updated manual

1 files changed, 103 insertions, 137 deletions

diff --git a/single-photon-interference.tex b/single-photon-interference.tex
index 350d5aa..53881da 100644
--- a/single-photon-interference.tex
+++ b/single-photon-interference.tex
@@ -1,7 +1,7 @@
 \documentclass[./manual.tex]{subfiles}
 \begin{document}
 
-\chapter{Single and Double-Slit Interference, One Photon at a Time}
+\chapter*{Single and Double-Slit Interference, One Photon at a Time}
 
 \noindent
  \textbf{Experiment objectives}: Study wave-particle duality for photons by measuring
@@ -23,40 +23,37 @@ There is a rich historical background behind the experiment you are about to per
  \caption{\label{young.fig} Young's double slit experiment. Top: The experimental setup.  Bottom: Young's interpretation of the result as interference, analogous to the corresponding phenomenon in water waves. The figure was drawn by Young himself.  {\it Images from Wikimedia.} }
 \end{figure}
 
-Young discovered that the light appeared as light and dark fringes on the screen, a fact that seemed impossible to explain in terms of corpuscles, but was very naturally explained in terms of waves, much like the interference observed in water and sound waves.  In particular, wave theory predicts that the intensity of light $I(x)$ from double slit interference should be distributed as:
+Young discovered that the light appeared as light and dark fringes on the screen, a fact that seemed impossible to explain in terms of corpuscles, but was very naturally explained in terms of waves, much like the interference observed in water and sound waves.  In particular, wave theory predicts that the intensity of light $I_{ds}(x)$ from double slit interference should be distributed as:
 
 \begin{equation} \label{2slit_wDif}
-I(x)= 4 I_0 \cos^2\left(\frac{\pi d}{\lambda}\frac{x}{\ell} \right)\left[\frac{\sin (\frac{\pi
+I_{ds}(x)= 4 I_0 \cos^2\left(\frac{\pi d}{\lambda}\frac{x}{\ell} \right)\left[\frac{\sin (\frac{\pi
 a}{\lambda}\frac{x}{\ell})}{\frac{\pi a}{\lambda}\frac{x}{\ell}} \right]^2
 \end{equation}
 
-Here, $x$ is the position on the viewing screen, with $x=0$ corresponding to the center of the interference pattern. The light wavelength is $\lambda$, the slits have a width of $a$, and they are separated by a distance $d$. The distance between the double slit and the viewing screen is $\ell$.
-
-Interference can also occur if there is only one slit. In this case the intensity distribution is described by Fraunhofer diffraction (see Fig.~\ref{interference.fig} and the derivations in the Appendix):
+Here, $x$ is the position on the viewing screen, with $x=0$ corresponding to the center of the interference pattern. The light wavelength is $\lambda$, the slits have a width of $a$, and they are separated by a distance $d$. The distance between the double slit and the viewing screen is $\ell$. In this expression the first $\cos$ term represents the interference of the two waves emerging from the two slits, while the last term (in square brackets) is due to the diffraction of light on a single slit, described by Fraunhofer diffraction (see Fig.~\ref{interference.fig} and the derivations in the Appendix B):
 \begin{equation} \label{1slit}
-I(x) = I_0 \frac{(\sin (\frac{\pi a}{\lambda}\frac{x}{\ell}))^2}{(\frac{\pi
+I_{ss}(x) = I_0 \frac{(\sin (\frac{\pi a}{\lambda}\frac{x}{\ell}))^2}{(\frac{\pi
 a}{\lambda}\frac{x}{\ell})^2}
 \end{equation}
 
 
-Young's experiment not only suggested that such `light waves' existed but was also able to determine the wavelength of light.  You should spend some time thinking about the latter point. How would the interference pattern change if we change $\lambda$? Given a particular interference pattern how might we determine $\lambda$? % refer to the calculation later in the lab
+%Young's experiment not only suggested that such `light waves' existed but was also able to determine the wavelength of light.  You should spend some time thinking about the latter point. How would the interference pattern change if we change $\lambda$? Given a particular interference pattern how might we determine $\lambda$? % refer to the calculation later in the lab
 
