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a/v0.1/em_waves.tex b/v0.1/em_waves.tex index 6822134436fa2c7708d4b44fe8ee9a43c377703f..72b8040877a33d0f555dd375524ae895a1835206 100644 --- a/v0.1/em_waves.tex +++ b/v0.1/em_waves.tex @@ -1133,7 +1133,7 @@ Weiterhin k\"onnen wir explizit noch aufschreiben: Aus den ersten beiden Gleichungen ergibt: $$\sin\alpha=\sin\alpha^\prime \leftrightarrow \alpha=\alpha^\prime,$$ zusammen mit der letzten Gleichung haben wir das Snell'sche Gesetz abgeleitet: - +\index{Snell'sches Gesetz} \begin{wichtig}[Snell'sches Gesetz] $$n_1\,\sin\alpha = n_2\sin\beta.$$ \end{wichtig} @@ -1148,6 +1148,7 @@ besagt, dass im dichteren Medium die ebene Welle zur Fl\"achennormalen gebrochen \subsection{Optisch anisotrope Medien: Doppelbrechung} \section{Geometrische Optik} + \subsection{Abbildungen: Lochkamera, Planspiegel, Hohlspiegel, Parabolspiegel} \subsection{Sammellinsen} \subsection{Zerstreuungslinsen} diff --git a/v0.1/figures/Elliptischer-sphaerischer-Spiegel.png b/v0.1/figures/Elliptischer-sphaerischer-Spiegel.png index 126779befff7a6f56d1580dd65b6743ceccb64fd..d6f1da157fd7d0e34527d6098136cc7bba69a12b 100644 Binary files a/v0.1/figures/Elliptischer-sphaerischer-Spiegel.png and b/v0.1/figures/Elliptischer-sphaerischer-Spiegel.png differ diff --git a/v0.1/figures/Lochkamera.png b/v0.1/figures/Lochkamera.png index 347b938ff9b7ef32eb75a4a02e2e75e2ea2dd07b..0c1505a1eeb572ff5986bc29c17c5dc810eff19b 100644 Binary files a/v0.1/figures/Lochkamera.png and b/v0.1/figures/Lochkamera.png differ diff --git a/v0.1/figures/mirror_image.png b/v0.1/figures/mirror_image.png new file mode 100644 index 0000000000000000000000000000000000000000..6bd03ee0c11339b435ffdaee64a0ba53478510bb Binary files /dev/null and b/v0.1/figures/mirror_image.png differ diff --git a/v0.1/figures/strahlenoptik.png b/v0.1/figures/strahlenoptik.png new file mode 100644 index 0000000000000000000000000000000000000000..3dc9b25562720ee2fc6f644421cfdd32e497743c Binary files /dev/null and b/v0.1/figures/strahlenoptik.png differ diff --git a/v0.1/geometrical_optics.tex b/v0.1/geometrical_optics.tex index 3f2b90bf9fd0ad653aae22867b6feec4b9a6b5eb..8e7b9f2a73385729f9574381521e46cc53e49123 100644 --- a/v0.1/geometrical_optics.tex +++ b/v0.1/geometrical_optics.tex @@ -2,17 +2,14 @@ In diesem Abschnitt besch�ftigen wir uns mit der \textbf{geometrischen Optik} \index{Geometrische Optik}. Die geometrische Optik ist eine N�herung der Wellenoptik: die Ausbreitung des Lichtes wird durch Strahlen und Lichtb�ndel dargestellt, d.h. der Lichtweg wird auf rein geometrische Weise beschrieben. +Ph�nomene, bei denen die Phasenlage der Wellen eine Rolle spielen, werden im anschlie�enden Abschnitt \textit{Wellenoptik} behandelt. -%In diesem Abschnitt konzentrieren wir uns auf die Ausbreitungsrichtung des Lichtes, und de �nderung dieser durch \textbf{abbildende Systeme} wie Spiegel und Linsen. +Unter welchen Bedingungen ist die Beschreibung des Lichtes durch die Strahlenoptik gerechtfertigt ? -\subsection{Annahmen / G�ltigkeit} - -Unter welchen Bedingungen ist die Beschreibung des Lichtes durch die Strahlenoptik gerechtfertigt ?