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% ask Stefan, DW: 141723
% Laser Current 45.1 mA \pm 0.1 mA
% Camera: Frame rate: 10..15 frames/s, Exposure time: 100 ms \pm 0.2 ms
% Coupling Fiber - Resonator contact: should be at 0.05 mm .. 0.2 mm
% Fabry-Perot: 1 m cavity, thus mode spacing 100 MHz. [in doc]
% Connected fiber has a section with only 500 nm in diameter.
% DFB laser around 850 nm, with a short-term (< 5 μs) linewidth of 400 kHz (Toptica Photonics, DL 100 DFB 1028).
% Laser frequency can be tuned over a max of 1.1 THz via temperature modulation.
% Laser fine tuning over a range of 20 GHz via input current.
% Laser power < 40 μW
% Fiber: n = 1.49896 (calculated)
% Laser current should be around 20 mA.
% Real laser current is 50.7 mA (on the first try) and 41.6 deg C.
% FWHM calculated using:
%   gnuplot> f(x) = a + b*(1 - (1/pi)*(g/2)/((x - x0)**2 + (g/2)**2)) [in doc]
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{\LARGE \textbf{Protokoll} \\ \vspace{0.2 cm}
 Praktikum Quantenphysik}\\[0.2\baselineskip] % Title

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\scshape % Small caps
{\Large Bottle Resonator}  \\ % Tagline(s) or further description
[\baselineskip] % Tagline(s) or further description
\large 12. Dezember 2014\par % Location and year

\vspace*{2\baselineskip} % Whitespace between location/year and editors

{\small Gruppe von:} \\[\baselineskip]
{\Large Danny Milosavljevic \small (Verfasser) \par}
{\Large Daryoush Nosraty Alamdary \par}
 % Editor list
\vspace*{0.5 cm}
{\itshape{\large  Universität Wien \\ Fakultät für Physik}\par} % Editor affiliation

\vfill % Whitespace between editor names and publisher logo

{\scshape 2015} \\[0.3\baselineskip] % Year published
{\large Wien, Österreich}\par % Publisher

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\begin{document}
\titleGP
\newpage
\tableofcontents
\newpage
\title{Evanescent Fields and Bottle Resonators}
\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 power (out of the Bottle Resonator or bypassing it).
\end{abstract}
\section{Introduction}
\subsection{Theory}
\subsubsection{Evanescent Fields}

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.

Snell's law is

\begin{eqnarray*}
n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
\end{eqnarray*}

where $\theta_1$ and $\theta_2$ are measured from the normal to the
interface plane.

If $ n_2 < n_1 $, the refracted wave only exists for angles $ \theta_1 $
smaller than some critical angle:

\begin{eqnarray*}
\sin(\theta_1) = \frac{n_2}{n_1} \sin(\theta_2) \\
%\sin(\theta_1) \le \frac{n_2}{n_1} \\
%\theta_1 \le \arcsin\left(\frac{n_2}{n_1}\right) \\
%\sin(\theta_1) \ge -\frac{n_2}{n_1} \\
-\arcsin\left(\frac{n_2}{n_1}\right) \le \theta_1 \le \arcsin\left(\frac{n_2}{n_1}\right)
\end{eqnarray*}

If $ \theta_1 $ is bigger than this critical angle (for $ n_1 = 1.5 $, $ n_2 = 1
$, that critical angle is $ \approx 42\degree $), then no refracted wave
into the second medium exists.\par

