From a98220753a61b4e766c1830142a0bea2a77317a1 Mon Sep 17 00:00:00 2001
From: Dieter Horns <dieter.horns@uni-hamburg.de>
Date: Mon, 19 May 2025 17:15:28 +0200
Subject: [PATCH] added a figure, up to 9.6.1, beginning of lecture 26
starting with 9.6.2 in lecture 26
---
.../drawing_parallelkonstruktion.png | Bin 0 -> 36545 bytes
.../drawing_parallelkonstruktion.svg | 636 ++++++++++++++++++
v0.1/geometrical_optics.tex | 100 ++-
3 files changed, 713 insertions(+), 23 deletions(-)
create mode 100644 v0.1/Graphiken/drawing_parallelkonstruktion.png
create mode 100644 v0.1/Graphiken/drawing_parallelkonstruktion.svg
diff --git a/v0.1/Graphiken/drawing_parallelkonstruktion.png b/v0.1/Graphiken/drawing_parallelkonstruktion.png
new file mode 100644
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diff --git a/v0.1/geometrical_optics.tex b/v0.1/geometrical_optics.tex
index 2fcb89b..33a780b 100644
--- a/v0.1/geometrical_optics.tex
+++ b/v0.1/geometrical_optics.tex
@@ -445,8 +445,8 @@ Man nennt eine gekrümmte Linsenfläche \textbf{konvex}, wenn die Linse zwischen
Diese Linsentypen finden sich in vielen optischen Geräten wieder, insbesondere ist die bikonkave Linse für Brillengläser sehr gut geeignet
(überlegen Sie sich bitte, warum das so ist).
-\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69625/5072/5486}}
\subsection{Abbildung mit einer dünnen Linsen}
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69625/5072/5486}}
Wir bestimmen zunächst die Fokallänge einer bikonvexen Linse (siehe Abbildung~\ref{fig:Linsenmachergleichung-Herleitung}) der Dicke $d$.
Bei einer \textbf{dünnen Linse} werden wir annehmen, dass der maximale Abstand $d=\overline{O_1 O_2}$
@@ -484,9 +484,9 @@ Die entsprechenden Abstände sind in Abbildung \ref{fig:Linsenmachergleichung-He
\end{figure}
%
%
-Der Punkt $A$ würde in den Punkt $B_1$ abgebildet werden, falls rechts von der linken Fläche nur das Medium mit der Brechzahl $n$ wäre.
-Aufgrund der zweiten Abbildung durch die rechte Grenzfläche erfolgt eine zweite Brechung und der Strahl kreuzt die optische Achse im Punkt $B$.
-Die Position des Punktes $B$ lässt sich konstruieren, indem wir den Punkt $B_1$ als Ausgangspunkt für Strahlen betrachten,
+Der Punkt $A$ würde in den Punkt $B_1$ abgebildet werden, falls rechts von der linken Fläche das Medium die Brechzahl $n$ hätte.
+Aufgrund der Brechung an der rechten Grenzfläche kreuzt der resultierende Strahl die optische Achse im Punkt $B$.
+Die Position des Punktes $B$ lässt sich konstruieren, indem wir den Punkt $B_1$ als Ausgangspunkt für zurücklaufende Strahlen betrachten,
die an der zweiten Grenzfläche gebrochen werden: $n_1=n,\:n_2=1$ und $a_2=-\left(b_1-d\right)$;
der entsprechende Krümmungsradius beträgt $-R_2$, so dass wir
\begin{equation}
@@ -494,25 +494,31 @@ der entsprechende Krümmungsradius beträgt $-R_2$, so dass wir
\end{equation}
erhalten. Addiert man die beiden Abbildungsgleichungen \ref{lens1} und \ref{lens2}, so erhalten wir
-\[\frac{1}{a_1}+\frac{1}{b_2}=\left(n-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)+\frac{n\cdot d}{b_1\left(b_1-d\right)},\]
-bzw. mit $a=a_1+\frac{d}{2}$ und $b=b_2+\frac{d}{2}$ gilt für dünne Linsen ($d\ll a,\:d\ll b$) die \index{Linsenmachergleichung} \textbf{Linsenmachergleichung}:
+\[\frac{1}{a_1}+\frac{1}{b_2}=\left(n-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)+\frac{n\cdot d}{b_1\left(b_1-d\right)}.\]
+Diese Gleichung ist auch für Linsen der Dicke $d$ allgemein gültig.
