typos fixed thanks to Michael

1 files changed, 15 insertions, 3 deletions

diff --git a/faraday_rotation.tex b/faraday_rotation.tex
index b1c443c..772f234 100644
--- a/faraday_rotation.tex
+++ b/faraday_rotation.tex
@@ -10,7 +10,9 @@
 \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 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 
 \begin{equation}
 I=I_0 \cos^{2}\theta
 \end{equation}
@@ -87,7 +89,12 @@ The experimental setup is shown in Fig.~\ref{fig:setup}.
 \begin{description}
 \item[Laser \& photodiode setup] The amplifier includes the laser power supply on the back. Plug the laser in, {\bf being careful to match colors between the cable and the power supply's connectors}. Align the laser so it travels down the center of the solenoid, through the glass rod, and into the center of the photodiode. Set the photodiode load resistor to $\unit[1]{k\Omega}$. Plug the photodiode into a DMM and measure DC voltage.
 
-\item[Calibration: intensity vs $\mathbf{\theta}$] We want to understand how the angle between the polarization vector of the laser light 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 which 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?
+\item[Calibration: intensity vs $\mathbf{\theta}$] We want to understand
+	how the angle between the polarization vector of the laser light
+	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?
 
 \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.
 
@@ -106,7 +113,12 @@ The experimental setup is shown in Fig.~\ref{fig:setup}.
 	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[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 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.
 \end{description}