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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 Helium-Neon Laser}  \\ % Tagline(s) or further description
[\baselineskip] % Tagline(s) or further description
\large 27. Jan 2015\par % Location and year

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

{\small Gruppe von:} \\[\baselineskip]
{\Large Danny Milosavljevic \small (Verfasser) \par}

{\itshape{\large  Technische Universit\"at Wien\par}} % Editor affiliation
\vspace*{0.5 cm}

{\Large Daryoush Nosraty Alamdary \par}
 % Editor list
%\vspace*{0.5 cm}
{\itshape{\large  Universität Wien \\ Fakultät für Physik}\par} % Editor affiliation
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{\scshape 2015} \\[0.3\baselineskip] % Year published
{\large Wien, Österreich}\par % Publisher

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\begin{document}
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%\title{Helium-Neon Laser}
% 26.01.2015 erstes Experiment bis ausschließlich Etalon.
%\maketitle
%\begin{abstract}
%We enable He-Ne lasing in different ways. We demonstrate wavelength
%selection by a Birefringent Crystal and mode selection by an Etalon.
%\end{abstract}
\begin{figure}[htbp]
\centering
\begin{gnuplot}[terminal=epslatex,terminaloptions=color]
plot [0:2*pi] sin(x) title 'Sine', cos(x) title 'Cosine'
\end{gnuplot}
\end{figure}
\section{Introduction}
\subsection{Theory of He-Ne Lasing}

\subsubsection{Population Inversion}
Usually, in thermal equilibrium, the number of particles per state (say $N_1$ and $N_2$, with $N_2$ for the higher-energy state) are Boltzmann distributed.\par
\begin{eqnarray*}
\frac{N_2}{N_1} = \exp\left({\frac{-(E_2 - E_1)}{k\,T}}\right)
\end{eqnarray*}
If the temperature is high enough, $N_2 = N_1$. If it is lower, $N_2 < N_1$.\par

For Lasers, we want to have $N_2 > N_1$. This is only possible if the system is not in thermal equilibrium.\par

Helium has metastable states, for example $2^1 S_0$, $2^3 S_0$. Because of the selection rules for optical transitions, from there, it is not possible to transition to the ground state.\par
These states can be populated by electron collisions (of the second type - the one where the collision causes a transition to take place) via gas discharge.\par
Such a Helium atom can collide with a Neon atom (atomic collision of the second type), the Helium atom reaches the ground state because its energy has been used to excite the Neon atom. The Neon atom reaches the $3s$ state (for example).\par
The Neon atom can transfer to $2p$ via an optical transition.\par
Eventually, the Neon atom will end up in $1s$ via spontaneous emission.\par
Because the lifetimes of the $3s$ states of Neon are 10 times longer than those of the $2p$ states, the necessary population inversion is achieved.\par
This enables Light Amplification by Stimulated Emission of Radiation.\par
Other Helium and Neon states are possible and cause different wavelengths of light to be emitted.

%\subsubsection{Line Broadening}
%The laser light that is observed has a much wider line width in the frequency domain than the natural line width.\par
%That is because the photons are emitted from moving sources, the atoms.\par
%The number of atoms in a velocity range (v, v + dv) are described by a Boltzmann distribution.\par
%In this way the doppler shift due to the movement with velocity v can be used to calculate the observed wavelength.
%All such observed wavelength from each atom in total form the observed laser spectrum.

% TODO atomic vs. electron collisions of the second order
\subsection{Experimental Setup}
We use a rail in order to fix devices we require for the particular experiment. Usually the laser tube is placed in the middle of the rail. Other devices are placed further outside along the rail, constructing our experimental setup.\par
If necessary, we have a pilot laser and iris in order to make a coarse alignment of the devices possible.\par
After alignment, we power the laser tube and adjust the devices in order to get the laser to operate.\par
We then optimize the laser output power by adjusting the devices and record a spectrum of the outgoing light using our spectrograph.\par

\subsubsection{Laser tube with Brewster Window}
The Laser tube contains the Helium-Neon gas in a capillary, and a cathode and anode at opposite ends of the tube.\par
The windows at the ends of the tube are tilted by Brewster's angle, reducing the gain of the $\sigma$-polarized light and minimizing reflections. However, the $\pi$-polarized light will pass through without loss.\par
Ingniting the laser takes about $8\,\unit{kV}$ (applied between anode and cathode), continuous operation takes about $2\,\unit{kV}$.\par
The voltage will cause a gas discharge and lasing will commence.\par
The tube we used has optimal output in the fundamental mode with a purely Gaussian beam.\par% and spectral purity in single mode operation.

