\chapter{Optical levers for tilt motion reduction}
\label{oplevs}

The sensors dedicated to measure the seismic motion need to account for horizontal, vertical and tilt displacements in all degrees of freedom in order to be efficient: the technology for their improvement is currently pushing and competing on sensing as low seismic motion as possible. On an interferometric detector, seismic motion affects the stabilization of the supports where the optics lie. This produces unwanted noise at low frequencies (< 30 Hz), which reduces the sensitivity of the detector.\\
During the first year of my PhD studies, I investigated the use of optical levers to reduce tilt motion: a device has been built at UoB, and tested at the Albert Einstein Institute (AEI) in Hannover in June 2019.\\
The content of this chapter has been re-adapted from my Mid-term report \cite{mca}. A poster about this project has been presented at the LVK meeting in Maastricht (September 2018) \cite{poster2}.

\section{Inertial sensors affected by tilt-coupling}
There are many contributions affecting aLIGO sensitivity at low frequency. One of the most investigated is the tilt of HAM vacuum chamber of ISI platforms, which dominates above 1 Hz \cite{lantz}.\\
For the rotational degrees of freedom, getting a good estimate of ground motion is not trivial because no rotational sensors capable of measuring the ground motion in rotation at low frequencies have been installed yet on aLIGO ISI platforms \cite{cooper}.\\
However, there is a possible way to measure angular displacements of the benches very precisely (10$^{-12}$ rad/$\sqrt{Hz}$) and to actively control them: this could be done by optical levers. In the following, we will analyse the main contributions to tilt motion and we will see how optical levers could be useful to suppress this motion.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images/HAMoplev.PNG}
\caption[Example of tilt-coupling contributions at LHO]{Plot of the contributions to the Suspension point L motion at LHO HAM5. The pitch (RX) contribution dominates above 1\,Hz (Figure taken from \cite{lantz}).}
\end{figure}

\paragraph*{Horizontal sensors}
One of the most important problems, in order to achieve good isolation, is the sensitivity of the horizontal sensors to rotation. If we could independently measure the rotation, we could calculate the true translational motion.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=1.6]{images/hor.pdf}
\caption{Basic sketch of horizontal sensor tilting.}
\label{a}
\end{figure}

\noindent
Referring to Fig. \ref{a}, when a rotation around the center of mass of the sensor occurs, an additional term to the equation of motion (F$_{tilt}$) appears:

\begin{equation}
\centering
F_{tilt} = mg\sin\theta
\end{equation}

\noindent
where m is the mass and g = 9.8 m/s$^2$ is the gravitational acceleration.\\
\noindent
So we have the following situation:

\begin{equation}
\centering
m\ddot{x} = -kw - b\dot{w} + F_{tilt},
\end{equation}

\noindent
where $x$ is the direction of motion, k is stiffness and b is damping of the spring.\\
We assume that the angle is very small, in such a way $\sin \theta$ $\simeq$ $\theta$. So the equations of motion are:

\begin{equation}
\centering
m\ddot{x} = -b(\dot{x} - \dot{y}) - k(x-y) + mg\theta,
\end{equation}

\noindent 
where $x$ is the displacement of the mass and $y$ is the displacement of the support of the sensor.\\
Since we want to study the system in the frequency domain, we substitute $w = x-y$ and apply the Laplace transform:

\begin{equation}
\centering
m(W+Y)s^2 = -bWs -kW + mg\Theta. 
\end{equation}

\noindent
After some manipulations, we obtain:

\begin{equation}
\centering
W = \frac{ms^2}{ms^2 + bs +k} \left(-Y + \frac{g}{s^2}\Theta\right).
\end{equation}

\noindent
where W is the measured quantity, Y and $\Theta$ are the quantities we are interested in. Remembering that s = i$\omega$ in a steady-state situation, we have:

\begin{equation}
\centering
W(\omega) = \frac{-m\omega^2}{-m\omega^2 + ib\omega +k} \left(-Y - \frac{g}{\omega^2}\Theta\right).
\end{equation}

\noindent
The relative sensitivity to translation and tilt are included in the second term in brackets. We expected this result, as the general one is that, for a horizontal seismometer, the ratio of the sensitivity to rotation (seismometer signal per radian of angle) to the sensitivity to horizontal motion (seismometer signal per meter of translation) at a particular frequency $\omega$ is:

\begin{equation}
\centering
\frac{rotation \; response}{translation \; response} = \frac{g}{\omega^2}.
\end{equation}

