sowie unter Ausnutzung von }}\psi _{1,2}\ll {\tfrac {\pi }{2}}\\&\Rightarrow &1-{\textstyle {\frac {1}{2}}}\psi _{1}^{2}&=1-\delta -{\textstyle {\frac {1}{2}}}\psi _{2}^{2}\\&\Rightarrow &\psi _{2}&={\sqrt {\psi _{1}^{2}-2\delta }}\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afba3e8621936c9407153e7bf90d247e096a0fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:105.729ex; height:16.509ex;" alt="{\displaystyle {\begin{array}{lcrl}n_{1}\sin \alpha =n_{2}\sin \beta &\Rightarrow &\sin({\tfrac {\pi }{2}}-\psi _{1})&=(1-\delta )\sin({\tfrac {\pi }{2}}-\psi _{2})\quad \\&\Rightarrow &\cos \psi _{1}&=(1-\delta )\cos \psi _{2}&{\text{, sowie unter Ausnutzung von }}\psi _{1,2}\ll {\tfrac {\pi }{2}}\\&\Rightarrow &1-{\textstyle {\frac {1}{2}}}\psi _{1}^{2}&=1-\delta -{\textstyle {\frac {1}{2}}}\psi _{2}^{2}\\&\Rightarrow &\psi _{2}&={\sqrt {\psi _{1}^{2}-2\delta }}\end{array}}}" loading="lazy">

Damit entwickelt man die Fresnelschen Formeln unter streifenden Einfall für Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</mstyle>

</mrow>

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

</semantics>

<dl><dt>Senkrechte Polarisation (TE)</dt></dl>

<msub>

<mrow>

<mfrac>

<msub>

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</mfrac>

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<mfrac>

<mrow>

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</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

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</mrow>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}}}={\frac {2\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{2})\cos({\textstyle {\frac {1}{2}}}\pi -\psi _{1})}{\sin {({\textstyle {\frac {1}{2}}}\pi -\psi _{1}+{\textstyle {\frac {1}{2}}}\pi -\psi _{2})}}}={\frac {2\cos \psi _{2}\sin \psi _{1}}{\sin(\psi _{1}+\psi _{2})}}\approx {\frac {2\psi _{1}}{\psi _{1}+\psi _{2}}}={\frac {2\psi _{1}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d20457d6a2f5df64945cd53b059d5a4d6c89173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:110.364ex; height:9.176ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{s}=t_{s}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}}}={\frac {2\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{2})\cos({\textstyle {\frac {1}{2}}}\pi -\psi _{1})}{\sin {({\textstyle {\frac {1}{2}}}\pi -\psi _{1}+{\textstyle {\frac {1}{2}}}\pi -\psi _{2})}}}={\frac {2\cos \psi _{2}\sin \psi _{1}}{\sin(\psi _{1}+\psi _{2})}}\approx {\frac {2\psi _{1}}{\psi _{1}+\psi _{2}}}={\frac {2\psi _{1}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}=-{\frac {\sin {(\alpha -\beta )}}{\sin {(\alpha +\beta )}}}=-{\frac {\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{1}-{\textstyle {\frac {1}{2}}}\pi +\psi _{2})}{\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{1}+{\textstyle {\frac {1}{2}}}\pi -\psi _{2})}}={\frac {\sin(\psi _{1}-\psi _{2})}{\sin(\psi _{1}+\psi _{2})}}\approx {\frac {\psi _{1}-\psi _{2}}{\psi _{1}+\psi _{2}}}={\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5ba1232e820f6dd21c783e44a0cbb97e7dbe4ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:108.215ex; height:10.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{s}=r_{s}=-{\frac {\sin {(\alpha -\beta )}}{\sin {(\alpha +\beta )}}}=-{\frac {\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{1}-{\textstyle {\frac {1}{2}}}\pi +\psi _{2})}{\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{1}+{\textstyle {\frac {1}{2}}}\pi -\psi _{2})}}={\frac {\sin(\psi _{1}-\psi _{2})}{\sin(\psi _{1}+\psi _{2})}}\approx {\frac {\psi _{1}-\psi _{2}}{\psi _{1}+\psi _{2}}}={\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}}" loading="lazy"></dd></dl>

Bemerkung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi -\psi _{1}+\psi _{2})=-\sin(\psi _{1}-\psi _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \sin(\pi -\psi _{1}+\psi _{2})=-\sin(\psi _{1}-\psi _{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f18902aee5ae997c0478e09693478b327b8555b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.746ex; height:2.843ex;" alt="{\displaystyle \sin(\pi -\psi _{1}+\psi _{2})=-\sin(\psi _{1}-\psi _{2})}" loading="lazy">

