From 8eef41c44344db27d854987f567d72256da9db93 Mon Sep 17 00:00:00 2001
From: bai2795 <dieter.horns@desy.de>
Date: Mon, 8 Jul 2024 22:53:43 +0200
Subject: [PATCH] added figures and changed wellenoptik.tex

added the fresnel mirror, give a definition of the coherence
---
 v0.1/figures/abbildung_2thin_lenses.pdf | Bin 0 -> 15375 bytes
 v0.1/figures/abbildung_2thin_lenses.png | Bin 28427 -> 89371 bytes
 v0.1/figures/double_slit.pdf            | Bin 0 -> 3282 bytes
 v0.1/figures/double_slit_far.pdf        | Bin 0 -> 3282 bytes
 v0.1/figures/drawing_2thinlenses.svg    |  12 +-
 v0.1/figures/drawing_double_slit.svg    | 459 ++++++++++++++++++++++++
 v0.1/geometrical_optics.tex             |   2 +-
 v0.1/pictures/fresnel.png               | Bin 0 -> 142749 bytes
 v0.1/skript.tex                         |   2 +-
 v0.1/wellenoptik.tex                    | 133 +++++--
 10 files changed, 568 insertions(+), 40 deletions(-)
 create mode 100644 v0.1/figures/abbildung_2thin_lenses.pdf
 create mode 100644 v0.1/figures/double_slit.pdf
 create mode 100644 v0.1/figures/double_slit_far.pdf
 create mode 100644 v0.1/figures/drawing_double_slit.svg
 create mode 100644 v0.1/pictures/fresnel.png

diff --git a/v0.1/figures/abbildung_2thin_lenses.pdf b/v0.1/figures/abbildung_2thin_lenses.pdf
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diff --git a/v0.1/geometrical_optics.tex b/v0.1/geometrical_optics.tex
index 2a1703c..1337424 100644
--- a/v0.1/geometrical_optics.tex
+++ b/v0.1/geometrical_optics.tex
@@ -563,7 +563,7 @@ Für den Fall, dass die Linsen in geringem Abstand zueinander stehen
  $m$ für das zusammengesetzte System - welches Vorzeichen erhalten Sie?})
 
  \begin{figure}
-	 \includegraphics[width=\linewidth]{abbildung_2thin_lenses.png}
+	 \includegraphics[width=\linewidth]{abbildung_2thin_lenses.pdf}
 	 \caption{\label{fig:2lenses} Abbildungskonstruktion für zwei dünne Sammellinsen.}
  \end{figure}
 
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diff --git a/v0.1/skript.tex b/v0.1/skript.tex
index 051f1bd..e1a8cb7 100644
--- a/v0.1/skript.tex
+++ b/v0.1/skript.tex
@@ -17,6 +17,7 @@
 \declaretheorem[name=Beispiel]{example}
 \declaretheorem[name=Definition]{definition}
 
+\usepackage{graphicx}
 \usepackage[pdftex,colorlinks,linkcolor=blue,citecolor=blue,backref]{hyperref}
 \hypersetup{
 	pdftitle={Skript Experimentalphysik II Dieter Horns},
@@ -31,7 +32,6 @@
 	pdfpagemode=UseOutlines,    % this is the option you were lookin for
 	pdfpagelayout=TwoPageRight
 }
-\usepackage{graphicx}
 %\usepackage{caption}
 %\usepackage{subfigure}
 
diff --git a/v0.1/wellenoptik.tex b/v0.1/wellenoptik.tex
index b03e707..a14a52c 100644
--- a/v0.1/wellenoptik.tex
+++ b/v0.1/wellenoptik.tex
@@ -10,7 +10,7 @@ Das \index{Huyghen'sche Prinzip} Huyghen'sche Prinzip erlaubt uns,
 eine jede Welle als Überlagerung von einzelnen Kugelwellen (sogenannten Elementarwellen \index{Elementarwellen}) zu verstehen. 
 
