Mostly changes to figures.

Author: jvdoorn

chapters/conventions.tex

@@ -1,3 +1,9 @@
 \chapter*{Conventions}
 \label{chap:conventions}
 In this thesis we regularly have to project 3D structures onto a 2D plane. To clarify the x-, y- and z-directions they have been color coded. This is done consistently throughout the thesis and matches with the colors assigned to the axes by COMSOL. The x-direction is colored in \textcolor{x_axis_color}{red}, the y-direction in \textcolor{y_axis_color}{green} and the z-direction in \textcolor{blue}{blue}. Additionally an attempt has been made to optimize figures for colorblind readers based on Paul Tol's color schemes\cite{paul_tol}.
+
+Furthermore, when a damping rate or Q-factor is given, we use the following definition:
+\begin{equation*}
+    Q = \frac{\text{energy stored}}{\text{energy dissipated per cycle}}
+\end{equation*}
+Another common way of expressing the Q-factor is in terms of the oscillations amplitude. This Q-factor is a factor $???$ higher than the Q-factor defined above.

chapters/method.tex

@@ -9,37 +9,52 @@ \section{Sample fabrication}
 
 The particle is magnetised by putting the whole sample in a magnetic field of approximately \qty{1.3}{\tesla} for several minutes at room temperature and pressure.
 
-A summary of the dimensions of the sample is shown in \autoref{tab:sample-dimensions}. For a schematic illustration of the sample see TODO.
-\begin{table}
+A summary of the dimensions of the sample is shown in \autoref{tab:sample-dimensions}. For a schematic illustration of the sample see \autoref{fig:sample-dimensions}.
+\begin{SCtable}
     \centering
     \begin{tabular}{lcc}
         \toprule
         \textbf{Parameter} & \textbf{Symbol} & \textbf{Value} \\
         \midrule
         Track thickness & $d$ & \qty{500}{\nm} \\
-        Inner track width & $w$ & \qty{50}{\um} \\
-        Inner track width & $w$ & \qty{100}{\um} \\
-        Inner loop radius & $r_1$ & \qty{85}{\um} \\
-        Outer loop radius & $r_2$ & \qty{170}{\um} \\
+        Outer track width & $w_1$ & \qty{50}{\um} \\
+        Inner track width & $w_2$ & \qty{100}{\um} \\
+        Inner loop radius & $r_1$ & \qty{60}{\um} \\
+        Outer loop radius & $r_2$ & \qty{120}{\um} \\
         \ce{Si}/Glass hole diameter & & \qty{100}{\um} \\
         \ce{Si}/Glass hole depth & & \qty{15}{\um} \\
         \bottomrule
     \end{tabular}
-    \caption{Dimensions of the sample. The radii of the inner and outer loops are the radii at the center of the track.}
+    \caption{Dimensions of the sample. The radii of the inner and outer loops are their inner radius. Refer to \autoref{fig:sample-dimensions} for further clarification.}
     \label{tab:sample-dimensions}
-\end{table}
+\end{SCtable}
+
+\begin{figure}
+    \centering
+    \begin{subfigure}[c]{.5\textwidth}
+        \centering
+        \includegraphics{figures/sample/trap_optical_microscope.pdf}
+    \end{subfigure}%
+    \begin{subfigure}[c]{.5\textwidth}
+        \centering
+        \includegraphics{figures/sample/trap_geometry.pdf}
+    \end{subfigure}%
+    \caption{An optical microscope image of the sample (\textbf{left}) and a schematic of the sample geometry (\textbf{right}). A couple features can be seen in the image 1) in the bottom left is a discoloration due to the epoxy 2) the dark streak in the bottom right is the edge of the cover glass 3) the dark discoloration is the \ce{Pt} layer of the glass. In the center of the trap you can clearly see the \ce{NdFeB} particle. Furthermore, you can see that their is a small misalignment between the hole in the \ce{Si} and hole in the cover glass. A definition of the symbols in the schemtic and their numerical value can be found in \autoref{tab:sample-dimensions}.}
+    \label{fig:sample-dimensions}
+\end{figure}
 