\\ - -\subsubsection{Lichtstrahlen und Lichtb�ndel} +\subsection{Lichtstrahlen und Lichtb�ndel} In isotropen Medien ist die Ausbreitungsrichtung einer Welle durch die Normale der Phasenfl�che bestimmt, welche in der geometrischen Optik als \textbf{Lichtstrahlen}\index{Lichtstrahlen} bezeichnet werden. Ein Lichtstrahl ist eine geometrische Gerade mit Querschnitt null als Idealisierung, die bei der graphischen Darstellung der Lichtausbreitung in optischen Systemen n�tzlich ist. @@ -23,15 +20,16 @@ Die Beschreibung einer r \begin{figure}[htbp] \centering - \includegraphics[width=0.90\textwidth]{figures/Lichtstrahlen.png} - \caption{Zur Definition eines Lichtb�ndels als r�umlich quer zur Ausbreitungsrichtung begrenzte Welle, deren Normale auf der Phasenfl�che die Ausbreitungsrichtung des Lichtstrahls angibt, dargestellt in einem homogenen Medium.} + \includegraphics[width=0.90\textwidth]{figures/strahlenoptik.png} + \caption{Zur Definition eines Lichtb�ndels: Mehrere parallel Strahlen werden + an einer Be deren Normale auf der Phasenfl�che die Ausbreitungsrichtung des Lichtstrahls angibt, dargestellt in einem homogenen Medium.} \label{fig:Lichtstrahlen} \end{figure} \textbf{Merkregel:}\\ Bei Wellenl�ngen von $\lambda=0,5\:\mu\text{m}$ m�ssen Lichtb�ndel einen Durchmesser von $D>10\:\mu\text{m}$ haben, um Beugungseffekte vernachl�ssigen zu k�nnen. -\subsubsection{Axiome der Optik} +\subsection{Grundlagen der geometrischen Optik} F�r die Ausbreitung von Lichtstrahlen in optischen Medien gelten die folgenden experimentellen Beobachtungen, die sich auch theoretisch herleiten lassen: @@ -45,47 +43,51 @@ F \end{itemize} -\subsubsection{Fermat�sches Prinzip} +\subsection{Fermat�sches Prinzip} -Die beiden ersten Aussagen lassen sich aus dem \index{Fermatschen Prinzip}\textbf{Fermat'schen Prinzip} herleiten, welches besagt dass das Licht zwischen zwei Punkten $P_1$ und $P_2$ stets jenen Weg zur�cklegt f�r den die Lichtlaufzeit minimal wird. In seiner allgemeinen Formulierung f�r inhomogene Medien, mit �rtlich ver�nderlichem Brechungsindex, lautet es +Die beiden ersten Aussagen lassen sich aus dem \index{Fermatschen Prinzip}\textbf{Fermat'schen Prinzip} herleiten, welches besagt dass das Licht zwischen zwei Punkten $P_1$ und $P_2$ stets jenen Weg zur�cklegt f�r den die Lichtlaufzeit extremal (meistens minimal) wird. In seiner allgemeinen Formulierung f�r inhomogene Medien, mit �rtlich ver�nderlichem Brechungsindex, lautet es \[\delta\int_{P_1}^{P_2}n \:ds=0,\] -wobei $\delta$ eine infinitesimale Variation des optischen Weges gegen�ber des k�rzesten Weges beschreibt. +wobei $\delta$ eine infinitesimale Variation des optischen Weges gegen�ber den +extremalen Weges beschreibt. -\section{Die optische Abbildung} +\subsection{Die optische Abbildung} Eine \index{optische Abbildung}\textbf{optische Abbildung} beschreibt in der Optik die Erzeugung eines Bildpunktes $P^{\prime}$ von einem Gegenstandspunkt $P$ durch Vereinigung aller Lichtstrahlen die von einem Gegenstandspunkt ausgehen, mittels eines optischen Systems. Ein \textbf{Bild} beschreibt die Gesamtheit aller einzelnen Bildpunkte, welche alle Gegenstandspunkte repr�sentieren. Man unterscheidet zwischen \index{reelles Bild}\textbf{reellen} und \index{virtuelles Bild}\textbf{virtuellen} Bildern. Im Falle eines reellen Bildes kann dieses auf einen Schirm aufgefangen werden, d.h. die Lichtstrahlen die von einem Gegenstand kommen, vereinigen sich tats�chlich in einem Bild. Ein virtuelles Bild scheint dagegen im Raum zu schweben, d.h. bei der strahlenoptischen Konstruktion des Bildes scheint das Licht aus virtuellen Bildpunkten zu kommen. -\subsubsection{Spiegel} +\subsubsection{Ebener Spiegel} Ein (ebener) Spiegel ist ein Beispiel f�r ein optisches System, das als optische Abbildung ein virtuelles Bild erzeugt. Genauer ist der ebene Spiegel das \textbf{einzige} optische Element, das eine