 Mit Hilfe des Huyghen'schen Prinzips lassen sich Phänomene erklären, die im
-Rahmen der geometrischen Optik nicht vorhersagbar sind. Ein einfaches Beispiel
+Rahmen der geometrischen Optik nicht erwarten werden. Ein einfaches Beispiel
 ist der Schattenwurf (siehe Abbildung \ref{figure:huygens}): In der Näherung
 der geometrischen Optik erzeugt ein Hindernis (z.B. eine Schneide) einen scharf
 umrissenen Schatten: kein Strahl eines parallel einfallenden Bündels erreicht
@@ -36,36 +36,39 @@ Zeigefinger zusammen. Was beobachten Sie im Schattenwurf kurz bevor sich Daumen
 \section{Koh\"arenz und Interferenz}
 \index{Kohärenz} \index{Interferenz}
 %
+Eine Welle ist \textit{kohärent} 
+\begin{itemize}
+		 \item während der \textit{Kohärenzzeit} $t_c$: an einem festen Ort ändert sich die Phase der Welle während einer Periode 
+			 um $2\pi$;
+		 \item über eine \textit{Kohärenzlänge} $\ell_c$: die Amplituden haben zu allen Zeiten zwischen zwei Punkten innerhalb der Kohärenzlänge einen zeitlich konstanten 
+			 Phasenunterschied.
+\end{itemize}
+Es gilt für eine Welle mit Ausbreitungsgeschwindigkeit $\ell_c = c t_c$.  Eine ideale ebene Welle hat $t_c, \ell_c \rightarrow \infty$.
+
+
 Aufgrund der linearen Eigenschaft der Wellengleichung lassen sich mehrere Wellen addieren.
 	Für den Spezialfall zweier Wellen \textit{gleicher}
-	Amplitude,\textit{gleicher} Frequenz und gleicher Polarisation ergibt
+	Amplitude, \textit{gleicher} Frequenz und gleicher Polarisation ergibt
 	sich je nach \textit{relativer} Phase $\Delta \phi$ eine
 	\textbf{konstruktive Interferenz} für $\Delta \phi=0,2\pi,4\pi,\ldots ,
 	m\cdot 2\pi$ und eine \textbf{destruktive
 	Interferenz}\index{Interferenz} für $\Delta
 	\phi=\pi,3\pi,5\pi,\ldots,\frac{2m+1}{2}\pi$. 
 
-Für die Intensität (Betrag des Poynting-Vektors) der resultierenden Welle ergibt sich entsprechend:
-
-$$ I=({E}_1+{E}_2\cos\Delta\phi)^2=E_1^2+E_2^2\cos^2\Delta\phi+2E_1 E_2 \cos \Delta \phi=
-	\begin{cases}
-	4 E^2 & E=E_1=E_2, \Delta \phi=m2\pi\\
-	0     & E_1=E_2, \Delta \phi=(2m+1)\pi/2.
-   \end{cases}$$
+Wir nehmen zwei Wellen an mit $E_1 = a\sin(\omega t)$ und $E_2 = a\sin(\omega t + \Delta\varphi)$.
+Für die Summe 
+$$ E = a \sin(\omega t)(1+\cos\Delta\varphi) + a\cos(\omega t)\sin\Delta\varphi.$$
+Die \textit{zeitlich gemittelte} quadratische Amplitude ergibt sich zu
+\[
+	\langle E^2\rangle = \frac{a^2}{2}( (1+\cos\Delta\varphi)^2 +  \sin^2\Delta\varphi)=a^2(1+\cos\Delta\varphi) = (\langle E_1^2\rangle + \langle E_2\rangle^2)(1+\cos\Delta\varphi) .
+\]
+Für eine vollständig kohärente Überlagerung kann es mit $\Delta \varphi=0,\pm 2\pi,\ldots$ zu \textit{konstruktiver} Interferenz kommen. Bei einem Phasenunterschied
+der beiden Wellen um $\Delta\varphi=\pm \pi/2, \pm 3\pi/2,\ldots$ kommt es zu \textit{destruktiver} Interferenz mit vollständiger Auslöschung der beiden Wellen!
 