<dl><dt>Parallele Polarisation (TM)</dt></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}\cos {(\alpha -\beta )}}}={\frac {2\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{2})\cos({\textstyle {\frac {1}{2}}}\pi -\psi _{1})}{\sin {(\pi -\psi _{1}-\psi _{2})}\cos {(-\psi _{1}+\psi _{2})}}}\approx {\frac {2\cos \psi _{2}\sin \psi _{1}}{\sin(\psi _{1}+\psi _{2})}}\approx {\frac {2\psi _{1}}{\psi _{1}+\psi _{2}}}={\frac {2\psi _{1}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}=t_{s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e65b4d3093148ed45b814fb94536e64ec31d0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:130.346ex; height:9.176ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)_{p}=t_{p}={\frac {2\sin {\beta }\cos {\alpha }}{\sin {(\alpha +\beta )}\cos {(\alpha -\beta )}}}={\frac {2\sin({\textstyle {\frac {1}{2}}}\pi -\psi _{2})\cos({\textstyle {\frac {1}{2}}}\pi -\psi _{1})}{\sin {(\pi -\psi _{1}-\psi _{2})}\cos {(-\psi _{1}+\psi _{2})}}}\approx {\frac {2\cos \psi _{2}\sin \psi _{1}}{\sin(\psi _{1}+\psi _{2})}}\approx {\frac {2\psi _{1}}{\psi _{1}+\psi _{2}}}={\frac {2\psi _{1}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}=t_{s}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {\tan {(\alpha -\beta )}}{\tan {(\alpha +\beta )}}}={\frac {\tan {(-\psi _{1}+\psi _{2})}}{\tan {(\pi -\psi _{1}-\psi _{2})}}}={\frac {-\tan {(\psi _{1}-\psi _{2})}}{-\tan {(\psi _{1}+\psi _{2})}}}\approx {\frac {\psi _{1}-\psi _{2}}{\psi _{1}+\psi _{2}}}={\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}=r_{s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/052f3e8a7056e233e00b4c1ec109e4553c8ff5c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:106.854ex; height:10.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)_{p}=r_{p}={\frac {\tan {(\alpha -\beta )}}{\tan {(\alpha +\beta )}}}={\frac {\tan {(-\psi _{1}+\psi _{2})}}{\tan {(\pi -\psi _{1}-\psi _{2})}}}={\frac {-\tan {(\psi _{1}-\psi _{2})}}{-\tan {(\psi _{1}+\psi _{2})}}}\approx {\frac {\psi _{1}-\psi _{2}}{\psi _{1}+\psi _{2}}}={\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}=r_{s}}" loading="lazy"></dd></dl>

Bemerkung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {({\textstyle {\frac {1}{2}}}\pi -\psi _{1}-{\textstyle {\frac {1}{2}}}\pi +\psi _{2})}=\cos {(\psi _{2}-\psi _{1})}\approx 1}">

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<annotation encoding="application/x-tex">{\displaystyle \cos {({\textstyle {\frac {1}{2}}}\pi -\psi _{1}-{\textstyle {\frac {1}{2}}}\pi +\psi _{2})}=\cos {(\psi _{2}-\psi _{1})}\approx 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fecfa91b325db89e5095d00f864c812f717c014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.586ex; height:3.509ex;" alt="{\displaystyle \cos {({\textstyle {\frac {1}{2}}}\pi -\psi _{1}-{\textstyle {\frac {1}{2}}}\pi +\psi _{2})}=\cos {(\psi _{2}-\psi _{1})}\approx 1}" loading="lazy">.

Die Fresnelschen Formeln stimmen sowohl für senkrechte als auch parallele Polarisation überein. Man braucht die Polarisationsrichtung für Röntgenstrahlung unter streifendem Einfall nicht berücksichtigen!

Für genügend hohe Einfallswinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}>\psi _{g}={\sqrt {2\delta }}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1}>\psi _{g}={\sqrt {2\delta }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62a78d095c2b17f9e73e83b37faf9e4d816ba01e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.446ex; height:3.343ex;" alt="{\displaystyle \psi _{1}>\psi _{g}={\sqrt {2\delta }}}" loading="lazy"> gilt:

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<annotation encoding="application/x-tex">{\displaystyle r_{s}=r_{p}={\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}={\frac {1-{\sqrt {1-2\delta /\psi _{1}^{2}}}}{1+{\sqrt {1-2\delta /\psi _{1}^{2}}}}}\approx {\frac {1-(1-\delta /\psi _{1}^{2})}{1+(1-\delta /\psi _{1}^{2})}}\approx {\frac {\delta }{2\psi _{1}^{2}}}={\frac {\psi _{g}^{2}}{4\psi _{1}^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebbc94e443542d5c7a66f3d6d657b405ef220d45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:80.605ex; height:10.509ex;" alt="{\displaystyle r_{s}=r_{p}={\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}={\frac {1-{\sqrt {1-2\delta /\psi _{1}^{2}}}}{1+{\sqrt {1-2\delta /\psi _{1}^{2}}}}}\approx {\frac {1-(1-\delta /\psi _{1}^{2})}{1+(1-\delta /\psi _{1}^{2})}}\approx {\frac {\delta }{2\psi _{1}^{2}}}={\frac {\psi _{g}^{2}}{4\psi _{1}^{2}}}}" loading="lazy"></dd></dl>

Beim kritischen Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}=\psi _{g}={\sqrt {2\delta }}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1}=\psi _{g}={\sqrt {2\delta }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f9eca727b72f0d8ded31c7f439d1ba746417cc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.446ex; height:3.343ex;" alt="{\displaystyle \psi _{1}=\psi _{g}={\sqrt {2\delta }}}" loading="lazy"> ist mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{p}=t_{s}=2}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc06b0eaf6e774139d88c3414266aa1a2d41969d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.101ex; height:2.843ex;" alt="{\displaystyle t_{p}=t_{s}=2}" loading="lazy"> die Amplitude der elektrischen Feldstärke im Medium doppelt so hoch, wie die einfallende Amplitude. Die Intensität vervierfacht sich. Dies lässt sich durch die Entstehung einer stehenden Welle an der Grenzfläche verstehen. Gemäß der Wikipedia-Seite zum <a href="Snelliussches_Brechungsgesetz" title="Snelliussches Brechungsgesetz">Snelliusschen Brechungsgesetz</a> überlagern sich an der Grenzfläche die einfallende Welle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}_{e}(t)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5024f8a73903e45a355e69a70238fe7a64f66c83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.343ex;" alt="{\displaystyle {\vec {E}}_{e}(t)}" loading="lazy"> und die reflektierte Welle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}_{r}(t)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82904ea7eb102c529cea7c9753c74c6396665143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.398ex; height:3.343ex;" alt="{\displaystyle {\vec {E}}_{r}(t)}" loading="lazy">