In the following, the case where $ \theta_1 > 42\degree $ will be considered.\par
In particular, what happens to the part that would have been the refracted wave will be considered.\par
Let $ E_2 = E\indices{_0_2} \exp\left(i \left(\vec{k_2}\cdot\vec{r} - ω t\right)\right) $.
Split $ \vec{k} $ (with $ k_2 := |\vec{k_2}| = 2 \pi \frac{n_2}{λ_0} $) into a component $ k_x = k_2 \sin(\theta_2) $ parallel to the interface and a component
$ k_y = k_2 \cos(\theta_2) = k_2 \sqrt{1 - (\sin(\theta_2))^2} = k_2 \sqrt{1 - \left(\frac{n_1}{n_2} \sin(\theta_1)\right)^2} $ describing the propagation normal to the
interface.\par
By precondition, $ \sin(\theta_1) > \frac{n_2}{n_1} $. Therefore, the argument of the square root above is certainly negative.\par
$ k_y = k_2 i \sqrt{\left(\frac{n_1}{n_2} \sin(\theta_1)\right)^2 - 1} $.\par
Let $ β = k_2 \sqrt{\left(\frac{n_1}{n_2} \sin(\theta_1)\right)^2 - 1} $
which is real.\par
\begin{eqnarray*}
k_y = i β \\
E_2 = E\indices{_0_2} \exp\left(i \left(k_x r_x + k_y r_y - ω t\right)\right) \\
%E_2 = E\indices{_0_2} \exp\left(i \left(k_x r_x + i β r_y - ω t\right)\right) \\
%E_2 = E\indices{_0_2} \exp\left(i k_x r_x + i i β r_y - i ω t\right) \\
%E_2 = E\indices{_0_2} \exp\left(i k_x r_x - β r_y - i ω t\right) \\
E_2 = E\indices{_0_2} \exp(- β r_y) \exp\left(i (k_x r_x - ω t)\right) \\
\end{eqnarray*}
$ E_2 $ exponentially decays with distance to the interface and is called "evanescent field".\par
$β$ is:\par
\begin{eqnarray*}
%β = 2 \pi \frac{n_2}{λ_0} \sqrt{\left(\frac{n_1}{n_2} \sin(\theta_1)\right)^2 - 1} \\
β = \frac{2 \pi}{λ_0} \sqrt{\left(n_1 \sin(\theta_1)\right)^2 - n\indices{_2^2}}
\end{eqnarray*}
Light from one medium can be coupled into another medium if the distance between the media is small enough so that the evanescent fields overlap. This phenomenon is called "frustrated internal reflection".

\subsubsection{Fabry P\'erot Resonator}
A Fabry P\'erot Resonator consists of two mirrors, both somewhat transmissive. The mirrors are oriented so that most light is captured between them and reflected from them.
The light between the two mirrors will form a (Gaussian) standing wave because of the reflection.
Let $l$ be the length of the space between the two mirrors, $i$ be a natural number, $ i > 0 $, $n$ be the refractive index of the medium, $c$ be the speed of light.
Then a standing wave is formed if:
\begin{eqnarray*}
l = i \frac{λ_0}{2\,n}
\end{eqnarray*}
At the resonance frequencies $ν_i$ there's constructive interference between the incoming light and the light between the mirrors.
\begin{eqnarray*}
ν_i = i \frac{c}{2\,n\,l}
\end{eqnarray*}
Let $τ$ be the time constant of the decay of the energy between the mirrors in the absense of a driving light field, $T$ be the reciprocal of the optical resonance frequency.
Then the quality factor $Q$ is defined as:
\begin{eqnarray*}
Q = 2 π \frac{τ}{T} = ω_0\,τ
\end{eqnarray*}
Let $N$ be the number of round-trips. 
Then the finesse $\mathcal{F}$ is defined as:
\begin{eqnarray*}
\mathcal{F} = 2 π N
\end{eqnarray*}

\subsubsection{Bottle-Resonator Coupling}
In a thin strand of optical fiber (called "Bottle Resonator") similar wave modes can also be observed.
Our setup uses such an ultrathin fiber which is not connected to anything by cable.
It's interesting how much energy is stored in the Bottle Resonator mode. According to \cite{Rausch2011}, a system (Bottle Resonator, Light Field) is described by:
\begin{eqnarray*}
\frac{d}{dt} a = i\,(ω_0 - ω)\, a - \frac{1}{2}\,\left(\frac{1}{τ_0} + \frac{1}{τ_{fiber}}\right) a + \frac{1}{τ_{fiber}^2} s
\end{eqnarray*}
... where $s$ is the input field amplitude and $t$ the output field amplitude.
The power transmitted through the coupling fiber is then proportional to $p^{*}\,p$, although the \emph{energy} stored in the resonator is proportional to $a^{*}\,a$.\par
The transfer of optical energy between the mode of the coupling fiber and the bottle mode is described by the characteristic time constant $τ_{fiber}$.\par
The intrinsic resonator energy loss is described by $τ_0$.\par
$ω_0$ is the resonance frequency of the bottle mode.\par
In the absence of a driving light field, the equation above is solved for energy as:
\begin{eqnarray*}
\frac{d}{dt} W = a^{*} \frac{d}{dt} a + a \frac{d}{dt} a^{*} = -\left(\frac{1}{τ_0} + \frac{1}{τ_{fiber}}\right) W \\
\frac{d}{dt} W = -\left(\frac{1}{τ_0} + \frac{1}{τ_{fiber}}\right) W \\
\frac{1}{τ_{load}} := \frac{1}{τ_0} + \frac{1}{τ_{fiber}} \\
W = W_0 \exp\left(-t\,\left(\frac{1}{τ_{load}}\right)\right) \\
Q_{load} := ω_0\,τ_{load}
\end{eqnarray*}

\subsubsection{WGM in Bottle Resonator}
Let there be a driving light field.\par

The light path of a Bottle Resonator mode (WGM) is generally circular and can have multiple rings.
$Q$, $ν_i$ are similar to the Fabry P\'erot resonator, although $l$ is the circumference of the fiber now.
Differences are: modes with multiple rings are possible, $ν_i$ depends on the number of rings (indirectly by $i$).

