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\chapter{First detection}
\label{A}
On 14th September 2015 the two LIGO antennas observed for the first time a signal from a gravitational wave produced by the merger of two black holes. This was the very first time that a merger of such massive and elusive objects could be observed.\\
The gravitational-wave signal has been named GW150914 and has been emitted by 2 black hole of masses of 36 $M_{\odot}$ and 29 $M_{\odot}$, which merged at a distance of 410 Mpc (z = 0.09)and produced a final BH of 62 $M_{\odot}$. The remaining 3 $M_{\odot}$ have been radiated in gravitational waves. Fig. \ref{gwsig} shows the signal detected from LIGO Hanford and LIGO Livingston.\\
This detection has been the result of a wide scientific collaboration which efforts made possible a discovery that deserved the Nobel Prize in Physics in 2017 to the pioneers of gravitational wave hunting \textit{'for decisive contributions to the LIGO detector and the observation of gravitational waves'}.
\begin{figure}
\centering
\includegraphics[scale=0.7]{images/outreach.png}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.77]{images/GWsignal.png}
\caption[First detection of a gravitatonal wave signal.]{First detection of a gravitational wave signal \cite{first}. The event is shown for both observatories at the time of observation 09:50:45 UTC on 14th September 2015. The top row is the gravitational wave amplitude for Hanford (H1) and Livingston (L1). In the L1 panel, there is a visual comparison of the two signals: the wave passe through L1 first, H1 signal (in orange) is shifted by the 6.9 ms of difference, and inverted due to their mutual orientation. The second row shows the consistency of the measured signal with expectations independently computed. Third row shows the residuals after subtraction of the measured time series and the numerical waveform. Bottom row is the same signal in frequency vs time, where it is evident the increase of frequency with time.}
\label{gwsig}
\end{figure}
\begin{figure}
\includegraphics[scale=0.8, angle=90]{images/logos.pdf}
\end{figure}
\ No newline at end of file
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\chapter{Control loops}
\label{B}
\section{Control theory}
The theory of controls is a branch of science and technology which studies how to drive a given dynamical system. This is generally characterized by inputs and outputs and the former can be manipulated to obtain a desired output, which is chosen by a reference setpoint. The design and the technology involved depends on the system, but in general they imply a sensing section, a software section which can modify some features of the input and a feedback section to check that the manipulation of the input signal gave the output as set by the reference.
\subsection{Principles: control loops}
The collection of all the sections forms a control loop and controls the behaviour of a given variable under exam. Control loops can be:
\begin{itemize}
\item{Open-loop}: the control action is independent from the output.
\item{Closed-loop}: the control action depends on the desired output conditions. This kind of loop uses feedback loops, that assure the process is correctly going on, i.e. the value of the variable under exam is that of setpoint.
\end{itemize}
\noindent
In order to design a control loop, we need to build all the subsystems up: the main one is the \textbf{plant}, which is the physical parameter to be controlled; the plant is measured by one or more \textbf{sensors}, which detect a deviation (if any) in the signal from the reference setpoint. An error signal is then produced and processed by a \textbf{controller} into a correction signal that is sent to an \textbf{actuator}. The actuator applies the correction to the plant.
\subsection{How to: block diagrams}
Block diagrams are useful graphical instruments to describe, study and build a control loop. Each element of the control system is represented by a block and each block is joined by lines with arrows showing the sequence of controls. Following the logical steps stated before, we can then draw the block diagram of a basic system to be controlled:
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/block.png}
\caption[Basic block diagram of a control loop.]{Basic block diagram of a control loop. The setpoint is injected as an input.}
\end{figure}
\noindent
Every variable under interest at the output of each block can be evaluated by \textit{solving} the diagram.
\section{Control analysis}
Once the control loop has been schematically drafted, it needs to be finalized: the software section implies instructions. These are given by a computation of the transfer functions of the whole system, which gives the response in the frequency domain of the output to a given input. The computed (and measured) transfer function will then be modified with suitable filters to make the output adjust to the reference setpoint.
\subsection{Transfer functions}
Every dynamical system is characterized by equations of motion in the time domain. In order to study the system in the frequency domain, it is possible to apply a Laplace transform. So, for any function f(t) in the time domain, the Laplace transform is defined as:
\begin{equation}
\centering
F(s) = \int e^{-st}f(t)dt
\end{equation}
\noindent
where s = a + i$\omega$.\\
A transfer function is defined as the ratio between the output and the input, in the frequency domain. For any input X(s) and output Y(s), the transfer function is:
\begin{equation}
\centering
T(s) = \dfrac{Y(s)}{X(s)}
\end{equation}
\noindent
and so the output is characterized by the product between the transfer function and the input signal in the Laplace domain, which is the Laplace transform of the convolution of the two functions in the time domain\footnote{An interesting demonstration of this statement can be found ....}.\\
Since, in general, a function can be written as a product of polynomials, the transfer function is also in the form:
\begin{equation}
\centering
T(s) = C \Pi_{i,j} \dfrac{(s-z_i)^j}{(s-p_i)^j}
\end{equation}
\noindent
where z and p are the $i$th zeros and poles of the polynomial, of order $j$ and C is the amplitude. This form is useful to study the stability of the system:
\begin{itemize}
\item{Re(p) $<$ 0}: the system is asymptotically stable;
\item{Re(p) = 0 with j = 1}: the system is marginally stable;
\item{Re(p) = 0 with j $>$ 1 or Re(p) $>$ 1}: the system is unstable.
\end{itemize}
\noindent
The gain of the system is defined as the radio between the amplitudes of the input and output, i.e. it's the absolute value of the transfer function:
\begin{equation}
\centering
G = \dfrac{\mid Y(s)\mid}{\mid X(s)\mid} = \mid T(s)\mid
\end{equation}
\noindent
and the phase is $\varphi$ = arg(T(s)).\\
In the frame of control loops, the transfer function is given by the gain contributions of all the subsystems of the loop.
\subsection{Phase and magnitude interpretation: the Bode plot}
The Bode plot is a graph representing the response in frequency of the magnitude and phase of the system under exam. It is largely used to define the marginal conditions for the stability of the loop. The magnitude in expressed in dB = 20$\log_{10} (x)$ and it is computes as the absolute value of the transfer function:
\begin{equation}
\centering
\mid T(s) \mid = \sqrt{T \cdot T^*}.
\end{equation}
\noindent
The phase is expressed in degrees (deg) and it is comoputed as:
\begin{equation}
\centering
\varphi = -\arctan \left(\dfrac{ImT(s))}{Re(T(s))}\right).
\end{equation}
\noindent
In the frame of loops, the closed-loop gain is given by:
\begin{equation}
\centering
G_{CL} = \dfrac{G_{OL}}{1+G_{OL}},
\end{equation}
\noindent
where G$_{OL}$ is the open-loop gain and also a pole for this relation. This means that if G$_{OL}$ = -1, G$_{CL}$ diverges and the loop is unstable. On the phase plot, we will have that $\varphi$ = -180$^{\circ}$. In general, when the trace on the phase plot approaches this value at certain frequencies, it means that the loop that we are building is unstable in that region.
\subsection{Spectral density}
Spectral densities are views of a signal in a frequency spectrum. It is a useful instrument to detect effects on the signal during processing, like peaks due to harmonics, or resonances. The physical parameter used in this study is the power spectral density, which measures the power of a signal as a function of frequency and has units of W/Hz$^{-1}$. When there is no direct power associated to the measurement (like in case of Volts) the units are in terms of the square of the signal per Hz. In some cases, an Amplitude Spectral Density (ASD), defined as the square root of the power spectral density, is used when the shape of the signal is quite constant; in this case the units are in the form of 1/Hz$^{-1/2}$ and the variations in the ASD will then be proportional to the variations of the signal itself.
\subsection{Coherence}
The coherence is a statistic relation between two signals or data sets x and y. It is defined as the ration between the cross spectral density of the two functions and the product of the spectral densities of each function:
\begin{equation}
\centering
C_{xy}(f) = \dfrac{\mid S_{xy}(f) \mid ^2}{S_{xx}(f)S_{yy}(f)}.
\end{equation}
\noindent
Coherence is a useful parameter to estimate the correlation between the two signals. In control loops, it can be used to verify how much the output can be predictable by the input. The values of the coherence lies between 0 and 1 and two signals are considered optimally correlated if their coherence approaches 1.
\section{The French fry factory}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.35]{images/fries.jpg}
\caption[The French fry factory.]{The French fry factory: basic principles of a control loop as explained by Dr. Jenne Driggers during a private conversation at LIGO Hanford. Original photo.}
\end{figure}
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%\documentclass[a4paper,12pt]{book}
%\usepackage[a4paper,top=2cm,bottom=2cm,left=2cm,right=2cm]{geometry}
%\usepackage[utf8]{inputenc}
%\usepackage[english]{babel}
%\usepackage{graphicx}
%\usepackage{subfigure}
%\usepackage{array}
%\usepackage{color}
%\usepackage{colortbl}
%\usepackage{verbatim}
%\usepackage{lscape}
%\usepackage{amsmath,amssymb}
%\linespread{2}
%%\title{CH.4 Reducing differential motion of aLIGO seismic platforms}
%\date{}
%
%\begin{document}
%\begin{figure}
%\centering
%\includegraphics[scale=0.5]{images/LIGO_LOGO.PNG}
%\end{figure}
\chapter{Reducing differential motion of aLIGO seismic platforms}
\noindent
During 2019, I spent some months working on LIGO Hanford site (Washington, USA). This experience allowed me to be critically involved in the complicated life of a gravitational-wave interferometer. In particular, I was given the opportunity to study how to improve LIGO performances at low-frequency, focussing on the reduction of seismic motion of the platforms where the optics live.\\
In this chapter and in the following I am going to show you in detail how we can modify the software set up of LIGO in order to obtain a different and possibly better performance. This work has been developed in collaboration with LIGO Hanford and LIGO Livingston laboratories, Stanford University, MIT and UoB, and this chapter is partially including some technical notes I shared with LIGO collaboration.