\noindent
If we know the size of our system, it is possible to calculate  the angle $\theta$.\\
\noindent
Since we have a factor $\omega^2$ at the denominator, it has more contributions at low frequencies: the contribution given by the tilt is decoupled and summed to the transfer function.\\
\noindent
When the seismometer is tilted, its sensitivity to angles increases as $g / \omega^2$. So, if we have some sort of seismic system measuring ground motion with horizontal seismometers, we could in principle measure this contribution and remove it by subtracting from the transfer function.

\paragraph*{Vertical sensors}
If we are dealing with vertical sensors, referring to Fig. \ref{v}, in presence of tilt we have:

\begin{equation}
\centering
m\ddot{x} = -b(\dot{x} - \dot{y}) - k(x-y) + mg\cos\theta.
\end{equation}
\noindent
Since $cos\theta$ $\sim$ 1 - $\theta^{2}/2$, the $\theta^{2}$ is second order and negligible for changes of angles.

\begin{figure}[h!]
\centering
\includegraphics[scale=1.6]{images/vert.pdf}
\caption[Tilting of vertical sensor]{Tilting of vertical sensor.}
\label{v}
\end{figure}

\section{Optical levers}
In general, an optical lever is a convenient device that makes use of a beam light and a position sensor to measure a small displacement and thus to make possible an accurate measurement of angles. This method is a very useful approach in sensitive non-contacting measurements. A light source, typically a laser, impinges on an optic reflecting the beam on a position device, which records any displacement of the beam, i.e. of the optic.\\
\noindent
When the optic is tilted by an angle $\theta$, we have the situation illustrated in Fig. \ref{opt2}: if all the distances are known, we can compute the angle $\theta$.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/opt2.pdf}
\caption{Concept of the optical lever working principle: when the optic is tilted by a known angle, the displacement is detected by the photodiode.}
\label{opt2}
\end{figure}

\noindent
What if we have both horizontal and vertical seismometers on the same bench, as on aLIGO? In this case, we have two instruments that are sensitive to horizontal and vertical ground motion at the same time. When the bench is tilted, they are tilted at the same time of the same angle, but they are not affected in the same way, as we have seen, but we are not able to deduce the tilt motion at low frequency because of the limitations given by the sensor noises.\\
\noindent
If we could measure both vertical and horizontal motions and decouple the contribution of the tilt for the horizontal one, we could know exactly the amount of corrections the actuators have to perform.\\
With optical lever systems we can measure the angle of the tilt, even if it is extremely small: in this way we could be able to directly measure the tilt angle $\theta$ and apply corrections to the horizontal sensor through a dedicated active system.\\

\noindent
The device described in this chapter should involve sensing and actuation for the seismic motion on aLIGO. The position device can not be set on the same bench where the other sensors are, because it would be affected by the same ground motion. So it has to be placed on another bench, at some distance L, and an actuation system is associated to it in order to adjust the tilt of the bench under exam. The angular sensitivity increases with the distance L. Moreover, even the bench where the position device is set needs to be stable: another optical lever could be placed on it, with the associated actuation, mirroring the first one and keeping both platforms stable. A basic picture of the whole system is shown in Fig. \ref{z}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images/opt3.pdf}
\caption{Basic principle of the optical lever used for sensing and actuation for seismic isolation.}
\label{z}
\end{figure}

\noindent
The purpose when thinking of interferometers is to help reducing the RX motion on the HAM chambers that propagates into the suspensions, proving a displacement of the suspension point and hence introducing noise in the cavities.

\section{Experiment design}
In order to understand the feasibility of the project in terms of performance, we have to estimate the noise budget and the sensitivity of the system.\\
\noindent
Let's start from the block diagram of the system, in Fig. \ref{BD}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/BD.PNG}
\caption{Block diagram of the optical lever system.}
\label{BD}
\end{figure}

\noindent
In the block diagram all the noises we have to deal with are described: the most relevant in terms of contributions are the shot and the thermal noises; then there are all the noises related to the electronics, like dark current, flicker and op-amp noises, usually given in the datasheet of the devices.\\
Beyond them, we have to consider the relative intensity noise (RIN), due to instabilities in the laser intensity: this kind of noises reduces the signal-to-noise ratio, limiting the performance of the electronic transmission. This may be reduced by making the signal positions independent of illumination intensity.\\
The translation coupling noise due to the motion of the platform where sensors are set is also considered: this gives a contribution in the measurement in terms of linear displacement, while we are measuring the angular motion of the platforms.