Let $ν_0$ be the optical resonance frequency of the mode, $\Delta ν$ be the spectral linewidth.
The loaded quality factor $Q_{load}$ is \cite[Section 2.1]{Rausch2011}:
\begin{eqnarray*}
Q_{load} = \frac{ν_0}{\Delta ν}
\end{eqnarray*}
And the finesse is:
\begin{eqnarray*}
\mathcal{F} = \frac{Δν_{FSR}}{Δν}
\end{eqnarray*}

\subsubsection{Bottle Resonator Free Spectral Range $Δν_{FSR}$}

We assume the diameter of the resonator is $60\,\unit{\micro m}$. Then its circumference is $l = \pi\,188.5\,\unit{\micro m}$.\par
Then $Δν_{FSR} = \frac{c}{2 n l} = \frac{299792458\,\unit{m}/\unit{s}}{2\cdot\,n\,\pi\,\cdot60\,\unit{\micro m}}$ (from the resonance condition).\par
Because in the Fabry-P\'erot cavity, a similar equation $100\,\unit{MHz} = \frac{c}{2 n L}$ holds, we can calculate $n = 1.49896$.\par
Next, we assume that the Fabry-P\'erot Resonator and the Bottle Resonator use the same material (with the same refractive index).\par
Therefore, $Δν_{FSR} \approx 530.5\,\unit{GHz}$ for the Bottle Resonator.\par
% 530.5164769729845 in GHz.

\subsubsection{Bottle Resonator Loss Mechanisms}

A photon in the resonator can be lost by:\par
\begin{itemize}
\item Absorption by erbium ions (causing green fluorescence)
\item Scattering on the surface of the coupling fiber (intrinsic loss)
\item Scattering away at the resonator-fiber coupling point (parasitic loss)
\end{itemize}\par

We introduce the coupling parameter $K$ defined by:\par
\begin{eqnarray}
K = \frac{K_{fiber}}{K_{para} + K_0}
\end{eqnarray}
where $K_{fiber} = 1/τ_{fiber}$ describes the rate of coupling between the coupling
fiber mode and the resonator mode, $K_0 = 1/τ_0$ describes the rate of
coupling to intrinstic loss channels and $K_{para} = 1/τ_{para}$ describes
the rate of coupling to parasitic loss channels.\par
Then the probability that a photon stored in the resonator is coupled back into the fiber (rather than lost) is:\par
\begin{eqnarray}
E_{photon} = \frac{K_{fiber}}{K_{para} + K_0 + K_{fiber}} = \frac{1}{1 + 1/K}
\end{eqnarray}

\subsubsection{Bottle Resonator Coupling Regimes}
It's common to classify different coupling regimes dependent on the ratio $\frac{τ_0}{τ_{fiber}}$:

\begin{itemize}
\item Under-coupled regime: $τ_{fiber} > τ_0$. For a large gap x, the transmission $T_{res}$ is close to 1 and increases with distance. Only little power is transferred to the resonator - where it is dissipated.
\item Critical coupling: $τ_{fiber} = τ_0$. For one particular gap the power that goes into the resonator is just as much as is lost by intrinsic losses. While the light field leaving the resonator couples back into the coupling fiber it has a phase shift of $π$ and thus there is almost zero transmission to the coupling fiber.
\item Over-coupled regime: $τ_{fiber} < τ_0$. For even smaller gaps x, the relative transmission $T_{res}$ will become close to 1 again.
\end{itemize}

\subsection{Experimental Setup}
A monochromatic tunable laser is used as light source to feed light into both a reference Fabry P\'erot Resonator 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, the evanescent waves couple by 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 power at the end of the coupling fiber using a photodiode 1. We also measure the Fabry P\'erot Resonator's output power using a photodiode 2.
The Fabry P\'erot Resonator will be used as reference in order to be able to read frequencies off the graph.\par
Both results are then fed into a oscilloscope. We use a scan module to scan the frequency of the laser and trigger the oscilloscope. 
The horizontal axis corresponds to the laser frequency, the vertical axis corresponds to 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.\par

