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diff --git a/manual_source/chapters/fabry-perot.tex b/manual_source/chapters/fabry-perot.tex new file mode 100644 index 0000000..d74745f --- /dev/null +++ b/manual_source/chapters/fabry-perot.tex @@ -0,0 +1,298 @@ +%\chapter*{Fabry-Perot Interferometer and the Sodium Doublet} +%\addcontentsline{toc}{chapter}{Fabry-Perot Interferometer} +\documentclass{article} +\usepackage{tabularx,amsmath,boxedminipage,epsfig} + \oddsidemargin 0.0in + \evensidemargin 0.0in + \textwidth 6.5in + \headheight 0.0in + \topmargin 0.0in + \textheight=9.0in + +\begin{document} +\title{Fabry-Perot Interferometer and the Sodium Doublet} +\date {} +\maketitle + + +\noindent + \textbf{Experiment objectives}: Assemble and align Fabry-Perot interferometer, + and use it to measure differential wavelength for the Na doublet. + + \section*{Theory} + +\subsection*{The Fabry-Perot Interferometer} + +Any interferometer relies on interference between two or more light field. In a Fabry-Perot configuration input +light field bounces between two closely spaced partially reflecting surfaces, creating a large number of +reflections. Interference of these multiple beams produces sharp spikes in the transmission for certain light +frequencies. Thanks to the large number of interfering rays, this type of interferometer has extremely high +resolution, much better than, for example, a Michelson interferometer. For that reason Fabry-Perot +interferometers are widely used in telecommunications, lasers and spectroscopy to control and measure the +wavelengths of light. In this experiment we will take advantage of high spectral resolution of the Fabry-Perot +interferometer to resolve two very closely-spaces emission lines in Na spectra by observing changes in a +overlapping interference fringes from these two lines. +\begin{figure}[h] +\begin{center} +\includegraphics[width=0.8\linewidth]{./pdf_figs/pfig1} +\caption{\label{fpfig1}Sequence of Reflection and +Transmission for a ray arriving at the treated inner surfaces $P_1 \& P_2$.} +\end{figure} +\end{figure} + +A Fabry-Perot interferometer consists of two parallel glass plates, flat to better than 1/4 of an optical +wavelength $\lambda$, and coated on the inner surfaces with a partially transmitting metallic layer. Such +two-mirror arrangement is normally called an {\it optical cavity}. The light in a cavity by definition bounces +back and forth many time before escaping; the idea of such a cavity is crucial for the construction of a laser. +Any light transmitted through such cavity is a product of interference between beams transmitted at each bounce +as diagrammed in Figure~\ref{fpfig1}. When the incident ray arrives at interface point $A$, a fraction $t$ is +transmitted and the remaining fraction $r$ is reflected, such that $t + r = 1$ ( this assumes no light is lost +inside the cavity). The same thing happens at each of the points $A,B,C,D,E,F,G,H\ldots$, splitting the initial +ray into parallel rays $AB,CD,EF,GH,$ etc. Between adjacent ray pairs, say $AB$ and $CD$, there is a path +difference of : +\begin{equation} + \delta = BC+CK +\end{equation}%eq1 + where $BK$ is normal to $CD$. In a development +similar to that used for the Michelson interferometer, you can show +that: +\begin{equation} + \delta = 2d\cos\theta +\end{equation}%eq.2 + If this path difference produces +constructive interference, then $\delta$ is some integer multiple of , +$\lambda$ namely, +\begin{equation} + m\lambda = 2d\cos\theta %eq.3 +\end{equation}%eq.3 + +This applies equally to ray pairs $CD$ and $EF, EF$ and $GH$, etc, so that all parallel rays to the right of +$P2$ will constructively interfere with one another when brought together. + +Issues of intensity of fringes \& contrast between fringes and dark background +are addressed in Melissinos, {\it Experiments in Modern Physics}, pp.309-312. + +\subsection*{The Sodium Doublet} + + In this lab you will measure the separation between the two + famous ``sodium doublet'' lines, the two closely spaced lines + which occur at 589 $nm$ and 589.59 $nm$, respectively. This ``doublet'' + emission is evidence that the atomic electron has the property + of intrinsic angular momentum, or spin S. As you are learning + in Modern Physics 201, the discrete spectral lines in atomic + emission are due to the quantization of electron energies in + the atom. As Niels Bohr postulated, electrons in atoms are + only allowed to absorb and emit energy in discrete + quantities. When an electron moves from one orbit to another + in an atom, a well-defined