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+%\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}