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</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</mrow>

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<msup>

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<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

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<mover>

</mover>

</mrow>

</mrow>

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</mtd>

<mtd>

<msub>

<mover>

</mover>

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</mfrac>

</mrow>

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {E}}_{e}(t)&={\vec {E}}_{e}{\text{e}}^{{\text{i}}[n_{1}{\frac {\omega }{c}}(x\cos \delta _{1}-y\sin \delta _{1})-\omega t]}\approx {\vec {E}}_{e}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(x\psi _{1}-y)-\omega t]}\\{\vec {E}}_{r}(t)&={\vec {E}}_{r}{\text{e}}^{{\text{i}}[n_{1}{\frac {\omega }{c}}(-x\cos \delta _{1}-y\sin \delta _{1})-\omega t]}\approx {\vec {E}}_{r}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(-x\psi _{1}-y)-\omega t]}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0a711062de4800363cdedb8d5628d3157b902e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.063ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\vec {E}}_{e}(t)&={\vec {E}}_{e}{\text{e}}^{{\text{i}}[n_{1}{\frac {\omega }{c}}(x\cos \delta _{1}-y\sin \delta _{1})-\omega t]}\approx {\vec {E}}_{e}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(x\psi _{1}-y)-\omega t]}\\{\vec {E}}_{r}(t)&={\vec {E}}_{r}{\text{e}}^{{\text{i}}[n_{1}{\frac {\omega }{c}}(-x\cos \delta _{1}-y\sin \delta _{1})-\omega t]}\approx {\vec {E}}_{r}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(-x\psi _{1}-y)-\omega t]}\end{aligned}}}" loading="lazy"></dd></dl>

zu

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)={\vec {E}}_{e}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(x\psi _{1}-y)-\omega t]}+{\vec {E}}_{r}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(-x\psi _{1}-y)-\omega t]}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e3a66340f7acfc37a0a8fce0bad7456ea4a7363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.231ex; height:3.509ex;" alt="{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)={\vec {E}}_{e}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(x\psi _{1}-y)-\omega t]}+{\vec {E}}_{r}{\text{e}}^{{\text{i}}[{\frac {\omega }{c}}(-x\psi _{1}-y)-\omega t]}}" loading="lazy"></dd></dl>

Ohne Einschränkung gilt für senkrechte Polarisation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}_{e}=E_{0e}{\vec {e}}_{z}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}_{e}=E_{0e}{\vec {e}}_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc6c2aed312a915025b4b6034c286b1a142bda8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.633ex; height:3.176ex;" alt="{\displaystyle {\vec {E}}_{e}=E_{0e}{\vec {e}}_{z}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}_{r}=E_{0r}{\vec {e}}_{z}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}_{r}=E_{0r}{\vec {e}}_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00f1911f861b49ee34deb49e834dc0c819e46cc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.584ex; height:3.176ex;" alt="{\displaystyle {\vec {E}}_{r}=E_{0r}{\vec {e}}_{z}}" loading="lazy">

<msub>

<mover>

</mover>

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</mrow>

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<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

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<msub>

<mover>

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</mrow>

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<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

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<mfrac>

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<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mfrac>

<mrow>

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</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

<mrow>

<msub>

</mrow>

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<msqrt>

</mrow>

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</msubsup>

</msqrt>

</mrow>

</mfrac>

</mrow>

<msup>

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</mrow>

<mfrac>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left[{\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}+\left({\frac {E_{0r}}{E_{0e}}}\right)_{s}{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right]=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left({\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}+{\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63509c0977605e2001b830ce6b081d1b665d6307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:115.402ex; height:10.509ex;" alt="{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left[{\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}+\left({\frac {E_{0r}}{E_{0e}}}\right)_{s}{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right]=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left({\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}+{\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right)}" loading="lazy"></dd></dl>

Für Einfallswinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cfdde1da54e02a016fe2a230c58b25dfcc014d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}" loading="lazy"> beim kritischen Winkel der Totalreflexion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}=\psi _{g}={\sqrt {2\delta }}}">

<msub>

</mrow>

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<msub>

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</msub>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{1}=\psi _{g}={\sqrt {2\delta }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f9eca727b72f0d8ded31c7f439d1ba746417cc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.446ex; height:3.343ex;" alt="{\displaystyle \psi _{1}=\psi _{g}={\sqrt {2\delta }}}" loading="lazy"> überlagern sich das einfallende und reflektierte Feld konstruktiv zu:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/213e0470b225b17632e4a4f9a8834a4efaf12596.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:84.968ex; height:4.843ex;" alt="{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left({\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}+{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\cdot 2\cos \left({\frac {\omega }{c}}x\psi _{g}\right)}" loading="lazy"></dd></dl>

und es bildet sich bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> ein Wellenbauch mit doppelter Amplitude und vierfacher Intensität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim \cos ^{2}\varphi \sim \cos 2\varphi }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d81aa583113f36d407acb46a0aa0620050e16b25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.805ex; height:3.176ex;" alt="{\displaystyle \sim \cos ^{2}\varphi \sim \cos 2\varphi }" loading="lazy">. Der nächste Wellenbauch vor der Grenzfläche entsteht falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\varphi =2{\frac {\omega }{c}}x\psi _{g}=\pi }">

