typos fixed thanks to Michael

1 files changed, 18 insertions, 9 deletions

diff --git a/faraday_rotation.tex b/faraday_rotation.tex
index c359ab3..b1c443c 100644
--- a/faraday_rotation.tex
+++ b/faraday_rotation.tex
@@ -10,7 +10,7 @@
 \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.
 
 \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 which 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 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 which 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 
 \begin{equation}
 I=I_0 \cos^{2}\theta
 \end{equation}
@@ -44,13 +44,22 @@ pass through a second polarizer with the transmission axis at an angle $\theta$
 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}{2}+\cos2\theta - \phi \sin2\theta  \right] \qquad (\phi^{2}\ll 1)\\
-  & = & I_0\left[\frac{1}{2}+\cos2\theta - C_V L B \sin2\theta  \right] \label{eq:Ifinal}
+  & \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}
 \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 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 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 $C_V$.
+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 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
+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 $C_V$.
 
 \section*{Experimental Setup}
 The setup consists of:
@@ -89,16 +98,16 @@ The experimental setup is shown in Fig.~\ref{fig:setup}.
 \subsection*{Measuring Faraday Rotation}
 
 \begin{description}
-\item[Choice of $\theta$] You need to pick an angle $\theta$, which may seem arbitrary. But, 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, and 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 wobble to the photodiode signal.   Can you see any wobble? 
+\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 wobble to the photodiode signal.   Can you 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 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 on the scope screen (and, in its memory bank). The scope has a feature which 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 scopes 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[Remove 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 on the scope screen (and in its memory bank). The scope has a feature which 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.
 \end{description}
 
 \begin{figure}[h]
@@ -109,7 +118,7 @@ output of the photodiode, AC coupled and averaged over 128 traces. The blue
 curve is the trigger output from the function generator and is being used
 to trigger the scope readout so that it's in phase with the changing
 magnetic field. The maximum and minimum amplitudes of the photodiode signal
-are measured with the scopes cursors.}
+are measured with the scope's cursors.}
 \end{figure}
 
 \subsection*{Data Analysis}