Picture of the Bottle Resonator, already glowing with light from the compling fiber (which originally came from the laser):
\FloatBarrier
\begin{figure}[h]
 \centering
 \includegraphics[width=3.0in, height=2.4in]{probesnapshot.png}
 \caption{Bottle Resonator}
 \label{with low}
\end{figure}
\FloatBarrier
% TODO Describe how to measure the transmission efficiency of the coupling fiber,
% TODO Describe how to calculate how much power is coupled into the resonator
\section{Results}
\subsection{Measurements and Data Analysis}
First we determined the Fabry-P\'erot mode spacing. For this, for each oscillator file we did the following:
\begin{itemize}
\item We found the beginning first-in-time Lorenzian, $t_{min}$
\item We found the end of the last-in-time Lorenzian, $t_{max}$
\item We counted the number of Lorenzians (peaks), $c$
\item We calculated the mode spacing by $ n_i := (t_{max} - t_{min})/c $.
\end{itemize}
Let $n$ be the union of all the $ n_i $ intervals.
Thus the Fabry-Perot mode spacing $n$ is (units will cancel through the division g/n but they are formally seconds):
\begin{eqnarray*}
\input{fabry.v}
\end{eqnarray*}
The value $n$ corresponds to the known Fabry-Perot mode spacing $100\,\unit{MHz}$ and we later use it to convert the $g$ linewidth of the bottle resonator to $\unit{MHz}$.

%TODO effect of light polarization on the resonator modes\par
%TODO power of the laser beams \par
%Sources of error\par
%We analyze the resonator properties.

\subsubsection{Critical Coupling}
For Critical Coupling, the transmission comes closest to zero because all the power is dissipated in the resonator. 
The light leaving the resonator couples back into the coupling fiber but is phase-shifted by $\pi$. \par
The corresponding quality factor is:
\begin{eqnarray*}
Q_{crit} = \frac{Q_0}{2}
\end{eqnarray*}
% TODO relative transmission
\oszismall{58} % TODO plot the mode spectrum (freq vs. "relative" transmission).\par
% FIXME state what power full transmission corresponds to.\par
For critical coupling, we determined the linewidth $Δν$, loaded quality factor $Q_{load} = \frac{ν_0}{Δν}$ (where $ν_0$ is the mode resonance frequency which we approximated using $c = ν_0 λ_0$ with $λ_0 = 852\,\unit{nm}$, one of the used wavelengths of our laser. We assumed that the extra error by not using a more variable wavelength is negilible).\par
We determine the intrinsic quality factor $Q_0$ by $Q_0 = 2\,Q_{load}$:\par
\begin{eqnarray*}
Q_0 = 6520000 \pm 110000
\end{eqnarray*}

%FIXME photon lifetime FIXME\par

\subsubsection{Resonant Transmission vs Coupling Gap}
Next we studied how transmission changes with gap, starting at a distance of $1.3 \unit{\micro m}$, moving to less distance.
The $x$ coordinates in the following plots are relative to $13.85\,\unit{\micro m}$ and using only Measurement 2 (we did not plot Measurement 1 because it is extremely similar and the $x$ coordinate ranges are overlapping).
The conversion from piezo voltages to distances used was: $Δx = ΔU_{piezo}\,0.25\,\unit{\micro m}/\unit{V}$.\par
% (overcoupled regime: $T_{res} = 0.96$, so $T_{trans} = 0.04$)
% impossible to do with our data: TODO useful: horizontal axis origin at fiber contact position.
%TODO Plot $T_{res}$ for $x$ range.\par
Our dips were quite small (dip height 40\% of the maximum signal). Therefore, we renormalized the data for $T_{res}$ to the interval 0..1.
We call the normalized data $T_{norm}$.\par
\FloatBarrier
\begin{figure}[H]
\input{fits/transmission_vs_x}
\end{figure}
Fitting this to the theroetically expected formula gave us large uncertainities:
\begin{eqnarray*}
a :=    1.586, a_{min} := -8.065\cdot 10^6, a_{max} := 8.065\cdot 10^6\\
b :=   -0.006795, b_{min} := -8.349\cdot 10^5, a_{max} := 8.349 \cdot 10^5\\
c :=    0.1642, c_{min} := 0.1403, c_{max} := 0.188\\
T_{norm}(x)=\left(\frac{1-a\,\exp(-(x-b)/c)}{1+a\,\exp(-(x-b)/c)}\right)^2
\end{eqnarray*}
\FloatBarrier
\subsubsection{Quality factor vs. Coupling Gap}
%\FloatBarrier
\begin{figure}[H]
\input{fits/Q_vs_x}
\end{figure}
\FloatBarrier
\hrulefill