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\varphi =2{\frac {\omega }{c}}x\psi _{g}=\pi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/245970d4a0112eae5fc04c750ef0f3d1862a43d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.52ex; height:4.676ex;" alt="{\displaystyle 2\varphi =2{\frac {\omega }{c}}x\psi _{g}=\pi }" loading="lazy"> und damit bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {\pi c}{2\omega \psi _{g}}}={\frac {\lambda }{4\psi _{g}}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\frac {\pi c}{2\omega \psi _{g}}}={\frac {\lambda }{4\psi _{g}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66cb92393ce5775c01cfbf1be16fae3319e3734b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.039ex; height:6.176ex;" alt="{\displaystyle x={\frac {\pi c}{2\omega \psi _{g}}}={\frac {\lambda }{4\psi _{g}}}}" loading="lazy"> mit der Kreisfrequenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi {\frac {c}{\lambda }}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =2\pi {\frac {c}{\lambda }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46efc103900b4f9d8d59c94c25b279812796b5b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.23ex; height:4.843ex;" alt="{\displaystyle \omega =2\pi {\frac {c}{\lambda }}}" loading="lazy">.

Im Grenzfall kleiner Einfallswinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq \psi _{1}\ll \psi _{g}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\neq \psi _{1}\ll \psi _{g}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43eee6b00a1b3738846feaa6385271e6731820d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.977ex; height:2.843ex;" alt="{\displaystyle 0\neq \psi _{1}\ll \psi _{g}}" loading="lazy"> geht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}\rightarrow -1}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}\rightarrow -1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b81ec596c95de18293fad1ed3dbc87390e955ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.771ex; height:10.509ex;" alt="{\displaystyle {\frac {\psi _{1}-{\sqrt {\psi _{1}^{2}-2\delta }}}{\psi _{1}+{\sqrt {\psi _{1}^{2}-2\delta }}}}\rightarrow -1}" loading="lazy"> und das einfallende und reflektierte Feld löschen sich aus:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left({\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}-{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\cdot 2{\text{i}}\sin \left({\frac {\omega }{c}}x\psi _{1}\right)\approx E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\cdot 2{\text{i}}\left({\frac {\omega }{c}}x\psi _{1}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bab5436ba59b3377322e8addb1171cf71b8ddaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:116.375ex; height:4.843ex;" alt="{\displaystyle {\vec {E}}_{e}(t)+{\vec {E}}_{r}(t)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\left({\text{e}}^{{\text{i}}{\frac {\omega }{c}}x\psi _{1}}-{\text{e}}^{-{\text{i}}{\frac {\omega }{c}}x\psi _{1}}\right)=E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\cdot 2{\text{i}}\sin \left({\frac {\omega }{c}}x\psi _{1}\right)\approx E_{0e}{\vec {e}}_{z}{\text{e}}^{-{\text{i}}[{\frac {\omega }{c}}y+\omega t]}\cdot 2{\text{i}}\left({\frac {\omega }{c}}x\psi _{1}\right)}" loading="lazy"></dd></dl>

Es bildet sich bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}=0}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d41cebee4bcc27762855da7e021d7f8af983d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.828ex; height:2.509ex;" alt="{\displaystyle \psi _{1}=0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}=\psi _{g}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27b27e75709e4ebcba26277af4e4b043d3c562c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.2ex; height:2.843ex;" alt="{\displaystyle \psi _{1}=\psi _{g}}" loading="lazy">.

<dl><dt>Totalreflexion für senkrechte Polarisation (TE) und parallele Polarisation (TM)</dt></dl>

Für Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cfdde1da54e02a016fe2a230c58b25dfcc014d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}" loading="lazy"> geringer als der Grenzwinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{g}={\sqrt {2\delta }}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{g}={\sqrt {2\delta }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17a7d6ac854527c997978b18215f0c793c190168.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.78ex; height:3.343ex;" alt="{\displaystyle \psi _{g}={\sqrt {2\delta }}}" loading="lazy"> tritt für Röntgenstrahlung gemäß der Wikipedia-Seite <a href="Totalreflexion" title="Totalreflexion">Totalreflexion</a> auf! Die Quadratwurzel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\psi _{1}^{2}-2\delta }}={\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\psi _{1}^{2}-2\delta }}={\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffa57ca2d5ae9e5346bec16219397ab58cac6f2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.631ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\psi _{1}^{2}-2\delta }}={\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}" loading="lazy"> wird komplex.