+
+Bei \textit{dünnen} Linsen lässt sich dieser Ausdruck weiter vereinfachen mit
+ $a=a_1+\frac{d}{2}$ und $b=b_2+\frac{d}{2}$ gilt für ($d\ll a,\:d\ll b$) die \index{Linsenmachergleichung} \textbf{Linsenmachergleichung}:
\begin{wichtig}[Linsenmachergleichung]
\[\frac{1}{a}+\frac{1}{b}=\left(n-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)= \frac{1}{f}.\]
\end{wichtig}
-Dies ist die Gleichung für eine \textit{dünne} Linse ($d\ll f$). Für eine dünne Linse kann die zeichnerische Konstruktion der Linsenabbildung durch die Brechung an der Mittelebene der Linse, statt der Brechung an beiden Grenzflächen, erfolgen (siehe Abbildung \ref{fig:Sammellinse-Charakteristische-Strahlen}).
-\begin{figure}[htbp]
- \centering
- \includegraphics[width=0.90\textwidth]{figures/Sammellinse-Charakteristische-Strahlen.png}
- \caption{Zeichnerische Konstruktion der Abbildung durch eine dünne Linse}
- \label{fig:Sammellinse-Charakteristische-Strahlen}
-\end{figure}
+\textbf{Spezialfall:}\\
+Eine bikonvexe Linse mit identischen Krümmungsradien erfüllt $R_1=R=-R_2$ und damit ergibt sich für die Brennweite
+
+\[f=\frac{R}{2(n-1)}.\]
+
+\textbf{Zum Vergleich:}\\
+Die Brennweite eines sphärischen Hohlspiegels beträgt $f=\frac{R}{2}$.\\
+
-Bei achsenparallel einfallendem Strahl ist in der Linsenmachergleichung $a=\infty$, welcher auf der Bildseite durch den Brennpunkt $F$ gehen muss, daher gilt $b=f$ und damit erhält man für die \textbf{Brennweite einer dünnen Linse}:
+Bei achsenparallel einfallendem Strahl ist in der Linsenmachergleichung $a=\infty$. Auf der Bildseite muss jeder Strahl durch den Brennpunkt $F$ gehen,
+daher gilt $b=f$ und damit erhält man für die \textbf{Brennweite einer dünnen Linse}:
\[f=\frac{1}{n-1}\left(\frac{R_1\cdot R_2}{R_2-R_1}\right).\]
-Setzt man diesen Ausdruck für die Brennweite in die Linsenmachergleichung ein, so erhält man \index{Linsengleichung-dünne Linsen}\textbf{Abbildungsgleichung für dünne Linsen}:
+Setzt man diesen Ausdruck für die Brennweite in die Linsenmachergleichung ein, so erhält man die
+\index{Linsengleichung-dünne Linsen}
\begin{wichtig}[Abbildungsgleichung für dünne Linsen]
Für Linsen der Dicke $d\ll f$ gilt: \[\frac{1}{g}+\frac{1}{b}=\frac{1}{f}\]
\end{wichtig}
@@ -520,17 +526,48 @@ Für Linsen der Dicke $d\ll f$ gilt: \[\frac{1}{g}+\frac{1}{b}=\frac{1}{f}\]
Anmerkung: Für dicke Linsen gilt die selbe Gleichung mit dem Unterschied,
dass bei einer dicken Linse zwei Hauptebenen definiert werden und $g$ und $b$ die Abstände zu der jeweiligen Hauptebene sind.
