From 1e6bd48a487bd0345894b409eb1a8136e9e1a273 Mon Sep 17 00:00:00 2001
From: Irina Novikova <inovikova75@gmail.com>
Date: Tue, 13 Oct 2020 23:48:23 -0400
Subject: Faraday lab changed

---
 faraday_rotation.tex | 142 ++++++++++++++++++++++++++++-----------------------
 1 file changed, 78 insertions(+), 64 deletions(-)

(limited to 'faraday_rotation.tex')

diff --git a/faraday_rotation.tex b/faraday_rotation.tex
index 9745b78..e473c97 100644
--- a/faraday_rotation.tex
+++ b/faraday_rotation.tex
@@ -1,86 +1,99 @@
 \documentclass[./manual.tex]{subfiles}
 \begin{document}
 
-\chapter{Faraday Rotation}
+\chapter*{Faraday Rotation}
 
 %\date {}
 %\maketitle \noindent
  
 
-\textbf{Experiment objectives}: Observe the {\it Faraday Effect}, the rotation of a light wave's polarization vector in a material with a magnetic field directed along the wave's direction. Determine the relationship between the magnetic field and the rotation by measuring the so-called {\it Verdet constant} of the material. Become acquainted with some new tools: an oscilloscope, a function generator and an amplifier, and a new technique: phase-locking.
+\textbf{Experiment objectives}: Observe the {\it Faraday Effect}, the rotation of a light wave's polarization vector in a material in the presence of a magnetic field directed along the light propagation direction. Determine the relationship between the magnetic field and the rotation angle by measuring the so-called {\it Verdet constant} of the material. Become acquainted with some new tools: an oscilloscope, a function generator and an amplifier, and a new technique: phase-locking.
 