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)=t={\frac {2\psi _{1}}{\psi _{1}+{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}\quad \Rightarrow \quad T=|t|^{2}={\frac {2\psi _{1}}{\psi _{1}+{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}{\frac {2\psi _{1}}{\psi _{1}-{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}={\frac {4\psi _{1}^{2}}{\psi _{1}^{2}+2\delta -\psi _{1}^{2}}}=4\cdot {\frac {\psi _{1}^{2}}{\psi _{g}^{2}}}\quad \Rightarrow \quad |t|\sim \psi _{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2687189bf95f0e25f4bf384be1fe2f523ec4af0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:126.973ex; height:8.676ex;" alt="{\displaystyle \left({\frac {E_{0t}}{E_{0e}}}\right)=t={\frac {2\psi _{1}}{\psi _{1}+{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}\quad \Rightarrow \quad T=|t|^{2}={\frac {2\psi _{1}}{\psi _{1}+{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}{\frac {2\psi _{1}}{\psi _{1}-{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}={\frac {4\psi _{1}^{2}}{\psi _{1}^{2}+2\delta -\psi _{1}^{2}}}=4\cdot {\frac {\psi _{1}^{2}}{\psi _{g}^{2}}}\quad \Rightarrow \quad |t|\sim \psi _{1}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)=r={\frac {\psi _{1}-{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}{\psi _{1}+{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}\quad \Rightarrow \quad R=|r|^{2}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9730beaf3cb1fe229ebb5267f99afa036dac45cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:52.651ex; height:10.509ex;" alt="{\displaystyle \left({\frac {E_{0r}}{E_{0e}}}\right)=r={\frac {\psi _{1}-{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}{\psi _{1}+{\text{i}}{\sqrt {2\delta -\psi _{1}^{2}}}}}\quad \Rightarrow \quad R=|r|^{2}=1}" loading="lazy"></dd></dl>

Der Betrag der transmittierten Amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|}">

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa9b1439497e4de838a6b1bcf724ef7a8fe48147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.133ex; height:2.843ex;" alt="{\displaystyle |t|}" loading="lazy"> nimmt linear mit dem Einfallswinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}">

<msub>

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

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Unterhalb des Grenzwinkels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{g}}">

<msub>

</mrow>

</msub>

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4edc7735beaeb2ceb700dd4f336133cf6b72576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.535ex; height:2.843ex;" alt="{\displaystyle \psi _{g}}" loading="lazy"> ist das Reflexionsvermögen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=1}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/459c50749e4be896d8474f0cea4ebc26108c7024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R=1}" loading="lazy">. Der totalreflektierende Spiegel ohne Absorption ist also ein idealer Spiegel! Aber auch oberhalb reflektiert der Spiegel mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{s}={\frac {\delta }{2\psi _{1}^{2}}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle r_{s}={\frac {\delta }{2\psi _{1}^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a725e9e68bc879b0c64bc1faa37fe2730239a3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.716ex; height:6.343ex;" alt="{\displaystyle r_{s}={\frac {\delta }{2\psi _{1}^{2}}}}" loading="lazy"> noch und zwar umso mehr je größer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

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

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</math><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"> ist. Die Absorption der Strahlung im Spiegelmaterial reduziert das Reflexionsvermögen.

Auf der Totalreflexion der Röntgenstrahlung an Materie beruhen eine Reihe von Anwendungen:

<ul><li>Röntgenstrahlen lassen sich mithilfe gekrümmter Spiegel fokussieren. Das ist eine umso interessantere Möglichkeit, als es keine Linsen für Röntgenstrahlung gibt. Das <a href="Wolter-Teleskop" title="Wolter-Teleskop">Wolter-Teleskop</a> ist solch ein Röntgenteleskop<a href="#cite_note-3">[3]</a>.</li></ul>

<ul><li>Die Totalreflexion kann dazu benutzt werden um den Brechungsindex von Materie im Röntgenbereich zu bestimmen.</li></ul>

<ul><li><a href="R%C3%B6ntgenbeugung" title="Röntgenbeugung">Röntgenbeugung</a> und Röntgenabsorption werden bei Totalreflexion inhärent oberflächenempfindlich.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_mit_Reflexions-_und_Transmissionsgrad">Zusammenhang mit Reflexions- und Transmissionsgrad</h2></div>

Man betrachte ein <a href="Strahlenb%C3%BCndel" title="Strahlenbündel">Strahlenbündel</a>, das auf die Grenzfläche eines isotropen, nicht-magnetischen Materials der Fläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> einfällt. Die Strahlquerschnitte des einfallenden, reflektierten bzw. transmittierten Strahls sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cos \alpha }">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45534d449461fdca6020bcb9c0f915f552b74ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.116ex; height:2.176ex;" alt="{\displaystyle A\cos \alpha }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cos \alpha }">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45534d449461fdca6020bcb9c0f915f552b74ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.116ex; height:2.176ex;" alt="{\displaystyle A\cos \alpha }" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cos \beta }">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69981bd010458188501b04f6647f6f36f75293aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.96ex; height:2.509ex;" alt="{\displaystyle A\cos \beta }" loading="lazy">. Die Energie, die während einer Zeitspanne durch eine Fläche fließt, deren Normale parallel zur Energieflussrichtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {S}}}">

<mover>

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</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c71a6b104c40975c738d5f0e22d445ebd509eb81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.538ex; height:3.009ex;" alt="{\displaystyle {\vec {S}}}" loading="lazy"> (bei isotropen Medien gleich Ausbreitungsrichtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}">