 
 
 
-\begin{figure}[h]
+
+
+ Light waves became even more acceptable with dynamical theories of light, such as Fresnel's and Maxwell's, in the 19th century, until it seemed that the wave theory of light was incontrovertible. And yet the discovery of the photoelectric effect, and its explanation in terms of light quanta by Einstein, threw the matter into dispute again.  The explanations of blackbody radiation, of the photoelectric effect, and of the Compton effect seemed to point to the existence of `photons', quanta of light that possessed definite and indivisible amounts of energy and momentum.  These are very satisfactory explanations so far as they go, but they throw into question the destructive-interference explanation of Young's experiment.  Does light have a dual nature, of waves and of particles?  And if experiments force us to suppose that it does, how does the light know when to behave according to each of its natures?
+ \begin{figure}[h]
 \centering
 \includegraphics[width=0.4\linewidth]{./pdf_figs/ambigram} \caption{\label{ambigram.fig} ``Light is a Particle / Light is a Wave'' oscillation ambigram (from \emph{For the Love of Line and Pattern}, p. 30).}
 \end{figure}
 
- Light waves became even more acceptable with dynamical theories of light, such as Fresnel's and Maxwell's, in the 19th century, until it seemed that the wave theory of light was incontrovertible. And yet the discovery of the photoelectric effect, and its explanation in terms of light quanta by Einstein, threw the matter into dispute again.  The explanations of blackbody radiation, of the photoelectric effect, and of the Compton effect seemed to point to the existence of `photons', quanta of light that possessed definite and indivisible amounts of energy and momentum.  These are very satisfactory explanations so far as they go, but they throw into question the destructive-interference explanation of Young's experiment.  Does light have a dual nature, of waves and of particles?  And if experiments force us to suppose that it does, how does the light know when to behave according to each of its natures?
-
-It is the purpose of this experimental apparatus to make the phenomenon of light interference as concrete as
-possible, and to give you the hands-on familiarity that will allow you to confront wave-particle duality in a
-precise and definite way.  When you have finished, you might not fully understand the mechanism of duality --
+It is the purpose of this experimental apparatus to allow you to explore the phenomenon of light interference in two cases: strong laser field (expected to behave as a wave) and a light so weak that only one photon is detected at a time (expected to behave as a particle). Comparing the interference patterns in these two limits will allow you to confront wave-particle duality in a
+precise and definite way.  When you are finished, you might not fully understand the mechanism of duality --
 Feynman asserts that nobody really does -- but you will certainly have
-direct experience with the actual phenomena
-that motivates all this discussion.
+direct experience with the actual phenomena.
 
 \newpage
 
@@ -80,19 +77,19 @@ You will need the Teachspin ``Two-slit interference'' apparatus, an oscilloscope
  As long as the top is off the rod must remain down and the HV must remain off. Finally, when the HV is on the laser/off/bulb switch at the far left end of the interference channel must be on bulb mode, or off. The laser is far too intense for the PMT. Make sure you have read this, understood, and got sign-off from the instructor before proceeding.
 \end{framed}
 
-\begin{figure}
+\begin{figure}{h!}
 \centering
 \includegraphics[width=0.8\linewidth]{./pdf_figs/tsisetup} \caption{\label{tsifig1.fig}The double-slit interference apparatus.}
 \end{figure}
 
 \begin{figure}
 \centering
-\includegraphics[width=0.8\linewidth]{./pdf_figs/bnc_and_banana_annotated} \caption{\label{bnc.fig} RG58 coaxial cable with a BNC connector and a BNC$\leftrightarrow$banana connector.}
+\includegraphics[width=0.4\linewidth]{./pdf_figs/bnc_and_banana_annotated} \caption{\label{bnc.fig} RG58 coaxial cable with a BNC connector and a BNC$\leftrightarrow$banana connector.}
 \end{figure}
 