 \section*{Introduction}
-The term polarization refers to the direction of the electrical field in a
-light wave. Generally, light is not polarized when created (e.g., by atomic
-deexcitations) but can be made so by passing it through a medium that transmits electric fields oriented in one direction  and absorbs all others. Imagine we create a beam of light traveling in the $z$ direction. We then polarize it in the $x$ direction ($\vect{E}=\vect{\hat x}E_0\cos(kz-\omega t)$) by passing it through a polarizer and then pass it through a second polarizer, with a transmission axis oriented at an angle $\theta$ with respect to the $x$ axis. If we detect the light beam after the second polarizer, the intensity is 
+The term \emph{polarization} refers to the direction of the electrical field in a
+light wave. Most natural sources of light are not polarized, but we can use a special devices called \emph{polarizers} that transmits electric fields oriented in one direction  and absorbs all others. Thus, a well-defined polarization direction, along the polarizer axis. Also, the output of many lasers is polarized.
+
+Imagine a beam of light traveling in the $z$ direction. We then polarize it in the $x$ direction ($\vect{E}=\vect{\hat x}E_0\cos(kz-\omega t)$) by passing it through a polarizer and then pass it through a second polarizer, with a transmission axis oriented at an angle $\theta$ with respect to the $x$ axis. If we detect the light beam after the second polarizer, the intensity is given by the Malus law:
 \begin{equation}
-I=I_0 \cos^{2}\theta
+I=I_0 \cos^{2}\theta.\label{eq:maluslaw}
 \end{equation}
-
-In 1845, Michael Faraday discovered that he could create polarized light and then rotate the direction of the polarization using a magnetic field. The field, created in our laboratory by a solenoid, points in the direction of the light beam.  The Faraday effect is caused by a combination of factors:
-\begin{enumerate}
-\item We can describe the polarization vector in terms of right- and left-handed components. In this description, the electric field of the right-handed component rotates clockwise around the $z$ axis as the wave travels. The left-handed component rotates counter-clockwise. 
-\item At the particle level, the right and left hand components correspond to photons with angular momentum $\hbar=h/2\pi$ directed parallel (right) or anti-parallel (left) to the $z$ axis.
-\item When an atom is placed in a magnetic field, single atomic energy levels may divide into multiple levels, each with slightly different energies. This is called the {\it Zeeman effect} and qualitatively it occurs because the moving atomic electrons may be thought of as a current $\vect{J}$ which interacts with the external magnetic field $\vect{B}$. 
-\item The atomic current $\vect{J}$ contains components with different angular momenta, and those components interact differently with right and left handed photons. 
-\item The macroscopic effect is that a material in a magnetic field can have a different index of refraction $n$ for left and right handed light. Since the phase velocity of light $c'$ depends on $n$ as $c'=c/n$, the relative phase between the left and right handed components changes as the light travels a distance $L$ through the material.
-\item That phase change causes a rotation in the polarization vector.
-\end{enumerate}
-
-The polarization vector rotates in proportion to the length of the material, the magnitude of the magnetic field, and a material dependent constant $C_{V}$ called the {\it Verdet constant}:
+It is easy to see that light is completely blocked if the two polarizers are oriented at $90^o$ with respect to each other - we usually refer to such polarizers as being ``crossed''. 
+
+In everyday life most transparent materials do not change the polarization of light. However, in 1845 Michael Faraday discovered that he could rotate the direction of the light polarization using a magnetic field, directed along the light propagation - we now call this phenomena \emph{Faraday effect}. 
+%  The Faraday effect is caused by a combination of factors:
+%\begin{enumerate}
+%\item We can describe the polarization vector in terms of right- and left-handed components. In this description, the electric field of the right-handed component rotates clockwise around the $z$ axis as the wave travels. The left-handed component rotates counter-clockwise. 
+%\item At the particle level, the right and left hand components correspond to photons with angular momentum $\hbar=h/2\pi$ directed parallel (right) or anti-parallel (left) to the $z$ axis.
+%\item When an atom is placed in a magnetic field, single atomic energy levels may divide into multiple levels, each with slightly different energies. This is called the {\it Zeeman effect} and qualitatively it occurs because the moving atomic electrons may be thought of as a current $\vect{J}$ which interacts with the external magnetic field $\vect{B}$. 
+%\item The atomic current $\vect{J}$ contains components with different angular momenta, and those components interact differently with right and left handed photons. 
+%\item The macroscopic effect is that a material in a magnetic field can have a different index of refraction $n$ for left and right handed light. Since the phase velocity of light $c'$ depends on $n$ as $c'=c/n$, the relative phase between the left and right handed components changes as the light travels a distance $L$ through the material.
+%\item That phase change causes a rotation in the polarization vector.
+%\end{enumerate}
+The polarization vector rotates in proportion to the length of the material $L$, the magnitude of the magnetic field $B$, and a material dependent constant $C_{V}$ called the {\it Verdet constant}:
 
 \begin{equation} \label{eq:faraday_rotation}
-\phi = C_V B L
+\phi = C_V B L.
 \end{equation}
-
+%
 Typically $C_V$ depends on the wavelength of the light and has a value of a
-few $\mathrm{rad}/\mathrm{T}\cdot\mathrm{m}$. For the solenoid we'll use in
-this lab, the field at the center is $B=\unit[11.1]{mT/A}$, and our
-material, a special sort of glass, is $\unit[10]{cm}$ long. For a current
-of \unit[0.1]{A}, we expect a rotation of a few$\times10^{-4}$ radians. This is a pretty small angle and it will require a special technique to detect.  
+few $\mathrm{rad}/(\mathrm{T}\cdot\mathrm{m})$. To put it into prospective, the Earth magnetic field is $5\cdot 10^{-5}$T, and the magnetic field of a strong refrigerator magnet is around $10^{-3}$T. Thus, it is easy to estimate that unless very strong magnets are used, the induced polarization rotation is very small, typically well below a milliradian. This is a pretty small angle and it will require a special technique to detect.  
+
+The field, created in our laboratory by a solenoid, points in the direction of the light beam, and the magnitude of the magnetic field at the center is . The length of the  material, which is a special sort of glass, is  long. For a current
+of \unit[0.1]{A}, we expect a rotation angle of a few $10^{-4}$ radians. 
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.5\textwidth]{./pdf_figs/faradayeffect} \caption{\label{fig:faradayeffect} The experimental arrangement for the Faraday rotation measurement. Note that the polarization rotation angle $\phi$ due to Faraday effect is much smaller than shown.}
+\end{figure}
 