ideale Abbildung in dem Sinne erzeugt, dass jeder Punkt $P$ des Raumes in einem anderen Punkt $P^{\prime}$ abgebildet wird, siehe Abbildung \ref{fig:Spiegel}. Dabei gibt es zu jedem Punkt $P$ einen virtuellen Punkt $P^{\prime}$ aus dem die Strahlen zu kommen scheinen. +Zum Nachdenken: Warum erscheint ein virtuelles Spiegelbild zwar seitenverkehrt aber nicht auf dem Kopf? \begin{figure}[htbp] \centering - \includegraphics[width=0.90\textwidth]{figures/Spiegel.png} + \includegraphics[width=0.90\textwidth]{figures/mirror_image.png} + \caption{Ein Spiegel erzeugt eine optische Abbildung von einem Punkt $P$ des Raumes (links des Spiegels) ein virtuelles Bild $P^{\prime}$, das Spiegelbild das von der rechten Seite des Spiegels zu kommen scheint.} \label{fig:Spiegel} \end{figure} + Andere Beispiele sind \textbf{elliptische} Spiegel (siehe Abbildung \ref{fig:Elliptischer-sphaerischer-Spiegel} oben), die nur zwei Punkte ineinander abbilden, n�mlich die beiden Brennpunkte. Der \textbf{Kugelspiegel} oder sph�rische Spiegel (siehe Abbildung \ref{fig:Elliptischer-sphaerischer-Spiegel} unten) als Spezialfall davon, bildet nur einen Punkt, den Mittelpunkt $M$, in sich ab. \begin{figure}[htbp] \centering \includegraphics[width=0.90\textwidth]{figures/Elliptischer-sphaerischer-Spiegel.png} - \caption{Oben. Ein elliptischer Spiegel bildet die beiden Brennpunkte $F_1$ und $F_2$ ineinander ab. Unten: Ein sph�rischer Spiegel bildet den Mittelpunkt $M$ in sich selber ab.} + \caption{Links: Ein sph�rischer Spiegel bildet den Mittelpunkt $M$ in sich selber ab. Rechts: Ein elliptischer Spiegel bildet die beiden Brennpunkte $F_1$ und $F_2$ ineinander ab. } \label{fig:Elliptischer-sphaerischer-Spiegel} \end{figure} -\section{Reelle Abbildungen mit der Lochkamera} +\subsubsection{Reelle Abbildungen mit der Lochkamera} -Eine Lochkamera (siehe Abbildung \ref{fig:Lochkamera}) ist eine geometrische Anordnung zur n�herungsweisen Abbildung beliebiger Punkte eines beleuchteten oder selbstleuchtenden Gegenstands. Dabei wird der Gegenstand in der Ebene $A$ durch einen undurchl�ssigen Schirm mit einer Lochblende auf die Beobachtungsebene $B$ abgebildet. +Eine Lochkamera (siehe Abbildung \ref{fig:Lochkamera}) ist eine geometrische Anordnung zur n�herungsweisen Abbildung beliebiger Punkte eines beleuchteten oder selbstleuchtenden Gegenstands. Dabei wird der Gegenstand in der Ebene $A$ durch einen undurchl�ssigen Schirm mit einer Lochblende auf die Beobachtungsebene $B$ abgebildet. Die Abbildung ist seitenverkehrt und kopf�ber. -Sei $d$ der Durchmesser der Blenden�ffnung, $a$ bzw. $b$ der jeweilige Abstand zwischen der Ebene $A$ und $B$, dann werden alle Strahlen, die von einem Punkt $P$ kommen, in eine elliptische Scheibe um $P^{\prime}$ abgebildet, deren Durchmesser $d^{\prime}$ sich mithilfe des Strahlensatzes bestimmen l�sst: +Sei $d$ der Durchmesser der Blenden�ffnung, $a$ bzw. $b$ der jeweilige Abstand zwischen der Ebene $A$ und $B$, dann werden alle Strahlen, die von einem Punkt $P$ kommen, in eine Scheibe um $P^{\prime}$ abgebildet, deren Durchmesser $d^{\prime}$ sich mithilfe des Strahlensatzes bestimmen l�sst: \[d^{\prime}=\frac{a+b}{a}\:d.