 \begin{figure}
 \centering
-\includegraphics[width=0.8\linewidth]{./pdf_figs/shutter_up_down} \caption{\label{shutter.fig} The apparatus's shutter. When down it protects the PMT and exposes the photodiode to light in the interference channel. When up, the PMT is exposed to the light. The shutter must be kept down when the box is open.}
+\includegraphics[width=0.4\linewidth]{./pdf_figs/shutter_up_down} \caption{\label{shutter.fig} The apparatus's shutter. When down it protects the PMT and exposes the photodiode to light in the interference channel. When up, the PMT is exposed to the light. The shutter must be kept down when the box is open.}
 \end{figure}
 
 
@@ -124,7 +121,7 @@ The experiment consists of three steps:
 \end{description}
 
 
-\subsection*{Observing interference and finding dial settings}
+\subsection*{Step 1: Observing interference and finding dial settings}
 \begin{framed}
 {\center{\large \bf \textcolor{red}{Important personal safety information}\\} }
 The apparatus has a 5-mW diode laser with an output wavelength of $670 \pm 5$~nm. It could potentially harm your eyes if you are not careful. Don't observe the beam directly (i.e., by staring into it) and be careful of reflections off of reflective surfaces.
@@ -157,7 +154,7 @@ Record the dial setting for each. Make sure that you are fully in the single and
 
 
 
-\subsection*{Measuring interference with an intense source}
+\subsection*{Step 2: Measuring interference with an intense source}
 
 In this part of the experiment you will use a photodiode to measure the intensity distribution of the interference pattern by varying the position of the detector slit.  You will do this using the red laser with the shutter down.  
 
@@ -187,18 +184,18 @@ In this part of the experiment you will use a photodiode to measure the intensit
 
 \item[Take a datapoint] Now you are ready to take data. Move the detector slit to one end of the interference pattern. Collect one datapoint by recording the detector slit position (by reading the micrometer) and the voltage from the photodiode. You will be taking a lot of datapoints and it is wise to plot them as you collect them. Therefore you should record the readings in a spreadsheet and do not need to put them in your lab book. 
 
-\item[Uncertainty -- Part 1] Estimate your uncertainty from the micrometer and multi-meter precision and stability. Record your uncertainty and any comments you have about it in your lab book. We will come back to this later.
+\item[Uncertainty -- Part 1] Estimate your uncertainty from the micrometer and multimeter precision and stability. Record your uncertainty and any comments you have about it in your lab book. We will come back to this later.
 
-\item[Collect more points] Move the detector slit by some small interval, collecting additional datapoints. You want to map out the interference pattern with sufficient details that you can easily resolve the light and dark fringes. For double-slit interference, taking data every $\unit[100]{\mu m}$ should be OK.  The single slit pattern is less detailed and you need only take data every $\unit[200]{\mu m}$.
+\item[Collect more points] Move the detector slit by some small interval, collecting additional datapoints. You want to map out the interference pattern with sufficient details that you can easily resolve the light and dark fringes. You may want to choose the different step for recording the double-slit and the single slit interference pattern. 
 
-\item[Plot the data] Plot your data as you take it to verify that you are seeing an interference pattern. Remake the plot every 20 steps or so. (This is easiest to do as a scatterplot in a spreadsheet).
+\item[Plot the data] Plot your data as you take it to verify that you are seeing an interference pattern. 
 
-\item[The goal] The goal is to take about 70 points in double slit mode and 35 in single slit mode. This should allow you to see multiple maxima and minima in double slit, and the central maximum in single slit mode, along with (hopefully) a secondary maximum on at least one side. \textcolor{red}{Note: before moving from double to single-slit mode you should estimate your uncertainties by following the procedure described below.}
+\item[The goal] The goal is to take enough data point to be able to clearly see multiple maxima and minima in double slit, and the central maximum in single slit mode, along with (possibly) a secondary maximum on at least one side. \textcolor{red}{Note: before moving from double to single-slit mode you should estimate your uncertainties by following the procedure described below.}
 \end{description}
 