\subsubsection{Mirrors}
The gain is proportional to $(N_2 - N_1)$. If the gain compensates for the losses, a standing wave (mode) will be built up in the resonator.\par
The mode must fulfil the condition\par
\begin{eqnarray*}
L = n\,\frac{\lambda}{2} = n\,\frac{c}{2\,\nu}
\end{eqnarray*}
where $L$ is the length of the resonator, $\lambda$ the wavelength of the mode, $c$ the speed of light, $\nu$ the frequency of the generated light and $n$ an integer.\par
Thus a mode has frequency
\begin{eqnarray*}
\nu(n) = n\,\frac{c}{2\,L}
\end{eqnarray*}
The spectral distance between neighbouring modes is:
\begin{eqnarray*}
\Delta\nu = \frac{c}{2\,L}
\end{eqnarray*}
The mirrors are tilted using adjustment screws until the laser ignites.

\subsubsection{Littrow Prism}
For wavelength selection, it's possible to use a Littrow prism. This is a special type of dispersive prism, usually triangular. A Littrow prism has a reflective coating on its back side, causing only one wavelength (selection by the angle of incidence) to be reflected back into the resonator.

\subsubsection{Birefringent Crystal}
In a Birefringent Crystal, the refractive index depends on the direction of propagation, causing an extraordinary beam (which seems to be violating Snell's law) to be emitted in addition to the ordinary beam.\par
It is an anisotropic crystal. It causes a phase shift of the extraordinary beam with respect to the ordinary beam, slowing one component down compared to the other.\par
Also, the incoming beam's polarization is changed in such a way that if the orginary and extraordinary beam were to be superposed, an elliptically polarized beam would result
(how exactly depends on the orientation of the birefringent crystal with respect to the optical axis).\par
In our laser setup, the birefringent crystal is used for wavelength selection.\par
The birefringent plate is placed under Brewster's angle into the resonator (Brewster's angle to minimize reflection losses).\par
The laser can only oscillate in a direction of polarisation given by the Brewster window.\par
The birefringent plate only doesn't change the polarisation if a phase shift of $2 \pi$ occurs between the ordinary and extraordinary beam (after passing through it twice).\par
For all other phase shifts, the polarisation will change and there will be reflection losses at the Brewster window, stopping the oscillation there.\par
In this way, we can select for a specific wavelength because all the beam contributions for other wavelengths will be attenuated very much.

% doppelbrechende Element: schiebt Phase zwischen eo Strahl und o Strahl. OK.
%                          Dreht die Polarisation des einfallenden Strahls OK.
%                               abhängig von dessen Polarisation relativ zur Orientierung der Achsen des doppelbrechenden Elements. OK.
\subsubsection{Etalon}
An Etalon can be used for mode (frequency) selection. It's a small glass plate acting as an additional resonator.
\begin{eqnarray*}
d := 1\,\unit{cm} \\
λ_m = \frac{2\, d}{m} \sqrt{n^2 - \sin(α)}
\end{eqnarray*}
where $λ_m$ is the wavelength with maximal transmission, $m$ is the order of the mode, $d$ the thickness of the etalon, $α$ the angle the etalon is tilted against the incoming beam, $n$ is the refractive index.
% 4% reflection on each side

% optical current for 632 nm line is 5 mA.

We calculate the number of modes that can oscillate in our laser with an emission wavelength of $λ_0 = 632.8\,\unit{nm}$:
\begin{eqnarray*}
L := 60\,\unit{cm} \\
λ_0 := 632.8\,\unit{nm} \\
c := 299792458\frac{\unit{m}}{\unit{s}} \\
k := 1.38\cdot 10^{-23} \frac{\unit{m}^2\,\unit{kg}}{\unit{s}^2\,\unit{K}} \\
T := 293.15 \unit{K} \\
m_{Ne} := 20.10\,\unit{u} = 3.34988\cdot 10^{-26} \unit{kg} \\
v_w := \sqrt{\frac{2\,k\,T}{m_{Ne}}} \approx 491\,\frac{\unit{m}}{\unit{s}} \\
λ_0 ν_0 = c \\
Δν_{Doppler} := 2\,\sqrt{\ln(2)}\,\frac{v_w}{c}\,ν_0 \\
%Δν_{Doppler} = 2\,\sqrt{\ln(2)}\,\frac{v_w}{c}\,\frac{c}{λ_0} \\
Δν_{Doppler} = 2\,\sqrt{\ln(2)}\,\frac{v_w}{λ_0} \approx 1.29\,\unit{THz} \\
Δν = \frac{c}{2\,L} \approx 250\,\unit{MHz} \\
N_{modes} := \frac{Δν_{Doppler}}{Δν} \approx 5
\end{eqnarray*}