\subsection{Quadrant Position Devices}
The Quadrant PhotoDiodes (QPD) are the position devices usually involved with optical levers. They consist of four distinct and identical quadrant-shaped photodiodes that are separated by a small gap (typically, $\sim$0.1 mm) and together form a circular detection area capable of providing a 2D measurement of the position of an incident beam.\\
When light is incident on the sensor, a photocurrent $I$ is detected by each quadrant Q in Fig. \ref{j}.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{images/quad.PNG}
\caption[QPD segmented details for beam position detection]{View of the segmented photodiode. Each quadrant Q receives a photocurrent which is the signal responsible for any displacement detection: depending on which quadrant is receiving more or less photocurrent, it is possible to derive the position of the beam on the active area.}
\label{j}
\end{figure}

\noindent
The normalized coordinates (X, Y) for the beam's location depend on the detected photocurrents and are given by the following equations:\\

\begin{equation}
\centering
X = \frac{(I_2+I_3)-(I_1+I_4)}{I_1+I_2+I_3+I_4}
\end{equation}
\begin{equation}
\centering
Y = \frac{(I_1+I_2)-(I_3+I_4)}{I_1+I_2+I_3+I_4}
\end{equation}

\noindent
If a symmetrical beam is centred on the sensor, four equal photocurrents will be detected, resulting in null difference signals and, hence, the normalized coordinates will be (X, Y) = (0, 0). The photocurrents will change if the beam moves off center, producing different signals that are related to the beam displacement from the center of the sensor.

%\subsection{Structure of the device}
%As we have seen, a QPD is formed by 4 photodiodes placed in such a way to form a circle. There are several way to build the circuit to convert the quadrant outputs into x and y position signals. In Fig. \ref{qpd} it is shown a schematic example of a circuit of the detector.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.6]{images/QPD.PNG}
%\caption{Example of a QPD position extraction circuit.}
%\label{qpd}
%\end{figure}
%
%\noindent
%In this example, after pre-amplification, each adjacent pair of quadrant signals is fed to a differential amplifier. These signals then give partial information about motion in the x or y axis. The signals from each axis are then summed by final stage of amplification, giving the x and y position signals.\\
%Note that also, it may be useful to generate a total intensity signal by summing all of the quadrants; this may be used to normalise the position signals to make them independent of illumination intensity (\cite{qpd}).

\subsubsection{Spot position and displacement}
At the light of what we have seen about QPDs, we have to compute where the beam is on the photodiode: the coordinates of the beam depend on the photocurrents. If we are dealing with a Gaussian beam, they are proportional to the Gaussian intensity:\\

\begin{equation}
\centering
I(x) = \frac{P_0}{\pi w_{x}^2}e^{-2\left(\frac{x}{w_{x}}\right)^2},
\end{equation}

\noindent
for coordinate x, the same for y; w$_{x}$ is the beam size (radius) in x direction and represents the distance from the x axis to which the amplitude reduces by 1/e and the intensity by 1/e$^2$; P$_0$ is the input power.\\
\noindent
If we want to obtain the signal in terms of power, we should integrate the Gaussian intensity. However, if the spot displacement is small, and assuming that the variation of the spot size is negligible with respect to the spot displacement ($\Delta w \ll \Delta x$), we can apply a linear approximation. So we have:

\begin{equation}
\centering
P_{x} = \frac{P_0}{\pi w_{x}^2} \int^R_0 e^{-2\left(\frac{x}{w_{x}}\right)^2} dx,
\end{equation}

\noindent
where R is the radius of the detector.\\
The integral of the Gaussian function is the Error Function, defined as:

\begin{equation}
\centering
{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} dt.
\end{equation}

\noindent
So we have:

\begin{equation}
\centering
P_{x} = \frac{P_0}{\pi w_{x}^2} \int^R_0 e^{-2\left(\frac{x}{w_{x}}\right)^2} dx = \frac{P_0}{\pi w_{x}} \frac{\sqrt{\pi}}{2} \sqrt{2} w_{x} {erf}(x).
\end{equation}