\section{Motivation: Duty cycle on LIGO}
Lock loss events are the main sources of preventing continuous observations for long periods of time: when light loses resonance in the cavities, a lock loss happens and the control system of the optical cavities are under effort to restore stabilization. This means that during lock loss the interferometer is no longer able to be stable and the observing time is interrupted.\\
Duty cycle is one of the main topic where commissioners focus on before starting an observing run. It is needed not only to observe more gravitational waves, but also to identify noise sources and improve sensitivity.\\
Since the number of detected events over a time period N(t) is proportional to the volume of Universe under observation V, the observing time t and the rate R of astrophysical sources that can occur in a certain volume:\\
\begin{equation}
\centering
N(t) = R\cdot V\cdot t,
\end{equation}\\
it can be seen that increasing the observing time towards a given direction, will increase the number of detected events.\\
\noindent
Other ways to spend time to improve duty cycle is instead to increase the observable volume: this can be achieved by spending time on hardware to improve sensitivity on a given frequency bandwidth.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/duty_cycle.png}
\caption[LHO duty cycle during O3b]{Example of duty cycle for Hanford Observatory, during O3b.}
\label{duty}
\end{figure}
\subsection{Differential motion between chambers}
It has been seen that among the noise sources which contribute to lock loss events there is the ground motion (CITA), including earthquakes and microseismic events. \\
In particular, during O3 run, it was observed that the chambers in the corner station (CS) show differential motion with respect to each other. It is reasonable to think that if the chambers have a synchronized motion, the whole interferometer will move following the ground motion, and not being affected by it. This would in principle help the cavities to be stable and maintain resonance. In case of lock losses due to large earthquakes or high wind, stable resonance could be achieved in shorter times.\\
On another side, reducing the differential motion between the chambers means to reduce a source of noise at low frequency (5-30 Hz), as we will show in the next section: this would improve the sensitivity of the interferometer.
\subsection{ISI stabilization}
Differential motion affects the ISI of the HAM and BSC chambers in the CS: these are then the platforms that we want to stabilize. Several sensors are responsible for sensing the seismic motion, in all degrees of freedom of each stage. They are T240, L4C, GS13, OSEMs and CPS.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images/isi.png}
\caption[Example of ISI scheme]{Example of ISI scheme.}
\label{isi}
\end{figure}
\noindent
In particular, CPS sensors are placed in every chambers at all stages: it is easy to compare motion between HAM and BSC chambers through the signal of a device sensing the same motion on every chamber.\\
The idea which should stabilize ISIs to follow the ground motion is to lock the chambers to each other, in order to make them move on a synchronized way, following a common motion given by a driver chamber (or block of chambers).
\subsection{Role of the mode cleaner}
We started our design on chambers of x arm. Along this direction, the Input Mode Cleaner (IMC) lies totally on HAM2 and HAM3 platforms: it can be used as a reference, or witness, of the motion between chambers, once they are locked together.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/IMC.png}
\caption[Optical layout of the HAM2 and HAM3 chambers]{Optical layout of the HAM2 and HAM3 chambers.}
\label{imc}
\end{figure}\\
\noindent
In the next section we will demonstrate that CPS are good witnesses to sense differential motion and they also can be used to lock the chambers with each other.
\section{CPS Differential motion and locking}
\subsection{Sensing differential motion via CPS}
Capacitive Position Sensors (CPS) measure the relative motion between two stages of the isolation system. On HAM chambers, they are are set between HEPI and ground, and between Stage 1 and HEPI. On BSC chambers, they also measure the relative motion between Stage 1 and Stage 2. Plots is Fig. \ref{diff} show the differential motion seen by the CPS between BSC and HAM chambers: the sensors are reliable for this measurement, and they put in evidence that the HAM chambers have a more synchronized motion with respect to the motion between HAM and BSC and BSCs only. This means that the block of HAM chambers on x arm is more relatively stable and can be used as driver for the other chambers, with the mode cleaner acting as witness.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/ham2_ham3.PNG}
\includegraphics[scale=0.3]{images/bs_itmx.PNG}
\includegraphics[scale=0.3]{images/bs_ham2.PNG}
\caption[CPS differential motion]{CPS differential motion between the HAM and BSC chambers along x axis. ISIs move in common, particularly in the same building. This can be confirmed by noting that the difference between two chambers is much lower than individual chambers.}
\label{diff}
\end{figure}
\noindent
We projected the CPS of the x axis chambers to the suspension point in order to obtain PRCL and IMCL traces sensed by the CPS. For BSCs, we decided to sum the contributions of the CPSs on stage 1 and stage 2 and to project this sum.\\
One of the main differences between the behaviour of CPS IMCL and CPS PRCL, is that the former is obviously involving only the HAM chambers. Since HAM2 and HAM3 have a very good common motion, IMCL can be considered more stable with respect to PRCL, which instead involves also BSCs. Indeed, CPS PRCL is following the only BSCs at frequencies below 0.02 Hz.\\
Fig. \ref{sus} shows plots of PRCL and ICML by CPS projection to suspoint. These projections indicate that reducing the differential motion as seen by the CPSs will help to reduce the residual motion seen by the optical cavities.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/CPS_SUSPOINT_IMCL.PNG}
\includegraphics[scale=0.3]{images/CPS_SUSPOINT_PRCL.PNG}
\caption[CPS suspoint projections]{CPS suspoint projections: IMCL and PRCL. The calibrated PRCL trace used as comparison has been de-whitened.}
\label{sus}
\end{figure}
\subsection{Locking chambers via CPS}
In the previous section we demonstrated that CPSs are good sensors for differential motion and can be used to monitor chamber motions at lower frequencies. Given that and remembering the aim of stabilize the motion of the chambers making them moving in sync, it is possible to use the CPSs to lock HAM2 and HAM3 together, HAM4 and HAM5 together, BSCs in the Corner Station together and BSCs hosting the ETMs together. This will stabilize the ISI differential motion with respect to a driving chamber.\\
Since we saw that HAM2 and HAM3 show a very good common motion and that we can use the IMC as a witness of it, our first step has been to lock the HAM2 and HAM3 chambers together by feeding HAM3 a calculated differential CPS signal. This is done in practice with an additive offset to the setpoint of the HAM3 isolation control loop.\\
The block diagram in Fig. \ref{ham2b}, shows the structure of HAM2, where the signals from $d_{2}$ and $i_{2}$ represent the offsets given by CPS and inertial sensors.\\
At low frequency the CPS noise is negligible because its contribution is about 10$^3$ times lower than the microseismic peak.\\
General block diagrams notations are listed in Tab. \ref{tab1}\\
\begin{table}[h!]
\centering
\begin{tabular}{|c|| c| }
\hline
P & Plant\\
\hline
S & Sensor correction\\
\hline
C & Control\\
\hline
L & Low pass filter \\
\hline
H & High pass filter\\
\hline
$N_{g}$ & ground noise\\
\hline
$N_{i}$ & inertial noise\\
\hline
$x_{g}$ & ground motion\\
\hline
$x_{p}$ & plant motion\\
\hline
\end{tabular}
\caption{Notations used in the block diagrams.}
\label{tab1}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/ham2B.PNG}
\caption[HAM2 simplified block diagram]{HAM2 simplified block diagram.}
\label{ham2b}
\end{figure}
\noindent
We can compute the signal $\textit{$d_{2}$}$ which will be the CPS offset to send to HAM3 chamber. In this case, HAM2 will drive HAM3 to follow its motion. Defining K = PC:\\
\begin{equation}
\centering
d_{2} = x_{p_{2}} - x_{g} =
\end{equation}
\begin{equation*}
\centering
\begin{split}
&= K[L_{2}(SN_{g} + Sx_{g} + d_2) + H_2N_{1_2} + H_2(x_g + d_2)] + Px_g - x_g\\
&= KL_{2}S(N_{g} + x_{g}) + KL_{2}d_2 + KH_2N_{i_2} + KH_2x_g + KH_2d_2 + Px_g - x_g.
\end{split}
\end{equation*}\\
Since L$_2$ + H$_2$ = 1, we get:\\
\begin{equation*}
\centering
d_2(1-K) = KL_{2}S(N_{g} + x_{g}) + KH_2N_{i_2} + KH_2x_g + Px_g - x_g,
\end{equation*}\\
which will lead to:\\
\begin{equation}
\centering
d_2 = \frac{K}{1-K} [L_2S(N_g + x_g) + H_2N_{i_2} + H_2x_g] + P\frac{x_g}{1-K} - \frac{x_g}{1-K}.
\label{d2}
\end{equation}\\
The platform motion of HAM2 can be computed following the block diagram in a similar way:\\
\begin{equation}
\centering
x_{p_{2}} = K\{L_{2}[S(N_g + x_g) + x_{p_2} - x_g] + H_{2}N_{i_{2}} + H_2x_{p_2}\} + Px_g.
\end{equation}\\
After some manipulations, and remembering that L$_2$ + H$_2$ = 1, we obtain:\\
\begin{equation}
\centering
x_{p_{2}} = \frac{K}{1-K}[L_2S(N_g + x_g) - L_2x_g + H_2N_{i_2}] + P\frac{x_g}{1-K}.
\end{equation}\\
The result in Eq. \ref{d2} is the signal to subtract to HAM3 in order to feed HAM3 a CPS differential motion; it is added to HAM3 as in the block diagram in Fig. \ref{ham3b} for HAM3. In the original configuration, without any feeding into HAM3 the block diagrams for both chambers would be identical. In this new configuration instead, there is no sensor correction and ground noise on HAM3 because they both come from the contribution from HAM2, which is the offset $d_{2}$ added to HAM3.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/ham3B.PNG}
\caption[HAM3 simplified block diagram with HAM2 offset]{HAM3 simplified block diagram for the new configuration: d$_{2}$ is the offset coming from HAM2.}
\label{ham3b}
\end{figure}
\noindent
We then want to know the reaction on HAM3 plant in this configuration, in order to compute differential motion between plants on both chambers.\\
Following the usual notations and the block diagram of HAM3:\\
\begin{equation*}
\centering
x_{p_{3}} = K[L_3d_2 + L_3x_p{_3} - L_3x_g + H_3N_{i_3} + H_3x_{p_3}] + Px_g,
\end{equation*}
\begin{equation}
\centering
x_{p_{3}} = \frac{K}{1-K}[L_3(d_2 - x_g) + H_3N_{i_3}] + P\frac{x_g}{1-K}.
\end{equation}\\
The differential motion will be:\\
\begin{equation}
\centering
x_{p_{3}} - x_{p_{2}} = \frac{K}{1-K}[L_3(d_2 - x_g) + H_3N_{i_3}] + P\frac{x_g}{1-K} -\frac{K}{1-K}[L_2S(N_g + x_g) - L_2x_g + H_2N_{i_2}] - P\frac{x_g}{1-K}.
\end{equation}\\
In the approximation where k$\rightarrow \infty$, the terms we computed become:\\
\begin{equation}
\centering
x_{p_{2}} = H_{2}N_{i_{2}} + L_{2}S(N_{g} + x_{g}) - L_{2}x_{g},
\label{xp2}
\end{equation}
\begin{equation}
\centering
d_{2} = H_{2}N_{i_{2}} + L_{2}S(N_{g} + x_{g}) + H_2x_{g},
\label{d2}
\end{equation}
\begin{equation}
x_{p_{3}} = H_{3}N_{i_{3}} + L_{3}(d_{2}-x_g).