\section{Results}
% some results are also in the original docs

\subsection{Mirror-Mirror}
We set up the rail with a mirror, the laser tube and a mirror, in that order.\par
Then we adjusted the mirrors using a pilot laser and an iris.\par
Then we enabled power to the laser tube, fine-tuned the mirrors until we accomplished lasing.\par
After further tuning to maximize laser output power, we got 12.80\,\unit{V} on the photodetector and the following spectrum (mind the clipping because of saturation):\par
\screenshot[Spectrum when using Mirror-Mirror; integration time: 100]{./2015-02-02/Mirror-Mirror/1}
%Here:\par
%\input{2015-02-02/Mirror-Mirror/1.tex}
\screenshot[Spectrum when using Mirror-Mirror; integration time: 3]{./2015-02-02/Mirror-Mirror/2_3ms}
%\screenshot[Spectrum when using Mirror-Mirror; integration time: 3]{./2015-02-02/Mirror-Mirror/3_3ms}
%dupe: \screenshot[; integration time: 3]{./2015-02-02/Mirror-Mirror/4_3ms}
\hline
Subtracting the noise floor we get:\par
\screenshot[Spectrum when using Mirror-Mirror; integration time: 3]{./2015-02-02/Mirror-Mirror/5_mit_offset_hand}
% \screenshot{./2015-02-02/Mirror-Mirror/7_mit_offset}
% \screenshot{./2015-02-02/Mirror-Mirror/7_mit_offsetb}

\subsection{Littrow Prism}
Note that from this point on, we subtracted the background light intensity in the graphs.\par
We then set up the laser using the Littrow Prism as described above and maximized the power.\par

%  - Tilt the prism a little by spinning the adjustment screw (A) until the red laser beam fades out.
%  - Spin the screw further until an orange laser beam occurs.
%  - If the orange beam does not occur make sure that the intensity of the beam is maximized and the Brewster's windows of the tube and the LPT are cleaned.
%  - maximize the intensities of the orange and red beam while operating the laser tube with 4 mA by fine tuning the prism, mirror, tube. Measure the maximized intensity.
%  - measure the wavelengths of both beams with a spectrometer. Compare the results with the spectral lines of the Helium-Neon gas in the tube.

Maximizing the output power gave us $12.84\,\unit{V}$ at the photodiode (on the left side).

%   LPT:
%     LPT: 12.83 V, 12.84 V.
%     photodiode measured on the left side.
%     max intensity 12.71 V without prism.
%     nach Verschieben Prism: 12.69 V, 12.70 V, 12.71 V.
%     mit Littrow: max. 12.75 V rot
%\screenshot[Spectrum when using Littrow Prism; integration time: 3]{./2015-02-02/LPT/1}
% TODO \screenshot{./2015-02-02/LPT/2_3ms_b}
\screenshot[Spectrum when using Littrow Prism; integration time: 3]{./2015-02-02/LPT/2_3ms}
% TODO \screenshot{./2015-02-02/LPT/4_3ms_schraeg}

\hline

We couldn't find the orange line at $λ = 611.8\,\unit{nm}$.

\subsection{Birefringent Crystal}
%   Birefringent Tuner:
%      power 12.70 V, 12.66 V, 12.53 V.
%      Birefringent 7grad, 8grad.
%      wavelength 632.8 nm with (12.72..12.76) V
%      how many times do the series of different colored lines occur when tuning? 7 times for 632.8 nm
We set up the laser using the birefringent crystal as described above and maximized the power.\par
The strongest lines are at $632.8\,\unit{nm}$ (and $640.1\,\unit{nm}$ - spectra not shown) wavelength, the photodiode voltage was $12.7\,\unit{V}$.\par
The other lines are now strongly attenuated.
%For $632.8\,\unit{nm}$, we saw the pattern repeat 7 times.