\noindent
The first term of the Taylor expansion of the error function is erf($x$) $\approx$ $\frac{2}{\sqrt{\pi}}x$, so we have:

\begin{equation}
\centering
P_{x} = \frac{P_0}{\pi w_{x}^2} \frac{\sqrt{\pi}}{2} \sqrt{2} w_{x} {erf}(x) \approx  \sqrt{2}\frac{P_0}{\pi w_{x}} x.
\end{equation}

%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.45]{images/Pw.PNG}
%\caption{Plot of the computed error function (blue) and linear approximation (red).}
%\end{figure}

%\noindent
%The variation of power onto the device is given by:
%
%\begin{equation}
%\centering
%\Delta P_x  \approx  \sqrt{2}\frac{P_0}{\pi w_{x}} \Delta erf(x),
%\end{equation}
%
%\noindent
%and plotted in Fig. \ref{q}.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.5]{images/dP.PNG}
%\caption{Variation of power.}
%\label{q}
%\end{figure}

\noindent
Using the linear approximation, the displacement in x is given by:

\begin{equation}
\centering
\Delta x  \approx  \frac{\pi w_{x}}{\sqrt{2}P_0} \Delta P_{x};
\end{equation}

\noindent
and the ratio between the variation of the power and the displacement in the x direction is given by:

\begin{equation}
\centering
\frac{\Delta P_{x}}{\Delta x} \approx  \sqrt{2}\frac{P_0}{\pi w_{x}} [{W}/{m}].
\end{equation}

\noindent
The same computation gives the result for the coordinate y:

\begin{equation}
\centering
\frac{\Delta P_{y}}{\Delta y} \approx  \sqrt{2}\frac{P_0}{\pi w_{y}} [{W}/{m}].
\end{equation}

\noindent
In order to estimate the resolution of the device and provide an estimate of its performance, we need to account for the noises coming from the QPD and external sources.

\subsection{Photon shot noise}
\label{sn}
Because of the fact that the working principle of the QPD is based on tracking the motion of the centroid of power density, it is useful to compute the contribution of the shot noise.\\
The shot noise is the fluctuation of the photon counting on the photodetector. This fluctuation obeys the Poisson statistics, but for a large mean number of photons ($<N> \gg 1$), it approaches the Gaussian one, with standard deviation $\sigma$ = $\sqrt{<N> }$.\\
If P$_{0}$ is the input power and $\omega$ is the frequency, the number of photons on the photodiode in a given time interval t is:

\begin{equation}
\centering
<N> = \frac{P_{0}t}{\hbar \omega};
\end{equation}

\noindent
the fractional fluctuation of the number of photons is then:

\begin{equation}
\centering
\frac{\sigma}{<N>} = \frac{1}{\sqrt{<N>}} = \sqrt{\frac{2\pi \hbar c}{P_{0} t \lambda}}.
\end{equation}

\noindent
The fractional fluctuation of the input power $\sigma_p$ is given by the fractional fluctuation of the number of photons:

\begin{equation}
\centering
\frac{\sigma_p}{P_{0}} = \frac{\sigma}{<N>},
\end{equation}

\begin{equation}
\centering
\sigma_p = P_{0}\frac{\sigma}{<N>} = P_{0}\sqrt{\frac{2\pi \hbar c}{P_{0} t \lambda}},
\end{equation}
For t=1 s:

\begin{equation}
\centering
\sigma_p = \sqrt{\frac{2\pi \hbar c P_{0}}{\lambda}}.
\end{equation}
\noindent
So the fractional fluctuation of the power scales as the square root of the input power. Since QPDs are sensitive to shape and density distribution of the incident beam, a beam which does not have a Gaussian power distribution will be centred based on the power, rather than the geometric center of the beam, so it will be more affected by shot noise.\\
With the laser wavelength $\lambda$ = 1064 nm and an input power P$_0$ = 1 mW that we chose for this experiment, we obtain:

\begin{equation}
\centering
\sigma_{p} = 1.4 \times 10^{-11} \frac{W}{\sqrt{Hz}}.
\end{equation}
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.5]{shot.PNG}
%\caption{Plot of the shot noise with respect to a given range of frequency.}
%\end{figure}


%\noindent
%APPROFONDISCI FACCENDA DEL QUADRANTE DIVISO!\\
%Note that the QPD is composed by 4 photodiodes, so we should consider for each section 1/4 of the shot noise previously computed. However, we can consider the device as a unique device because...