+Für nicht-kohärente Wellen kommt es zu keiner Interferenz. Bei gleichförmiger Verteilung des Phasenunterschieds $\Delta \phi$ verschwindet bei der zeitlichen Mittelung der
+Term $\langle \cos\Delta\varphi\rangle=0$. 
 
-Wir haben dabei nicht berücksichtigt, dass der Phasenunterschied
-$\Delta \phi$ zeitlich oder räumlich variieren kann: $\Delta\phi(t,\bm{r})$. 
-Im allgemeinen ist der Phasenunterschied nur innerhalb eines charakteristischen Länge bzw. über ein
-endliches Zeitintervall konstant.
 
-Wir charakterisieren entsprechend die \textit{Kohärenz} \index{Kohärenz} 
-einer Lichtquelle als ein Maß für die Konstanz von $\Delta \phi$, 
-indem wir eine \textit{Kohärenzzeit} $\Delta t_c$ angeben, 
-über die die Wellen einer Lichtquelle (gemessen an einem Ort)
-in ihrer relativen Phase  um $\approx 2\pi$ variieren. 
-Analog hierzu können wir die \textit{Kohärenzlänge} $\Delta s_c=c\Delta t_c$ 
-als diejenige Länge angeben, über die zu einer festen Zeit die relative Phase entsprechend um $2\pi$ variiert.
 \paragraph{Linienbreite und Kohärenz}
  Wir betrachten  Wellen der Frequenz $\omega$, die aus einer punktförmigen
  Quelle stammen, sich aber um $\Delta\omega\ll\omega$ in der Frequenz
@@ -79,11 +82,24 @@ als diejenige Länge angeben, über die zu einer festen Zeit die relative Phase
 
 \textit{Beispiele:} 
 \begin{itemize}
- \item Sichtbares Licht: Hier ist $\Delta \nu\approx 10^{14}~$Hz, entsprechend ist $\Delta l_c\approx3~\mu$m,
-  \item isoliertes Atom: $\Delta \nu \approx 10^{8}$~Hz, so dass $\Delta l_c\approx 3~$m,
-  \item Laser: $\Delta \nu \approx $kHz für sehr gut stabilisierte Laser, damit ist $l_c\approx 300$~km. 
+ \item Sichtbares Licht: Hier ist $\Delta \nu\approx 10^{14}~$Hz, entsprechend ist $\Delta s_c\approx3~\mu$m,
+  \item isoliertes Atom: $\Delta \nu \approx 10^{8}$~Hz, so dass $\Delta s_c\approx 3~$m,
+  \item Laser: $\Delta \nu \approx $kHz für sehr gut stabilisierte Laser, damit ist $s_c\approx 300$~km. 
 \end{itemize}	
 