\end{equation}
\begin{equation}
\centering
\begin{split}
x_{p_{3}} - x_{p_{2}} &= L_3d_2 - L_3x_g + H_3N_{i_3} - L_2S(N_g + x_g) + L_2x_g - H_2N_{i_2}\\
& = L_3L_{2}S(N_{g} + x_{g}) + L_3H_{2}N_{i_{2}} + L_3H_2x_{g}-\\
&L_3x_g + H_3N_{i_3} - L_2S(N_g + x_g) + L_2x_g - H_2N_{i_2}.
\end{split}
\end{equation}\\
If both chambers are mainly driven by the low pass filters (H $\rightarrow 0$), the differential motion becomes:\\
\begin{equation*}
\centering
\begin{split}
x_{p_{3}} - x_{p_{2}} & = L_3d_2 - L_3x_g - L_2S(N_g + x_g) + L_2x_g\\
& = L_3\cdot L_2S(N_g + x_g) - L_3x_g - L_2S(N_g + x_g) + L_2x_g,
\end{split}
\end{equation*}
\begin{equation}
x_{p_{3}} - x_{p_{2}} = L_2S(N_g + x_g)(L_3 -1) + x_g(L_2 - L_3).
\end{equation}
\section{Analysis of feasibility}
Next step is to study how to modify the low and high pass filters in order to obtain the best performances from each one in the new configuration of chambers. To do this, we are going to change the blending filters, i.e. those filters whose combination gives the best performance of the set low+high pass filters.\\
If by definition we have L+H=1, we can write it also as:
\begin{equation}
\centering
L_{lh}+H_{lh}=\frac{(\alpha + b)^{l+h-1}}{(\alpha + b)^{l+h-1}}.
\end{equation}\\
According to the values of l and h we have different order of magnitudes of the binomials, which can be solved for the real part.\\
\noindent
In our case, we have two main contributions given by inertial sensors and CPS. We will apply the high-pass filter to the GS13 and the low-pass one to the CPS.\\
To do this, we need the specific contributions for each chamber to be specified, with all the components well defined. So for example, in the case of the CPS contribution, we need to define the tilt component, the CPS noise and the ground motion, which will take part into the platform motion as seen by the CPS sensor. This is because these components are independent from each other and will need to be summed in quadrature.\\
Besides, as we saw in the previous computations, we will need to apply filters: the Sensor Correction filter will be the one used on LIGO and shown in Fig. \ref{SC}; the high- and low- pass filters will be find through blending several possible filters across a certain number of l and h order of magnitude, as introduced before. The best blended filter will be given by a combination of two l and h values at a specific blending frequency.\\
At the end of the analysis for each chamber (HAM and BSC) in isolation, we will connect the chambers via CPS and look at the results.\\
All this analysis has been performed through Matlab software.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/SCbode.png}
\includegraphics[scale=0.3]{images/SC.png}
\caption[Sensor correction filter]{Sensor correction filter.}
\label{SC}
\end{figure}
\subsection{Contributions from CPS and inertial sensors}
To calculate the CPS signal contribution, we need the ground motion and we used the ITMY STS signal on X direction: this is going to be the same motion for every chamber, since there is only one sensor in the Corner Station to measure it, because it has been found that the ground motion is the same everywhere in the Corner Station (cita!!). From this signal, we separate the contribution given by the tilt ($\theta_g$) from the microseismic frequency (0.08 Hz). Then we subtract the tilt to obtain the ground motion $x_g$ from the STS:\\
\begin{equation}
x_g = sts - \theta_g.
\end{equation}\\
The CPS signal has been then computed summing in quadrature the contributions given by tilt, ground motion and CPS noise, and applying the sensor correction filter:\\
\begin{equation}
CPS_{inj}=\sqrt{(\theta_g\cdot SC)^2+(x_g\cdot (1-SC))^2+(N_{cps})^2}.
\end{equation}\\
Figure \ref{cps_inj} shows the CPS signal and all its contributions:
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/cpsinj.png}
\caption[CPS contributions]{CPS contributions.}
\label{cps_inj}
\end{figure}
\noindent
To calculate the platform motion of the BSC, we used data from the ITMX ISI along x direction. This is the signal from the T240 sensor. As before, we separate the tilt contribution ($BSC\theta_p$) from the signal and to obtain the inertial sensor contribution for the BSC chambers we sum in quadrature the contributions fro tilt and T240 noise:
\begin{equation}
T240_{inj} = \sqrt{{BSC\theta_p}^2+{N_{T240}}^2}.
\end{equation}\\
Figure \ref{t240_inj} shows the T240 signal and its contributors:
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/t240inj.png}
\caption[BSC contributions]{BSC contributions..}
\label{t240_inj}
\end{figure}
\noindent
The inertial contribution for HAM chambers is computed in a similar way: the sensor in this case is the GS13 and we used data from the ISI of HAM2 in x direction. Calling variables in the usual way:
\begin{equation}
GS13_{inj}= \sqrt{{HAM\theta_p}^2+{N_{GS13}}^2}.
\end{equation}\\
Figure \ref{gs13_inj} shows the GS13 signal and its contributors:
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/gs13inj.png}
\caption[HAM contributions]{HAM contributions..}
\label{gs13_inj}
\end{figure}
\subsection{Blending filters}
In order to compute platform motions for single chambers in isolation and, later, locked together via CPS, we need low- and high- pass filters. Many possible blended filters have been found for different combinations of order of magnitude and blending frequency: plots in Fig. \ref{blend} show the velocity rms for every combination.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/bscblend.png}
\includegraphics[scale=0.3]{images/hamblend.png}
\caption[Blending filters]{Plots of all possible costs built with different combinations of blending filters. Orders of magnitude are between l =[1,4] and h = [1,4].}
\label{blend}
\end{figure}
\noindent
The best combination has been found computing the orders and the blending frequency which give the minimum of the cost. The optimized blending filter has been then built using the best values of l and h orders and blending frequency. The cost is given by:
\begin{equation}
cost = \sqrt{(cps_{inj}\cdot LP)^2 + (T240_{inj}\cdot HP)^2}
\end{equation}\\
and a similar equation for the HAM chamber.\\
Fig. \ref{cost} shows the cost and its rms obtained with the best blending filters for BSC and HAM chambers.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/bscrms.png}
\includegraphics[scale=0.3]{images/hamrms.png}
\caption[HAM and BSC velocity rms and their contributions]{HAM and BSC velocity rms and their contributions. For BSC, the best blending filter is given by l = 1, h = 4 at a blending frequency of fb = 0.13 Hz. For Ham, the best indeces are l = 1, h = 4 and fb = 1.14 Hz.}
\label{cost}
\end{figure}
\subsection{Locking chambers}
With these elements, we can proceed with the analysis of the behaviour of the chambers when locked via CPS. We refer to HAM2 and HAM3 chambers, since in the previous document we made the computations for these chambers. Reminding the equations we need:
\begin{equation}
\centering
x_{p_{2}} = H_{2}N_{i_{2}} + L_{2}SC(N_{g} + x_{g}) - L_{2}x_{g},
\label{xp2}
\end{equation}
\begin{equation}
\centering
d_{2} = H_{2}N_{i_{2}} + L_{2}SC(N_{g} + x_{g}) + H_2x_{g},
\label{d2}
\end{equation}
\begin{equation}
x_{p_{3}} = H_{3}N_{i_{3}} + L_{3}(d_{2}-x_g),
\end{equation}\\
where $x_{p_{2}}$ is HAM2 platform motion, $d_{2}$ is the signal from HAM2 to send to HAM3 and $x_{p_{3}}$ is HAM3 motion when attached to HAM2 via CPS. The differential motion was:
\begin{equation}
x_{p_{3}} - x_{p_{2}} = L_2S(N_g + x_g)(L_3 -1) + x_g(L_2 - L_3).
\end{equation}\\
For the analysis of this section, we need to know which terms of these equations are coherent, in order to separate them from the incoherent ones, which will need to be summed in quadrature. Since we know that the ground motion is the same everywhere in the Corner Station, the terms involving $x_g$ are coherent. Noises are instead, by definition, independent from each other.\\
The previous equations then become:
\begin{equation}
\centering
x_{p_{2}} = \sqrt{(H_{2}N_{i_{2}})^2 + (\theta_g\cdot SC\cdot L_2)^2} + (x_g\cdot SC\cdot L_2) - (x_g\cdot L_2),
\end{equation}
\begin{equation}
\centering
d_{2} = \sqrt{(N_{i_{2}}{H_2})^2 +(\theta_g\cdot SC\cdot L_2)^2} + (x_g\cdot SC\cdot L_2) + (x_g\cdot H_2),
\end{equation}
\begin{equation}
x_{p_{3}} = \sqrt{(N_{i_{2}}{H_2})^2 + [L_3\sqrt{(N_{i_{2}}{H_2})^2 +(\theta_g\cdot SC\cdot L_2)^2}]^2} + (x_g\cdot SC\cdot L_2 + x_g\cdot H_2)\cdot L_3 - (x_g\cdot L_3),
\end{equation}\\
Since L$_3$ = L$_2$ and H$_2$=H$_3$:
\begin{equation}
x_{p_{3}} - x_{p_{2}} = |\sqrt{(L_2\cdot SC\cdot \theta_g\cdot L_2)^2 - (L_2\cdot SC\cdot \theta_g)^2} +(x_g\cdot SC\cdot {L_2}^2)-(x_g\cdot SC\cdot L_2)|.
\end{equation}.\\
Plot in Fig. \ref{diffham} shows the differential motion of HAM2 and HAM3 in isolation, and Fig. \ref{cpsdiff} shows motions of the chambers when locked to each other and their differential motion.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/diffham.png}
\caption[Ham chambers in isolation]{Ham chambers in isolation: motion of HAM2 as a reference.}
\label{diffham}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/cpsdiff.png}
\caption[HAM chambers in CPS locking condition]{HAM chambers in CPS locking condition: the plot show the motion of each chamber, where HAM3 depends on HAM2, through CPS locking, and the differential motion between them.}
\label{cpsdiff}
\end{figure}
\section{Inertial sensors locking}
\subsection{Locking chambers via inertial sensors}
In the frame of ISI stabilization from ground motion, the inertial sensors could also be used for the same purpose to lock chambers, in addition to CPS locking. This means that HAM3 will be fed by the inertial sensors signal coming from HAM2.\\
Signals from inertial sensors are represented by i$_2$ and i$_3$ notations in Fig. \ref{ham2b} and \ref{ham3b}. However, adding the signal i$_2$ to HAM3 implies that the block diagram will be modified as in Fig. \ref{ham3bi}.