 \subsubsection*{\textcolor{blue}{Uncertainties}}
 
-Are your uncertainties really just due to the measurement precision of the micrometer dial and DVM?  It is wise to test this by repeating some measurements in both single and double slit modes.. To do so:
+Are your uncertainties really just due to the measurement precision of the micrometer dial and DVM?  It is wise to test this by repeating some measurements in both single and double slit modes. To do so:
 
 \begin{description}
 \item[Back to the beginning]
@@ -253,129 +250,23 @@ You don't need to repeat datapoints to establish an uncertainty. The dominant co
 
 You should be careful not to wait to record ``round numbers'' or other aesthetically pleasing readings from the counter. The best way of doing this is to establish a routine. For example, turn the micrometer dial and then look at the counter. Record the second reading that you see since the first might be corrupted by the move. You need only record the rate as an integer (i.e., you can neglect any numbers after the decimal place).
 
-\subsubsection*{Qualitative observations}
-
-In both the one and two slit cases make some qualitative observations. Do the patterns appear to agree with Equations.~\ref{2slit_wDif}-\ref{1slit}?  What would happen if we increased or decreased $d$, $a$, $\lambda$ and $\ell$.  Record your thoughts in your labbook, and use these insights in setting the initial parameters when fitting your data.
-
-%The plots of your experimental data are clear evidence of particle-wave duality for photons. You've made contact with the central question of quantum mechanics: how can light, which so clearly propagates as a wave that we can measure its wavelength, also be detected as individual photon events?  Or alternatively, how can individual photons in flight through this apparatus nevertheless `know' whether one, or both, slits are open, in the sense of giving photon arrival rates which decrease when a second slit is opened?  Discuss these issues in your lab report.
 
-\vspace{1cm}
-\hrule
-
-{\huge Your instructor may have told you to do a ``brief writeup'' of this experiment. If so, please see the instructions starting on the next page (page~\pageref{pag:briefwriteup}).}
-
-\hrule
-
-\newpage
-
-\section*{Instructions for a brief writeup}\label{pag:briefwriteup}
-
-In this lab you have collected 4 independent datasets (red laser and green bulb with single or double slit) and the analysis will require fits to these data.  Therefore, a formal lab report with abstract, introduction, conclusion, etc. is \textbf{not} required.  For this brief writeup: you only need to include the tables and figures requested below, as well as answering the questions listed in this section. {\bf Please restate each one so it's easy for us to follow along.} {\it Refer to the relevant section earlier in the manual or later in Appendices for additional discussion of the analysis procedure. This is just a bullet point summary of the deliverables.}
-
-%\subsection*{How does the interference patterns depend on  $d$, $a$, and $\lambda$?}
-%
-%What would happen if we increased or decreased $d$, $a$, and $\lambda$?  Explore this by finding the locations of the minima of Eqs.~(\ref{1slit}) and %(\ref{2slit_wDif}). Then look at what happens to those locations as you change $d$, $a$, and $\lambda$. Record the results in your lab book and report. Knowing how the pattern changes as you change the variables is useful as you pick starting values for those parameters to use in your fits.
-
-\subsection*{Data Tables and Uncertainties}
-
-\begin{itemize}
-\item Tabulate your data for each of the four interference patterns and include them (with labels and captions).
-\item Describe how uncertainties on position, photodiode voltage, and photon counting rate were determined, and report your estimated uncertainty values.
-\end{itemize}
-
-\subsection*{Data Analysis and Fitting}
+\section*{Data Analysis and Fitting}
 