\subsection{Thermal noise}
\label{tn}
The other, important noise affecting the measurements is the thermal noise due to the transimpedance resistor of the photodiode R. It is given by:

\begin{equation}
\centering
V_{th} = \sqrt{\frac{4k_{B}T}{R}} \frac{A}{\sqrt{Hz}},
\end{equation}

\noindent
where k$_B$ = 1.38 $\times$ 10$^{-23}$ J/K is the Boltzmann constant, T is the temperature. In order to obtain the thermal noise in units of W/$\sqrt{Hz}$ we divide by the responsivity $\rho$ (in A/W). For a 1064 nm laser wavelength the responsivity is typically 0.2 A/W.\\
To compute R, consider that the output voltage from the photocurrent is given by:

\begin{equation}
\centering
V_{ph} = P_0 \rho R,
\end{equation}

\noindent
and because the output voltage is limited by the range imposed by the interferometer itself, i.e. [+10, -10] V, we have that R = 5 $\times$ 10$^4$ $\Omega $.\\
So, considering T = 300 K at room temperature, we have:

\begin{equation}
\centering
T = 1.47 \times 10^{-12} \frac{W}{\sqrt{Hz}}.
\end{equation}

\subsection{Resolution}
Now that we have extracted the noise contributions to our system, we can determine the sensitivity $\alpha$ of the sensor. This means that we want to know the efficacy of our system in measuring angles (in rad/$\sqrt{Hz}$).\\
So, according to the block diagram in Fig. \ref{BD}, to obtain the angular measurement we have that:

\begin{equation}
\centering
\alpha = {shot \; noise} \times \frac{1}{signal} \times \frac{1}{Length},
\end{equation}

\begin{equation}
\centering
\alpha = \sigma_{p} \times \frac{1}{\sqrt{2}\frac{P_0}{\pi w_{y}}} \times \frac{1}{L},
\end{equation}

\begin{equation}
\centering
\alpha = 1.4 \times 10^{-11} \times \frac{1}{2.22} \times \frac{1}{10},
\end{equation}

\begin{equation}
\centering
\alpha = 3 \times 10^{-12} \frac{rad}{\sqrt{Hz}}.
\end{equation}

\noindent
This value is of the order of magnitude of the sensitivity of optical levers anticipated earlier.

\subsection{Estimated sum of contributions}
In order to obtain a plot of the noise budget for the optical lever prototype, we need to take into account some more elements to add to the ones just computed:

\begin{itemize}
\item The motion along z of the platforms is used as noise: however, at low frequency, most sensors are not sensitive to this motion so what we need is a differential motion between HAM chambers (say HAM4 and HAM5 for this derivation); the best estimation we have is the platform z motion measured by GS13s. This motion is given by channels of LIGO Livingston data;

\item The best performance of current tested optical levers is the one tested at the AEI and shown in Fig. 4.4 of reference \cite{sina};

\item The ground $\theta$ motion of the chambers is given by the Beam Rotation Sensors (BRS) and used as noise source. This motion is taken from channels of LIGO Livingston data;

\item The RX motion is given by the CPS on HAM4 and HAM5 and it is used for comparison with the optical lever performance. This motion is taken from channels of LIGO Livingston data.

\end{itemize}

\noindent
The optical lever performance reported in \cite{sina} takes into account the motion along the x axis, the spot displacement of the beam on the photodiode and the displacement of the photodiode itself.\\
The differential Z motion is given by the difference between the z motion measured by the GS13 sensors on HAM4 and HAM5:

\begin{equation}
\centering
\Delta Z = (GS13^{HAM5}_z - GS13^{HAM4}_z).
\end{equation}