 We are going to take the polarized laser light and direct it through the glass
-rod, which is inserted into the center of the solenoid. The beam will then
-pass through a second polarizer with the transmission axis at an angle $\theta$ with respect to the initial polarization of the laser. The intensity of the transmitted light will then depend on the sum of the angle $\theta$ and the additional rotation $\phi$ caused by the magnetic field:
+rod, which is inserted into the solenoid. Electric current induces magnetic field in the center of the solenoid at the rate of $\unit[11.1]{mT/A}$, that result in the rotation of the laser polarization by a small angle $\phi$. The beam then
+pass through a second polarizer with the transmission axis at an angle $\theta$ with respect to the initial polarization of the laser, as shown in Fig.\ref{fig:faradayeffect}. The intensity of the light after the second polarizer then depend on the sum of the angle $\theta$ and the additional rotation $\phi$ caused by the magnetic field:
 \begin{eqnarray}
-I & = & I_0 \cos^{2}(\theta+\phi)\\
-  & = & I_0 \frac{1 + \cos(2\theta + 2\phi)}{2} \\
-  & = & \frac{I_0}{2} + \frac{I_0}{2}\left[\cos2\theta\cos2\phi-\sin2\theta\sin2\phi \right] \\
-  & \approx & I_0\left[\frac{1+\cos2\theta}{2} - \phi \sin2\theta  \right] \qquad (\phi^{2}\ll 1)\\
-  & = & I_0\left[\frac{1+\cos2\theta}{2} - C_V L B \sin2\theta  \right] \label{eq:Ifinal}
+I & = & I_0 \cos^{2}(\theta+\phi)
+   =  I_0 (1 + \cos(2\theta + 2\phi))/2 \\ \nonumber
+  & = & \frac{I_0}{2} + \frac{I_0}{2}\left[\cos2\theta\cos2\phi-\sin2\theta\sin2\phi \right],
 \end{eqnarray}
-
-We'll see the Faraday effect by observing changes in the intensity of light as we vary the magnetic field. But, there is a problem. The term involving $\phi$ is much smaller than the other terms and it can easily be buried in noise from a photo-detector. We'll use an experimental technique, called phase-locking, to get around the problem. We will then relate the changes in the intensity to changes in the angle $\phi$ using a special calibration dataset. Since we know the corresponding changes in $B$ we can use Eq.~\ref{eq:faraday_rotation} to extract $C_V$.
-
-The phase-locking technique works in the following way. We'll vary the
-magnetic field periodically with time as a sine wave, and then observe the
-signal from the photodiode as a function of time.  The signal will look
-like a large constant with a small sine wave "wobble" on it, along with some random
-noise with a similar magnitude to the wobble. However, we can subtract off
-the non time-varying portion of the signal, using a high pass filter. Then,
-since we know the period and phase of the magnetic field, we can time our
+where $I_0$ is the laser intensity after the first polarizer.
+As it is mentioned above, the expected polarization rotation angle $\phi$ is very small, in order of a few $10^{-4}$ radians. Thus, we can safely assume that $\sin2\phi\approx 2\phi$ and $\cos2\phi \approx 1$. Then, the expression for the transmitted laser intensity becomes 
+ \begin{equation}\label{eq:Ifinal}
+ I = I_0\frac{1+\cos2\theta}{2} - \phi I_0 \sin2\theta. 
+ \end{equation}
+Here the first term describes the laser intensity after the second polarizer at zero magnetic field, and the second term, proportional to $\phi = C_V B L$, shows the change in the transmission due to the Faraday rotation. 
+
+To find the value of the Verde constant $C_V$ one obviously needs to find the linear dependence of the polarization rotation angle on the applied magnetic field, according to Eq.~\ref{eq:faraday_rotation}. However, this is harder than it may seem. In reality, the term involving $\phi$ in Eq.(\ref{eq:Ifinal}) is much smaller than the first term, and it can easily be buried in noise from a photo-detector. Thus, to measure this small variation in transmission, we will use an experimental technique, called \emph{phase-locking}. 
+
+The phase-locking technique works in the following way. We will vary the
+magnetic field periodically with time as a sine wave $B(t)=B_0 \sin(2\pi ft)$, causing the light polarization rotation to ``wiggle'' 
+at the same frequency. Then, if we observe the
+signal from the photodiode as a function of time, accoreding to Eq.(\ref{eq:Ifinal}) it will look
+like a large constant with a small sine wave ``wobble'' on it, along with some random
+noise with a similar magnitude to the wobble. Oscilloscope can help us to remove 
+the non-time-varying portion of the signal, using a high pass filter (AC coupling). To get
+rid of the noise, we can take advantage of the fact that we can predict how our expected signal
+looks in time. Since we know the period and phase of the magnetic field, we can time our
 observations to be exactly in sync with the magnetic field, and use this to
-average out the noise. What's left over is the wobble in $I$, essentially
-the $C_V L B \sin2\theta$ term. Knowing $B,L$ and $\theta$, we can determine $C_V$.
+average out the noise. To do that, we start the oscilloscope acquisition at exactly the same time 
+with respect to the magnetic field variation (using triggering function), and average multiple traces.
+What is left over is the wobble (our useful signal) in $I$, essentially
+the $\phi(t) = C_V L B(t) \sin2\theta$ term. Knowing $B,L$ and $\theta$, we can now determine $C_V$.
 