<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ccd4b98d198d6538010ae815ee1199baabd3493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}" loading="lazy">) steht, ist gegeben durch den komplexen <a href="Poynting-Vektor" title="Poynting-Vektor">Poynting-Vektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\underline {S}}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\underline {S}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271b84849ead596452dc3a15a0ee6af6b7b4e4ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-left: -0.048ex; margin-bottom: -0.776ex; width:1.549ex; height:4.009ex;" alt="{\displaystyle {\vec {\underline {S}}}}" loading="lazy">:<a href="#cite_note-Damask-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\underline {S}}}={\vec {\underline {E}}}\times {\vec {{\underline {H}}^{*}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/153269dbc276aba5215181630779a4dda7847381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-left: -0.048ex; margin-bottom: -0.776ex; width:12.383ex; height:4.843ex;" alt="{\displaystyle {\vec {\underline {S}}}={\vec {\underline {E}}}\times {\vec {{\underline {H}}^{*}}}}" loading="lazy"></dd></dl>

Die mittlere <a href="Energieflussdichte" class="mw-redirect" title="Energieflussdichte">Energieflussdichte</a> erhält man durch zeitliche <a href="Mittelwert" title="Mittelwert">Mittelwertbildung</a><a href="#cite_note-Damask-4">[4]</a> und einige Umformungen:

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=\left\langle S\right\rangle ={\frac {1}{2}}\Re \left\lbrace {\vec {\underline {E}}}\times {\vec {{\underline {H}}^{*}}}\right\rbrace ={\frac {1}{2}}\Re \left\lbrace {\sqrt {\frac {\underline {\varepsilon }}{\underline {\mu }}}}{\vec {\underline {E}}}\times {\vec {{\underline {E}}^{*}}}\right\rbrace ={\frac {1}{2}}\Re \left\lbrace {\sqrt {\frac {\underline {\varepsilon }}{\underline {\mu }}}}\right\rbrace \left|E_{0}\right|^{2}={\frac {\varepsilon _{0}c}{2}}\Re \left\lbrace N\right\rbrace \left|E_{0}\right|^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b4c0e81560a7b05590415da6c23178fb903bd65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:90.654ex; height:8.509ex;" alt="{\displaystyle I=\left\langle S\right\rangle ={\frac {1}{2}}\Re \left\lbrace {\vec {\underline {E}}}\times {\vec {{\underline {H}}^{*}}}\right\rbrace ={\frac {1}{2}}\Re \left\lbrace {\sqrt {\frac {\underline {\varepsilon }}{\underline {\mu }}}}{\vec {\underline {E}}}\times {\vec {{\underline {E}}^{*}}}\right\rbrace ={\frac {1}{2}}\Re \left\lbrace {\sqrt {\frac {\underline {\varepsilon }}{\underline {\mu }}}}\right\rbrace \left|E_{0}\right|^{2}={\frac {\varepsilon _{0}c}{2}}\Re \left\lbrace N\right\rbrace \left|E_{0}\right|^{2}}" loading="lazy"></dd></dl>

Die mittlere Energie, die pro Zeitspanne vom Strahlenbündel transportiert wird (mittlere Leistung, die auf Fläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{e}A\,\cos \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/997811117240d9281e47026fee011b557df4607b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.525ex; height:2.509ex;" alt="{\displaystyle I_{e}A\,\cos \alpha }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{r}A\,\cos \alpha }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{r}A\,\cos \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b429d1a640369117642d4cf1b841528e80dafc3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.5ex; height:2.509ex;" alt="{\displaystyle I_{r}A\,\cos \alpha }" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{t}A\,\cos \beta }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{t}A\,\cos \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d22de53504effd1c9a6bcc0dd90753953b299f7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.197ex; height:2.509ex;" alt="{\displaystyle I_{t}A\,\cos \beta }" loading="lazy">.</dd></dl>

Allgemein (unpolarisiertes Licht) wird der <a href="Reflexionsgrad" title="Reflexionsgrad">Reflexionsgrad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> (oft auch mit ρ bezeichnet) folgendermaßen definiert:

<mfrac>

<mtext>reflektierte Leistung</mtext>

<mtext>eingestrahlte Leistung</mtext>

</mfrac>

</mrow>

<mfrac>

<msub>

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</msub>

<msub>

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<mrow>

<mfrac>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<mrow>

<msub>

</munder>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</munder>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R={\frac {\text{reflektierte Leistung}}{\text{eingestrahlte Leistung}}}={\frac {P_{r}}{P_{e}}}=\left|{\frac {A\left\langle {\vec {S}}_{r}\right\rangle \cdot {\vec {n}}}{A\left\langle {\vec {S}}_{e}\right\rangle \cdot {\vec {n}}}}\right|=\left|{\frac {I_{r}A\cos \alpha }{I_{e}A\cos \alpha }}\right|=\left|{\frac {\Re \left\lbrace {\underline {N}}_{1}\right\rbrace \cos \alpha }{\Re \left\lbrace {\underline {N}}_{1}\right\rbrace \cos \alpha }}\right|\cdot \left|{\frac {E_{0r}}{E_{0e}}}\right|^{2}=\left|{\frac {E_{0r}}{E_{0e}}}\right|^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fc134dcfdf881dc1461c59208c0434743a3e724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:101.092ex; height:10.176ex;" alt="{\displaystyle R={\frac {\text{reflektierte Leistung}}{\text{eingestrahlte Leistung}}}={\frac {P_{r}}{P_{e}}}=\left|{\frac {A\left\langle {\vec {S}}_{r}\right\rangle \cdot {\vec {n}}}{A\left\langle {\vec {S}}_{e}\right\rangle \cdot {\vec {n}}}}\right|=\left|{\frac {I_{r}A\cos \alpha }{I_{e}A\cos \alpha }}\right|=\left|{\frac {\Re \left\lbrace {\underline {N}}_{1}\right\rbrace \cos \alpha }{\Re \left\lbrace {\underline {N}}_{1}\right\rbrace \cos \alpha }}\right|\cdot \left|{\frac {E_{0r}}{E_{0e}}}\right|^{2}=\left|{\frac {E_{0r}}{E_{0e}}}\right|^{2}}" loading="lazy"></dd></dl>

und als <a href="Transmissionsgrad" class="mw-redirect" title="Transmissionsgrad">Transmissionsgrad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> (oft auch mit τ bezeichnet):