\noindent
The contribution from the GS13 needs to be manipulated to give rad/$\sqrt{Hz}$: this is done converting the measured velocity to displacement.
\noindent
All the noise sources are divided by the lever arm, in order to obtain an estimation in radians.\\
A low pass filter (LP) at 1 Hz is applied to the BRS motion and a high pass filter (HP) is applied to the $\Delta Z$ motion at 0.1 Hz: we use the filters to estimate the motion, since we know that the measured signals are limited by noise where we are applying the filters.\\
Since all noises are independent from each other, summing all the elements in quadrature, we have the total noise performance of the optical lever, which is shown in the plot in Fig. \ref{oplevnoise}: the plot shows that the improvements that optical levers can give are limited to a restricted range of frequencies and that they suffer the differential Z motion contribution below 0.1 Hz. Given the technical difficulties of developing and installing optical levers (see following sections), the effort would only be worthwhile if the Z motion was improved via better sensors.

\begin{equation}
\centering
OpLev_{noise} = \sqrt{(OpLev_{AEI})^2 + (\theta^{BRS}_g)^2 + (Thermal)^2 + (Shot)^2 + (\Delta Z)^2}.
\end{equation}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/oplevsum.png}
\caption[Optical lever noise budget]{Optical lever total noise budget. This plot shows that our model predicts significant signal-to-noise ratio improvement from $\sim$0.15 Hz to 1 Hz (Vs CPS), but we are limited by the differential Z motion over a wide band.}
\label{oplevnoise}
\end{figure}

\section{Design of the prototype}
The optical design has been simulated, taking into account some general constraints of the sensor: generally, the QPD diameter is around 10 mm, so the beam size should not exceed 1-3 mm; gaps in quadrant photodiodes are of the order of tens $\mu$m. Moreover, it is ideal for the setup to be compact: the prototype has been design to lie on a [75 mm $\times$ 36 mm] platform.\\
\noindent
The chosen light source is a 1064 nm wavelength fiber-coupled Nd:YAG solid-state laser. Because of the fact that the beam size impinging on the photodiode has to be around 1 mm, a fiber collimator is used at the fiber output, and a plano-convex lens is used to focus the beam at the photodiode. In this way, with the chosen collimator, the beam size at its output is 1.38 mm. This is considered the starting point for the free propagation of the laser beam. The use of the collimator ensures that the beam size enlargement after a length L of propagation is minimized: according to the simulated free propagation, after 10 m the beam size is 2.8 mm.
\noindent
The focussing lens of focal length 150 mm is inserted 10 cm before the photodiode. In this way, the beam size impinging on the lens surface is 2.79 mm $\ll$ 12.7 mm of lens diameter. The beam size at the photodiode is 0.95 mm. The basic sketch of the optical system is shown in Fig. \ref{syst}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images/syst.pdf}
\caption{Basic sketch of the lever optical design (not in scale).}
\label{syst}
\end{figure}

\noindent
The prototype and its own pre-amplifying electronics have been built at UoB (Fig. \ref{oplev20}) and tested in air and in vacuum at the AEI. The purpose is to calibrate the prototype with the electronics from UoB in vacuum conditions and test the sensitivity of the device to angular displacements. This first step is necessary for a good sensing system characterization.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{images/OpLev20.jpg}
\caption[Photo of the optical lever prototype]{Photo of the optical lever prototypes as built at UoB. In this picture, the devices are not connected to electronics. Each platform hosts a laser source and a sensor. Each sensor is covered by a tube to avoid spurious light on the active area, and the focusing lens is placed at the suitable distance from it.}
\label{oplev20}
\end{figure}

\section{Test at the AEI}
The aim of the collaboration was to test the optical lever prototype in vacuum. We used the South bench of the 10 m prototype at AEI in Hannover.\\
The device and part of its electronics have been adjusted in order to match the requirements for a measurements using a Control and Data System (CDS) and facilities at AEI.\\
The pin configuration of the QPDs has been reset because the AEI electronics is set on a different one. It has been changed to the following:\\

\begin{itemize}
\item Q1: PIN 1 to PIN 1
\item Q2: PIN 2 to PIN 2
\item Q3: PIN 3 to PIN 6
\item Q4: PIN 4 to PIN 7
\item BIAS: PIN 5 to PIN 4
\end{itemize}

\noindent
Two adaptor cables have been built to connect the UoB boxes to the QPDs with the new pin configurations.\\
\noindent
To isolate the QPD, a small shield of plastic has been added to the QPD mount and the related metal screws have been changed with peek screws. Because of the presence of the new plastic layer, the height of all other components of the platforms has been adjusted.\\
Every component has been vacuum-cleaned using an ultra-sonic bath.