% FIXME Habt ihr 640 nm auch gefunden.
%\begin{alltt}
%- set up laser using Birefringent Tuner. 
%  - Maximize intensity. 
%  - Measure wavelength of the beam.
%  - strongest lines: 632.8 nm, 640.1 nm.
%  - measure the occuring wavelengths and the intensity. (632.8 nm: 12.72 V..12.76 V)
%  - how many times: 7 times for 632.8 nm.
%\end{alltt}
Tuning the birefringent crystal, we saw the same spectrum repeated seven times. Two of them follow:
\screenshot[Spectrum when using Birefringent Crystal; integration time: 10]{./2015-01-26/BFC/1}
%same \screenshot[Spectrum when using Birefringent Crystal; integration time: 10]{./2015-01-26/BFC/1b}
%same \screenshot[Spectrum when using Birefringent Crystal; integration time: 10]{./2015-01-26/BFC/2}
% \screenshot{./2015-01-26/BFC/2b}
 \screenshot[Spectrum when using Birefringent Crystal; integration time: 3]{./2015-01-26/BFC/3_3ms}
% \screenshot{./2015-01-26/BFC/3_3msb}

\subsection{Etalon}
We used the Etalon as described above. We found multiple modes around the maximum intensity ($12.79\,\unit{V}$) at $λ = 632.5\,\unit{nm}$.
The linewidth is now very narrow.\par
%Two modes had $m > 1$.
%The central mode was very instable.

% FIXME Wie viele Moden habt ihr gefunden.

% most likely speed of Neon atoms

% FIXME Habt ihr Fotos von den Moden gemacht?

% 2.2.2015: gemessen:
%   mehrere Moden, aber alle mit demselben λ = 632.5 \unit{nm}.
%   rechter Spiegel getuned ohne Etalon: 12.66 V.
%   2 Moden mit m > 1, horz. vert.
%   0th Zentralmode sehr instabil
%   v3.File: rechte Spalte (!!)
%   Fenster geputzt, linker Spiegel festgeschraubt, Voltage 12.79 V mit Spiegel-Spiegel. Nach Justage 12.80 V.
%   nach Anrempeln: 12.79 V.
%   Etalon:
%     12.66 V mit Etalon. Max bei 632.5 nm.
%--
%FIXME
%\begin{alltt}
%- set up laser using Etalon. Find the zero order and measure the intensity of the beam.
%  - spectrometer
%  - find the first and second order by tilting the etalon
%  - measure corresponding intensities with photodet and spectrometer
%  - identify the light points of the higher orders by using the iris
%  - calculate how many modes can oscillate in a He-Ne laser with a resonator length of 60 cm and an emission wavelength of 632.8 nm.
%  - (formulas in part II).
%  - discuss the practical relevance of your calculation. Can you achieve the calculate number of modes. Why/why not?
%\end{alltt}

%The zero order mode was very instable.\par
After some seeking, we found the zero order mode:\par
\FloatBarrier
\input{./2015-02-02/Etalon/Central_Mode}
\FloatBarrier
\hline
After very minimal tilting (i.e. can happen by the air without anyone touching it) the zero order mode will turn off:\par
\input{./2015-02-02/Etalon/Central_Mode_2}
\FloatBarrier

Spectra of the other orders are very weak (for example the following spectrum was taken when the photodiode voltage was $1.4\,V$):
\screenshot[Spectrum when using Etalon; integration time: 3]{./2015-02-02/Etalon/2_3ms_only_1_4_V}\par
\FloatBarrier
\hline
Putting the Etalon straight, we got:\par
\screenshot[Spectrum when using Etalon; integration time: 3]{./2015-02-02/Etalon/3_3ms_gerade}
\FloatBarrier
\hline
Tilting the Etalon, we got:\par
\screenshot[Spectrum when using Etalon; integration time: 3]{./2015-02-02/Etalon/4_3ms_schraeg}
\FloatBarrier
\hline
% same \input{./2015-02-02/Etalon/Central_Mode_3}
%\FloatBarrier
%\begin{thebibliography}{9}
%\bibitem{Schumm2013} Dr. Thorsten Schumm - Praktikum Quantenoptik
%\end{thebibliography}

\end{document}
