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\begin{document}
\title{Evanescent Fields and Bottle Resonators}
\author{xxx}
\maketitle
\begin{abstract}
We research the storage of light in a Bottle Resonator, an extremely thin disconnected strand of fiber. We change the distance of a coupling fiber to the Bottle Resonator and the laser frequency and analyze the effect of distance and laser frequency on stored and transmitted (out of the Bottle Resonator or bypassing it) power.
\end{abstract}
\section{Introduction}
TODO Introduction to 
1) evanescent fields and how they arise\par

At the interface between two media with different optical indices, in general an incoming light wave can be reflected and/or refracted. 
It is possible to choose the incident angle in such a way (large angle with respect to the normal to the interface, that is a small angle with respect to the interface) that no refracted wave results from the incoming wave. However, then the electric field decays exponentially beyond the interface.

2) the WGM bottle resonator and some main differences to the Fabry P\'erot resonator

Modes in the Bottle Resonator are the possible configurations (TODO) where light stays in the Bottle Resonator. The light path in such a mode is generally circular and can have multiple rings.
TODO: same: Q, mode freq, resonance condition.

\section{Experimental Setup}
A monochromatic tunable laser is used as light source to feed light into both a reference Fabry P\'erot Microresonator and our Bottle Resonator.
The bottle resonator is standalone, but is in close proximity to a coupling fiber that is transporting the light from the source. 
The distance between bottle resonator and coupling fiber can be varied (in one direction only) by a stepper motor (for coarse tuning) and a piezo (for fine tuning).
The light is sent into the fiber in such a way that total reflection happens. Then, an evanescent wave still exists outside the (thin) coupling fiber. Because the Bottle Resonator is also very thin, this evanescent wave can continue into the Bottle Resonator. That is called frustrated internal reflection.
(It is also possible for the wave in the Bottle Resonator to continue into the coupling fiber again, usually there will be a phase shift of $\pi$ then)\par

We measure the flux TODO at the end of the coupling fiber using a photodiode 1. We also measure the Fabry P\'erot Microresonator's output using a photodiode 2. \par
Both results are then fed into a oscilloscope. We use a scan module to scan the frequency of the laser. The horizontal axis is the laser frequency, the vertical axis is photodiode voltage. Graphs are of 1) voltage of photodiode 1, 2) voltage of photodiode 2\par
Note that the goal is to judge the power stored in the Bottle Resonator, so that would be the complement of photodiode 1's signal.

Picture of the Bottle Resonator, already glowing with light from the compling fiber (which originally came from the laser):
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TODO Describe how to measure 
the transmission efficiency of the coupling fiber
how to calculate how much power is coupled into the resonator
\section{Data Analysis}
TODO Sec 6 Sec 6.1 Sec 5.3 transmission efficiency of the coupling fiber effect of light polarization on the resonator modes power of the laser beams Q-factor of the resonator mode etc Sources of error
\section{Conclusion and Summary}
TODO mention what you learned
main results
error estimates
\section{Appendix}
TODO all plots of the mode spectra
Laser was set up with:
\begin{tabular}{|l|l|}
\hline
Name & Value \\
\hline
Temperature & 41.6 grad C \\
Wavelength & 852 nm \\
Current & 50.7 mA \\
Scanning TODO Amplitude & 5.1 \\
\hline
\end{tabular}

TODO Reference Fabry-P\'erot-Freq = ?
TODO Anderes Oszilloskop dokumentieren.

\subsection{Measurement 1}
Stepper position = $44.0002\,\overline{\mu m}$, TODO error\par

With polarization 0, 0.\par

\begin{equation}
60\,\overline{V} = 15\,\overline{\mu m} \\
\overline{V} = 0.25\,\overline{\mu m}
\end{equation}

\begin{tabular}{|l|l|l|l|}
\hline
Time & Piezopos/$\overline{V}$ & Piezopos/$\overline{\mu m}$ (calc.) & File \\
\hline
21:08 & 48.0 & 12.0 & 44 \\
21:10 & 47.4 & 11.85 & 45 \\
? & 46.8 & 11.7 & 46 \\
21:11 & 46.4 & 11.6 & 47 \\
21:13 & 46.0 & 11.5 & 48 \\
21:14 & 45.6 & 11.4 & 49 \\
21:15 & 45.2 & 11.3 & 50 \\
21:17 & 45.0 & 11.25 & 51 \\
21:18 & 44.6 & 11.15 & 52 \\
21:19 & 44.3 & 11.075 & 53 \\
\hline
\end{tabular}
\par
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\subsection{Measurement 2}

Stepper position = $42.2699\,\overline{\mu m}$, TODO error\par

With polarization 210 for the $\lambda/2$ polarisator and 140 for the $\lambda/4$ polarisator in order to maximize dip height:\par

\begin{tabular}{|l|l|l|l|}
\hline
Time & Piezopos/$\overline{V} $ & Piezopos/$\overline{\mu m}$ (calc.) & File\\
\hline
21:26 & 57.7 & 14.425 & 54 \\
21:28 & 56.9 & 14.225 & 55 \\
21:29 & 56.2 & 14.05 & 56 \\
21:30 & 55.8 & 13.95 & 57 (crit.) \\
21:31 & 55.4 & 13.85 & 58 \\
21:32 & 55.0 & 13.75 & 59 \\
21:33 & 54.4 & 13.6 & 60 \\
21:34 & 54.2 & 13.55 & 61 \\
21:35 & 53.9 & 13.475 & 62 \\
21:36 & 53.6 & 13.4 & 63 \\
21:38 & 53.3 & 13.325 & - (touches) \\
\hline
\end{tabular}
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\begin{equation} x \end{equation}
\subsection{A}

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% TODO references

\end{document}