 \section*{Experimental Setup}
-The setup consists of:
+The experimental setup is shown in Fig.~\ref{fig:setup}. Its main components are: 
 \begin{itemize}
-\item A solenoid to produce the magnetic field.
-\item A glass rod inside the solenoid. You'll measure $C_V$ of this material.
-\item A polarized HeNe laser, to shine light through the field.
-\item A second polarizer, the ``analyzer'', to define the angle $\theta$.
-\item A photodiode to detect the laser light after the coil and the analyzer.
+\setlength\itemsep{0em}
+\item A solenoid to produce the magnetic field. The conversion factor between the magnetic field and the solenoid current is $\unit[11.1]{mT/A}$.
+\item A glass rod inside the solenoid. The length of the rod $L= \unit[10]{cm}$. You'll measure $C_V$ of this material.
+\item A polarized HeNe laser.
+\item A second polarizer, the ``analyzer''.
+\item A photodiode to measure the laser intensity after the coil and the analyzer.
 \item A function generator and amplifier to supply current to the solenoid.
 \item A digital multimeter (DMM) to measure the solenoid current.
-\item A digital oscilloscope to read out the photodiode.
+\item A digital oscilloscope to read out the photodiode voltage.
 \end{itemize}
 
-The experimental setup is shown in Fig.~\ref{fig:setup}.
-
 \begin{figure}[h]
 \centering
-\includegraphics[width=\textwidth]{./pdf_figs/faraday_setup} \caption{\label{fig:setup} The experimental setup.}
+\includegraphics[width=0.6\textwidth]{./pdf_figs/faraday_setup} \caption{\label{fig:setup} The experimental setup.}
 \end{figure}
 
 \section*{Experimental Procedure}
@@ -94,32 +107,32 @@ The experimental setup is shown in Fig.~\ref{fig:setup}.
 	and the polarizer direction affects the intensity.  Vary the angle
 	of the analyzing polarizer and use a white screen (e.g., piece of
 	paper) to observe how the intensity of the transmitted light
-	changes. Find the angles that give you  maximum and minimum transmission. Then, use the DMM to measure the photodiode output as a function of $\theta$, going between the maximum and minimum in $5^\circ$ steps. Tabulate this data. Graph it. What functional form should it have? Does it?
+	changes. Find the angles that give you  maximum and minimum transmission. Then, use the DMM to measure the photodiode output as a function of $\theta$, going between the maximum and minimum in small steps (use your first pre-lab exercise to determine the size of the step). Plot the resulting graph and verify that it follows Eq.(\ref{eq:maluslaw}).
 
