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\begin{document}
\title{Faraday Rotation measurement of the Verdet Constant of SF-57 glass at 650 nm: Third Draft}
\author{Lucy Liang}
\affiliation{Smith College}
\author{Alisha Vira}
\affiliation{Affiliation not available}
%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy
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%\keywords{Suggested keywords}%Use showkeys class option if keyword
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\maketitle
\section{Aims} % Section titles are automatically converted to all-caps.
% Section numbering is automatic.
The Faraday Rotation of Polarized light is the rotation of the plane of polarization of light propagating through a medium due to the presence of a magnetic field. In this experiment, we aim to measure the Verdet constant of a glass rod (SF-57) through the Faraday Effect.
\section{Introduction}
By 1845, it was understood that different materials change the polarization of light. Michael Faraday was searching for evidence that the electric force affects the polarization of light. At the time, the experimental methods were not sensitive enough to measure this change of polarization; however, when light passes through various substances Faraday was able to observe this change. Therefore, this is called the Faraday effect. To quote Faraday in his daily journals, "but when the contrary magnetic poles were on the same side, there was an effect produced on the polarized ray, and thus magnetic force and light were proved to have relation to each other."
The Faraday Effect measures the polarization of light going through an active medium inside a magnetic field. The Faraday effect causes light to rotate as it goes through a magnetic field. And this rotation of polarization is linearly proportional to the component of the magnetic field.The degree of rotation depends on the color of light, the length of the magnetic field (L), and the properties of the medium the light goes through.
A polarizer is an optical filter that passes light of a specific polarization and blocks other waves based on the angle of the rotation. For instance, when the polarized light has the same polarization angle as the filter (equivalently, think of two linear filters parallel to each other), all of the light passes through. However, when the relative polarization angle between the light and the polarizer is at $\frac{\pi}{2}$ (two linear filters perpendicular to each other), none of the light passes through.
Using the basic knowledge of linear filters, we can now apply that to circular rotation of light through a magnetic field. In our experimental set up, see Fig \ref{fig:Apparatus}, the intensity of light that passes through the polarizer is measured by a photodiode.
In our experiment, the light from the output of the laser is at least partially polarized, and the laser has a small polarizer attached to the output so that the light is fully linearly polarized in a particular direction.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/faraday-rotation/faraday-rotation}
\caption{{\label{fig:Apparatus}
Our general apparatus consists of a laser wavelength of 650nm supplied by a current of 25mA, a solenoid, polarizer, and photodiode. The solenoid has a rod of SF-57 glass with a length of 0.1m inserted into the solenoid.%
}}
\end{center}
\end{figure}
Light intensity $I$ after passing through two polarizers at angles $\theta_1$ and $\theta_0$, measured by the photodiode, is predicted to be %
\begin{equation}
% \label{eq:Light_Intensity}
I = I_0 \cos^2[\theta_1 - \theta_0]
\end{equation}
where $I_0$ is the original light intensity. Since it is easier to think in terms of a $\cos(x)$ function, we can use a trig identity to simplify the intensity of light equation. Using $\Delta\theta=\theta_1 - \theta_0$, we can use a substitute the trig identity.
\begin{equation}
% \label{eq:trigidentity}
\cos^2[\Delta\theta]=\frac{1}{2}+\frac{\cos[2\Delta\theta]}{2}
\end{equation}
Since we are studying when the rate of change of the intensity is at a maximum, we need to take the derivative of Eq. \ref{eq:trigidentity}.
\begin{equation}
% \label{eq:sensitivity}
\frac{d\cos^2[\Delta\theta]}{d[\Delta\theta]}=-\sin[2\Delta\theta]
\end{equation}
Now, we can see that the rate of change of the intensity of light will be a maximum in relation to the polarization angle. The light intensity is at a maximum when 2$\Delta\theta$ is equal to $\frac{\pi}{2}$ because we are looking at a sine function. Therefore, using simple algebra, $\Delta\theta$ would be equal to $\frac{\pi}{4}$. Sensitivity $\frac{dV_{photodiode}}{d\theta}$ is defined to measure the largest change in voltage for a given change in $\theta$. This means that the sensitivity will be at a maximum when $\Delta\theta=\frac{\pi}{4}$. The sensitivity is directly related to Verdet constant,
\begin{equation}
% \label{eq:verdetequation}
v_c=\frac{1}{L}\frac{d\theta}{dV}\frac{dV}{dB}
\end{equation}
Following Melissinos and Napolitano \cite{Melissinos_2003} pages 204 - 210, we set the sensitivity to be at a maximum to study Verdet constant.