 For each of the four interference patterns, address the following points:
 
 \begin{enumerate}
 \item Plot the measurements for the photodiode voltage or photon counting rate (y-axis) vs position (x-axis) with uncertainties and fit to Eqs.~(\ref{2slit_wDif}) and (\ref{1slit}) for Double and Single Slit, respectively.  Include a figure with the plot of your data and the resulting fit in your report (with labels and captions).
-\item Is it a good fit: in terms of the observed $\chi^2$ and number of degrees of freedom?
-\item What are the best fit parameters and uncertainties?
+\item Is it a good fit? Discuss how closely the experimental data follow the theory expectations, and possible reasons for deviations.
+\item Compare the slit parameters ($a$ and $d$) obtained from different fits. Are they consistent with each other within the experimental uncertainty? 
 \item Is B consistent with what you expect from the background (\textit{i.e.} pedestal or dark rate) measurements you collected during lab?  Comment on any discrepancies. 
 \end{enumerate}
 
-\noindent Finally, quantitatively compare your measurements of $a$ and $d$ between the four datasets.  Are they consistent?
-
-\subsection*{Did we really see single photon interference?}
-
-To conclude that photons interfere with themselves, rather than with other nearby photons, we have to be sure that the rate of photons is low enough that only one of them is in the vicinity of the double slit at any given time.  You need to work through the steps in the Appendix ``Computing the photon rate" below, writing numbers in your lab book. In this brief writeup summarize the results in a paragraph or two, including the numbers you calculate at each stage.  Write a nice bit of narrative text, not just a bunch of bullets with numbers.
-
-\newpage 
-\section*{Appendix: Computing the photon rate}
-
-{\em This is a so-called back of the envelope calculation.}
-
-\begin{enumerate}
-
-%\item Begin by roughly estimating the number of photons per second arriving to the detector. 
-\item Begin by computing the amount of electrical power that is converted to light by the light bulb. Assume we operate the bulb at \unit[6]{VDC} and it draws \unit[0.2]{A} of current.  From this you can get the power drawn by the bulb. % Ans: 1.2W
-
-\item Most of the power goes into heating the bulb. Assume 5\% of it is converted to light evenly distributed from 500-1500~nm. 
- 
-\item The bulb has a green filter that blocks the light outside the range 541-551~nm.  From this you can get the amount of power radiated as green light. %1.2W*0.05*10nm/1500nm = 6e-4W
-
-\item You can convert the power (energy/time) into a rate of photons using the fact that $E = \frac{hc}{\lambda}$.  From this you can find the rate of green photons emitted from the bulb+filter.  % One 546nm photon has E=6.626e-34 J sec * 3e8 m/sec / 546e-9m  = 3.64e-19 J 
-% (6e-4 J/sec) / (3.64e-19 J/photon) = 1.65e15 photons/sec
 