amount of energy is emitted as + light at a fixed wavelength. Later in this class we will + explore the spectra of various atomic gases. +\begin{figure}[h] +\centerline{\epsfig{height=5cm, file=fpfig2.eps}} \caption{\label{fpfig2.fig}Fine Structure Splitting in sodium +giving rise to the sodium doublet lines} +\end{figure} + For many + atoms, {\bf atomic levels are further split}, for example, + by interactions of electrons with each other (Russell-Saunders + coupling), external magnetic fields (Zeeman effect), and even + the interaction between the spin of an electron and the + magnetic field created by its orbital angular momentum + (spin-orbit coupling). This is known as fine structure + splitting (FSS). The fine structure splitting for the sodium + 3P state is due to spin-orbit coupling, and is illustrated in + Figure \ref{fpfig2.fig}. The "3P" state refers to sodium's +valence electron + which has a principal quantum number of $n=3$ and an orbital + quantum number of $l=1$ (a P-state). Further, the electron has + an intrinsic spin (like a top), described by a spin quantum + number $S$, which is either +1/2 or -1/2. The electron has a + magnetic moment due to its intrinsic spin, given by $m_S$. Due to +its orbital angular momentum around a charged nucleus, it + senses a magnetic field ${\mathbf H}$. The energy of interaction of a + magnetic moment in a field is given by $E = -\mu\cdot {\mathbf H}$. +This gives + rise to the splitting and two spectral emission lines. + + +\section*{Procedure} + +\subsection*{Set Up} +\textbf{Equipment needed}: Pasco precision interferometry kit, Na lamp, +adjustable-hight platform. + +\begin{figure} +\centerline{\epsfig{width=0.7\linewidth,file=fpfig3new.eps}} \caption{\label{fpfig3.fig}The Fabry-Perot +Interferometer} +\end{figure} + +The interferometer layout is shown in Figure \ref{fpfig3.fig}. The inner spacing $d$ between two +partially-reflecting mirrors ($P1$ and $P2$) can be roughly adjusted by loosening the screw that mounts $P2$ to +its base. It is important that the plates are as closely spaced as possible. Move the plates to within $1.0 - +1.5$~mm of each other, but make sure the mirrors do not touch! + +\subsection*{Data acquisition} + +\begin{enumerate} +\item \textbf{Turn on the sodium lamp as soon as you arrive. It should warm up for about 20 minutes +before starting}. +\item Turn the micrometer close to or at 0.00. +\item Remove the diffuser sheet from in front of the lamp. Look through +plate $P2$ towards the lamp. If the plates are not parallel, you will see +multiple images of the lamp filament. Adjust the knobs on $P1$ until the images +collapse into one. At this point, you should see faint circular interference +fringes appear. +\item Place the diffuser sheet in +front of the lamp so you will now only see the fringes. Continue to adjust the +knobs on one plate (not the knobs to move the plate back and forth, but the +ones to bring it parallel) to get the best fringe pattern. It is the most +convenient to view the interference picture directly. +\item +Seek the START condition illustrated in Fig.(\ref{fpfig4.fig}), in which all bright fringes are evenly spaced. +You do this by moving the micrometer. Note that alternate fringes may be of somewhat different intensities, one +corresponding to fringes of $\lambda_1$, the other to those for $\lambda_2$. If you do not see this condition, +try moving the mirror $P2$ across the range of micrometer screw. If you still cannot find them, you can also +move the whole plate by loosening one plate and sliding it a little. +\item Practice going through the fringe conditions as shown in Fig.(\ref{fpfig4.fig}) +by turning the micrometer and viewing the fringes. Do not be surprised if you +have to move the micrometer quite a bit to go back to the original condition. +\item Find a place on the micrometer ($d_1$) where you +have the ``START'' condition for fringes shown in Fig.(\ref{fpfig4.fig}). Now +advance the micrometer rapidly while viewing the fringe pattern ( NO COUNTING +OF FRINGES IS REQUIRED ). Note how the fringes of one intensity are moving to +overtake those of the other intensity (in the manner of +Fig.(\ref{fpfig4.fig})). Keep turning until the ``STOP'' pattern is achieved +(the same condition you started with). Record the micrometer reading as $d_2$. +\emph{Remember that 1 tick mark is 1 micrometer ($10^{-6}m$). That means if you +read 1.24, your really move 124 $\mu m$.