 \section{Experimental setup}
 The sample is mounted on a printed circuit board (PCB) with a thick (\qty{1}{\mm}) copper baseplate. The top of the sample is exactly flush with the top of the PCB. Additionally the pads for the wirebonds on the sample and the PCB are aligned and directly next to each other. This allows for very short wirebonds to be used. This significantly reduces the resistance of the wirebonds and thus the maximum current that can be applied to the sample. The wirebonds are made of \qty{25}{\um} thick \ce{Pt}.
 
 The PCB is then placed inbetween two Helmholtz coils with a diameter of \qty{30}{\mm} and 825 windings. They Helmholtz coils provide the uniform magnetic field to align the sample and the gradient magnetic field to control the vertical position of the particle. A dc current on the order of \qty{1}{\ampere} is sent through the coils by two power supplies (Tenma 72-2540).
 
-A microscope objective with a $10\times$ magnification and large working distance is used to provide a visual image of the sample. Additionally a laser (\qty{635}{\nm}, \qty{1}{\milli\watt}) is coupled in. The reflection of the laser is then imaged on a photodiode (ThorLabs PDA36A2). The photodiode is connected to a lock-in amplifier (Zurich Instruments MFLI) allowing for the detection of the motion of the particle.
+A microscope objective with a $10\times$ magnification and large working distance is used to provide a visual image of the sample. Additionally a laser (\qty{635}{\nm}, \qty{1}{\milli\watt}) is coupled in. The reflection of the laser is then imaged on a photodiode (ThorLabs PDA36A2). The photodiode is connected to a lock-in amplifier (Zurich Instruments MFLI) allowing for the detection of the motion of the particle. In addition to the photodiode, we can use the camera feed to analyse the motion of the particle. This is done using Python and OpenCV using the CSRT tracker. This tracker is quite robust at the cost of being computationally expensive. As such it cannot be used in real-time.
 
 \section{Simulations}
 There are two assumptions in the theory: infinitely thin tracks; and perfectly symmetric loops. Because of this the ideal current ratio will be different in practice. A simulation in COMSOL is used to guide our choice of the current ratio. The simulation models the \ce{Si} wafer, the \ce{Au} tracks and a air box surrounding them. The simulation is performed in 3D using a stationary dc-analysis.
 
 \subsection{Q-factor}
+\label{subsec:q-factor}
 In addition to this we estimate our effective Q-factor. A finite Q-factor is caused by damping. Below we discuss several sources of damping.
 
 \subsubsection{Gas damping}

chapters/results.tex

@@ -6,22 +6,23 @@ \chapter{Results}
 
 \begin{figure}
     \centering
-    \includegraphics{figures/magnetic_field_curvature.pdf}
+    \includegraphics{figures/data/magnetic_field_curvature.pdf}
     \caption{the \textbf{top} figures from left to right show $z$-component of the magnetic field evaluated along a line parallel to the $x$-, $y$- and $z$-axis respectively through the origin. The gray zones indicate the boundaryies of the trap. The \textbf{bottom} figure shows the curvature of the magnetic field for these calculations evaluated in the extrema. The simulations were performed for $i_1=\qty{0.5}{\ampere}$ and $\xi$ between \numrange{-0.5}{-5}. The curves have been labelled with the corresponding value for $\xi$.}
 \end{figure}
 