-\item The photons emitted by the bulb get diluted by the apertures in the interference channel.  The first is a single slit about $R=\unit[10]{cm}$ from the bulb with a width of \unit[0.1]{mm} and a height of \unit[1]{cm}.  Assuming that the bulb radiates evenly onto a spherical surface of radius $R$, compute the fraction of the light that would pass through the slit. % slit area/4 pi R^2=7.96e-6
-% rate passing slit is 7.96e-6 * 1.65e15 photons/sec = 1.31e10 photons/sec
-
-\item After the single slit the beam is diffracted into a single slit interference pattern and impinges on the double slit. If you had the apparatus in front of you, you could estimate the area that the interference pattern covers. A good guess is $\unit[1]{cm^2}$. Assume this and that the double slits have an area twice that of the single slit. You can compute the rate of photons passing through the double slits. % 1.31e10 photons/sec * 2 * 7.96e-6 = 2.1e5 photons/sec
-
-\item We know that a photon travels at a speed $c=\unit[3\times 10^8]{m/sec}$. Divide that by the number of photons per second that pass through the double slits. This gives you the typical space between photons in meters/photon. % 1428 m/photon
-
-\item Invert the number from the previous bullet point and multiply by the 0.5~m between the double slit and detector slit to determine how many photons are in the setup on average.  Is it a number much less than one?  If so, there is a good reason to believe that the photons must be interfering with themselves to form the light and dark fringes you see.  % 3.5e-4 photons in the half meter downstream of the double slit
-
-\end{enumerate}
-
-\newpage 
-\section*{Appendix: Fitting the interference data}
-
-You need to fit your data with Eqs.~(\ref{2slit_wDif}) and (\ref{1slit}).  A detailed example is provided in ``Help on the double slit analysis'' in the Matlab folder on Blackboard or ``Fitting Help: Single and Double Slit" in the Python folder on Blackboard. {\em Please utilize this, it will save you a lot of time!}
-
-A few fitting tips and notes:
-
-\begin{itemize}
-\item Make sure that units of all your measured values are self-consistent - the program will go crazy if you mix meters and micrometers together!
-\item Try to plot your function using estimated values of the parameters before doing a fit with it. This catches many silly errors.
-\item The fitting function has a term to account for the overall normalization and also for the fact that the maximum is not set at $x=0$. In other words, we substitute $x \rightarrow x-x_0$ when coding the fit function. Estimate both of these (for the normalization, look at what happens when $x-x_0 = 0$) but include them as free parameters in the fit.
-\item The fitting function has a term to account for the ``pedestal'' (for the photodiode) or ``dark rate'' (for the PMT) you observed when the lights were off. The estimated value should be changed to agree with what you observed. 
-\item Just a note: fitting functions should include the minimal number of parameters. If I have parameters $a$ and $b$ but they always appear as $ab$ in my function, then I am much better off including a term $c=ab$ when doing the fit. Otherwise the fit is underconstrained. Adjusting $a$ up has the same effect as adjusting $b$ down so it's impossible to converge on unique values for $a$ and $b$. Fitting algorithms really dislike this situation.
-\item If the program has trouble fitting all the parameters, first hold the values of the parameters you know fairly well (such as the light wavelength, maximum peak intensity, background, etc.). Once you determine the approximate values for all other parameters, you can release the fixed ones, and let the program adjust everything to make the fit better.
-\end{itemize}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Appendices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newpage 
-\section*{Appendix: Two-Slit interference with atoms\footnote{Special thanks to Prof. Seth Aubin for providing the materials for this section}}
-
-\emph{According to quantum mechanics, the wave-particle duality must be applied not only to light, but to any
-``real'' particles as well. That means that under the right circumstance, atoms should behave as waves with
-wavelength $\lambda_{\mathrm{atom}}=h/\sqrt{2mE}=h/p$ (often called de Broglie wavelength), where $h$ is Planck's
-constant, $m$ is the mass of the particle, and $E$ and $p$ are respectively the kinetic energy and the momentum
-of the particle. In general, wave effects with ``massive'' particles are much harder to observe compare to
-massless photons, since their wavelengths are much shorter. Nevertheless, it is possible, especially now when
-scientists have mastered the tools to produce ultra-cold atomic samples at nanoKelvin temperatures. As the energy
-of a cooled atom decreases, its de Broglie wavelength becomes larger, and the atom behaves more and more like
-waves. For example, in several experiments, researchers used a Bose-Einstein condensate (BEC) -- the atomic
-equivalent of a laser -- to demonstrate the atomic equivalent of the Young's double-slit experiment. As shown in
-Fig.~\ref{BECinterferfometer.fig}(a), an original BEC sits in single-well trapping potential, which is slowly
-deformed into a double-well trapping potential thus producing two phase-coherent atom wave sources. When the
-trapping potential is turned off, the two BECs expand and interfere where they overlap, just as in the original
-Young's double-slit experiment.}
-%
-\begin{figure}[h]
-\centering
-\includegraphics[width=0.8\linewidth]{./pdf_figs/BECinterferfometer} \caption{\label{BECinterferfometer.fig}
-Atom interferometry version of Young's double-slit experiment: \emph{(a)} schematic and \emph{(b)}
-experimentally measured interference pattern in an ${}^{87}$Rb Bose-Einstein condensate.}
-\end{figure}
 
-\emph{Fig.~\ref{BECinterferfometer.fig}(b) shows the resulting interference pattern for a ${}^{87}$Rb BEC. Atom
-interferometry is an area of active research, since atoms hold promise to significantly improve interferometric
-resolution due their much shorter de Broglie wavelength compared to optical photons. In fact, the present most
-accurate measurements of accelerations, rotations, and gravity gradients are based on atomic interference. }
+\newpage
 