<mfrac>

<mtext>transmittierte Leistung</mtext>

<mtext>eingestrahlte Leistung</mtext>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<mrow>

<msub>

</munder>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</munder>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T={\frac {\text{transmittierte Leistung}}{\text{eingestrahlte Leistung}}}={\frac {P_{t}}{P_{e}}}=\left|{\frac {A\left\langle {\vec {S}}_{t}\right\rangle \cdot {\vec {n}}}{A\left\langle {\vec {S}}_{e}\right\rangle \cdot {\vec {n}}}}\right|=\left|{\frac {I_{t}A\cos \beta }{I_{e}A\cos \alpha }}\right|=\left|{\frac {\Re \left\lbrace {\underline {N}}_{2}\right\rbrace \cos \beta }{\Re \left\lbrace {\underline {N}}_{1}\right\rbrace \cos \alpha }}\right|\cdot \left|{\frac {E_{0t}}{E_{0e}}}\right|^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb4abcb73515079f10cf1edff994d5f24a8c80d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:92.315ex; height:10.176ex;" alt="{\displaystyle T={\frac {\text{transmittierte Leistung}}{\text{eingestrahlte Leistung}}}={\frac {P_{t}}{P_{e}}}=\left|{\frac {A\left\langle {\vec {S}}_{t}\right\rangle \cdot {\vec {n}}}{A\left\langle {\vec {S}}_{e}\right\rangle \cdot {\vec {n}}}}\right|=\left|{\frac {I_{t}A\cos \beta }{I_{e}A\cos \alpha }}\right|=\left|{\frac {\Re \left\lbrace {\underline {N}}_{2}\right\rbrace \cos \beta }{\Re \left\lbrace {\underline {N}}_{1}\right\rbrace \cos \alpha }}\right|\cdot \left|{\frac {E_{0t}}{E_{0e}}}\right|^{2}}" loading="lazy"></dd></dl>

Die beiden Werte lassen sich nun mit Hilfe der fresnelschen Formeln berechnen, sie sind das Produkt des entsprechenden Reflexions- bzw. Transmissionsfaktors mit dessen <a href="Komplexe_Konjugation" title="Komplexe Konjugation">konjugiert komplexem</a> Wert.

<msub>

</mrow>

</msub>

<msup>

<mrow>

<msub>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{i}=\left|\left({\frac {E_{0r}}{E_{0e}}}\right)_{i}\right|^{2}=r_{i}\cdot {\bar {r}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9730e7e9badbfab32ee29be9b48831d48171cd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.32ex; height:6.676ex;" alt="{\displaystyle R_{i}=\left|\left({\frac {E_{0r}}{E_{0e}}}\right)_{i}\right|^{2}=r_{i}\cdot {\bar {r}}_{i}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

<mrow>

<msub>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{i}=\left|\Re {\biggl (}{\frac {\left\lbrace N_{2}\right\rbrace \cos \beta }{\left\lbrace N_{1}\right\rbrace \cos \alpha }}{\Biggr )}\right|\cdot \left|\left({\frac {E_{0t}}{E_{0e}}}\right)_{i}\right|^{2}=\left|\Re {\biggl (}{\frac {\left\lbrace N_{2}\right\rbrace \cos \beta }{\left\lbrace N_{1}\right\rbrace \cos \alpha }}{\Biggr )}\right|t_{i}\cdot {\bar {t}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a293b71bfdb68968a2952325548fefee927d5f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.282ex; height:7.509ex;" alt="{\displaystyle T_{i}=\left|\Re {\biggl (}{\frac {\left\lbrace N_{2}\right\rbrace \cos \beta }{\left\lbrace N_{1}\right\rbrace \cos \alpha }}{\Biggr )}\right|\cdot \left|\left({\frac {E_{0t}}{E_{0e}}}\right)_{i}\right|^{2}=\left|\Re {\biggl (}{\frac {\left\lbrace N_{2}\right\rbrace \cos \beta }{\left\lbrace N_{1}\right\rbrace \cos \alpha }}{\Biggr )}\right|t_{i}\cdot {\bar {t}}_{i}}" loading="lazy"></dd></dl>

Für ideale Dielektrika, die keine Absorption und daher nur reellwertige Brechungsindizes aufweisen, vereinfachen sich die Gleichungen zu:

<msub>

</mrow>

</msub>

<msup>

<mrow>

<msub>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{i}=\left|\left({\frac {E_{0r}}{E_{0e}}}\right)_{i}\right|^{2}=r_{i}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6de11f65dd2ca3f548c18ddffc8135859104b4d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.804ex; height:6.676ex;" alt="{\displaystyle R_{i}=\left|\left({\frac {E_{0r}}{E_{0e}}}\right)_{i}\right|^{2}=r_{i}^{2}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<msub>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{i}={\frac {n_{2}}{n_{1}}}{\frac {\cos \beta }{\cos \alpha }}\left|\left({\frac {E_{0t}}{E_{0e}}}\right)_{i}\right|^{2}={\frac {n_{2}}{n_{1}}}{\frac {\cos \beta }{\cos \alpha }}t_{i}^{2}={\frac {\tan \alpha }{\tan \beta }}t_{i}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d9901789052f30d988cb15c287351bd117cff1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.465ex; height:6.676ex;" alt="{\displaystyle T_{i}={\frac {n_{2}}{n_{1}}}{\frac {\cos \beta }{\cos \alpha }}\left|\left({\frac {E_{0t}}{E_{0e}}}\right)_{i}\right|^{2}={\frac {n_{2}}{n_{1}}}{\frac {\cos \beta }{\cos \alpha }}t_{i}^{2}={\frac {\tan \alpha }{\tan \beta }}t_{i}^{2}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> für die s- bzw. p-polarisierte Komponente.