 \section{Measurements at atmospheric pressure}
 \label{sec:measurements-at-atmospheric-pressure}
+TODO: change this to only mention the x-mode? As we did not drive the y-mode separately.
 At atmospheric pressure we determined the dependence of $\omega_{x,y}$ on $\Omega$ and $i_1$. In these measurements the ratio $\xi$ was kept constant. The spectra are obtained using the lock-in amplifier connected to the photodiode. Due to the low Q-factor at atmospheric pressure, it is not possible to tell the $x$- and $y$-modes apart. The results are shown in \autoref{fig:xy-mode-dependence-1bar}. The $z$-mode is not visible in these measurements. In addition to this the dependence of the rotational mode on $B_0$ is shown in \autoref{fig:rotational-mode-dependence-1bar}. We note three curves with a apparently linear dependence on $B_0$. The two additional curves are modulations of the resonance peak with the trapping frequency. A fourth curve can be seen in the bottom right. Its origin is unclear. A fit has not been made due to the sparsity of the data. $\omega_\alpha$ has not been observed.
 
 \begin{figure}
     \centering
     \begin{subfigure}{0.45\textwidth}
-        \includegraphics{figures/xy_mode_dependence_on_driving_frequency.pdf}
+        \includegraphics{figures/data/xy_mode_dependence_on_driving_frequency.pdf}
         %\label{fig:xy-mode-dependence-on-driving-frequency-1bar}
     \end{subfigure}
     \begin{subfigure}{0.45\textwidth}
-        \includegraphics{figures/xy_mode_dependence_on_inner_current.pdf}
+        \includegraphics{figures/data/xy_mode_dependence_on_inner_current.pdf}
         %\label{fig:xy-mode-dependence-on-inner-current-1bar}
     \end{subfigure}
     \caption{The dependence of $\omega_{x,y}$ on $\Omega$ (\textbf{left}) and $i_1$ (\textbf{right}) at atmospheric pressure. The dashed lines are a fit following the theory of $\omega_{x,y}(\Omega) \sim 1 / \Omega$ and $\omega_{x,y}(i_1) \sim i_1$. The corresponding prefactors are $2\pi \cdot \qty{377.33170211685874\pm9.030914540546724}{\kilo\hertz\per\hertz}$ and \qty{626.5282946695123\pm20.984120764687987}{\hertz\per\ampere}. The normal value for $\Omega = 2\pi \cdot \qty{2.5}{\kilo\hertz}$ and $i_1 = \qty{200}{\milli\ampere}$ when they are not part of the sweep, the ratio $\chi = 2$ is always kept constant.}
@@ -30,7 +31,18 @@ \section{Measurements at atmospheric pressure}
 
 \begin{figure}
     \centering
-    \includegraphics{figures/rotational_mode_dependence_on_B0.pdf}
+    \includegraphics{figures/data/rotational_mode_dependence_on_B0.pdf}
     \caption{The dependence of $\omega_{\gamma,\tilde\beta}$ on $B_0$ at atmospheric pressure.}
     \label{fig:rotational-mode-dependence-1bar}
 \end{figure}
+
+\section{Q-factor dependence on pressure}
+\label{sec:q-factor-dependence-on-pressure}
+Using an external coil we are able to excite the $x$- and $y$-mode seperately. Ringdown measurements are performed to determine the pressure dependence on the Q-factor. The Q-factor in this case is defined as $Q = \omega_{x,y} \cdot \tau / 2$ where $\tau$ is the decay time of the oscillation ($\sim \exp\left(t / \tau\right)$). We qualitatively observed a strong dependence of the Q-factor on the driving amplitude. When the driving amplitude is too high we observe non-linear time dependent behaviour. The results in \autoref{fig:q-factor-pressure-dependence} are obtained with a sufficiently low driving amplitude.
+
+\begin{figure}
+    \centering
+    \includegraphics{figures/data/q_factor_pressure_dependence.pdf}
+    \caption{The pressure dependence of the Q-factor for the $x$-, $y$- and $z$-mode. The dashed lines are a prediction based on the theory from \autoref{subsec:q-factor}. The experimental Q-factors are determined using the ringdown method. The $z$-mode has not been observed experimentally. The $x$- and $y$- mode were driven sinusoidally at their respective resonance frequencies.}
+    \label{fig:q-factor-pressure-dependence}
+\end{figure}