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/ham3Bi.png}
\caption[Inertial sensor locking setup]{New setup of HAM3 chamber fed by CPS and inertial sensor signals from HAM2.}
\label{ham3bi}
\end{figure}
\noindent
Following a similar path for the math in this configuration of i$_2$ signal from HAM2:\\
\begin{equation}
\centering
i_2 = x_{p_2} + N_{i_2} = \frac{K}{1-K}[L_2S(N_g+x_g) - L_2(N_{i_2}+x_g)] + \frac{Px_g}{1-K}+\frac{N_{i_2}}{1-K}.
\end{equation}\\
This is the signal from HAM2 inertial sensors to inject into HAM3 as shown in Fig. \ref{ham3bi}. Then, the platform motion of HAM3 becomes:\\
\begin{equation*}
\centering
x_{p_3} = K[L_2d_2 +L_3x_{p_3} - L_3x_g +H_3i_2 + H_3N_{i_3} + H_3x_{p_3}] + Px_g,
\end{equation*}
\begin{equation}
\centering
x_{p_3} = \frac{K}{1-K}[L_3(d_2-x_g) + H_3(i_2+N_{i_3})] + \frac{Px_g}{1-K}.
\end{equation}\\
In the approximation where K $\rightarrow$ $\infty$:\\
\begin{equation}
\centering
i_2 = L_2S(N_g+x_g) - L_2(n_{i_2}+x_g),
\end{equation}
\begin{equation}
\centering
x_{p_3} = L_3(d_2-x_g)+ H_3(i_2+N_{i_3}).
\end{equation}\\
Remembering Eqs. \ref{xp2} and \ref{d2} for x$_{p_2}$ and d$_2$, we can compute the differential motion:\\
\begin{equation*}
\centering
\begin{split}
x_{p_3} - x_{p_2} & = L_3d_2-L_3x_g+ H_3i_2+H_3N_{i_3} - H_{2}N_{i_{2}} - L_{2}S(N_{g} + x_{g}) + L_{2}x_{g}\\
= & L_3L_2S(N_{g} + x_{g}) + L_3H_2N_{i_2} + L_3H_2x_g - L_3x_g + H_3L_2S(N_{g} + x_{g}) -\\
&H_3L_2(N_{i_2}+x_g) + H_3N_{i_3}- L_2S(N_{g} + x_{g}) + L_2x_g - H_2N_{i_2},
\end{split}
\end{equation*}
\begin{equation}
\centering
x_{p_3} - x_{p_2} = x_g(L_3H_2 - H_3L_2 + L_2 - L_3) + N_{i_2}(L_3H_2 - H_3L_2 -H_2) +N_{i_3}H_3.
\end{equation}\\
The computation of the differential motion between HAM2 nad HAM3 in the conditions where the two ISIs are connected both via CPS and inertial sensors, shows that there is no contribution from the sensor correction and from the ground noise.\\
Besides, it is worth notice that if L$_2$ = L$_3$, also H$_2$=H$_3$ by definition and the differential motion is:\\
\begin{equation}
\centering
\begin{split}
x_{p_3} - x_{p_2} &= N_{i_3}H_3 - N_{i_2} H_2\\
&= H(N_{i_3}-N_{i_2}),
\end{split}
\end{equation}\\
which is exactly the solution that we would obtain if the differential motion was computed without any feeding.
\section{CPS locking set up on LIGO}
\section{Beyond: LSC signals optimization on LIGO}
A positive consequence of this work might be the improvement of the LSC signals from LIGO cavities. Among them, DARM is particularly important, because it represents the gravitational wave signal. During the 2019 commissioning break, in collaboration with LIGO Livingston Observatory, we tried to apply the new CPS configuration in order to obtain improvements in LSC signals at LIGO Hanford.\\
\subsection{LSC offloading}
We saw that the cavities (and the optical signals) in LIGO are affected by the ISI motion, simply because they lie on them. Given the work done with the CPSs to suppress the ISI motion, we should see an improvement on LSC signals. This is not immediate, though, nor trivial, because the optics are just set on the optical bench, without any communication with the ISI. The motion of the optics on the chambers due to other factors than seismic is not seen by the platforms: if we could connect this motion to the platform via software, this would make the optics and the platform more dependent on each other. This means that we can control the stabilization of the cavity lengths also with the ISIs.\\
What we expect is a faster reach of locking and a longer state of lock of the interferometer during observing runs.\\
\noindent
This work has been performed on LIGO during the commissioning break between O3a and O3b observing runs, in October 2019. The reason of this choice is that we needed the interferometer to \textit{not} be observing, since we were going to modify some software structure of the instrument.\\
\noindent
Through CPSs locking, we reduced the differential motion of HAM2 and HAM3 chambers and made them to move in sync. So they can be considered as a whole block. The IMC is entirely lying on HAM2 and HAM3, and it is straightforward to use it as a witness: to make this real, we need to feed the HAM2-HAM3 block with IMCL. This will lock the cavity signal to the HAM2-HAM3 block. The same feeding will be performed with PRCL, SRCL, DARM and MICH cavities, which optics lie on the other chambers, in and out the corner station. Fig. \ref{chamb} illustrates the chambers and the locations of the cavities.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/chambs.jpg}
\caption{Sketch of the blocks and the locations of the cavities.}
\label{chamb}
\end{figure}
\noindent
To lock the LSC signals to ISIs, we need to do something similar to what we did with the HAM chambers: we need to connect via software two different setups which do not talk to each other. We decided to start from the Power Recycling Cavity Length (PRCL) because we locked HAM2 and HAM3 chambers, so it was natural to start to lock the cavities on the x axis. The same work is foreseen to be done for the other cavities: the very short period of time available during the commissioning break allowed us to modify only the software for PRCL, since the job involved the request of permissions to modify the structure of the interferometer and the synchronization with the job of other people working on different parts of LIGO. Moreover, during commissioning break, time is also used to work on the chambers, profiting of the out-of-lock mode. This means that, for every attempt of software modification, a locking trial was needed, to see if the new configuration of the instrument was giving better performances and, also, if it was affecting negatively other sides of the instrument. To try to lock LIGO, we needed people not to work besides the chambers. This was a huge and collaborative work, which involved many people on site, and their time.
\paragraph{The Power Recycling Cavity Length (PRCL)}
We need to connect the ISI to the cavity and to do it we need to know how the PR cavity works.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/PRCLfeed.png}
\caption{Block diagram of PRCL locked to ISI.}
\label{prcl}
\end{figure}
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\chapter{The low frequency window}
\label{LF}
\section{Hidden GW sources}
\ No newline at end of file
main/LIGO.tex
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......@@ -139,7 +139,7 @@ The HAMs provide five levels of isolation, among which there is the Internal Sei
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/HAM.png}
\includegraphics[scale=0.9]{images/HAM.png}
\caption[Advanced LIGO HAM chamber design]{Schematic (a) and CAD model (b) of a HAM chamber \cite{mat}. Suspensions of auxiliary optics provide levels of passive isolation above 10 Hz. The ISI platforms where the suspensions live are optical tables actively isolated via low noise inertial sensors at low frequency ($\sim 0.1 Hz$). The hydraulic attenuators of the \textit{Hydraulic External Pre-Isolator} (HEPI) and the geophones gives isolation from ground motion.}
\label{ham}
\end{figure}
......@@ -149,7 +149,7 @@ The BSCs have a similar design as the HAMs, but they have two stages of ISI to s
\begin{figure}
\centering
\includegraphics[scale=1]{images/BSC.png}
\includegraphics[scale=0.9]{images/BSC.png}
\caption[Advanced LIGO BSC chamber design]{Schematic (a) and CAD model (b) of a BSC chamber \cite{mat}. The active isolation is similar to the one exposed for HAM chambers. The two ISIs provide two stages of isolation while and the suspensions are design to be quadruple pendulums, for a total of seven levels of isolation.}
\label{bsc}
\end{figure}
......
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%\documentclass[a4paper,12pt]{book}
%\usepackage[a4paper,top=2cm,bottom=2cm,left=2cm,right=2cm]{geometry}
%\usepackage[utf8]{inputenc}
%\usepackage[english]{babel}
%\usepackage{graphicx}
%\usepackage{subfigure}
%\usepackage{array}
%\usepackage{color}
%\usepackage{colortbl}
%\usepackage{verbatim}
%\usepackage{lscape}
%\usepackage{amsmath,amssymb}
%\linespread{2}
%%\title{CH.5 LSC signal optimization on LIGO}
%\date{}
%
%\begin{document}
\chapter{LSC signals optimization on LIGO}
\label{LSCdiff}
\noindent
In the previous chapter I collaborated to enhance the performances of LIGO at low frequencies, modifying the software set up of the ISIs to reduce the motion of the chambers. As I anticipated, a consequence of this work could be the improvement of the LSC signals from LIGO cavities. Among them, DARM is particularly important, because it represents the gravitational wave signal. In this chapter I am going to show you how we tried to apply the new CPS configuration in order to obtain improvements in LSC signals at LIGO Hanford. This work has been performed during the 2019 commissioning break, in collaboration with LIGO Livingston Observatory.
\section{LSC signals in LIGO cavities}
Length to Signal Control is a crucial part of LIGO: as we know, gravitational waves can be seen by the interferometer if its sensitivity is high and stable enough. Optical signals in the cavities are very important in this frame:
\section{Optimization: LSC offloading}
We saw that the cavities (and the optical signals) in LIGO are affected by the ISI motion, simply because they lie on them. Given the work done with the CPSs to suppress the ISI motion (described in the previous chapter), we should see an improvement on LSC signals. This is not immediate, though, nor trivial, because the optics are just set on the optical bench, without any communication with the ISI. The motion of the optics on the chambers due to other factors than seismic is not seen by the platforms: if we could connect this motion to the platform via software, this would make the optics and the platform more dependent on each other. This means that we can control the stabilization of the cavity lengths also with the ISIs.\\
What we expect is a faster reach of locking and a longer state of lock of the interferometer during observing runs.