\] @@ -102,12 +104,12 @@ Eine Lochkamera erzeugt damit keine exakte punktweise Abbildung, die Sch \label{fig:Lochkamera} \end{figure} -Das optische Aufl�sungsverm�gen einer Lochkamera ist beugungsbedingt begrenzt, d.h. man kann die Lochblende nicht beliebig verkleinern. Sobald die Gr��e des zentralen Beugungsmaximums $d_B>d^{\prime}$ erf�llt, d.h. +Das optische Aufl�sungsverm�gen einer Lochkamera ist beugungsbedingt begrenzt. Sobald die Gr��e des zentralen Beugungsmaximums $d_B$ bei der Wellenl�nge $\lambda$ \[d_B=\frac{2b\cdot\lambda}{d}>\frac{a+b}{a}\:d=d^{\prime},\] - -wird die Bildsch�rfe wieder schlechter. Die Lochblende hat damit einen optimalen Blendendurchmesser von +vergleichbar wird mit $d^\prime$, +wird die Bildsch�rfe mit kleiner werdenden Blendendurchmesser wieder schlechter. Die Lochblende hat damit einen optimalen Blendendurchmesser von \[d_{\text{opt}}=\sqrt{\frac{a\cdot b}{a+b}\cdot 2\lambda}.\] @@ -116,29 +118,35 @@ Keine Abbildungsfehler, einfacher Aufbau (z.B. f geringe Lichtst�rke aufgrund kleiner Lochblendendurchmesser. Mit einer Lochblende (z.B. ein kleines Loch in Alufolie) ganz dicht vor dem Auge gehalten, lassen sich aus unmittelbarer N�he und bei ausreichendem Licht sehr kleinskalige Strukturen betrachten. -Um die Lichtst�rke zu verbessern, bedient man sich daher anderer abbildender Elemente wie Linsen und Hohlspiegel die gr��ere �ffnungen erlauben und das Bild eines Gegenstandes in jeder Entfernung erzeugen k�nnen. +Um die Lichtst�rke zu verbessern, bedient man sich daher anderer abbildender Elemente wie Linsen und Hohlspiegel, die Licht sammeln und das Bild eines Gegenstandes in jeder Entfernung erzeugen k�nnen. -\subsection{Hohlspiegel - paraxiale N�herung} +\subsubsection{Hohlspiegel - paraxiale N�herung} -In diesem Abschnit befassen wir uns mit \index{Hohlspiegel} Hohlspiegel, die im Gegensatz zum ebenen Spiegel keine verzerrungsfreien 1:1 Bilder von Gegenst�nden erzeugen. Aufgrund der gekr�mmten Spiegelfl�chen werden verkleinerte oder vergr��erte Bilder erzeugt, welche in der Regel nicht mehr v�llig verzerrungsfrei sind. +In diesem Abschnitt befassen wir uns mit \index{Hohlspiegel} Hohlspiegel, die im Gegensatz zum ebenen Spiegel keine verzerrungsfreien 1:1 Bilder von Gegenst�nden erzeugen. Aufgrund der gekr�mmten Spiegelfl�chen werden verkleinerte oder vergr��erte Bilder erzeugt, welche in der Regel nicht mehr v�llig verzerrungsfrei sind. \begin{figure}[htbp] \centering \includegraphics[width=0.90\textwidth]{figures/Kugelspiegel.png} - \caption{a) Sph�rischer Hohlspiegel mit Brennpunkt $F$, Kugelmittelpunkt $M$ und Brennweite $f=\overline{OF}\approx\frac{R}{2}$, b) Zur Abh�ngigkeit der Brennweite $f$ eines Kugelspiegels: Achsenfernere Strahlen besitzen eine kleinere Brennweite als achsennahe Strahlen.} + \caption{a) Sph�rischer Hohlspiegel mit Brennpunkt $F$, Kugelmittelpunkt $M$ und Brennweite $f=\overline{OF}\approx\frac{R}{2}$, b) Zur Abh�ngigkeit der Brennweite $f$ eines Kugelspiegels: Achsenfernere Strahlen besitzen eine kleinere Brennweite als achsennahe Strahlen (sph�rische Aberration + \index{sph�rische Aberration}).} \label{fig:Kugelspiegel} \end{figure} -Der Grund hierf�r ist in Abbildung \ref{fig:Kugelspiegel} zu sehen. In \ref{fig:Kugelspiegel} a) werden alle Strahlen zun�chst gem�� Reflexionsgesetz an der Spiegelfl�che reflektiert. Da das Dreieck $MFS$ gleichschenklig ist, gilt: +Der Grund hierf�r ist in Abbildung \ref{fig:Kugelspiegel} dargestellt. In \ref{fig:Kugelspiegel} a) werden alle Strahlen zun�chst gem�� Reflexionsgesetz an der Spiegelfl�che reflektiert. Da das Dreieck $MFS$ gleichschenklig ist, gilt: \[\overline{FM}=\frac{\frac{R}{2}}{\cos\alpha}\Rightarrow f=\overline{OF}=R\left(1-\frac{1}{2\cos\alpha}\right)=R\left[1-\frac{1}{2\cos\alpha}\right]=R\left[1-\frac{R}{\sqrt{R^2-h^2}}\right],\] wobei $\cos\alpha=\sqrt{1-\sin^2\alpha}$ und $\sin\alpha=\frac{h}{R}$ benutzt wurde. F�r achsennahe Strahlen (d.h. kleines $h$) gilt n�herungsweise $\cos\alpha\approx 1$ und damit $\overline{OF}=f=\frac{R}{2}$. Dies nennt man \textbf{paraxiale N�herung}. + +\parbox{0.4\linewidth}{ \textbf{Aber:}\\ Im Allgemeinen ist der Schnittpunkt $F$, und damit die Brennweite der reflektierten Strahlen abh�ngig vom Abstand $h$ der einfallenden Strahlen. Hinweis: Betrachten Sie das Bild am Boden einer Tasse - die charakteristische Form nennt sich \textit{Katakaustik} \index{Katakaustik}. �berlegen Sie sich (bevor Sie es googeln), wie die Form einer Katakaustik zustande kommen kann. +} +\parbox{0.05\linewidth}{\hfill} +\parbox{0.55\linewidth}{\includegraphics[width=0.7\linewidth]{pictures/IMG_0798.png}} \subsubsection{Abbildungen mit dem Hohlspiegel} @@ -165,10 +173,10 @@ F woraus die \index{Abbildungsgleichung-Hohlspiegel}\textbf{Abbildungsgleichung} f�r achsennahe Strahlen folgt: -\begin{center} -\fboxsep=5. mm -\fbox{$\frac{1}{g}+\frac{1}{b}\approx\frac{2}{R}\approx\frac{1}{f}$} -\end{center} + +\begin{wichtig}[Abbildungsgleichung: Hohlspiegel, paraxiale N�herung] + \[ \frac{1}{g} + \frac{1}{b} \approx \frac{2}{R}\approx \frac{1}{f} \] +\end{wichtig} Diese Gleichung verkn�pft unter den gemachten Annahmen die Gegenstandsweite $g$ mit der Bildweite $b$ und der Brennweite $f$. @@ -192,12 +200,17 @@ Es existieren drei charakteristische Strahlen, die man von $A$ aus zeichnen kann \item $S_3$ verl�uft durch den Kugelmittelpunkt und wird in sich selbst reflektiert. \end{itemize} -In der paraxialen N�herung schneiden sich alle drei Strahlen im Punkt $B$, welcher f�r $g=\overline{A^{\prime}O}>R=\overline{OM}$ zwischen $F$ und $M$ liegt. Da $B$ in diesem Fall auf der anderen Seite der Symmetrieachse liegt, erh�lt man umgekehrtes Bild. Aus Abbildung \ref{fig:Spiegel-Bildkonstruktion} folgt, wegen $\frac{\overline{AA^{\prime}}}{\overline{A^{\prime}O}}=\tan\beta=\frac{\overline{BB^{\prime}}}{\overline{B^{\prime}O}}$, f�r den Abbildungsma�stab: +In der paraxialen N�herung schneiden sich alle drei Strahlen im Punkt $B$, welcher f�r $g=\overline{A^{\prime}O}>R=\overline{OM}$ zwischen $F$ und $M$ liegt. Da $B$ in diesem Fall auf der anderen Seite der Symmetrieachse liegt, erh�lt man ein umgekehrtes Bild. Aus Abbildung \ref{fig:Spiegel-Bildkonstruktion} folgt, wegen $\frac{\overline{AA^{\prime}}}{\overline{A^{\prime}O}}=\tan\beta=\frac{\overline{BB^{\prime}}}{\overline{B^{\prime}O}}$, f�r den Abbildungsma�stab $m$: + + +\[m:=\frac{\overline{AA^{\prime}}}{\overline{BB^{\prime}}}=\frac{g-R}{R-b}=\frac{g}{b},\] +f�r die letzte Umformung ist die Abbildungsgleichung verwendet worden. -\[\frac{\overline{AA^{\prime}}}{\overline{BB^{\prime}}}=\frac{g-R}{R-b}=\frac{g}{b}.\] +Der Abbildungsma�stab + +Man erh�lt ein virtuelles Bild falls der Gegenstand $AA^{\prime}$ zwischen Brennpunkt $F$ und Spiegel liegt, da sich in diesem Fall die r�ckw�rtigen Verl�ngerungen (in paraxialer N�herung) in den Punkten $BB^{\prime}$ sammeln. Das Bild wird dann nicht mehr auf einen Schirm abgebildet, den man in die Bildebene des virtuellen Bildes stellt, sondern nur das hinter dem Spiegel erscheinende Spiegelbild des Gegenstandes. -Man erh�lt ein virtuelles Bild falls der Gegenstand $AA^{\prime}$ zwischen Brennpunkt $F$ und Spiegel liegt, da sich in diesem Fall die r�ckw�rtigen Verl�ngerungen (in paraxialer N�herung) in den Punkten $BB^{\prime}$ sammeln. Das Bild wird dann nicht mehr auf einen Schirm abgebildet, den man in die Bildebene des virtuellen Bildes stellt, sondern nur das hinter dem Spiegel erscheinende Spiegelbild des Gegenstandes.