We begin by assuming that the shift in polarization of the light $\phi(B)$ passing through a rod of material of length $L$ due to the application of an external co-axial magnetic field $B$ is linearly proportional to the applied magnetic field and the length of the material, so that
\begin{equation}
\phi(B) = v_c B L
\end{equation}
where $v_c$ is a (wavelength-dependent) `constant' known as the Verdet constant. The value of the Verdet constant depends on the type of material and the wavelength of the polarized light.
Suppose then that the original angle of polarization of the light emitted from the laser is $\theta_0$, the change in polarization angle due the application of the magnetic field (as the light passes through a rod-shaped material) is $\Delta \theta(B) = v_c L B$, the angle of the polarization of the light after passing through the rod is $\theta_1 = \theta_0 + \Delta \theta$, and the angle of the second polarizer though which the polarized light passes before being detected by the photodiode (but after exiting the rod) is $\theta_2$.
Then, as noted by \cite{Melissinos_2003} if the voltage output from the photodiode $V_{\textrm{ pd}}$ is proportional to the light intensity, the photodiode response is given by
\begin{equation}
% \label{eq:Diode_Response}
V_{\mathrm{pd}}(B) = V_0 \cos^2[\phi(B)]
\end{equation}
Using Eq. \ref{eq:trigidentity}, the trig identity could be substituted into the diode response equation, Eq. \ref{eq:Diode_Response}, to further simplify the equation. In order to study all the different functions that could occur for the photodiode response, we have to make the equation as general as possible. So, we introduced two additional variables: $\varphi$ to represent the phase shift associated with the angle and $O_{offset}$ to represent a possible offset that our data could have.
\begin{equation}
% \label{eq:Simplifed_Diode_Response_Equation}
V(\phi) = \frac{V_0}{2} \cos[2 \phi + \varphi] + O_{offset}
\end{equation}
The photodiode sensitivity $\eta$ = $dV_{pd}/dB$ is given by
\begin{eqnarray}
\eta & = & - 2 V_0 \cos[\phi]\sin[\phi] \frac{d\phi}{dB} \\
& = & -V_0 \sin[2\phi] \frac{d\phi}{dB} \\
& = & + v_c LV_0 \sin[2\phi] \\
\end{eqnarray}
since $\phi = \theta_{2} - \theta_{1} = \theta_{2} - \left(\theta_0 + \Delta \theta(B)\right)$.
An example of a fit to Eq.~\ref{eq:Diode_Response} is shown in Fig.~\ref{fig:MethodOneGraph}.
Note that the sensitivity $\theta$ has a maximum value at $\phi = \pi /4$; at that orientation,
\begin{equation}
% \label{eq:MaxSensitivity}
\left.\frac{dV_{\mathrm{pd}}(B)}{dB}\right|_{\textrm{max}} =\left.\left( \frac{dV_{\mathrm{pd}}}{d\phi}\right)\right|_{\pi/4} \left(\frac{d\phi}{dB}\right)= -\left(V_0 \left.\sin[2\phi]\right)\right|_{\phi = \pi/4})\left(\frac{d\phi}{dB}\right).
\end{equation}
The derivation above (with some change in notation) is from \cite{Melissinos_2003}. See pages 205 - 210 for additional details.