} +\end{enumerate} + +\noindent \fbox{\parbox{\linewidth}{\textbf{Experimental tip:} You may have to +``home in'' on the best START and STOP conditions. Let's say that the even +fringe spacing for the START condition ($d_1$) is not exactly in view. Now move +the micrometer, looking to see if the pattern moves toward even spacing. If so, +stop and read the micrometer for $d_1$. Move a bit more. If this second fringe +spacing looks better than the first, then accept this for $d_1$. The same +``homing in'' procedure should be used to select the reading for $d_2$. In +other words as you approach the even spacing condition of the STOP pattern, +start writing down the micrometer positions. Eventually you will favor one +reading over all the others.}} + +\section*{Analysis} + + Since the condition we are seeking above for ``START'' places + the bright fringes of $\lambda_1$ at the point of destructive + interference for $\lambda_2$, we can express this for the bull's eye + center ($\theta= 0 $) as: +\begin{equation} +2d_1=m_1\lambda_1=\left(m_1+n+\frac{1}{2}\right)\lambda_2 +\end{equation} + + Here the integer n accounts for the + fact that $\lambda_1 > \lambda_2$ , and the $1/2$ for the +condition of + destructive interference for $\lambda_2$ at the center. Since the + net action of advancing by many fringes has been to increment + the fringe count of $\lambda_2$ by one more than that of +$\lambda_1$ , + then we express the ``STOP'' condition as: +\begin{equation} +2d_2=m_2\lambda_1=\left(m_2+n+\frac{3}{2}\right)\lambda_2 +\end{equation} + Subtracting the + two interference equations gives us: +\begin{equation} +2(d_2-d_1)=(m_2-m_1)\lambda_1=(m_2-m_1)\lambda_2+\lambda_2 +\end{equation} + Eliminating $(m_2-m_1)$ + in this equation we obtain: + +\begin{equation} +2(d_2-d_1)=\frac{\lambda_1\lambda_2}{(\lambda_1-\lambda_2)} +\end{equation} + + Solving this for $\Delta \lambda = \lambda_1-\lambda_2$, and + accepting as valid the approximation that $\lambda_1\lambda_2\approx +\lambda^2$ ( where $\lambda$ is the + average of $\lambda_1$ and $\lambda_2 \approx 589.26 nm$ ), we obtain: +\begin{equation} +\boxed{\Delta\lambda=\frac{\lambda^2}{2(d_2-d_1)}} +\end{equation} + +Each lab partner should independently align the interferometer and make at least \textit{two} measurements of +``START'' and ``STOP'' positions. A sample table to record the data is shown below. \\{\large +\begin{tabular}{|p{27mm}|p{27mm}|p{27mm}|p{27mm}|} +\hline + $d_1$ $\pm \dots$ & $d _2$ $\pm \dots$& $(d_1-d_2)$ $\pm \dots$& +$\Delta \lambda(nm) $ $\pm \dots$\\ +\hline +&&&\\ +\hline &&&\\ \hline &&&\\ \hline &&&\\ \hline &&&\\ \hline +\end{tabular} +} + +\vspace{1cm} Calculate average value of Na doublet splitting and its standard deviation. Compare your result +with the established value of $\Delta \lambda_{Na}=0.598$~nm. + + +\begin{figure}[h] +\centerline{\epsfig{width=0.8\linewidth,file=fpfig4.eps}} \caption{\label{fpfig4.fig}The Sequence of fringe +patterns encountered in the course of the FSS measurements. Note false colors: in your experiment the background +is black, and both sets of rings are bright yellow.} +\end{figure} + +\end{document} + +\newpage +\noindent +Physics 251 Section:\\ +\hskip 4.5in Name:\\ +\hskip 4.5in Partners:\\ +\vskip 0.5in +\subsection*{The Fabry-Perot Interferometer} +1. Briefly describe how the Fabry-Perot interferometer gives and interference +pattern (in one or two sentences):\\ +\vskip 1.2in +2. How does the interferometer's resolving power of the fringes depend on the +reflectivity of plates, r ? That is, does the sharpness of the fringes increase +or decrease with r ? Consult Melissinos or Professor Kane's Mathview program. +(The reflectivity of the plates defines the {\it finess} of the cavity).\\ +\vskip 1in. + + +{\large +\noindent +Fill in: + +The sodium doublet lines arise because an atomic$\underline{\hskip 1.in}$ +is split into two by$\underline{\hskip 1.in}$ coupling. +The electron has +intrinsic $\underline{\hskip 1.in}$, like a top, with values of +$\underline{\hskip 1.in}$ or$\underline{\hskip 1.in}$. Because of +this, the electron has in intrinsic magnetic$\underline{\hskip 1.in}$ and +has magnetic +energy in a magnetic field given by E=$\underline{\hskip 1.in}$ +case comes from the electron's $\underline{\hskip 1.in}$ motion. + +} + +\subsection*{DATA:} +{\large +\begin{tabular}{|p{27mm}|p{27mm}|p{27mm}|p{27mm}|} +\hline + $d_1$ $\pm \dots$ & $D _2$ $\pm \dots$& $(d_1-d_2)$ $\pm \dots$& +$\Delta \lambda(nm) $ $\pm \dots$\\ +\hline +&&&\\ +\hline +&&&\\ \hline +&&&\\ \hline +&&&\\ \hline +&&&\\ \hline +\end{tabular} +} +\vskip .2in +$\Delta \lambda=$\hskip 1.5in nm\\ +\vskip .2in +Standard deviation= \hskip 1.5in nm +\newpage +\end{document} |