-\item[Function Generator Setup] Plug the function generator output and its trigger (a.k.a.~ pulse) output into different channels on the scope. Trigger the scope on the trigger/pulse output from the function generator and look at the function generator signal. Modify the function generator to provide a \unit[200]{Hz} sine wave with an amplitude of about \unit[1]{V}. There is no more need to touch dials on the function generator.
+\item[Function Generator Setup] Plug the function generator output and its trigger (a.k.a.~ pulse) output into different channels on the scope. Trigger the scope on the trigger/pulse output from the function generator and look at the function generator signal. Modify the function generator to provide a \unit[200]{Hz} sine wave with an amplitude of about \unit[1]{V}. You will be using these settings for the rest of the measurements, so do not touch the function generator dials.
 
-\item[Amplifier setup] Now, plug the output of the function generator into the amplifier. Use a coaxial$\leftrightarrow$double-prong connector to feed the amplifier output into the scope, still triggering on the generator. Vary the amplifier dial setting and observe the change in the output. At some point, the output will become clipped, as the amplifier reaches its power output. Record this dial setting, and the amplitude of the sine wave, just as the clipping sets in. This is the maximum useful output from the amplifier. Turn the dial all the way off, and hook the amplifier up to the solenoid, with the DMM in series to measure AC current. Go back to the maximum setting and measure the current flowing through the solenoid. This is the maximum useful current.  You now have a time varying magnetic field in the solenoid and you can control its magnitude with the amplifier. 
+\item[Amplifier setup] Disconnect the output of the function generator from the oscilloscope and plug it into the amplifier. Use a coaxial$\leftrightarrow$double-prong connector to feed the amplifier output into the oscilloscope, still triggering it on the generator. Vary the amplifier dial setting and observe the change in the output voltage. At some point, the output will become clipped, as the amplifier reaches its maximum power output. Record this dial setting, and the amplitude of the sine wave, just as the clipping sets in. This is the maximum useful output from the amplifier. Turn the dial all the way down, and hook the amplifier up to the solenoid, with the DMM in series to measure AC current. Go back to the maximum setting and measure the current flowing through the solenoid. This is the maximum useful current.  You now have a time varying magnetic field in the solenoid and you can control its magnitude with the amplifier. 
 