\section{Method}
The arrangement of our general apparatus that produces the signal to be analyzed is shown in Fig.~\ref{fig:Apparatus}. It is important for the laser, the solenoid and the glass rod (SF-57) inside, the polarizer and the photodiode to be aligned for a signal that is as large as possible to so that the signal will be larger than error as much as possible, and that the signal follows Eq.~\ref{eq:Light_Intensity}.\\
The laser used in this experiment has a wavelength of $650\textrm{nm}$. The solenoid provides the uniform magnetic field that will result in the rotation of the polarization of light, and the polarizer determines the angle at which light can pass through, both of which are needed to determine the Verdet constant.\\
The photodiode was used to measure light intensity. A photodiode receives light as a current, and this current goes through a resistor, for which, the voltage across was measured and is proportional to the light intensity. Note that we picked a $1\textrm{k}\Omega$ resistor for the photodiode because if the resistance is too high the $1\textrm{A}$ current signal that our photodiode receives is not sufficient, for example, for a $30\textrm{k}\Omega$ resistor and the voltage output will saturate as a consequence.\\
There were no particular actions taken to eliminate light sources other than the laser (stray light), but this will not be a big problem. A constant stray light will result in a constant offset for the directly measured $V_{pd}$, but can be corrected by a constant for the Direct Fit Method. This does not matter for the Slope Method since results shifted by a constant have no affect on the slope, $dV_{pd} (B)/dB$, which is what we will use in the calculation. Constant stray light also has no effect for the Lock-in Method since the lock-in amplifier will filter out none alternating signals. Stray light that drift/fluctuate in intensity, on the other hand, cannot be easily subtracted, and will affect the accuracy and precision of result for the Direct Fit Method. A drifting stray light will affect the slope, $dV_{pd} (B)/dB$, for the Slope Method. But any stray light with drift/fluctuation on time scales longer than one modulation period of the reference signal is eliminated by the lock-in amplifier in the Lock-in Method.
\subsection{Direct Fit Method}
The most straight forward way of measuring the rotation angle caused by the change in magnetic field is to measure directly. We can see from Eq.~\ref{eq:Diode_Response}, the voltage signal from the photodiode is a cosine wave with respect to angle. To create these waves, we found an angle at which light intensity is close to greatest when $\vec{B_1}=0\textrm{T}$, and incrementing in $10^\circ$ steps for a whole cycle of $360^\circ$, the voltage output from the photodiode was read. This is then repeated with a magnetic field $\vec{B_2}=33.3\textrm{T}$. Both of these waves are plotted on a single graph, shown in Fig.~\ref{fig:MethodOneGraph} for easy comparison. Measurements can be taken at several other $B$ values as well for more accurate $v_c$.
By fitting the two waves on the graph, the shift in angle can be read directly. This corresponds to the difference in $dB=\vec{B_2}-\vec{B_1}$. $d\phi / dB$ and $v_c$ can be then be calculated.
Using a modified Eq.~\ref{eq:Light_Intensity}, we have $V_{pd}$as a function of $\theta$, adapted for real experimental data:
\begin{equation}
V_{\mathrm{pd}}(\phi)=V_0cos^2(\phi+\varphi)+C\\
\end{equation}
$dV_{\mathrm{pd}}/d\phi$ was calculated to find the angle at which the $dV_{\mathrm{pd}}$ is most sensitive to phase shift due to $dB$. This will give us the same result as evaluating Eq.~\ref{eq:sensitivity}, $\phi=\pi/4$. This angle $\phi$ is carried into our second method, where data is only collected at one fixed angle for a varying magnetic field.
\subsection{Slope Method}
In this method, instead of changing the angle of the polarizer for a full period of $V_{pd}$ ($\frac{dV_{pd}}{d\phi}$ in Eq.~\ref{eq:13} a modification of Eq.~\ref{eq:MaxSensitivity}), we kept the angle to be at a constant and varied the magnetic field in the solenoid through changing the current for the solenoid ($\frac{dV_{pd}(B)}{dB}$). Since we are staying at a constant angle, we want to make sure that this at this angle, $V_{pd}$ is most sensitive to change in polarization angle. The method for finding this angle is mentioned at the end of Section 3.1 where the angle $\phi$ is subtracted from the polarization angle at which the light intensity is measured to be greatest.
Values of $V_{pd}$ were measured as magnetic field was changed by changing the current going through the solenoid from -3A to 3A ($\vec{B}$ from -33.3mT to 33.3mT, conversion Eq.~\ref{eq:ItoB}), with increments of 0.5A.
\begin{equation}
% \label{eq:13}
\left.\frac{dV_{\mathrm{pd}}(B)}{dB}\right|_{\textrm{max}} =\left.\left( \frac{dV_{\mathrm{pd}}}{d\phi}\right)\right|_{\pi/4} \left(\frac{d\phi}{dB}\right)= -\left(V_0 \left.\sin[2\phi]\right)\right|_{\phi = \pi/4} v_c L.