\noindent
This work has been performed on LIGO during the commissioning break between O3a and O3b observing runs, in October 2019. The reason of this choice is that we needed the interferometer to \textit{not} be observing, since we were going to modify some software structure of the instrument.\\
\noindent
During the commissioning break, we performed the experiment: through CPSs locking, we reduced the differential motion of HAM2 and HAM3 chambers and made them to move in sync. So they can be considered as a whole block. The IMC is entirely lying on HAM2 and HAM3, and it is straightforward to use it as a witness: to make this real, we need to feed the HAM2-HAM3 block with IMCL. This will lock the cavity signal to the HAM2-HAM3 block. The same feeding will be performed with PRCL, SRCL, DARM and MICH cavities, which optics lie on the other chambers, in and out the corner station. Fig. \ref{chamb} illustrates the chambers and the locations of the cavities.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/chambs.jpg}
\caption{Sketch of the blocks and the locations of the cavities.}
\label{chamb}
\end{figure}
\noindent
To lock the LSC signals to ISIs, we need to do something similar to what we did in the previous chapter: we need to connect via software two different set ups which do not talk to each other. We decided to start from the Power Recycling Cavity Lecgth (PRCL) because we locked HAM2 and HAM3 chambers, so it was natural to start to lock the cavities on the x axis. The same work is foreseen to be done for the other cavities: the very short period of time available during the commissioning break allowed us to modify only the software for PRCL, since the job involved the request of permissions to modify the structure of the interferometer and the synchronization with the job of other people working on different parts of LIGO. Moreover, during commissioning break, time is also used to work on the chambers, profiting of the out-of-lock mode. This means that, for every attempt of software modification, a locking trial was needed, to see if the new configuration of the instrument was giving better performances and, also, if it was affecting negatively other sides of the instrument. To try to lock LIGO, we needed people not to work besides the chambers. This was a huge and collaborative work, which involved many people on site, and their time.
\paragraph{The Power Recycling Cavity Length (PRCL)}
We need to connect the ISI to the cavity and to do it we need to know how the PR cavity works.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/PRCLfeed.png}
\caption{Block diagram of PRCL locked to ISI.}
\label{prcl}
\end{figure}
\section{Beyond}
%\begin{thebibliography}{9}
%
%\bibitem{phd} D. Tuyenbayev, \textit{Extending the scientfic reach of Advanced LIGO by compansating for temporal variations in the calibration of the detector.}, PhD thesis, University of Texas, 2017
%
%\end{thebibliography}
%
%\end{document}
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</diagram></mxfile>
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main/laserstab.tex 0 → 100644
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%\documentclass[a4paper,12pt]{book}
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%%\title{CH.6 Laser stabilization for 6D isolation injection}
%\date{}
%
%\begin{document}
\chapter{Laser stabilization for 6D isolation system device}
In this chapter I will introduce the 6D device, a new technology for inertial isolation. This project has been presented to the scientific community at the 10th ET Symposium in 2019 \cite{poster}. My contribution to the development of this technique focussed on the sensing side: a laser will be injected into the device and will need to be stabilized in frequency. We will see here why and how.\\
This work has been done entirely at UoB during the pandemic period: the design of the project has been conducted from home during the lockdown in 2020, while the experiment has been built and tested from September 2020, when the University accorded me the permission to return to the lab.
\section{6D inertial isolation system overview}
The 6D inertial isolation system is a device based on a new technology under development at University of Birmingham and at Vrije Univestiteit in Amsterdam, which could enable detection of gravitational waves at 10 Hz and below \cite{6d}. We have already seen the importance for this frequency window to be opened (chap 2): this facility can be installed on Earth-based interferometers of every type, on or under ground, allowing the different instruments to easily use the same device.\\
As the name reminds, the 6D investigates the motion of a reference mass in all 6 degrees of freedom, using 6 interferometers. In Fig. \ref{6d} it is shown a sketch of the design of the facility. \\
\begin{figure}[h!]
\centering
\includegraphics[scale=1.2]{images/6d.png}
\caption[6D design.]{Sketch of the 6D device (Figure taken from \cite{6d}). The working principle is based on a isolated, suspended reference mass which is monitored by position sensors detecting the relative motion between the ground and the platform; actuators apply corrections to the platform and the whole apparatus is in vacuum.}
\label{6d}
\end{figure}
\noindent
All six degrees of freedom are simultaneously low-noise, reducing the cross-coupling affecting low force-noise measurements.\\
The reference mass, suspended from a single, thin, fused-silica fibre and a metal spring, provides supports in the vertical (Z) degree of freedom. An interferometric readout provides control in the X and Y tilting degrees of freedom.\\
The interesting advantage is that this system provides isolation in all the degrees of freedom with the use of only one device: currently aLIGO is seismically isolated by three seismometers and twelve geophones \cite{lisa}.\\
\noindent
What we expect from 6D is isolation at low frequencies and reduction of fundamental noises: the thermal noise of the suspension is suppressed by the quasi-monolithic, fused-silica fibre; temperature gradients are kept under control thanks to the vacuum enclosure.\\
We expect a reduction of the platform motion at aLIGO and the bandwidth of the control loop by a factor of about 5 \cite{6d}.\\
The expected performance is shown in Fig. \ref{6dsens}: at low frequency 6D isolator provides two order of magnitude better results with respect to the current devices. The time of observation is expected to be increased from 66\% to 81\% for each detector.\\
The control noise will become negligible above 5Hz because the bandwidth for the control loops will be reduced to 0.5 Hz.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/6dsens.PNG}
\caption[6D sensitivity curve]{A comparison of the expected performance of the 6D isolator and that of current STS-2 seismometers (Figure taken from \cite{6d}).}
\label{6dsens}
\end{figure}
\subsection{HoQI technology}
The 6 interferometers dedicated to the sensing role of the facility are the HoQI devices (Homodyne Quadrature Interferometer) developed at UoB \cite{hoqi}.\\
HoQI is a compact, fibre-coupled interferometer with high sensitivity and large working range. In Fig. \ref{hoqi1} you can see the optical layout of the device:
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/hoqi1.png}
\caption[HoQI optical layout]{HoQI optical layout (Figure taken from \cite{hoqi}). The working principle is based on a Mach-Zender interferometer which uses two different beam splitters to recombine the beam independently. Moreover, the required differential phase shift is generated by a polarization scheme conveniently designed.}
\label{hoqi1}
\end{figure}
\noindent
This device has been designed to sense motion at low frequency, with a sensitivity of 2 $\times$ 10$^{-14}$ m/ $\sqrt(Hz)$ at 70 Hz and 7 $\times$ 10$^{-11}$ m/ $\sqrt(Hz)$ at 10 mHz \cite{hoqi}. This is a result obtained when combining HoQI devices to inertial sensors to create an "interferometric inertial sensor" \cite{sam}. In the frame of compact devices, HoQIs are designed to be very small in size, so can can easily be attached to sensors. They are then ideal for 6D purposes, not only for their high sensitivity, but also for their small size.
\noindent
The six HoQIs used for the 6D device need to be fed by a laser source that is sent into the vacuum chamber: my project focussed on this source, specifically how to stabilize it in frequency.
\section{Laser stabilization: requirements and key technology}
The laser chosen as source for 6D is a 1064 nm RIO ORION Laser Module (see Fig. \ref{rio}). This has been chosen for its low frequency noise, inexpensiveness and small size, relatively to other options. The key point in the stabilization of the frequency noise of this source is that the technology will be based on HoQIs: the same devices used by the 6D are sensitive enough to be installed also to stabilize the laser source. This solution is very convenient in terms of costs and presents practical advantages: the HoQIs are known devices, compact in size and, as we will see, they allow the setup to be moved easily (in vacuum or in air), according to the main 6D tank requirements.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/rio.jpg}
\caption[Rio Orion laser modules]{Picture of the RIO Orion laser mounted on a breadboard.}
\label{rio}
\end{figure}
\noindent
What we want from this source is a low-noise readout for the HoQIs inside the 6D tank, and thus the laser source needs to be as low noise in frequency fluctuations as possible at frequencies below 1Hz, because this is the range of frequencies where the 6D isolator is aimed to detect and control seismic noise: we are going to use two Rio Orion lase modules to obtain a frequency stabilization suitable for 6D requirements. Constraints to these requirements are mainly given by the HoQIs. For 6D readout, HoQIs are built in such a way that the arm length is L$_{6D}$ $<$ 3 mm. Limitations to this number are given by BOSEM size ($\pm$ 2 mm) and the ability to adjust it, once the devices are in vacuum. Another parameter to take into account is the ADC noise of HoQIs, which is ADC = 2 $\times$ 10$^{-14}$ m/$\surd{Hz}$ at about 10 Hz \cite{hoqi}. Frequency fluctuations depend on both these parameters and we want it to meet the following requirement:
\begin{equation}
\centering
\delta f_{6D} \ll f \times \frac{ADC}{L_{6D}} = 2000 \frac{Hz}{\surd{Hz}}.
\end{equation}\\
The technique we are going to adopt to stabilize the laser in frequency, as anticipated, is to use HoQIs, because we can associate frequency fluctuations to fluctuations of arm length:\\
\begin{equation}
\centering
\delta f = \frac{\delta L}{L}\cdot f,
\label{df}
\end{equation}\\
and this arm length can belong to a HoQI placed on the optical bench. The use of compact interferometers to stabilize solid-state lasers in frequency is new and allows the whole set up to be small in size. This technique, in combination with cheap laser sources, makes the set up competitive with other more expensive products.\\
We can then apply the same relation of. eq. \ref{df} to the arm length of the HoQI used for the laser stabilization, remembering that the requirement of $\delta$f $\ll$ 2000 Hz/$\surd{Hz}$ must remain valid. So, constraints to the arm length in this case are due also to the size of the bench and the whole set up.\\ We said we want a compact setup, but the arm length of this HoQI (say L$_{stab}$) can have a wider range of sizes to fit the requirement. For example, for L$_{stab}$ = 0.1 m we have:\\
\begin{equation}
\centering
\delta f_{stab} = f \times \frac{ADC}{L_{stab}} \simeq 55 \frac{Hz}{\surd{Hz}},
\end{equation}\\
which is still much lower than the threshold.\\
Since we want the setup to be as much compact as possible, we need to find the lowest possible L$_{stab}$ which gives an interesting $\delta f_{stab}$, compared to the current performances of RIO Orion and the best products available.\\
In the plot in Fig. \ref{perf} there is the analysis and comparison with two of the best products available.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/perf.png}
\caption{Analysis and comparison of RIO Orion laser with other products and with the configuration involving HoQIs.}
\label{perf}
\end{figure}
\noindent
It is evident that we cannot build a HoQI with L=10 m. If we want our device to be competitive even with the best product (ADJUSTIK X15, shown in blue line), we will need a L=30 cm. However, our purpose is to make the set up \textit{compact} and \textit{competitive} with most of the available products, so the best compromise is choosing L=10 cm. With this configuration, the device will still be competitive with ADJUSTIK X15 in terms of price.\\
Fig. \ref{free} shows a plot of the measured frequency noise of the Rio Orion laser modules and the level of stabilization required by the 6D with the chosen L$_{stab}$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/free.png}
\caption[Free running noise and requirements]{Measured free running frequency noise of the Rio Orion laser modules compared with the frequency noise of the Mephisto Light laser module. The dotted line is the threshold of the 6D requirements for stabilization of its laser source, which is the goal we aim to achieve with the proposed setup. This threshold has been built using L$_{stab}$=10 cm and the HoQI noise and ADC floors as in \cite{hoqi}.}
\label{free}
\end{figure}
\section{Experiment design}
The optical set up for the stabilization of the laser source is built on a 800 mm $\times$ 650 mm breadboard: this choice allows to adjust the position of the laser source easier when the light is sent to the 6D vacuum chamber. The setup includes two RIO Orion lasers with output power 12 mW each, and a double-check of the light signal through an optical heterodyne detection: the beat frequency is monitored to assure that the two light frequencies are as much similar as possible.\\
The frequency of the lasers is tunable via temperature and input modulation: in the first case, the Thermo-Electric-Controller (TEC) is driven by a software provided by the manufacturer, while in the second case the module can be integrated to any software code to apply a modulation voltage between +4 V and - 4 V.\\
To minimise airflows, the optical setup has been enclosed into a box made of foam.
