typos fixed thanks to Dean

1 files changed, 22 insertions, 5 deletions

diff --git a/faraday_rotation.tex b/faraday_rotation.tex
index f0ddd75..c359ab3 100644
--- a/faraday_rotation.tex
+++ b/faraday_rotation.tex
@@ -17,7 +17,7 @@ I=I_0 \cos^{2}\theta
 
 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 a 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 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. 
@@ -31,9 +31,15 @@ The polarization vector rotates in proportion to the length of the material, the
 \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 a few$\times10^{-4}$ radians. This is a pretty small angle and it will require a special technique to detect.  
+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.  
 
-We are going to take 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 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:
+We are going to take 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:
 \begin{eqnarray}
 I & = & I_0 \cos^{2}(\theta+\phi)\\
   & = & I_0 \frac{1 + \cos(2\theta + 2\phi)}{2} \\
@@ -85,14 +91,25 @@ The experimental setup is shown in Fig.~\ref{fig:setup}.
 \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[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 dials setting and notice how the amplitude of the wobble changes. You are seeing the Faraday effect.
+\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.
 \end{description}
 
 \begin{figure}[h]
 \centering
-\includegraphics[width=\textwidth]{./pdf_figs/faraday_scope_trace} \caption{\label{fig:trace} An example scope trace. The yellow curve is the 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 amplitude of the photodiode signal is measured with the scopes cursors.}
+\includegraphics[width=\textwidth]{./pdf_figs/faraday_scope_trace}
+\caption{\label{fig:trace} An example scope trace. The yellow curve is the
+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.}
 \end{figure}
 
 \subsection*{Data Analysis}