\end{equation}
This data was then plotted with Voltage(photodiode) on the y axis, and Magnetic Field on the x axis, which you can see in Fig.~\ref{fig:MethodTwoGraph}. A linear fit was applied to the graph to obtain $dV_{pd}(B)/dB$, and $dV_{pd}/d\phi$ was found evaluating Eq.~\ref{eq:Simplifed_Diode_Response_Equation} at $\phi=\pi/4$, which is just $-V_0$. Using Eq.~\ref{eq:13}, we calculated $v_c$.
Note that since we are changing the $\vec{B}$, if you are using a solenoid as the source of magnetic field, beware of the limit of the current that is allowed for the solenoid.
It came to our attention as we were collecting data that there is a difference in the rate of change of $V_{pd}$ with respect to $\vec{B}$ between sweeping up and sweeping down the field. The cause of such needs to be further investigated, but in this experiment, we used the sweep up data for its consistency over several measurements, and consistency with the other two methods.
\subsection{Lock-in Method}
This is perhaps the most complicated method in terms of setup, but it eliminates the most noise.
A function generator was used to generate an alternating voltage, a sine wave with a DC offset, and the voltage signal was converted into an alternating current with a Kepco Current source. This results in a field that will alternate the directing of shift of the polarization angle.
The setup for $V_{pd}$ is shown in Fig.~\ref{fig:lockin}. This output voltage that carries an alternating signal was sent to a preamplifier with AC coupling setting, allowing only the alternating part of the signal to pass through, and eliminating any non alternating noise. It was set up to have a gain of $100$. The signal was then sent through a bandpass filter, set to the modulation frequency at $100\textrm{Hz}$, eliminating noise signals with frequencies much different from our real signal. The filter had a gain of $5$ (test data for the bandpass filter was consulted to ensure that the signal will not be saturated). A lock-in amplifier was then used to lock-in to a reference signal with the same frequency and phase (adjusted with a phase shifter) as the output signal, ideally allowing only $V_{pd}$ with a gain to pass through. The lock-in was set to have a gain of $10$. And finally, a resulting $V_{pd}$ with a total gain of $5000$ was sent to a digital voltmeter for $V_{pd,RMS}$ to be read.
This measurement was made 3 times with different AC amplitudes, as shown in Fig.~\ref{fig:MethodThreeGraph}, to make sure the relationship in the size of oscillations between $V_{func.}$ and $V_{pd}$ is linear, and to pick an AC $V_{func.}$ ($60 \textrm{mV}$) that will not saturate. At this oscillating $V_{func.}$, data was collected 5 times for different DC offsets, which in plotted in Fig.~\ref{fig:MethodThreeGraphTwo}.
The alternating $V_{pd}$ corresponds to the alternating part of $B$ in this method. $dV_{pd}/dB$ can be directly calculated through the alternating $V_{pd}/B$ (where $\textrm{V}_\textrm{pd}$ is the original value without any gain). $dV_{pd}/d\phi$ is obtained in the same way as the seconde method, and $v_c$ can be calculated.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/Lock-in1/Lock-in1}
\caption{{\label{fig:lockin}
This is the set up for which the signal was amplified and the noise was filtered through. The bandpass filter allowed us to filter noise that are not in the frequency range of our signal, and the lock-in amplifier eliminated noise with frequency and/or phase different from our signal by locking in with a reference signal, and amplify the optimized signal at the same time.%
}}
\end{center}
\end{figure}
\section{Results}\label{sec:Results}
%\label{DispEqSection} You can label sections for reference
With all the information from the methods section, we can now look at our data.