\paragraph*{Opto-mechanical design}
The optical layout is shown in Fig. \ref{las}: the two lasers have a twin optical layout. There is a Faraday Isolator (FI) at each output and then a 1 to 4 fibre beam splitter (BS) which separates the beam into 4 outputs of equal power: 3 outputs go into the vacuum chamber (for a total of 6 laser inputs, one for each 6D HoQI into the vacuum chamber). The remaining output is sent through a fibre coupler to a Schafter-Kirchhoff collimator and gives an output of about 1.2 mW for each laser; this proceeds freely on the breadboard towards a 1 inch, 10/90 (R/T) beam splitter: 10$\%$ of the light is sent to a fast DC coupled 125-MHz photoreceiver (PD) acted to sense the beat-note of the two lasers; two 1 inch mirrors deviate one of the two laser beams towards the transmitting surface of another 1 inch beam splitter, which combines the light from both lasers towards the photoreceiver; the other 90$\%$ of it is sent to the HoQIs, one for each laser. The optical path lengths (OPL) have been set to be equal, to assure the same beam size from both lasers at the photoreceiver.\\
The photoreceiver has strict constraints about the beam size and the input power: a focussing lens in front of the active area assures that the beam size is suitable to fit the 0.3 mm active area. Damping filters are added along the OPL, because the maximum input power of the device is 55 $\mu$W.\\
The whole optical setup lies on the bradboard and it is relatively easy to align because all the optomechanical components have been manufactured to make the beams out of the collimator to travel at the same height as HoQI components and the photoreceiver, so that there is no need of pitch tuning.
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/layout.png}
\caption[Optical layout]{Optical layout of the laser stabilization experiment.}
\label{las}
\end{figure}
\paragraph*{HoQI design}
HoQIs for 6D laser stabilization have been built to fit the requirements, as shown previously: the adjustable arm length is of 10 cm; the photodiodes have a bigger active area with respect to the one of the HoQIs inside the 6D vacuum chamber, because the laser spot size is larger than the one travelling into the 6D device. Moreover, this type of HoQI is independent from any inertial sensor, so both the arms end with a steering mirror. Table \ref{hoqi} shows other small details that have been adapted for this experiment.\\
\begin{table}[h!]
\centering
\begin{tabular}{|c|| c| c| }
\hline
& \textbf{Original} & \textbf{HoQI stab}\\
\hline
Platform thickness & 6 mm & 1 cm\\
\hline
Size &65$\times$75mm &65$\times$200mm\\
\hline
Corner Cubes & 2 & 0 \\
\hline
PD active area &3.6$\times$3.6mm &5.8 $\times$5.8mm\\
\hline
Steering mirrors & 1 & 2\\
\hline
\end{tabular}
\caption{Main differences between the original HoQI and the one used for 6D laser stabilization.}
\label{hoqi}
\end{table}
\noindent
In Fig. \ref{chia} there is a photo of the HoQI built for this experiment.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/HoQIChia.png}
\caption[Photo of a HoQI]{Photo of one of the HoQIs built for the laser stabilization experiment.}
\label{chia}
\end{figure}
\noindent
The photodiodes are Hamamatsu S2386-8K and the optical layout is similar to the one shown in the figure \ref{hoqi1}: there are commercial 0.5 inch cubic beam splitters (three polarizing and 1 non polarising) mounted on custom cubic bases, two 0.5 inch mirrors on steering mounts, one 0.5 inch $\lambda$/2 waveplate and one 0.5 inch $\lambda$/4 waveplate, both mounted on a custom base allowing them to rotate for fine tuning. The whole optical set up is placed on a 75 mm $\times$ 200 mm $\times$ 10 mm baseplate.\\
The two HoQIs have been tuned to obtain the best fringe visibility, which is 0.8 for HoQI1 and 0.6 for HoQI2. The technique used to measure the fringe visibility is based on measured power on a sin vs cos plot, once the beams have been aligned to overlap in far field and produce interference, and tuning the steering mirrors for fine alignment and the waveplates for power adjustments (Fig. \ref{fringes}).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/fringes.png}
\caption[HoQI fringes alignment.]{Example of how we set the fringes of the HoQIs to obtain the desired alignment. Every photodiode detects the fringes independently from the others: to obtain the same response, we adjusted the offsets and the gains of each diode on the pre-amplifier and via software.}
\label{fringes}
\end{figure}
\section{AC-coupled control loop}
To acquire our data, we need to connect the HoQIs and the beat-note receiver to a data acquisition system. At UoB we have 3 CDS racks and one of them is dedicated to 6D. As shown in Fig. \ref{signals}, HoQIs will need a pre-amplifier, an Anti-Aliasing (AA) device and an Analog to Digital Converter (ADC) before connecting to CDS (Fig. \ref{signals}).\\
The sensing and control system of the experiment is based on the HoQIs: the software code manipulating the HOQIs signal and driving the input modulation is written with Matlab Simulink and controlled by the CDS. The controller filter has been built taking into account all the features of the loop and implemented into the CDS. The lasers can then be controlled via input modulation through a feedback control loop where HoQIs act as the sensors and feedback devices for frequency stability.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/signals.png}
\caption[Scheme of the signals]{Basic scheme of the row signals for laser frequency control: the power detected by the photodiodes is converted into $\mu$A and sent to a pre-amp, one for each HoQI; the pre-amps convert the signal in double-ended voltage to be sent to an ADC. The output of the CDS is a $\pm$ 10 V double-ended signal out of the DAC: since the lasers require a $\pm$ 4 V single-ended input, the double-ended signal is converted into single with a custom differential- to single-ended amplifier of gain 2.5. The calibration of the CDS gives 0.00061 V/counts, which allows to digit into the CDS the right counts divided by the gain of the differential- to single-ended amplifier to obtain the desired voltage to send to the lasers.}
\label{signals}
\end{figure}
\noindent
The beat-note receiver is 15 V powered and connected to a frequency counter, and then to the CDS. A detailed scheme of the electronics is shown in Fig. \ref{cables}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/cables.png}
\caption[Scheme of the electronics]{Detailed scheme of the electronics designed for this experiment. The different colors of the arrows represents different types of cables. Bacardi and Peapsy are the CDS and the computer controlling it, as we named them at UoB. Differential to single ended converters (Diff2SE) are needed because the CDS supports single ended outputs while the pre-amps are double ended. The Mokulab is the device used as an oscilloscope and/or as a spectrum analyser, connected to the beat-note fast photoreceiver. Green arrows indicate power supplies. The frequency counter can be connected to a computer to acquire data or a USB drive can be inserted to save data directly from the device. The temperature modulation requires the use of a software provided by the manufacturer and installed on computers. Each laser needs its own software connection. }
\label{cables}
\end{figure}
\noindent
Models of our system are designed with Symulink and the controller filters is designed using the tools provided by the CDS in Birmingham laboratory (Fig. \ref{filter}).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/filter.png}
\caption[Controller filter]{Bode plot of the controller filter installed into the CDS (blue) and of the closed-loop expected gain when this filter is applied (red). Since we want to lower the frequency noise below 1 Hz, the filter has been designed with a pole at 0.1 Hz: this design should push the gain from below 10 Hz, with the maximum gain at 0.1 Hz, assuring stability.}
\label{filter}
\end{figure}
\section{Noise hunting}
\paragraph*{Noise budget}
The performances of the setup depend strongly on the HoQIs because they are the sensing and feedback devices of the setup: the noise budget in Fig. \ref{noiseb} shows that the measured HoQI readut follows the free running frequency noise of the lasers detected by the frequency counter at low frequencies, and then it sits on the ADC noise (estimated as in \cite{hoqi}) at frequencies above 100 Hz. What we are interested in is getting the lowest possible frequency noise from the lasers, reducing the noises affecting the HoQIs. The improvement of HoQIs sensitivity is crucial to obtain the best performances in sensing frequency fluctuations, in order to provide the correct and stable control and feedback to the setup.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/noisebudget.png}
\caption[Noise budget]{Noise budget of the laser stabilization setup. The paper which provided the HoQI noise and the ADC noise is given by \cite{hoqi}.}
\label{noiseb}
\end{figure}
\paragraph*{Tested noise sources}
There are several noise sources to take into account: air currents and vibrations from electronics and cables have been reduced placing the optical setup into a foam box and moving the electronic devices suitably. Cables have been isolated from the table and the breadboard by rubber feet.\\
\noindent
The test in Fig. \ref{sound} shows that the setup is sensitive to acoustic noise: we injected a sound at 75 Hz and both HoQIs clearly detected it. Moreover, we found out that HoQI1 is detected some noise around 22 Hz that HoQI2 is not able to sense: the two peaks in the figure are present in every condition of the laboratory and part of the day. The source of this noise is still under investigation: it could be a permanent sound in the lab non audible by humans. The fact that only HoQI1 can detected could be due to its position with respect to the noise source: it might be closer to it than HoQI2. Imperfections in the optics and general setup of the HoQIs are also taken into account.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.33]{images/sound75hz.png}
\caption[Test of the setup to sound stimulation.]{Test of the setup to sound stimulation. The peaks at 75 Hz shows that both HoQIs are sensitive to acoustic stimulation; HoQI1 is also detecting another noise around 22 Hz, which is instead non-visible by HoQI2.}
\label{sound}
\end{figure}
\noindent
Temperature changes affected dramatically the measurements: the two lasers can be driven also via temperature modulation. This method has been used to move the beat-note peak along the frequencies and set it around 60 Hz, being this the setpoint we decided for it. However, both laser modules are sensitive to changes of the room temperature, which make the peak move out from the setpoint on large time scales (~hours): this affects long time measurements. The stabilization of the room temperature requires the use of the air conditioning, which in turn creates air currents visible by the setup below 1 Hz.\\
Temperature changes are also responsible for deformations of metals; this induces noises into the HoQI platforms because of the different materials they are built of: platform, screws, post holders expand in different ways with temperature changes, and this produces deformations and friction between the metals, which translate into displacement noise visible by the HoQIs. This issue has been reduce by inserting rubber rings between the junctions where different metals are mounted.\\
\noindent
Another noise we noticed from the datasheets of the laser modules is the intensity fluctuations: in particular, laser2 is more affected by this noise than laser1 (compare the datasheets reports in Fig. ...). This in part explains why HoQI2 is in general noisier than HoQI1, as shown in the following sections.