-\section*{Appendix: Fraunhofer Diffraction at a Single Slit and Two-Slit interference}
+\section*{Appendix A: Fraunhofer Diffraction at a Single Slit and Two-Slit interference}
 
 \textit{Diffraction at a Single Slit} We will use a \emph{Fraunhofer} diffraction model to  calculate the
 intensity distribution resulting from light passing a single-slit of width $a$, as shown in
@@ -451,5 +342,80 @@ $\theta$ between the normal to the plane containing the slits and the direction
 connect these equations to Eqs.~(\ref{1slit}) and(\ref{2slit_wDif}) we assume that $\sin\theta \simeq \tan\theta
 = x/\ell$ where $x$ is the distance to the point $P$ on the screen, and $\ell$ is the distance from the two slit
 plane to the screen.
+\newpage 
+\section*{Appendix B: Computing the photon rate}
+
+{\em This is a so-called back of the envelope calculation.}
+
+\begin{enumerate}
+
+%\item Begin by roughly estimating the number of photons per second arriving to the detector. 
+\item Begin by computing the amount of electrical power that is converted to light by the light bulb. Assume we operate the bulb at \unit[6]{VDC} and it draws \unit[0.1]{A} of current.  From this you can get the power drawn by the bulb. % Ans: 1.2W
+
+\item Most of the power goes into heating the bulb. Assume 5\% of it is converted to light evenly distributed from 500-1500~nm. 
+ 
+\item The bulb has a green filter that blocks the light outside the range 541-551~nm.  From this you can get the amount of power radiated as green light. %1.2W*0.05*10nm/1500nm = 6e-4W
+
+\item You can convert the power (energy/time) into a rate of photons using the fact that $E = \frac{hc}{\lambda}$.  From this you can find the rate of green photons emitted from the bulb+filter.  % One 546nm photon has E=6.626e-34 J sec * 3e8 m/sec / 546e-9m  = 3.64e-19 J 
+% (6e-4 J/sec) / (3.64e-19 J/photon) = 1.75e15 photons/sec
+
+\item Only the small fraction of the photons emitted by the bulb get through the double slits. Let's assume that slits are   about $R=\unit[50]{cm}$ from the bulb, and each of them has the width of \unit[0.1]{mm} and a height of \unit[1]{cm}.  Assuming that the bulb radiates evenly onto a spherical surface of radius $R$, compute the fraction of the light that would pass through the slits. % slit area/4 pi R^2=3.18e-7
+% rate passing slit is 3.18e-7 * 1.65e15 photons/sec = 5.56e8 photons/sec
+
+%\item After the single slit the beam is diffracted into a single slit interference pattern and impinges on the double slit. If you had the apparatus in front of you, you could estimate the area that the interference pattern covers. A good guess is $\unit[1]{cm^2}$. Assume this and that the double slits have an area twice that of the single slit. You can compute the rate of photons passing through the double slits. % 1.31e10 photons/sec * 2 * 7.96e-6 = 2.1e5 photons/sec
+
+\item We know that a photon travels at a speed $c=\unit[3\times 10^8]{m/sec}$. Divide that by the number of photons per second that pass through the slits. This gives you the typical space between photons in meters/photon. % 1428 m/photon
+
+\item Invert the number from the previous bullet point and multiply by the 0.5~m between the double slit and detector slit to determine how many photons are in the setup on average.  Is it a number less than one?  If so, there is a good reason to believe that the photons must be interfering with themselves to form the light and dark fringes you see.  % 3.5e-4 photons in the half meter downstream of the double slit
+
+\end{enumerate}
+
+\newpage 
+\section*{Appendix C: Fitting the interference data}
+
+You need to fit your data with Eqs.~(\ref{2slit_wDif}) and (\ref{1slit}).  A detailed example is provided in ``Help on the double slit analysis'' in the Matlab folder on Blackboard or ``Fitting Help: Single and Double Slit" in the Python folder on Blackboard. {\em Please utilize this, it will save you a lot of time!}
+
+A few fitting tips and notes:
+
+\begin{itemize}
+\item Make sure that units of all your measured values are self-consistent - the program will go crazy if you mix meters and micrometers together!