\\ Au�erdem spricht man von einem \textbf{konkaven} Spiegel (Hohlspiegel) falls $M$ auf der gleichen Seite wie der Gegenstand $A$ liegt, und von einem \textbf{konvexen} Spiegel (W�lbspiegel) falls $M$ und $A$ auf entgegengesetzten Seiten des Spiegels liegen. Im letzteren Fall erh�lt man nur virtuelle Bilder. Der Abbildungsma�stab $m$ beim Hohlspiegel ist aufgrund geometrischer Verh�ltnisse definiert als @@ -208,31 +221,40 @@ F \[\frac{1}{g}-\frac{1}{mg}=\frac{1}{f}\Rightarrow\frac{1}{mg}=\frac{1}{g}-\frac{1}{f}\Rightarrow m=\frac{f}{f-g}.\] -Aus dieser Gleichung geht hervor, dass man f�r $g>2f$ ein verkleinertes reelles Bild $\left(-1<m<0\right)$ erh�lt, und f�r $f<g<2f$ das Bild vergr��ert ist (d.h. $m<-1$). Wenn $g<f$ ist, so wird $m>0$, der Gegenstand als virtuelles Bild vergr��ert und mit gleichuer Orientiertung abgebildet. -\\ +Aus dieser Gleichung geht hervor, dass man f�r $g>2f$ ein verkleinertes reelles Bild $\left(-1<m<0\right)$ erh�lt, und f�r $f<g<2f$ das Bild vergr��ert ist (d.h. $m<-1$). Wenn $g<f$ ist, so wird $m>0$, der Gegenstand als virtuelles Bild vergr��ert und mit gleicher Orientiertung abgebildet. + -Bei einem W�lbspiegel gilt zwischen Kr�mmungsradius und Brennweite $f=\frac{R}{2}$, und da Kr�mmungsmittelpunkt und Brennpunkt hinter dem Spiegel liegen, haben wir es mit einer negativen Brennweite zu tun, d.h. $f<0$. Aus der Spiegelgleichung $\frac{1}{q}+\frac{1}{b}=\frac{1}{f}$ folgt damit, $f<b<0$, daher entsteht immer ein virtuelles, aufrechtes Bild beim W�lbspiegel. Die Vergr��erung ist stets kleiner als $1$. Dies sieht man auch aus der obigen Betrachtung des Hohlspiegels, denn f�r $f<0$ (wie beim W�lbspiegel der Fall) gilt stets $0<m<1$, d.h. man erh�lt man virtuelles, verkleinertes Bild. Aufgrund dieser Eigenschaft werden W�lbspiegel u.a. zur �berwachung ansonsten un�bersichtlicher Umgebungen eingesetzt, z.B. als Verkehrspiegel oder �berwachungsspiegel zur Vermeidung von Diebstahlsdelikten.\\ +Bei einem W�lbspiegel gilt der Zusammenhang zwischen Kr�mmungsradius $R$ und Brennweite $f=\frac{R}{2}$. Da Kr�mmungsmittelpunkt und Brennpunkt hinter dem Spiegel liegen, haben wir es mit einer negativen Brennweite zu tun, d.h. $f<0$. Aus der Spiegelgleichung $\frac{1}{g}+\frac{1}{b}=\frac{1}{f}$ folgt damit, $f<b<0$, daher entsteht immer ein virtuelles, aufrechtes Bild beim W�lbspiegel. Die Vergr��erung ist stets kleiner als $1$. Dies sieht man auch aus der obigen Betrachtung des Hohlspiegels, denn f�r $f<0$ (wie beim W�lbspiegel der Fall) gilt stets $0<m<1$, d.h. man erh�lt ein virtuelles, verkleinertes Bild. Aufgrund dieser Eigenschaft werden W�lbspiegel u.a. zur �berwachung ansonsten un�bersichtlicher Umgebungen eingesetzt, z.B. als Verkehrspiegel oder �berwachungsspiegel zur Vermeidung von Diebstahlsdelikten. \textbf{Anmerkung:}\\ -Das Problem der h�henabh�ngigen Brennweite achsenparalleler Strahlen bei sph�rischen Hohlspiegeln l�sst sich durch Verwendung eines \textbf{Parabolspiegels} vermeiden, bei der die Spiegelfl�che parabelf�rmig ist und die dazu f�hrt dass eine ebene Welle nach der Reflexion in einem Brennpunkt $F$ fokussiert wird. So entsteht aus einer ebenen Welle eine Kugelwelle. +Das Problem der h�henabh�ngigen Brennweite achsenparalleler Strahlen bei sph�rischen Hohlspiegeln l�sst sich durch Verwendung eines \textbf{Parabolspiegels} vermeiden, bei der die Spiegelfl�che parabelf�rmig ist. Eine ebene Welle wird nach der Reflexion in einem Brennpunkt $F$ fokussiert wird ohne sph�rische +Aberrationseffekte. +\begin{figure} + \includegraphics[width=0.6\linewidth]{pictures/IMG_0799.png} + \includegraphics[width=0.33\linewidth]{pictures/IMG_0801.png} + \caption{Links: Zwei Abbildungen eines Hohl- und W�lbspiegels (Suppenl�ffel), rechts: reales Bild f�r $g<f$.