\paragraph*{Loop performances}
The behaviour of the two HoQIs has been tested in loop and out of loop, to check if they are detecting and responding correctly to the injection of the controller filters through the input modulation of the laser modules. The expectation is that the HoQIs output in out-of-loop mode should show the injection of the gain: Fig. \ref{looptest} shows that the expectations are satisfied.\\
This test shows that HoQI2 is in general noisier than HoQI1, especially above 1 Hz: this affects laser stabilization measurement and loop stability, thus it has been deeply investigated. The higher intensity fluctuations of laser2 can partially explain the reason of HoQI2 noise.
\begin{figure}[h!]
%\centering
\includegraphics[scale=0.3]{images/hoqisOLCL.png}
\caption[In-loop test of HoQIs performances.]{In-loop test of HoQIs performances. The out-of-loop traces (cyan and purple) are following the free running frequency noise trace (blue) as expected, while when the loop is closed the HoQI outputs (green and red) show that the controllers are pushing the expected gain (orange). There is an evident un-match with the orange trace below 0.4 Hz and this is likely due to loop leakage.}
\label{looptest}
\end{figure}
\section{Laser stabilization: tests and results}
The tests have been performed measuring the stability of the beat-note peak around the 60 Hz setpoint: the frequency counter used for this measurements is a Keysight 53230A 350 MHz - 20 ps. The output of the fast photoreceiver is DC-coupled and can be directly connected to the counter. The measurements has been recorded on a USB drive: the data provided by the counter are in frequency (Hz).\\
Several tests have been taken in different conditions for noise hunting along the frequency range of interest. QUI CI VANNO I TEST COL COUNTER DI CUI PARLAVA CONOR: LO SCOPO E' DIMOSTRARE CHE IL SETUP E' STATO ADEGUATAMENTE TESTATO
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/test2807.png}
\caption{Last laser stabilization test}
\label{test}
\end{figure}
\section{Alternative test}
An alternative test has been made to make sure that the input modulation is effectively reducing the frequency noise of the lasers. Since we found that HoQI2 is noisier than HoQI1 and that laser2 has larger intensity fluctuations, we decided to use laser1 to feed both lasers. The new concept is to let only HoQI1 be the in-loop sensor, while HoQI2 will act as the out-of-loop sensor.\\
Results are shown in Fig. \ref{alt} and are encouraging: the frequency noise of the out-of-loop sensor is lowered by about one order of magnitude when the controller on laser1 is active. This means that the control loop works well and that there are still more external noise sources that are reducing the performances of the HoQIs, impacting also on the measurement through the heterodyne detection.\\
This test highlights also that HoQI2 is still noisier than HoQI1, despite the use of the laser with less intensity fluctuations: this clarify that HoQI2 noise arises from other external sources that HoQI1 is non-sensitive to or imperfections into the optics. A test in vacuum could solve the doubts about the external sources.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/CLOLtest2907.png}
\caption[Alternative stabilization test]{Alternative test for frequency stabilization: the out-of-loop sensor (HoQI2) is less noisy by an order of magnitude when the in-loop sensor is actuated by the controller filter. This is a proof that the frequency noise of the laser is reduced by the control loop.}
\label{alt}
\end{figure}
\section{Beyond: installation onto the 6D device}
%\begin{thebibliography}{9}
%
%\bibitem{poster} \textit{https://dcc.ligo.org/LIGO-G1900741}
%
%\bibitem{6d} C.M. Mow-Lowry, D. Martinov, \textit{A 6D interferometric inertial isolation system}, 2019 Class. Quantum Grav. 36, 24
%
%\bibitem{hoqi} S. Cooper et. al, \textit{A compact, large-range interferometer for precision measurement and inertial sensing}, 2018 Class. Quantum Grav. 35 095007
%
%\bibitem{sam} S. Cooper, \textit{Breaking the Seismic Wall: How to Improve Gravitational Wave Detectors at Low Frequency}, PhD thesis, University of Birmingham, 2019
%
%\bibitem{rio} $https://rio-lasers.com/1064-nm-laser-module/$
%
%\bibitem{adj} $https://www.photonicsolutions.co.uk/upfiles/ADJUSTIKFiberLaserLG13Dec17.pdf$
%
%\bibitem{mephi} $https://edge.coherent.com/assets/pdf/COHR\_ MephistoNPRO\_ WP\_ 9\_ 24\_ 19.pdf$
%
%
%\end{thebibliography}
%
%\end{document}
\ No newline at end of file
main/main.tex
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......@@ -2,7 +2,7 @@
\usepackage[a4paper,top=3cm,bottom=3cm]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{palatino}
\usepackage{bookman}
\usepackage[english]{babel}
\usepackage{graphicx}
\usepackage[font=small,hang]{caption}
......@@ -14,7 +14,7 @@
\usepackage{lscape}
\usepackage{amsmath,amssymb}
\linespread{2}
\title{Innovative perspectives for seismic isolation of gravitational waves detectors}
\title{Innovative perspectives for seismic isolation of gravitational-wave detectors}
\author{Myself}
\date{}
\titlehead{A Thesis submitted for the degree of Philosophiae Doctor}
......@@ -32,7 +32,7 @@
A brief summary of the project goes here, with main results.
\chapter{Introduction}
\chapter{Structure of this thesis}
An introduction to frame the work and structure of the thesis go here.\\
STRUCTURE OF THESIS [DRAFT]\\
......@@ -64,51 +64,75 @@ Solar mass:\\
$M_{\odot} = 10^{33}$ g
\chapter{Acronyms}
AA = Anti-Aliasing\\
ADC = Analogue-to-Digital Converter\\
AEI = Albert Einstein Institute\\
aLIGO = Advanced Laser Interferometric Gravitational-wave Observatory\\
AI = Anti-Imaging\\
ASD = Amplitude Spectral Density\\
BS = Beam Splitter\\
BSC = Basic Symmetric Chamber\\
BH = Black Hole\\
CARM = Common Arm length\\
CDS = Control and Data System\\
CP = Compensation Plate\\
CPS = Capacitive Position Sensors\\
CS = Corner Station\\
DAC = Digital-to-Analogue Converter\\
DARM = Differential Arm Length\\
DIFF2SE = Differential to Single-Ended\\
ETM = End Test Mass\\
ET = Einstein Telescope\\
FI = Faraday Isolator\\
FIR = Finite Impulse Response\\
HAM = Horizontal Access Module\\
HEPI = Hydraulic External Pre-Isolator\\
HoQI = Homodyne Quadrature Interferometer\\
HP = High Pass filter\\
IMC = Input Mode Cleaner\\
IMCL = Input Mode Cleaner Length\\
ISI = Internal Seismic Isolation\\
ITM = Input Test Mass\\
LHO = LIGO Hanford Observaotry\\
LLO = LIGO Livingston Observatory\\
LP = Low Pass filter\\
LSC = Length Sensing and Control\\
LVK = Ligo-Virgo-Kagra meeting\\
MCA = Mid-Course Assessment\\
MICH = Michelson length\\
MIT = Massachusetts Institute of Technology\\
ND = Neutral-Density (filter)\\
NPBS = Non-Polarizing Beam Splitter\\
NS = Neutron Star\\
OPL = Opltical Path Length\\
PBS = Polarizing Beam Splitter\\
PD = PhotoDiode\\
PR = Power Recycling\\
PRCL = Power Recycling Cavity Length\\
QPD = Quadrant Position Device\\
RIN = Relative Intensity Noise\\
SC = Sensor Correction\\
SR = Signal Recycling\\
SRCL = Signal Recycling Cavity Length\\
TEC = Thermo-Electric Controller\\
UoB = University of Birmingham\\
\mainmatter
\part{Gravitational astrophysics}
\part{Gravitational-wave frontiers}
\include{GW}
\part{Detectors and seismic isolation}
\include{LF}
\include{LIGO}
\include{oplevs}
\part{Lowering seismic noise}
\part{Lowering seismic motion}
\include{oplevs}
\include{CPSdiff}
%\include{LSCdiff}
\include{laserstab}
\appendix
%\include{A}
%\include{B}
\include{A}
\include{B}
\backmatter
......@@ -136,6 +160,34 @@ Beginning of Gravitational Wave Astronomy}
\bibitem{lsc} K. Izumi, D. Sigg, \textit{Advanced LIGO: length sensing and control in a dual recycled interferometric gravitational wave antenna}, 2017 Class. Quantum Grav. 34 015001
\bibitem{poster} \textit{https://dcc.ligo.org/LIGO-G1900741}
\bibitem{6d} C.M. Mow-Lowry, D. Martinov, \textit{A 6D interferometric inertial isolation system}, 2019 Class. Quantum Grav. 36, 24
\bibitem{lisa} L Barsotti et al, \textit{Alignment sensing and control in advanced LIGO} 2010 Class. Quantum Grav. 27 084026
\bibitem{hoqi} S. Cooper et. al, \textit{A compact, large-range interferometer for precision measurement and inertial sensing}, 2018 Class. Quantum Grav. 35 095007
\bibitem{sam} S. Cooper, \textit{Breaking the Seismic Wall: How to Improve Gravitational Wave Detectors at Low Frequency}, PhD thesis, University of Birmingham, 2019
\bibitem{rio} $https://rio-lasers.com/1064-nm-laser-module/$
\bibitem{adj} $https://www.photonicsolutions.co.uk/upfiles/ADJUSTIKFiberLaserLG13Dec17.pdf$
\bibitem{mephi} $https://edge.coherent.com/assets/pdf/COHR\_ MephistoNPRO\_ WP\_ 9\_ 24\_ 19.pdf$
\bibitem{phd} D. Tuyenbayev, \textit{Extending the scientfic reach of Advanced LIGO by compansating for temporal variations in the calibration of the detector.}, PhD thesis, University of Texas, 2017
\bibitem{intro} B. Lantz et al., \textit{Estimates of HAM-ISI motion for A+}, T1800066-v2, March 2018, https://dcc.ligo.org/LIGO-T1800066
\bibitem{intro2} S. Cooper et al., \textit{Ham ISI model}, Technical note, University of Birmingham, March 2018, https://dcc.ligo.org/LIGO-T1800092
\bibitem{ven} Venkateswara et al., \textit{Subtracting tilt from a horizontal-seismometer using a ground-rotation-sensor}, Bulletin of the Seismological Society of America (2017) 107 (2): 709-717
\bibitem{mca} C. Di Fronzo \textit{Optical sensors for improving low-frequency performance in GW detectors}, Mid-Course Assessment, University of Birmingham, 2018
\bibitem{poster} C. Di Fronzo et al., \textit{Optical Lever for interferometric inertial isolation}, poster, LVC meeting, Maastricht 2018, https://dcc.ligo.org/LIGO-G1801693
\end{thebibliography}
\chapter{Acknowledgements}
......
main/oplevs.tex 0 → 100644
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\chapter{Optical Levers for tilt motion reduction}
\label{oplevs}
In this chapter I will introduce the sensors dedicated to measure the seismic motion. They need to account for horizontal, vertical and tilt displacements in all degrees of freedom in order to be efficient and the technology for their improvement is currently pushing and competing on sensing as low as possible seismic motion. As we know from the previous sections, 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 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 MCA report \cite{mca}. A poster about this project has been presented at the LVK meeting in Maastricht (September 2018) \cite{poster}.