-\textbf{Spezialfall:}\\
-Eine bikonvexe Linse mit identischen Krümmungsradien erfüllt $R_1=R=-R_2$ und damit ergibt sich für die Brennweite
-
-\[f=\frac{R}{2(n-1)}.\]
+\marginpar{Experiment \qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/517/643}}
+\textbf{Luftlinsen:}\\ Bei der Linsenmachergleichung ändert sich das Vorzeichen der Fokallänge bei gleichen Krümmungsradien, wenn
+die Brechung an einer Grenzfläche die Medien vertausche (also von $n>1$ nach $n=1$ und umgekehrt breche).
-\textbf{Zum Vergleich:}\\
-Die Brennweite eines sphärischen Hohlspiegels beträgt $f=\frac{R}{2}$.\\
+\textbf{Brechkraft:}
Der Kehrwert der Brennweite $f$ wird als \index{Brechkraft}\textbf{Brechkraft} $\boldsymbol{D}$ bezeichnet, d.h. $D=\frac{1}{f}$ mit $\left[D\right]=1\:\text{dpt}=1\frac{1}{m}$; die Einheit der Brechkraft ist \textbf{Dioptrie}.\\
+
+
+
+
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/664/963}}
+Für die zeichnerische Konstruktion der Brechung an einer dünnen Linse benutzen wir die \textbf{Parallelkonstruktion} (siehe Video). Hiermit lässt sich der gebrochene
+Strahl für beliebige Strahlen von der Gegenstandsseite auf der Bildseite zeichnerisch ermitteln.
+In der Abbildung~\ref{fig:parallelkonstruktion} konstruieren wir
+den gesuchten axialen Punkt $A'$, indem wir eine Hilfslinie durch den Mittelpunkt der Linse konstruieren. Die Hilfslinie verläuft parallel zu dem Strahl,
+den wir brechen wollen. Da alle parallel einfallenden Strahlen in der paraxialen Näherung sich in der Fokalebene auf der Bildseite schneiden,
+erhalten wir so einen zweiten Punkt auf der Bildseite, den wir für die Konstruktion des gebrochenen Strahls nutzen können.
+\begin{figure}[htbp]
+ \centering \includegraphics[width=0.9\textwidth]{Graphiken/drawing_parallelkonstruktion.png}
+ \caption{Brechung eines einfallenden Strahls: Parallelkonstruktion}
+ \label{fig:parallelkonstruktion}
+\end{figure}
+
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/963/1599}}
+Für die Abbildung eines Gegenstandes $G$ können wir zeichnerisch die Position des Bildes $B$ mit dem Mittelpunktstrahl und Parallelstrahl
+(oder Brennpunktstrahl) konstruieren. Hieraus ergeben sich dann auch die Maßstäbe der Abbildung (siehe Abb.~\ref{fig:Sammellinse-Charakteristische-Strahlen}).
+
+
+
+
+\begin{figure}[htbp]
+ \centering
+ \includegraphics[width=0.90\textwidth]{figures/Sammellinse-Charakteristische-Strahlen.png}
+ \caption{Zeichnerische Konstruktion der Abbildung durch eine dünne Linse}
+ \label{fig:Sammellinse-Charakteristische-Strahlen}
+\end{figure}
+
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/53/267}}
+
Der \index{Abbildungsmaßstab} \textbf{Abbildungsmaßstab} $\mathbf{M}$ gibt das Verhältnis von Bild- zu Gegenstandsgröße an und kann aus dem Strahlensatz über ähnliche Dreiecke ($D_1$ und $D_2$: siehe Abbildung \ref{fig:Sammellinse-Charakteristische-Strahlen}) bestimmt werden
\footnote{
Zur Übung: auf diesem Wege lässt sich ebenfalls die Linsengleichung ableiten.}
@@ -579,6 +616,7 @@ $G_4$&$g<f$&$b<0$&$y^{\prime}>y$&$m>1$&virtuell, aufrecht\\\hline
\end{figure}
\subsubsection{Zerstreuungslinse}
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/1631/1707}}
Zerstreuungslinsen sind in der Regel bikonkave Linsen\footnote{Zerstreuungslinsen können auch konvex-konkav sein (siehe Abbildung~\ref{fig:Linsentypen}c).}, die eine negative Brechkraft $D<0$ bzw. eine negative Brennweite $f<0$ besitzen und damit nur \textit{virtuelle} Bilder erzeugen. Das heißt, dass achsenparallele Strahlen durch die Linse in divergentes Lichtbündel verwandelt werden (siehe Abbildung \ref{fig:Zerstreuungslinse}), dessen Ursprung in einem \textbf{virtuellen Brennpunkt} zu liegen scheint. Gegenstand und Bild liegen auf der Gegenstandsseite der Linse (siehe auch Abbildung~\ref{fig:Zerstreuungslinse}).