} +\end{figure} \subsection{Linsen (d�nne Linsen, Linsensysteme)} -Das wichtigste optische Element in der Entwicklungsgeschichte der Optik ist bzw. sind \index{optische Linsen}\textbf{optische Linsen} in ihren verschiedenen Ausf�hrungsformen. Historisch gesehen wurden Linsen zun�chst in Fernrohren verwendet, und u.a. von Galilei dazu benutzt erstmals die nach im benannten ersten vier Monde des Jupiter (Io, Europa, Ganymed und Callisto) zu beobachten.\\ +Das wichtigste optische Element in der Entwicklungsgeschichte der Optik ist bzw. sind \index{optische Linsen}\textbf{optische Linsen} in ihren verschiedenen Ausf�hrungsformen. -Neben dem Linsenfernrohr basieren viele weitere wichtige optische Ger�te (z.B. Brille, Lupe, Mikroskop, Projektoren, Fotoapparat) auf \textbf{Linsensystemen}, d.h. auf geeigneten Kombinationen verschiedener Linsen.\\ +Neben dem Linsenfernrohr basieren viele weitere wichtige optische Ger�te (z.B. Brille, Lupe, Mikroskop, Projektoren, Fotoapparat) auf \textbf{Linsensystemen}, d.h. auf geeigneten Kombinationen verschiedener Linsen. -Man unterscheidet grob zwischen \textbf{Sammellinsen} und \textbf{Zerstreuungslinsen} und zwischen dicken und d�nnen Linsen, wobei wir uns hier der Einfachheit halber nur auf den letzteren Fall konzentrieren.\\ +Man unterscheidet zwischen \textbf{Sammellinsen} und \textbf{Zerstreuungslinsen}. \subsubsection{Kr�mmungsradius und Linsentypen} +\todo{Zun�chst sollte die Brechung an einer gekr�mmten Fl�che betrachtet werden. +Dann muss der �bergang f�r d�nne Linsen durch die Darstellung einer Brechungsebene +ersetzt werden.} -Eine Linse besteht im Allgemeinen aus einem durchsichtigen Material mit Brechungsindex $n_2$, das auf beiden Seiten durch polierte Grenzfl�chen von einem anderen Medium mit Brechungsindex $n_1$. Die Abbildung mittels einer Linse beruht damit auf der Lichtbrechung gem�� Snelliuschem Brechungsgesetz an der Grenzfl�chen zweier optischer Medien mit unterschiedlicher Brechzahl.\\ +Eine Linse besteht im Allgemeinen aus einem durchsichtigen Material mit Brechungsindex $n_2$, das auf beiden Seiten durch polierte Grenzfl�chen von einem anderen Medium mit Brechungsindex $n_1$ umgeben ist. Der Verlauf der Strahlen l�sst sich mit dem Snell'schen Brechungsgesetz an den gekr�mmten Grenzfl�chen verfolgen. \textbf{Anmerkung:}\\ -Betrachtet man Luft als umgebendes Medium, so ist $n_1=1$ und $n_2=n$.\\ +Betrachtet man Luft als umgebendes Medium, so ist $n_1=1$ und $n_2=n$. -Wir unterscheiden Sammel-und Zerstreuungslinsen anhand des Vorzeichens der Kr�mmung der Linsenl�che an der die Lichtstrahlen gebrochen werden. Der \textbf{Kr�mmungsradius} $R$ einer gekr�mmten Fl�che ist die gerichtete Strecke von dieser Fl�che zum Kr�mmungsmittelpunkt $M$. Es ist $R>0$ (d.h. positive Kr�mmung), falls $R$ in den positiven Halbraum zeigt und $R<0$ (d.h. negative Kr�mmung) falls $R$ in den negativen Halbraum zeigt. In beiden F�llen kommt das Licht von links, d.h. es l�uft vom negativen in den positiven Halbraum.