+Wenn wir die Interferenz von elektromagnetischen Wellen untersuchen, ist die
+Kohärenzlänge einer Quelle eine wichtige Größe. Wenn wir z.B. mit einem
+Strahlteiler das Licht aufteilen und über einen zweiten Strahlteiler wieder
+zusammenführen, können wir prinzipiell konstruktive oder auch destruktive
+Interferenz beobachten, wenn sich die Phasen der beiden Wellen mit einem
+Phasenunterschied $\Delta \phi = 0,2\pi,4\pi,\ldots$ (konstruktiv) bzw. $\Delta
+\phi = \pi, 3\pi, 5\pi\,..$ (destruktiv) überlagern. Ist jedoch der
+Wegunterschied zwischen den beiden Lichtwegen größer als die Kohärenzlänge, ist
+kein Interferenzeffekt mehr beobachtbar. Umgekehrt lässt sich mit einem solchen
+Verfahren die Kohärenzlänge und damit die Linienbreite der Quelle untersuchen.
+
+
+
 \begin{figure}
 	\centerline{
 	\includegraphics[width=0.8\linewidth]{figures/coherence.png}}
@@ -91,13 +107,71 @@ als diejenige Länge angeben, über die zu einer festen Zeit die relative Phase
 	Der zeitliche Verlauf der ortsfesten Amplituden von zwei Wellen mit leicht unterschiedlicher Frequenz. Gezeigt ist der Verlauf bis
 	$t_c/2=\pi/\Delta \omega$ }
 \end{figure}
+\section{Interferenz von zwei Wellen}
+Um die Interferenz von zwei Wellen nachzuweisen, benötigen wir mindestens zwei Quellen von kohärenten Wellen. Wir nehmen  zwei punktförmige Quellen 
+im Abstand $d$ zueinander  an mit folgenden Eigenschaften:
+\begin{enumerate}[i)]
+	\item gleiche Amplitude,
+	\item gleiche Phase (bzw. zeitlich konstanter Phasenunterschied),
+	\item gleiche Frequenz,
+	\item gleiche Polarisation.
+\end{enumerate}
+Wenn wir uns das resultierende Wellenfeld (z.B. für zylinderförmige Wellen) zu einem festen Zeitpunkt nach Einschalten der beiden Quellen anschauen, erhalten wir
+die Anordnung in Abb.~\ref{nodal_antinodal}.
+Für einen Punkt $P$ im Abstand $r_1$ und $r_2$ zu den beiden Quellen ergibt sich folgende Bedingung für \textit{konstruktive Interferenz}:
+$$ r_2 - r_1 = 0, \pm \lambda, \pm 2\lambda, \ldots, \pm n\lambda.$$
+Für \textit{destruktive} Interferenz ist die Bedingung
+$$ r_2 - r_1 = \pm \frac{\lambda}{2}, \pm \frac{3\lambda}{2}, \ldots, \frac{2n+1}{2}\lambda.$$
+ Die beiden Gleichungen charakterisieren Hyperbeln, sogenannte Nodal-, bzw. Antinodallinien, entlang derer sich die Wellen konstruktiv bzw. destruktiv überlagern. 
+Das Wellenfeld lässt sich anschaulich mit Oberflächenwellen in einem Wellenbecken darstellen.
+
+Insgesamt finden wir $2d/\lambda$ Nodal/Antinodallinien für den gegebenen Spaltabstand $d$ bei einer Wellenlänge $\lambda$. 
+
+Als nächstes schauen wir uns die Intensitätsverteilung der beiden Wellen an einer
+Projektionsand im Abstand $L\gg d$ zu den Quellen an. 
+Die hyperbolischen Nodallinien schmiegen sich an Geraden an, die einen Winkel $\theta$ zur Horizontalen haben (siehe Abbildung~\ref{fig:nodal_antinodal}).
+
+Wir können jetzt die Winkelbedingung für konstruktive 
+$$ r_1 - r_2 = d \sin\theta = n\lambda$$
+und destruktive Interferenz 
+$$ r_1 - r_2 = d\sin \theta = \frac{2n+1}{2}\lambda$$
+aufschreiben. 
+Für große Abstände der Quellen zum Schirm nähern wir die Hyperbeln durch Geraden mit kleinen Winkeln
+$\theta\ll 1$ an, 
+so dass für die Positionen der Maxima auf dem Schirm im Abstand $L\gg d$ gilt:
+$$  x_i = \theta_i \cdot L = n \frac{\lambda}{d} L,$$
+für $n=0, \pm 1, \pm 2, \ldots $.  Die Position $x_i$ ist jeweils der Abstand
+zu dem Maximum für $n=0$. 
+Die Minima finden sich entsprechend bei
+$$ x_i = n \frac{2n+1}{2} \frac{\lambda}{d} L.$$
+
+Ein einfaches Verfahren zur Erzeugung von zwei kohärenten Quellen ist der sogenannte \textit{Fresnel}-Doppelspiegel. Hierbei werden zwei Spiegel um einen kleinen Winkel zueinander verkippt. Wenn wir jetzt mit einer Lichtquelle die beiden Spiegel ausleuchten, erzeugen die beiden Spiegel jeweils eine optische Abbildung der einfallenden Welle, die dann zur Überlagerung der beiden kohärenten Wellen führt (siehe Abbildung~\ref{fig_fresnel}).
+
+Die dabei auf dem Schirm im Abstand $s=R+d$ entstehenden Streifen haben  den Abstand $\Delta y\approx \lambda s/a$.
+
+
+
+\begin{figure}
+\includegraphics[width=\linewidth]{Graphiken/young_doppelspalt.png}
+	\caption{Zwei punktförmige Quellen von Wellen gleicher Wellenlänge, Amplitude und Phase erzeugen zwei sich überlagernde Wellen. Die einzelnen Wellen haben Maxima (durchgezogene Linien) und Minima (gestrichelte Linien). Es existieren Punkte, an denen sich Maxima und Minima jeweils aufaddieren. Diese Punkte sind entlang der \textit{Nodallinien} (in rot) angeordnet. Entlang der Nodallinien kommt es zu \textit{konstruktiver} Interferenz. Es existieren auch Punkte, an denen sich Maxima und Minima addieren. Diese \textit{destruktive Interferenz} findet entlang der \textit{Antinodallinie} (in blau) statt.\label{nodal_antinodal}}
+\end{figure}
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{double_slit.pdf}
+	\caption{Bei großen Abständen verlaufen die Nodallinien entlang der Geraden mit Winkel $\theta$ zur Horizontalen.\label{fig:asymptotic_nodal}}
+\end{figure}
+
+\begin{figure}
+	\centerline{	\includegraphics[width=0.7\linewidth]{fresnel.png}}
+	\caption{Fresnel'scher Doppelspiegel zur Erzeugung von Interferenzmustern: Die Lichtquelle $S$ leuchtet zwei leicht zueinander verkippte Spiegel aus, so dass
+	es zur Überlagerung von Wellen kommt, die scheinbar aus den Punkten $S_1$ und $S_2$ im Abstand $d$ zueinander stammen (Abbildung aus E. Hecht: Optik).
+	\label{fig_fresnel}}
+\end{figure}
 