\subsubsection{Photon Coupling Efficiency}
For the overcoupled regime:
\begin{eqnarray*}
K = \frac{1 + \sqrt{T_{norm}}}{1 - \sqrt{T_{norm}}}
\end{eqnarray*}
For the undercoupled regime:
\begin{eqnarray*}
K = \frac{1 - \sqrt{T_{norm}}}{1 + \sqrt{T_{norm}}}
\end{eqnarray*}
% below: FIXME $\frac{\sigma_K}{K} = \frac{\sigma_T}{\sqrt{2} T_{res}}$\par

\FloatBarrier
\begin{figure}[H]
\input{fits/logK_vs_x}
\end{figure}
% TODO auto-read from fit.log
$log(K)$ was fit with $g(x) := a\,x + b\,\unit{\micro m}$, with $a = -6.75751 \pm 0.5784$, $b = 0.630094 \pm 0.1854$.\par

\FloatBarrier
\hrulefill

$E_{photon} = \frac{1}{1 + 1/K}$.

\FloatBarrier
\begin{figure}[H]
\input{fits/E_vs_x}
\end{figure}
\FloatBarrier
\hrulefill

\subsection{Conclusion and Summary}
We reproduced the expected linear dependency of $\log(K)$ (with $K$ being the photon coupling efficiency) vs $x$.\par

The mode we used did not couple that well, our dips were just 40\% of the maximum, therefore our analysis has some extra error.

More sources of error:\par
\begin{itemize}
\item absorption
%\item Rayleigh scattering
\item scattering at the beam splitter
\item scattering at surface blemishes, dust
%\item 
\end{itemize}

% TODO mention what you learned
% TODO main results
%% TODO error estimates
\section{Appendix}
Laser was set up with:\par
\begin{tabular}{|l|l|}
\hline
Name & Value \\
\hline
Temperature & $ 41.6\,\unit{\degree C} $ \\
Wavelength & $ 852\,\unit{nm} $ \\
Current & $ 50.7\,\unit{mA} $ \\
Scanning Amplitude & $ 5.1 $ \\
\hline
\end{tabular}

Fabry-P\'erot-Frequency = $100\,\unit{MHz}$.\par
Fabry-P\'erot-Cavity-Length $L := 1\,\unit{m}$\par
% TODO Anderes Oszilloskop dokumentieren.

\subsection{Fit Function}
The function used for fitting the Lorentzian in the mode spectra (see appendix 3.2, appendix 3.3) was proportional to $(1 - \text{Lorentzian})$:\par
\begin{eqnarray*}
f(x) := \begin{cases} 0, g < 0 \\ a + b \left(1 - \frac{1}{\pi}\,\frac{g/2}{(x - x_0)^2 + (g/2)^2}\right), g \ge 0 \end{cases} \\
f(x_0) = \begin{cases} 0, g < 0 \\ a + b \left(1 - \frac{1}{\pi}\,\frac{1}{g/2}\right), g \ge 0 \end{cases}
\end{eqnarray*}
The center height h of the transmission dip is thus approximately:\par
\begin{eqnarray*}
h := f(\infty) - f(x_0) \\
h \approx \frac{b}{\pi\,g/2}
\end{eqnarray*}

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

With polarization 0, 0.\par

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

\begin{tabular}{|l|l|l|l|}
\hline
\input{positionsA.inc}
\hline
\end{tabular}
\par
The FWHM ($\Delta ν$) of the WGM mode (CH1) is as follows:\par
\oszismall{44}
\oszismall{45}
\oszismall{46}
\oszismall{47}
\oszismall{48}
\oszismall{49}
\oszismall{50}
\oszismall{51}
\oszismall{52}
\oszismall{53}
\par

\subsection{Measurement 2}

Stepper position = $42.2699\,\overline{\mu m}$\par % , 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
\input{positionsB.inc}
\hline
\end{tabular}
\par
The FWHM ($\Delta ν$) of the WGM mode (CH1) is as follows:\par
\oszismall{54}
\oszismall{55}
\oszismall{56}
\oszismall{57}
\oszismall{58}
\oszismall{59}
\oszismall{60}
\oszismall{61}
\oszismall{62}
\oszismall{63}
\par

\begin{thebibliography}{9}
\bibitem{Rausch2011} Prof. Dr. Arno Rauschenbeutel: Evanescent fields \& Bottle resonators
\end{thebibliography}
\end{document}