\subsection{Installing the device}
After cleaning, we installed the device into the South bench of the 10-m prototype. Due to the availability of the bench, only one fibre could be connected to one collimator; consequently, only one QPD has been connected.\\
\noindent
The lever arm has been set to be 20 cm: with the optical configuration foreseen for this lever arm the spot size on the photodiode is w $\simeq$ 1 mm. The power on that point is P = 3.5 mW.\\
\noindent
Summarizing, the prototype is ready for the test with the specifications listed in Tab \ref{oplevspec}.\\

\begin{table}[h!]
\centering
\begin{tabular}{c|lcl}
Beam size at QPD  & w = 1 mm\\
Power at output   & P = 3.5 mW\\
Displacement      & $\Delta$ x = 2.22 $\times$ 10$^{-3}$ m\\
Lens focal lenght & F = 150 mm\\
Shot noise                & SN = 75 nV/$\sqrt{Hz}$  \\
Responsivity Si @ 1064 nm & $\rho$ = 0.2 A/W      \\
Thermal noise             & T = 21 nV/$\sqrt{Hz}$  \\
Op-amp noise              & OP = 8,8 nV/$\sqrt{Hz}$
\end{tabular}
\caption{Specifications of the optical lever prototype tested at the AEI.}
\label{oplevspec}
\end{table}

%\begin{table}[h!]
%\centering
%\begin{tabular}{l|l|l|}
%\cline{2-3}
%                                     & UOB     & AEI     \\ \hline
%\multicolumn{1}{|l|}{Whitening}      & 30 Hz   & 100 Hz  \\ \hline
%\multicolumn{1}{|l|}{Transimpedance} & 27 k$\Omega$ & 33 k$\Omega$ \\ \hline
%\end{tabular}
%\end{table}

\paragraph*{Preliminary test in air}
To test if everything was set in the best way, we performed a first measurement in air, using one of the AEI pre-amp boxes connected to the CDS. Fig. \ref{inair} shows the trend of pitch P = (Q1+Q4)-(Q2+Q3) and yaw Y = (Q1+Q2)-(Q3+Q4).

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/inair_test.PNG}
\caption[OpLev test in air]{Preliminary test in air: the traces show the trend of the pitch and as from the output of the pre-amp built at UoB.}
\label{inair}
\end{figure}

\section{Test in vacuum}
We decided to set the vacuum in two steps: this idea allows to have a faster temperature gradient, decreasing the waiting time for temperature (and benches) to stabilize.\\
So for the first step we set the pressure at 30 mbar, the day after we set the pressure at 5 $\times$ 10$^{-3}$ mbar.\\
\noindent
Variables under examinations during the two steps of vacuum setting are: trend of temperature, pressure and position of the South bench along z axis.\\
Also, the alignment of the optical fibre has been checked during the process.\\
%The temperature trend shows that the two-step vacuum procedure was a good idea: it improved the temperature gradient by two times faster.\\

%\subsection{Temperature trend}

\subsection{30mbar test}

%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.3]{images/LVDT_Z.PNG}
%\caption[30 mbar LVDT test]{Motion along z axis of the South bench during vacuum pump to 30 mbar.}
%\label{LVDT}
%\end{figure}
\noindent
Fig. \ref{QPD} shows the measurements taken with the QPD. There are some peaks due to intensity fluctuations: we do not expect they disappear at lower pressure, because they are due to power fluctuation of the fibre itself.\\
Some peaks at lower frequencies may be due to bench motion: if the assumption is correct, at lower pressure and more stable temperature, these peaks should be less visible.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/QPD.PNG}
\caption[In vacuum QPD test: 30 mbar]{QPD signals during 30 mbar pressure conditions. It is noticeable a difference of two orders of magnitude with respect to the in-air condition. This is due to the reduced impact of air flows and dust on the beam path.}
\label{QPD}
\end{figure}
\noindent
The movement of the South bench along z axis is used as a reference to monitor the bench adjustments with temperature variations. The variable under examination is displacement tested by a Linear Variable Displacement Transformer (LVDT).