@@ -590,6 +628,7 @@ Zerstreuungslinsen sind in der Regel bikonkave Linsen\footnote{Zerstreuungslinse
\end{figure}
\subsubsection{Linsensystem}
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/1712/2471}}
Um die gewünschten optischen Eigenschaften zu erreichen, lassen sich Linsen zu \textit{Linsensysteme} kombinieren. Durch eine optimierte Kombination verschiedener Linsen lässt sich die Qualität der Abbildung erheblich verbessern und auch Abbildungsfehler korrigieren.
@@ -651,10 +690,16 @@ Für den Fall, dass die Linsen in geringem Abstand zueinander stehen
\end{enumerate}
Anschließend müssen wir noch den gemeinsamen Fokus finden und die dazugehörige Hauptebene $H$. Für den Fokus verlängern wir den Strahl $1$
durch $L_2$, so dass der Strahl durch den Bildpunkt (Spitze des Pfeils) geht. Der Schnittpunkt mit der optischen Achse ist der gemeinsame Fokus.
- Die Hauptebene identifiziern wir mit derjenigen Ebene, an der ein einlaufender paraller Strahl so gebrochen wird, dass er den Fokus schneidet. Hierzu suchen wir den Schnittpunkt des Brennpunktstrahls (vom Fokus $F$ ausgehend) mit dem verlängerten Strahl $1$. Die Hauptebene befindet sich im Abschnitt $f$ zu dem gemeinsamen Fokus.
+ Die Hauptebene identifizieren wir mit derjenigen Ebene, an der ein einlaufender paralleler Strahl so gebrochen wird, dass er den Fokus schneidet.
+ Hierzu suchen wir den Schnittpunkt des Brennpunktstrahls (vom Fokus $F$ ausgehend) mit dem verlängerten Strahl $1$.
+ Die Hauptebene befindet sich im Abschnitt $f$ zu dem gemeinsamen Fokus.
+\marginpar{Experiment \qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/2492/2621}}
+
\subsection{Abbildungsfehler}
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/2939/4142}}
+
Alle bisher gemachten Aussagen für Linsen gelten in der paraxialen Näherung, d.h. sie gelten für achsennahe Strahlen. Sobald man nicht mehr achsennahe Strahlen betrachtet oder die Strahlen die Linse asymmetrisch zur Achse durchlaufen, kommt es zu \index{Abbildungsfehler} \textbf{Abbildungsfehlern}. Diese führen dazu, dass Strahlen die von einem Punkt ausgehen sich nicht mehr in einem Punkt vereinigen, sondern nur noch in der Umgebung der Bildpunktes. Die Folgen sind Bildunschärfen oder Verzerrungen, welche für verschiedene Bildbereiche unterschiedlich groß sein können. Wir betrachten kurz die relevantesten Linsenfehler und deren Korrekturen.
\begin{itemize}
@@ -686,6 +731,10 @@ Der zweite Term berücksichtigt die Abweichung der Bildweite $b$ aufgrund der sp
Zur Verringerung der sphärischen Aberration lassen sich z.B. achsenferne Strahlen durch eine Blende unterdrücken (führt damit zu einer kleineren Apertur und damit Intensitätsverlust) - alternativ werden eine Kombinationen verschiedener Sammel- und Zerstreuungslinsen (zur sphärischen Korrektur) verwendet oder speziell optimierte, nicht-sphärische Linsen hergestellt. Dies erfordert allerdings einen technisch aufwändigen Schleifprozess - einfacher zu produzieren sind asphärische Linsen aus gepresstem Kunststoffglas (z.B. Acrylglas).