\subsection{Direct Fit Method}
For method 1, we have graphed the data of all the different voltage measurements, from the photodiode, verses the polarizer angle. Ideally, we wanted to start these measurements at $\frac{\pi}{4}$, so we took some preliminary measurements to find the angle at which the sensitivity reaches its maximum. At 340 degrees, we had a voltage output of 0.2078V, which seemed to be the highest the voltage could reach. \\
Based on Eq.~\ref{eq:Diode_Response}, we were able to confirm a cosine squared function.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/Screen-Shot-2015-10-22-at-9.45.51-pm1/Screen-Shot-2015-10-22-at-9.45.51-pm1}
\caption{{\label{fig:MethodOneGraph}
This plot displays the voltage verses the angle over 360 degrees both with and without an applied external magnetic field. The orange theoretical fit represents the experimental data taken without the magnetic field. Whereas, the blue theoretical fit represents the experimental data taken with the magnetic field of 3 amps. The variables with uncertainty from the best fit is given in the figure and in equations below.%
}}
\end{center}
\end{figure}
We have found all the values of the variables from Fig. \ref{fig:MethodOneGraph} for Eq. \ref{eq:Simplifed_Diode_Response_Equation}. \\\selectlanguage{english}
\begin{table}
\begin{tabular}{ c c c }
& Without the Magnetic Field (B=0mT) & With the Magnetic Field (B=33.3mT) \\
$V_o$ in V & 0.105 $\pm$ 0.00024 & 0.109 $\pm$ 0.00035 \\
$\phi$ in radians & 0.770 $\pm$ 0.0024 & 0.644 $\pm$ 0.0033 \\
\end{tabular}
\end{table}
\\
$\phi$=0.770-0.644=0.126 radians has a precision of 0.0024+0.0033=0.0058 radians but an accuracy of only 0.035 radians.
\subsubsection{Analysis of angle results}
For the preliminary analysis, to determine how many degrees off from the maximum sensitivity, we should look at the phase shift. When the magnetic field was off, the phase shift was 0.770 radians (22$^\circ$). Likewise, when the magnetic field was on, the phase shift was 0.644 radians (18$^\circ$). Whereas, we experimentally found the maximum to be around 340 degrees, which has a discrepancy of 342$^\circ$-340$^\circ$=2$^\circ$. This means that we have an uncertainty determination of $\pm$2$^\circ$=0.035 rad.
\subsection{Slope Method}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/Screen-Shot-2015-10-22-at-10.05.54-pm/Screen-Shot-2015-10-22-at-10.05.54-pm}
\caption{{\label{fig:MethodTwoGraph}
This plot displays the voltage from the photodiode over the full range of the magnetic field at a fixed angle.%
}}
\end{center}
\end{figure}
From the experimental data in Fig. \ref{fig:MethodTwoGraph}, we can fit a theoretical fit using Plot.ly to exact important information used to calculate Verdets constant.
\begin{equation}
% \label{eq:MethodTwoResults}
\frac{dV}{dB}= (4150 \pm 18) \cdot 10^-7\\
\frac{V_o}{2}= (1118 \pm 0.38) \cdot 10^-4\\
\end{equation}
From Fig. \ref{fig:MethodTwoGraph}, $\frac{dV}{dB}$ is the slope of the graph; whereas, $\frac{V_o}{2}$ represents the intercept of the graph, refer to Eq. \ref{eq:MaxSensitivity}.
The measurements for Fig. \ref{fig:MethodTwoGraph} were determined using a constant polarizer angle of $\frac{\pi}{4}$ because that is the point when we have a maximum sensitivity. When we stay at a constant angle, that would be equivalent to taking the derivative at a point in Fig \ref{fig:MethodOneGraph}. Therefore, it makes sense to have a linear fit, refer to Fig. \ref{fig:MethodTwoGraph}.
\subsection{Lock in Method}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/Screen-Shot-2015-11-23-at-8.41.19-pm/Screen-Shot-2015-11-23-at-8.41.19-pm}
\caption{{\label{fig:MethodThreeGraph}
This plot displays the voltage output from the solenoid verses the voltage from the sinusoidal output of the function generator. The plot was generated by connecting the function generator through a voltage to current power supply then to the solenoid. The purpose of plotting the values in the figure above was to verify a linear relationship.%
}}
\end{center}
\end{figure}
\subsubsection{Further Results for the lock in method}
We can proceed to pick one of the $V_{input,rms}$, which is the voltage coming from the function generator, to measure several times in order to study the uncertainty of the output direct current output.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/Screen-Shot-2015-11-23-at-8.51.39-pm/Screen-Shot-2015-11-23-at-8.51.39-pm}
\caption{{\label{fig:MethodThreeGraphTwo}
In the plot, the output voltage from the multimeter is displayed as the magnetic field of the solenoid varies.