Darüber hinaus sind der Reflexions- und Transmissionsgrad über folgende allgemeine Energiestrombilanz an einer Grenzfläche (keine Absorption, d. h. <a href="Absorptionsgrad" title="Absorptionsgrad">Absorptionsgrad</a> ist null) miteinander verknüpft:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b5270a5277bd5b65417caaf8d25568d431f0a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.822ex; height:2.509ex;" alt="{\displaystyle T_{i}+R_{i}=1}" loading="lazy">.</dd></dl>

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

<ul><li><a href="Wolfgang_Nolting_(Physiker)" title="Wolfgang Nolting (Physiker)">Wolfgang Nolting</a>: Grundkurs Theoretische Physik 3: Elektrodynamik. 7. Auflage. Springer, Berlin 2002, ISBN 3-540-20509-8.</li>

<li><a href="Wolfgang_Demtr%C3%B6der" title="Wolfgang Demtröder">Wolfgang Demtröder</a>: Experimentalphysik 2. Springer, Berlin 2004, ISBN 3-540-20210-2.</li>

<li><a href="John_David_Jackson_(Physiker)" title="John David Jackson (Physiker)">John David Jackson</a>: Klassische Elektrodynamik. de Gruyter, Berlin 2006, ISBN 3-11-018970-4.</li>

<li><a href="Karl_J._Ebeling" class="mw-redirect" title="Karl J. Ebeling">Karl J. Ebeling</a>: Integrierte Optoelektronik: Wellenleiteroptik, Photonik, Halbleiter. 2. Auflage, Springer. Berlin 1998, ISBN 3-540-54655-3.</li></ul>

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

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Fresnel_equations?uselang=de">Commons: Fresnelsche Formeln</a> – Sammlung von Bildern, Videos und Audiodateien</div>

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

<li id="cite_note-Bass-1">↑ <a href="#cite_ref-Bass_1-0">a</a> <a href="#cite_ref-Bass_1-1">b</a> vgl. M. Bass (Hrsg.): <cite style="font-style:italic">Handbook of Optics. Volume I - Geometrical and Physical Optics, Polarized Light, Components and Instruments</cite>. 3. Auflage. McGraw-Hill Professional Publishing, 2009, ISBN 978-0-07-162925-6, S. 12.6–12.9.<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:Fresnelsche+Formeln&rft.btitle=Handbook+of+Optics.+Volume+I+-+Geometrical+and+Physical+Optics%2C+Polarized+Light%2C+Components+and+Instruments&rft.date=2009&rft.edition=3.&rft.genre=book&rft.isbn=9780071629256&rft.pages=12.6-12.9&rft.pub=McGraw-Hill+Professional+Publishing" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Eugene Hecht: <cite style="font-style:italic">Schaum’s outline of theory and problems of optics</cite>. McGraw-Hill Professional, 1975, ISBN 0-07-027730-3, S. 40–50.<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:Fresnelsche+Formeln&rft.au=Eugene+Hecht&rft.btitle=Schaum%E2%80%99s+outline+of+theory+and+problems+of+optics&rft.date=1975&rft.genre=book&rft.isbn=0070277303&rft.pages=40-50&rft.pub=McGraw-Hill+Professional" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Matthias Bartelmann, Björn Feuerbacher, Timm Krüger, Dieter Lüst, Anton Rebhan, Andreas Wipf: <cite style="font-style:italic">Theoretische Physik 2. Elektrodynamik</cite>. 10. Auflage. Springer-Verlag, Berlin / Heidelberg / New York 2018, ISBN 978-3-662-56117-1, S. 213.<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:Fresnelsche+Formeln&rft.au=Matthias+Bartelmann%2C+Bj%C3%B6rn+Feuerbacher%2C+Timm+Kr%C3%BCger%2C+...&rft.btitle=Theoretische+Physik+2.+Elektrodynamik&rft.date=2018&rft.edition=10.&rft.genre=book&rft.isbn=9783662561171&rft.pages=213&rft.place=Berlin+%2F+Heidelberg+%2F+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-Damask-4">↑ <a href="#cite_ref-Damask_4-0">a</a> <a href="#cite_ref-Damask_4-1">b</a> Jay N. Damask: <cite style="font-style:italic">Polarization optics in telecommunications</cite>. Springer, New York 2005, ISBN 0-387-22493-9, S. 10–17.<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:Fresnelsche+Formeln&rft.au=Jay+N.+Damask&rft.btitle=Polarization+optics+in+telecommunications&rft.date=2005&rft.genre=book&rft.isbn=0387224939&rft.pages=10-17&rft.place=New+York&rft.pub=Springer" style="display:none">

</li>

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