+ \item \index{Öffnungsfehler}\textbf{Öffnungsfehler (Koma):}
+ Schräg einfallende parallele Strahlen haben keinen eindeutigen Schnittpunkt in der Fokalebene.
+
+
\item \index{Aberration, chromatische}\textbf{chromatische Aberration:}
Chromatische Aberration ist eine Folge der Dispersion: die Brechzahl $n\left(\lambda\right)$ des Linsenmaterials wird im Allgemeinen von der Wellenlänge $\lambda$ des Lichtes abhängen und damit auch die Brennweite $f\left(\lambda\right)$. Für eine bikonvexe Linse mit $R_1=R_2=R$ gilt
@@ -720,8 +769,10 @@ Astigmatismus tritt nicht nur bei Linsen auf, sondern auch wenn ein Lichtbündel
\end{itemize}
-\todo{Es fehlen noch Bildfeldwölbungen}
+%\todo{Es fehlen noch Bildfeldwölbungen}
\section{Optische Instrumente}
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/4144/4794}}
+
In diesem Abschnitt befassen wir uns kurz mit den wichtigsten optischen Instrumenten, die das Auflösungsvermögen des Auges erhöhen (z.B. Lupe, Mikroskop), die Lichtsammelfläche vergrößern (Spiegelteleskop) oder den Spektralbereich erweitern (z.B. Bildwandler).
@@ -760,6 +811,7 @@ bei entspanntem Muskel für die bildseitige Fokallänge $f_2=22$~mm (entspricht
Die deutliche Sehweite $s_0=0,2$~m ist dabei der kürzeste Abstand, den wir ohne merkliche Anstrengung betrachten können. Wir beziehen uns bei Angaben zur Vergrößerung immer auf die deutliche Sehweite.
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/4829/5014}}
\textbf{Weitere Fehlsichtigkeiten des Auges:}
\begin{itemize}
@@ -787,6 +839,7 @@ Verzerrungen feststellen (Astigmatismus), handelt es sich um entsprechende
Korrekturgläser. \end{itemize}
\subsection{Sehwinkel}
+\marginpar{\qrcode[hyperlink,height=1.5cm]{https://lecture2go.uni-hamburg.de/l2go/-/get/v/69647/5045/5431}}
Gemäß Abbildung \ref{fig:Sehwinkel} erscheint ein Gegenstand scheinbar umso
größer, je kleiner die Gegenstandsweite ist, d.h. der Winkel $\epsilon$
@@ -813,6 +866,7 @@ durch die Linsengleichung$^{\footnotemark}$
gegeben ist. Der Abstand $b$ ist durch die Geometrie des Auges fest vorgegeben, daher muss die Brennweite $f$ der Augenlinse durch Veränderung der Augenkrümmung derart angepasst werden, dass das Bild auf der Netzhaut scharf erscheint (d.h. $s=g$). Die Adaption des Auges funktioniert nur bis zu einem bestimmten Mindestabstand $s_{\text{min}}\approx 0,1$~m (altersabhängig). Weiter oben wurde die deutliche Sehweite $s=s_0$ eingeführt, der zugehörige Sehwinkel wird als $\epsilon_0=G/s_0$ bezeichnet
+
\subsection{Vergrößerung}
Mithilfe optischer Instrumente lässt sich der Sehwinkel $\epsilon$ vergrößern, ohne dabei die deutliche Sehweite $s_0$ des Auges zu unterschreiten.\\
--
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