\\ +Wir unterscheiden Sammel-und Zerstreuungslinsen anhand des Vorzeichens der Kr�mmung der Linsenl�che, an der die Lichtstrahlen gebrochen werden. Der \textbf{Kr�mmungsradius} $R$ einer gekr�mmten Fl�che ist die gerichtete Strecke von dieser Fl�che zum Kr�mmungsmittelpunkt $M$. Es ist $R>0$ (d.h. positive Kr�mmung), falls $R$ in den positiven Halbraum zeigt und $R<0$ (d.h. negative Kr�mmung) falls $R$ in den negativen Halbraum zeigt. In beiden F�llen kommt das Licht von links, d.h. es l�uft vom negativen in den positiven Halbraum. \begin{figure}[htbp] \centering diff --git a/v0.1/pictures/IMG_0789.png b/v0.1/pictures/IMG_0789.png new file mode 100644 index 0000000000000000000000000000000000000000..1ab7c18c597fa6a57debdd45e3343e6deb418ee1 Binary files /dev/null and b/v0.1/pictures/IMG_0789.png differ diff --git a/v0.1/pictures/IMG_0798.JPG b/v0.1/pictures/IMG_0798.JPG new file mode 100644 index 0000000000000000000000000000000000000000..2169fe42531f86b7081617fbad52520560375991 Binary files /dev/null and b/v0.1/pictures/IMG_0798.JPG differ diff --git a/v0.1/pictures/IMG_0798.png b/v0.1/pictures/IMG_0798.png new file mode 100644 index 0000000000000000000000000000000000000000..7a1218718cdc32d66ec75f2814837e22d061aa5c Binary files /dev/null and b/v0.1/pictures/IMG_0798.png differ diff --git a/v0.1/pictures/IMG_0799.png b/v0.1/pictures/IMG_0799.png new file mode 100644 index 0000000000000000000000000000000000000000..4448b14e82e3583e9ee451ef195b78a07abbb63a Binary files /dev/null and b/v0.1/pictures/IMG_0799.png differ diff --git a/v0.1/pictures/IMG_0801.png b/v0.1/pictures/IMG_0801.png new file mode 100644 index 0000000000000000000000000000000000000000..e8d07a1b99b41b0ed336a32b4dfb364416715c50 Binary files /dev/null and b/v0.1/pictures/IMG_0801.png differ diff --git a/v0.1/skript.pdf b/v0.1/skript.pdf index aae78d6bfbb2a448e4d777fec3df667ef5aedaa6..d0c3e7bf1b7f49310aa5cc96d112c779618f6166 100644 Binary files a/v0.1/skript.pdf and b/v0.1/skript.pdf differ diff --git a/v0.1/skript.tex b/v0.1/skript.tex index 8c37b817a552ebc06fcac7bbf5563f89845b058a..0220b60db27ed008576aba7ae805e9c3be9f6bde 100644 --- a/v0.1/skript.tex +++ b/v0.1/skript.tex @@ -47,7 +47,7 @@ \setlength{\parskip}{1.2ex} % ------------------------------ -% \makeindex +\makeindex \begin{document} @@ -56,7 +56,8 @@ \includegraphics[width = 80mm]{figures/logo.png}\\[8ex] \vfill \begin{center} -{\bfseries \Huge Begleitendes Skript f\"ur Physik II \\ WiSe 2017/2018}\\[+2cm] +{\bfseries \Huge Begleitendes Skript f\"ur die Vorlesung + Experimtalphysik II }\\[+2cm] {\large Prof. Dr. Dieter Horns} \vfill \def\zangle{-10} @@ -107,12 +108,12 @@ node[below, sloped, near end]{Reflektor}; %\end{abstract} \section*{Vorbemerkungen} Das vorliegende Skript ist \textit{work in progress}. Bitte seien Sie nachsichtig mit mir und helfen bei der Verbesserung des Skripts -- produktive Kommentare und Vorschl\"age sind immer willkommen. Falls Sie etwas konkret beisteuern oder verbessern wollen, sprechen Sie mich bitte an. Offensichtlich ist noch nicht alles abgedeckt - zur Zeit finden Sie nur einen Gro\3teil des Abschnitts zu elektromagnetischen Wellen. Die anderen Abschnitte sind in Vorbereitung (danke auch an Victor Danescu, der mir sehr viel geholfen hat). Denken Sie auch daran, dass dies kein Lehrbuch sein soll. Die Darstellung ist sehr eng mit der Vorlesung verkn\"upft und ist bewusst knapp gew\"ahlt. \\ -{\flushright{Prof. Dr. Dieter Horns, Januar 2018.}}\newpage +{\flushright{Prof. Dr. Dieter Horns, Mai 2018.}}\newpage \setcounter{chapter}{0} \input{sectionheaders1} \input{ac_circuit} \input{em_waves} \input{geometrical_optics} - % \printindex -\end{document} \ No newline at end of file +\printindex +\end{document}