- 
 \section{Interferenz an d\"unnen Schichten}
 \begin{figure}
-	\begin{center}
-	\includegraphics[width=0.7\linewidth]{Graphiken/thinlayer_slanted.png}
-	\end{center}
+	\centerline{
+		\includegraphics[width=0.7\linewidth]{Graphiken/thinlayer_slanted.png} }
       	\caption{Interferenz an dünner Schicht der Dicke $d$: 
 	Der einfallende Lichtstrahl wird an den beiden Grenzflächen jeweils gebrochen und reflektiert. 
 	Der Wegunterschied der beiden Strahlen  
@@ -106,11 +180,6 @@ als diejenige Länge angeben, über die zu einer festen Zeit die relative Phase
 	wobei $\Phi_{ij}$ den möglichen Phasensprung beim Übergang bzw. Reflexion an der Grenzschicht von dem Medium mit $n_i$ und $n_j$ angibt. }    
 \end{figure}
 \section{Interferometer}
-\section{Beugung an Spalt und Gitter}
-\subsection{Young'scher Doppelspalt}
-\begin{figure}
-\includegraphics[width=\linewidth]{Graphiken/young_doppelspalt.png}
-\end{figure}
 \section{Aufl\"osungsverm\"ogen optischer Instrumente}
 \label{section:resolution}
 \label{section:diffraction}
-- 
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