+\item Try to plot your function using estimated values of the parameters before doing a fit with it. This catches many silly errors.
+\item The fitting function has a term to account for the overall normalization and also for the fact that the maximum is not set at $x=0$. In other words, we substitute $x \rightarrow x-x_0$ when coding the fit function. Estimate both of these (for the normalization, look at what happens when $x-x_0 = 0$) but include them as free parameters in the fit.
+\item The fitting function has a term to account for the ``pedestal'' (for the photodiode) or ``dark rate'' (for the PMT) you observed when the lights were off. The estimated value should be changed to agree with what you observed. 
+\item Just a note: fitting functions should include the minimal number of parameters. If I have parameters $a$ and $b$ but they always appear as $ab$ in my function, then I am much better off including a term $c=ab$ when doing the fit. Otherwise the fit is underconstrained. Adjusting $a$ up has the same effect as adjusting $b$ down so it's impossible to converge on unique values for $a$ and $b$. Fitting algorithms really dislike this situation.
+\item If the program has trouble fitting all the parameters, first hold the values of the parameters you know fairly well (such as the light wavelength, maximum peak intensity, background, etc.). Once you determine the approximate values for all other parameters, you can release the fixed ones, and let the program adjust everything to make the fit better.
+\end{itemize}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Appendices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newpage 
+\section*{Appendix D: Two-Slit interference with atoms\footnote{Special thanks to Prof. Seth Aubin for providing the materials for this section}}
+
+\emph{According to quantum mechanics, the wave-particle duality must be applied not only to light, but to any
+``real'' particles as well. That means that under the right circumstance, atoms should behave as waves with
+wavelength $\lambda_{\mathrm{atom}}=h/\sqrt{2mE}=h/p$ (often called de Broglie wavelength), where $h$ is Planck's
+constant, $m$ is the mass of the particle, and $E$ and $p$ are respectively the kinetic energy and the momentum
+of the particle. In general, wave effects with ``massive'' particles are much harder to observe compare to
+massless photons, since their wavelengths are much shorter. Nevertheless, it is possible, especially now when
+scientists have mastered the tools to produce ultra-cold atomic samples at nanoKelvin temperatures. As the energy
+of a cooled atom decreases, its de Broglie wavelength becomes larger, and the atom behaves more and more like
+waves. For example, in several experiments, researchers used a Bose-Einstein condensate (BEC) -- the atomic
+equivalent of a laser -- to demonstrate the atomic equivalent of the Young's double-slit experiment. As shown in
+Fig.~\ref{BECinterferfometer.fig}(a), an original BEC sits in single-well trapping potential, which is slowly
+deformed into a double-well trapping potential thus producing two phase-coherent atom wave sources. When the
+trapping potential is turned off, the two BECs expand and interfere where they overlap, just as in the original
+Young's double-slit experiment.}
+%
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.8\linewidth]{./pdf_figs/BECinterferfometer} \caption{\label{BECinterferfometer.fig}
+Atom interferometry version of Young's double-slit experiment: \emph{(a)} schematic and \emph{(b)}
+experimentally measured interference pattern in an ${}^{87}$Rb Bose-Einstein condensate.}
+\end{figure}
+
+\emph{Fig.~\ref{BECinterferfometer.fig}(b) shows the resulting interference pattern for a ${}^{87}$Rb BEC. Atom
+interferometry is an area of active research, since atoms hold promise to significantly improve interferometric
+resolution due their much shorter de Broglie wavelength compared to optical photons. In fact, the present most
+accurate measurements of accelerations, rotations, and gravity gradients are based on atomic interference. }
+
 
 \end{document}