The AC variation in the magnetic field is represented as $B(t)=B_{DC}+B_{AC,RMS}\sin$(2$\pi$ft). In this case, the oscillator voltage is kept at a constant voltage of 60mV which is used to calculate the $dB$ (refer to \ref{sec:lock_in_method}).%
}}
\end{center}
\end{figure}
From Fig. \ref{fig:MethodThreeGraphTwo}, we can study the uncertainty of the DC voltage output. As discussed in Section \ref{sec:Analysis}, we are looking for the uncertainty of the output voltage. That uncertainty can be calculated by calculating standard deviation of the points in Fig. \ref{fig:MethodThreeGraphTwo}.
$\therefore dV= 0.6580 \pm 0.01785$
Now that we have an understanding of all the data, we could calculate Verdet's constant using the information provided in this section.
\section{Analysis}\label{sec:Analysis}
\subsection{Direct Fit Method}
Verdet's constant is defined as
\begin{equation}
% \label{eq:Verdet_constant}
v_c= \left(\frac{1}{L}\right) \left(\frac{d\theta}{dB}\right) .
\end{equation}
If we substitute into Eq.~\ref{eq:Verdet_constant} the values for polarization angle we determined from our data for $B = 33.3$ mT and $B=0$ , we find
\begin{eqnarray}
v_c & = &\left( \frac{1}{0.1 \textrm{m}}\right) \left(\frac{\frac{1}{2}(0.770 \textrm{ rad} - 0.644 \textrm{ rad})}{33.3 \textrm{ mT}}\right) \\
& = & 0.0189 \left[\frac{\textrm{rad}}{\textrm{mT} \cdot \textrm{m}}\right]
\end{eqnarray}
Note that since we doubled $\theta$ to use a trig identity to convert a $\cos^2(x)$ function into a $\cos(x)$ function, we have to divide our nominal value for $\Delta\theta$ by 2. In this case, $\Delta\theta = \frac{0.770 - 0.644}{2} = \frac{0.126}{2}$ = 0.063.
An uncertainty in this value can be calculated from the reported error in fit coefficients for polarization angle: $\delta(d\theta)=\delta(\theta_o)+\delta(\theta_B)=0.00243+0.00331=0.00574 \textrm{ radians}$. This is an uncertainty of $9 \%$, leading to a corresponding error in $v_c$ of $9 \%$ as well (assuming no error in $L$ or $B$). Thus $v_c = 0.0189 \pm 0.0017 \left[ \frac{rad}{mT \cdot m} \right]$.
\subsection{Slope Method}
\label{sec:slope_method}
In order to determine Verdet's constant, we are finding the value of $V_o$ at the constant angle that we stayed at while varying the magnetic field to find $\frac{d\theta}{dB}$. Therefore, using Eq. \ref{eq:MaxSensitivity}, we can substitute $\phi$ for $\frac{\pi}{4}$. In this case, we have $V_{pd}=\frac{V_o}{2} \text{ at } \phi = \frac{\pi}{4}$.
Using Eq.~\ref{eq:Verdet_constant} and the chain rule,
\begin{eqnarray}
% \label{eq:chainrule}
v_c & = & \frac{1}{L} \frac{d\theta}{dV} \frac{dV}{dB} \\
& = & \frac{1}{L} \frac{1}{V_0} \frac{dV}{dB}
\end{eqnarray}
where we have substituted $V_o$ for $\frac{dV}{d\theta} = V_0$ into Eq. \ref{eq:chainrule}.
The value of $V_0$ can be found from the intercept $b$ of the fit to the data in Fig.~\ref{fig:MethodTwoGraph}. Here, $V_0 = 2 b = 0.2236 \textrm{ Volts}$.
From the slope of the linear fit to the data in Fig 5, we find that $\frac{dV}{dB}$ = 0.00415 V/mT . Substituting into Eq. \ref{eq:Verdet_constant}, we calculate $v_c = 0.01856\left[ \frac{rad}{mT \cdot m} \right]$
The uncertainty for Verdet's constant can be calculated by estimating how many degrees off we were from the actual maximum sensitivity of our apparatus. After doing some experimental analysis, we found an uncertainty of 5$^\circ$= 0.984 radians. We determined this value by adding together all the uncertainties from the apparatus.