\subsection{Final vacuum set up}
The pressure has been set at 5 $\times$ 10$^{-3}$ mbar. What we expect is to find no variations in terms of the peaks we think are due to power fluctuations of the laser. Variations in LVDT trend can be due to temperature stabilization and related variations of pitch and yaw are then due to the more stable bench conditions (Fig. \ref{LVDT_FIN}).\\

\noindent
In this condition, also the signals from the L4C seismometers and accelerometers (Watt's Leakage) placed on the Central bench have been measured (Fig. \ref{central}). The plots with the UoB electronics show that there is some leakage below 10 Hz, probably due to saturation, in the measurement of the accelerometers.\\

\noindent
QPD performance is shown in the plots \ref{qpd_fin}. With AEI boxes we had expected results: no variations in the power fluctuation peaks and expected behaviour of pitch and yaw.\\
However, with UoB pre-amp the measurements do not seem consistent with what we expected: we think that some non-linearities in UoB pre-amp could be the cause of the problem. This is still under investigation at UoB.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.25]{images/LVDT_T.PNG}
\caption[Different pre-amps test: bench LVDT motion]{Bench motion along z axis during the vacuum pump from 30 mbar to 5 $\times$ 10$^{-3}$ mbar pressure conditions. Pressure has been set at 30 mbar at first stage to let temperature to stabilize faster. The two-step vacuum procedure was a good idea: it accelerated the lowering of temperature by two times.}
\label{LVDT_FIN}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/AEI_SOUTH.PNG}\\
\includegraphics[scale=0.3]{images/UOB_SOUTH.PNG}
\caption[Different pre-amps test: bench motion]{Motion of Central bench measured by L4Cs and accelerometers, with AEI and UoB pre-amps. These plots highlights that the UoB electronics is not performing well (due to loop leakage) below 10 Hz.}
\label{central}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/AEI_QPD_TEST.PNG}\\
\includegraphics[scale=0.3]{images/UOB_QPD_TEST.PNG}
\caption[In vacuum QPD test]{QPD performance, with AEI and UoB pre-amps. There is an evident difference between the measurements taken with the two different electronics: pitch signal is ~5x noisier than yaw with UoB electronics.}
\label{qpd_fin}
\end{figure}

\newpage
\paragraph*{Electronic noise}
Noise measurements of CDS with unplugged electronics have been taken, to check if there could be issues related to it. However, they do not show any unexpected behaviour:  CDS dominates nearly everywhere and the CDS noise is lower than any of our optical measurements everywhere, typically by at least a factor of 10.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/EL_NOISE.PNG}
\caption[Electronic noise]{Measurements of CDS noise and output of unplugged electronics.}
\label{noise}
\end{figure}

%\paragraph*{UOB and AEI comparison}
%The shot noise can be computed as:
%
%\begin{equation}
%\centering
%SN = \sqrt{\frac{2 \pi \hbar c P_0}{\lambda}},
%\end{equation}
%
%\noindent
%where P$_0$ = 3,5 mW is the output power, $\lambda$ = 1064 nm is the laser wavelength and c the speed of light. With the known parameters, we get:
%
%\begin{equation}
%\centering
%SN = 8,1 \time 10^{-11} \frac{W}{\sqrt{Hz}}.
%\end{equation}

\newpage
\section*{Conclusions}
\addcontentsline{toc}{section}{Conclusions}
The analysis of feasibility of this experiment showed that the optical lever can be in principle a good device to sense tilt motion over long lever arms. However, the noise budget indicated a small frequency window of good operation, while below 0.1 Hz the levers are limited by the ground motion along the z axis. It is anyway a good device to be tested.\\
During the test of the prototype, the measurements have shown that we had issues when calibrating the device due to problems highly related to electronics from UoB, since the tests with the AEI electronics showed that the optical setup was well built and aligned. The very short time of the visit did not allow to take more in-depth tests.\\
Other possible reasons to investigate for better performance might lie in the structure of the prototype: further tests might be useful to understand if the device can be improved by changing the position of the lens with respect to the QPD, and let the diode sit at the focus on the lens. This solution will concentrate the power and decrease the size of the beam.\\
The device is currently not suitable for the purposes we tested for, but it opened the way to further tests to improve the technology: since the pitch and yaw tests have shown that the optical lever might be sensitive to the vertical motion of the bench, a reduction of this motion might be of great impact to improve the sensitivity of the levers \cite{luise}. With a good sensing system of tilt motion, the addition of an actuation system able to reduce this motion will be crucially helpful to stabilize the suspension points of the optical chains and then of the whole cavity.