\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 benches, which dominates above 1 Hz \cite{intro}.\\
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 \cite{intro2}.\\
However, there could 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.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.85]{images/HAMoplev.PNG}
\caption{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{intro}).}
\end{figure}
\paragraph*{Horizontal sensors}
The most important problem, in order to achieve good isolation, is the sensitivity of the horizontal sensors to rotation (Fig. \ref{a}). If we could independently measure the rotation, we could calculate the true translation motion.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/hor.PNG}
\caption{Basic sketch of horizontal sensor tilting.}
\label{a}
\end{figure}
\noindent
When a rotation around the center of mass occurs, an additional term 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} = -kx - b\dot{x} + F_{tilt},
\end{equation}
\noindent
where $x$ is the direction of motion.\\
We assume that the angle is very small, in such a way $\sin \theta$ $\simeq$ $\theta$. So the equation 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.\\
We apply the Laplace transform and make some computations using $w = x-y$:
\begin{equation}
\centering
m(W+Y)s^2 = -bWs -kW + mg\theta
\end{equation}
\begin{equation}
\centering
W = \frac{ms^2}{ms^2 + bs +k} \left(-y + \frac{g}{s^2}\theta\right)
\end{equation}
\noindent
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 $g \theta/ \omega^2$. 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 sensor, 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
If $\theta \ll$ 1, $\cos \theta \rightarrow$ 1: this means that the vertical sensor is affected by the tilt contribution by a constant factor, if the angle is small.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/vert.PNG}
\caption{Tilting of vertical sensor (Figure taken from \cite{ven}).}
\label{v}
\end{figure}
%\section{LIGO inertial sensors}
%UNA CARRELLATA VELOCE DI COME VIENE ISOLATO LIGO
\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. There is a light source, typically a laser, impinging 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=0.5]{images/opt2.PNG}
\caption{Tilt of the optic.}
\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.\\
\noindent
If we are able to 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.\\
\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 longer L, the better the sensitivity to small angles. Moreover, the bench where the position device is set needs to be stable: another optical lever could be placed on it, with the associated actuation. A basic picture of the whole system is shown in Fig. \ref{z}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{images/opt3.PNG}
\caption{Basic principle of the optical lever used for sensing and actuation for seismic isolation.}
\label{z}
\end{figure}
\subsection*{Noise budget}
In order to understand the feasibility of the project in terms of performances, 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 performances 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.
\subsubsection{Quadrant Position Devices}
The Quadrant Position Devices (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 section in Fig. \ref{j}.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/quad.PNG}
\caption{Basic scketch of the segmented photodiode.}
\label{j}
\end{figure}
\noindent
The normalized coordinates (X, Y) for the beam's location are given by the following equations:\\
\begin{equation}
\centering
X = \frac{(I2+I3)-(I1+I4)}{I1+I2+I3+I4}
\end{equation}
\begin{equation}
\centering
Y = \frac{(I1+I2)-(I3+I4)}{I1+I2+I3+I4}
\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 difference 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}
\subsubsection{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 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.\\
If we have a laser wavelength $\lambda$ = 1064 nm and an input power P$_0$ = 1 mW, 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}
\bigskip
%\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...
\subsubsection{Thermal noise}
\label{tn}
The other, important noise affecting the measurements is the thermal noise due to the 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 0.77 A/W.\\
To compute R, consider that the output voltage is given by:
\begin{equation}
\centering
V = 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=1.3 $\times$ $10^4$ $\Omega $.\\
So, considering T=300 K at room temperature, we have:
\begin{equation}
\centering
T_{h}=1.47 \times 10^{-12} \frac{W}{\sqrt{Hz}}.
\end{equation}
\subsection*{Resolution}
Now that we have extracted the noise budget of our system, we can determine the sensitivity $\alpha$ of the sensor. This means that we want to know the efficiency of our system in measuring angles (in rad/$\sqrt{Hz}$).\\
So, according to the block diagram in Fig. \ref{BD}, to obtain the angle 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}
\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.\\
\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=0.7]{images/syst.PNG}
\caption{Basic sketch of the optical lever system (not in scale).}
\label{syst}
\end{figure}
\noindent
The prototype and its own pre-amplifying electronics has been built at UoB (Fig. \ref{oplev20} and tested in air and in vacuum at the AEI.
\begin{figure}
\centering
\includegraphics[scale=0.5]{images/OpLev20.jpg}
\caption[Photo of the optical lever prototype]{Photo of the optical lever prototype.}
\label{oplev20}
\end{figure}
\section{Test at the AEI}
The aim of the visit 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 the CDS and facilities at AEI. The main modifications are the following:
\paragraph*{PIN configuration}
The pin configuration of the QPDs has been re-adjusted because the AEI electronics is set on a different one. In Fig. there is a scheme of the original configuration; 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 build to connect the UoB boxes to the QPDs with the new pin configurations.
\paragraph*{Device modification}
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.\\
\noindent
Every component has been vacuum-cleaned using an ultra-sonic bath: every mount was dis-mounted and then re-mounted after cleaning.\\
To clean the collimators, it was sufficient to remove all the labels and wipe them with alcohol to remove the residuals of glue.
\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 (see Tech Note) the spot size on the photodiode is w $\simeq$ 1 mm. The power on that point is P = 3.5 mW.\\
%Test images have been taken for a rough beam profiling, using a Wincam camera.
%
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.6]{images/6cm.PNG} \includegraphics[scale=0.6]{images/14cm.PNG} \includegraphics[scale=0.6]{images/34.PNG}
%\caption{Wincam images of the beam: from left to right, the camera has been placed at 6 cm, 14 cm and 34 cm from the output of the collimator.}
%\end{figure}
\noindent
Summarizing, the prototype is ready for the test with the following specifications:\\
\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$^{-3}$\\
Lens focal lenght & F = 150 mm\\
Shot noise & SN = 75 nV/$\sqrt{Hz}$ \\
Responsivity Si @ 1064 nm & $\rho$ = 0.2 A/W \\
Thermal noise & Th = 21 nV/$\sqrt{Hz}$ \\
Op-amp noise & OP = 8,8 nV/$\sqrt{Hz}$
\end{tabular}
\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}
\subsection{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, Yaw and Sum of the QPD quadrants.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/inair_test.PNG}
\label{inair}
\caption[OpLev test in air]{Preliminary test in air: the traces show the trend of the pitch, yaw and the sum of all QPD quadrants.}
\label{inair}
\end{figure}
\section{Test is 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}
Fig. \ref{LVDT} shows the movement of South bench along z axis, that we use as a reference measurement for bench adjustments with temperature variations.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/LVDT_Z.PNG}
\label{LVDT}
\caption[30 mbar LVDT test]{Motion along z axis of the South bench during vacuum pump to 30 mbar.}
\end{figure}
\noindent
Fig. \ref{QPD} shows the measurements taken with the QPD. The 4 quadrants and the Pitch, Yaw and Sum values show an expected behaviour.\\
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}
\label{QPD}
\caption[In vacuum QPD test: 30 mbar]{QPD signals during 30 mbar pressure conditions.}
\end{figure}
\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. Variations in LVDT trend due to temperature stabilization and related variations of Pitch and Yaw due to the more stable bench conditions.\\
\noindent
We took measurements both with AEI and UOB electronics. We have two UOB pre-amps (serial numbers ending with 0604 ans 0519). Firstly, we decided to modify UOB box no. 0604, to better adapt to AEI electronics and avoid saturation. We substituted the R334 Dx non inverting gain (originally of 9.1 k$\Omega$) with DH 0.1\% "Metalschicht" of 6.8 k$\Omega$. Unfortunately, no measurements have been possible, because we saturated. So we used the other box, not modified.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/LVDT.PNG}\\
\includegraphics[scale=0.3]{images/ULVDT.PNG}
\label{LVDT_FIN}
\caption[Different pre-amps test: bench LVDT motion]{Bench motion long z axis at 5 $\times$ 10$^{-3}$ mbar pressure conditions. Blue curve is the trend measured by AEI pre-amp, red curve is measured with UOB pre-amp.}
\end{figure}
\noindent
In this conditions, also the signals from the L4C seismometers and accelerometers on Central bench have been measured.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/AEI_SOUTH.PNG}\\
\includegraphics[scale=0.3]{images/UOB_SOUTH.PNG}
\label{central}
\caption[Different pre-amps test: bench motion]{Motion of Central bench measured by L4Cs and accelerometers, with AEI and UOB pre-amps.}
\end{figure}
\noindent
QPD performance are shown in the following pictures. With AEI boxes we had expected results: no variations in the power fluctuation peaks and expected behaviour of Pitch and Yaw with single quadrants.\\
However, with UOB box (no. 0519) the measurements do not seem consistent with what we expected: despite the behaviour of each quadrant seems to follow the expected trend (even if differently from AEI trend), the curves of Pitch and Yaw do not match with the quadrants trend. We think that some non-linearities in UOB box could be the cause of the problem: this is still under investigation at UOB.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/AEI_QPD_TEST.PNG}\\
\includegraphics[scale=0.3]{images/UOB_QPD_TEST.PNG}
\label{qpd_fin}
\caption[In vacuum QPD test]{QPD performance, with AEI and UOB pre-amps.}
\end{figure}
\subsubsection{Noise measurements}
Noise measurements of CDS and unplugged electronics have been taken, to check if there could be issues related to it. However, they do not show any unexpected behaviour.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/EL_NOISE.PNG}
\label{noise}
\caption[Electronic noise]{Electronic noise measurements of CDS and AEI unplugged electronics.}
\end{figure}
\section{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}
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