To calculate the uncertainty, we need to find the maximum and minimum Verdet's constant to generate the length of the errors bars for our new Verdet constant. Using same method above, we can substitute in our new angles to calculate the uncertainty of Verdet's constant. We find $\delta(v_c)=v_{c,max} - v_{c,min}= 0.018352\left[ \frac{rad}{mT \cdot m} \right] - 0.018196\left[ \frac{rad}{mT \cdot m} \right]=0.000156\left[ \frac{rad}{mT \cdot m} \right]$ \\
\textbf{$\therefore v_c= 0.01856 \pm 0.00016\left[ \frac{rad}{mT \cdot m} \right]$ \\ \\
\subsection{Lock in Method}
\label{sec:lock_in_method}
Due to the problems associated with the output current from the Kepco DC to AC Current Supply based on the input voltage from a DC power supply or AC oscillator, we should use the expression $I_{coil,dc}=\beta V_{osc,dc}$ to convert $V_{osc,dc}$ to the current directly. \\\\
Experimentally $\beta$ is found to be 0.6137 $\frac{A}{V}$ (taken from Professor Nathanael Fortune's \textit{\href{https://plot.ly/~NathanaelFortune/347/rms-output-current-amplitude-through-solenoid-magnet-coil-as-a-function-of-rms-o/}{Kepco Current Source Voltage to Current Conversion Factor}}, verified at several voltages).\\
Using the conversion provided in the caption of Fig. \ref{eq:Diode_Response}, we used an oscillator voltage $V_{osc,dc}$=60mV.\\
\begin{equation}
% \label{eq:differentcurrent}
I_{coil,dc}=\beta V_{osc,dc}=(0.6137\frac{A}{V})(60 mV)\\
I_{coil,dc}=0.036822 A \\
\end{equation}
$\therefore dB= I\cdot11.1\frac{mT}{A}=0.40939 mT$ \cite{Reichert_1989}.
To recap all the variables we have, $V_{s,RMS}$ refers to RMS amplitude of the ac signal coming from the photodiode (as measured by the lock-in). Whereas, $V_{dc,out}$ is the voltage output from the lock-in, which is proportional to Vs,RMS. Eq. \ref{eq:magic} gives the constant of proportionality.
\begin{equation}
% \label{eq:magic}
V_{s,rms}=\frac{1.11}{Gain}V_{dc,out}
\end{equation}
The mean value of data provided in Fig. \ref{fig:MethodThreeGraphTwo} gives the value for $V_{dc,out}$. Substituting in our numerical values the gain $G = 5000$ and $V_{dc,out} = 0.658 V$ into the equation above, we find $V_{s,rms}=000416 V$.
If we substitute the value of maximum voltage from the photodiode $V_o=0.203 V$, $dB=0.40939mT$, and $dV=V_{s,rms}=000416 V$ into Eq. \ref{eq:chainrule}, we calculate the following Verdet's constant:
\begin{equation}
v_c=\frac{1}{v_o\cdot L}\cdot\frac{dV}{dB}= v_c=0.017665\left[ \frac{rad}{mT \cdot m} \right]\\
\end{equation}
For the uncertainty, we use the same method described in Section \ref{sec:slope_method}. Since our apparatus is the same, we found an uncertainty of 5$^\circ$= 0.984 radians.