 \end{description}
 
 \subsection*{Measuring Faraday Rotation}
 
 \begin{description}
-\item[Choice of $\theta$] You need to pick an angle $\theta$, which may seem arbitrary. There is a best choice. Examine Eq.~\ref{eq:Ifinal}.  Pick $\theta$ and be sure to tighten the thumbscrew.
-\item[Faraday rotation] Plug the photodiode output into the scope, set the scope so its channel is DC coupled, and make sure that the ``probe'' setting is at 1x. Turn the amplifier dial about halfway to the maximum setting you found. Observe the photodiode trace on the scope, perhaps changing the volts/div setting so you can see the trace more clearly. What is the voltage? Record it.  The changing magnetic field should be causing a change in the polarization angle of the laser light, which should cause a sinusoidal time dependence to the photodiode signal, referred to as the "wobble."   Can you see any wobble? 
+\item[Choice of $\theta$] Pick optimal $\theta$, based on your pre-lab exercise, and be sure to tighten the thumbscrew on the polarizer mount to make sure it does not change during the measurements.
+\item[Faraday rotation] Plug the photodiode output into the scope, set the scope so its channel is DC coupled, and make sure that the ``probe'' setting is at 1x. Turn the amplifier dial about halfway to the maximum setting you found. Observe the photodiode trace on the scope, perhaps changing the volts/div setting so you can see the trace more clearly. What is the voltage? Record it.  The changing magnetic field should be causing a change in the polarization angle of the laser light, which should cause a sinusoidal time dependence to the photodiode signal, referred to as the ``wobble''.   Check if  you can see any wobble. 
 \item[AC coupling] The wobble is riding atop a large constant (DC) signal.
 	The scope can remove the DC signal by ``AC coupling'' the
 	photodiode channel. This essentially directs the scope input
 	through a high pass filter.  Do this, and then set the photodiode
 	channel to the \unit[2]{mV} setting.  You should now see a wobble.
-	Vary the amplifier dial's setting and notice how the amplitude of the wobble changes. You are seeing the Faraday effect.
-\item[Remove the noise] The signal is noisy, but now we'll really benefit
+	Vary the amplifier dial setting and notice how the amplitude of the wobble changes. You are seeing the Faraday effect.
+\item[Reduce the noise] The signal is noisy, but now we'll really benefit
 	from knowing the waveform that the function generator is producing.
 	Because we trigger the scope on the function generator, the maxima
-	and minima will, neglecting random noise, occur at the same point
+	and minima for the useful signal will occur at the same point
 	on the scope screen (and in its memory bank). The scope has a
 	feature that allows you to average multiple triggers. Doing this mitigates the noise, since at each point on the trace we are taking a mean, and the uncertainty in a mean decreases as we increase the number of measurements $N$ as $1/\sqrt{N}$. Turn on the averaging feature by going to the ``Acquire'' menu. Observe how the averaged trace becomes more stable as you increase the number of traces being averaged. The larger the better, but $\sim$100 traces should be enough.
-\item[Take measurements]  You should now systematically measure the amplitude of the wobble as a function of the current in the solenoid. There should be a linear relationship, which can be fit to extract $C_V$. Take about 10 measurements, evenly separated between the smallest current for which there is a measurable wobble, and the maximum you found earlier.  In each case, you want to start acquisition (Run/Stop on the scope), let the averaged signal converge onto a nice sine wave, stop acquisition and measure the amplitude of the signal using the scope's cursors. One measurement is shown in Fig.~\ref{fig:trace}. Record the negative and positive amplitudes $V_\mathrm{low}$ and $V_\mathrm{high}$ ($\unit[\pm 640]{\mu V}$ in Fig.~\ref{fig:trace}) and the peak to peak voltage $\Delta V$ ($\unit[1.28]{mV}$), along with the current in the coil -- $I_\mathrm{coil}$ . Estimate the uncertainty in your measurements.
+\item[Take measurements]  You should now systematically measure the amplitude of the wobble as a function of the current in the solenoid. There should be a linear relationship, which can be fit to extract $C_V$. Take about 10 measurements, evenly separated between the smallest current for which there is a measurable wobble, and the maximum you found earlier.  In each case, you want to start acquisition (Run/Stop on the scope), let the averaged signal converge onto a nice sine wave, stop acquisition and measure the amplitude of the signal using the scope's cursors. One measurement is shown in Fig.~\ref{fig:trace}. Record the negative and positive amplitudes $V_\mathrm{low}$ and $V_\mathrm{high}$ ($\unit[\pm 640]{\mu V}$ in Fig.~\ref{fig:trace}) and the peak to peak voltage $\Delta V$ ($\unit[1.28]{mV}$), along with the current in the coil -- $I_\mathrm{coil}$ . Estimate the uncertainty in your measurements based on the noise in your averaged signal and/or the achievable accuracy of oscilloscope.
 \end{description}
 
 \begin{figure}[h]
@@ -147,8 +160,9 @@ Figuring out $\Delta\phi$ is a little more difficult. We need to use our calibra
 The factor of two is because we defined $\Delta V$ as the peak to peak voltage.
 %\begin{description}
 
-Compute $C_V$ for a few points, along with the uncertainty. Complete the
-analysis by fitting $\Delta\phi$ vs. $\Delta B$ to a straight line and then use Eq.~\ref{eq:faraday_fit} to extract $C_V$ and its uncertainty. Do you see a nice linear relationship? What should the intercept be? Is it what you expect? Are there any outlier points, particularly at the ends of your curve? You could do the same analysis using $V_\mathrm{low}$ or $V_\mathrm{high}$... does doing so yield the same answer for $C_V$?
+Complete the
+analysis by fitting $\Delta\phi$ vs. $\Delta B$ to a straight line and then use Eq.~\ref{eq:faraday_fit} to extract $C_V$ and its uncertainty (consider what the intercept should be).  
+Check the quality of your linear fit. If there are visible deviations, discuss possible reasons.  
 
 \end{document}
 
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