We found the range of Verdet's constant to be equal to
\begin{equation}
\delta(v_c)=v_{c,max} - v_{c,min}= 0.017665\left[ \frac{rad}{mT \cdot m} \right] - 0.017056\left[ \frac{rad}{mT \cdot m} \right]=0.000609\left[ \frac{rad}{mT \cdot m} \right]
\end{equation}
\textbf{$\therefore v_c= 0.017665 \pm 0.000609\left[ \frac{rad}{mT \cdot m} \right]$ \\ \\
\end{table}\section{Discussion}
\label{sec:Discussion}
To summarize all our Verdet constants and the different methods used:
\begin{table}
\label{table:final}
\begin{tabular}{ l l l }
Corresponding Plot Number & Method Type & Verdet Constant \\
1 & Direct Fit Method & 0.0189 $\pm$ 0.0017 $\left[ \frac{rad}{mT \cdot m} \right]$ \\
2 & Slope Method & 0.01856 $\pm$ 0.00016 $\left[ \frac{rad}{mT \cdot m} \right]$\\
3 & Lock in Method & 0.0177 $\pm$ 0.0006 $\left[ \frac{rad}{mT \cdot m} \right]$ \\
\end{tabular}
\caption{\textbf{The different Verdet constant values are displayed in the figure below. For instance, the direct fit method, number 1, is labeled as 1 along the 'method type' axes. }}
\end{table}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.95\columnwidth]{figures/Screen-Shot-2015-11-26-at-11.17.24-pm/Screen-Shot-2015-11-26-at-11.17.24-pm}
\caption{\label{fig:final}
The final calculated Verdets constant with uncertainties as shown in Table.~\ref{table:final} for each of the three methods are plotted. The dotted orange line is average value of the Verdet constant from the second method.%
}
\end{center}
\end{figure}
After careful calculation and analysis of uncertainty, our final Verdets constant data are shown in Fig.~\ref{fig:final}. \\
The huge range of uncertainty for the direct fit method could be because we did not take enough measurements. 37 different measurements is probably not enough measurements for a method that is not very sensitive. At the peaks and dips, the measured voltage is not sensitive to small changes in the polarization angle. \\
For the last two methods-- slope method and lock in method-- we took into account how carefully we set the angles by carefully considering an uncertainty of 5$^\circ$. \\
The Verdets constant from the lock in method is below the average value, as shown in Fig. \ref{fig:final}. During the third week of our experiment, our whole apparatus mysteriously came loose. One possible reason that could have caused the whole apparatus to come loose is due to wood changing shape and thickness with temperature and humidity. In our experiment, the change of the laser and polarizer placement affected our lock in method. We tried our best to compensate for this error by experimentally remeasuring the maximum intensity from the polarizer, $V_o$. However, this could still be the reason why the Verdets constant is slightly below the average value. It is quite possible that the apparatus change could result in less power output from our laser and that could be why our Verdet constant is below the average value. \\
Overall, our different Verdet constant values agree within two standard deviations but not within one standard deviation. This means that there might or might not be a systematic error in one of the methods. To determine that, we would need to improve our measurements. Since the data probably lies within two standard deviations, we have concluded a good experiment.
\section{Conclusion}
We studied the faraday rotation through a few different methods. The direct fit method was shown have the most amount of uncertainty; however, it contained the most lowest limit. The direct fit method obtained a Verdet constant of 0.0189 $\pm$ 0.00172 $\left[ \frac{rad}{mT \cdot m} \right]$. The slope method was shown to be the most precise method as it involved careful apparatus planning. Whereas, the slope method was able to obtain a Verdet constant of 0.01856 $\pm$ 0.000156 $\left[ \frac{rad}{mT \cdot m} \right]$. And the lock in method fell during an unfortunate time of our experiment. Verdet constant from the lock in method, 0.017665 $\pm$ 0.000609 $\left[ \frac{rad}{mT \cdot m} \right]$, was slightly below all the other Verdet constant.
\appendix %\appendix* % Omit the * if there's more than one appendix.
\section{Magnetic Solenoid}
% \label{sec:MagSol}
The magnetic solenoid have these following specifications: \\
Length: 150 mm \\
Turns\ layers: 140 \\
Numbers of layers: 10 \\
Wire Size: 18 double insulated \\
DC Resistance: 2.6 $\Omega$ \\
The approximate calibration constant for the unit, measured at the center of the solenoid is:
\begin{equation}
% \label{eq:ItoB}
B=(11.1\frac{mT}{A})I\\
\end{equation}
where I is in amperes and B is in millitesla.
The magnetic field does vary along the axis of the solenoid; however, such variation can be neglected due to the insignificance inside the coil. These variations could be more significant outside the coil. However, that was never considered through this experiment.
\begin{acknowledgments}
We are thankful for our class instructor, Professor Nathanael Fortune, for helping us develop our experiment, teaching us the necessary skills to analyze data, and Professor Nalini Easwar for taking the time to listen to us and giving us mental support. And we would like to thank the Smith College Physics Department for making this experiment possible by providing the equipments, the space, and faculty. The peers from our Experimental Physics class also provided us help. Without the support, this experiment would not have been possible.
\end{acknowledgments}
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