corrections ongoing

main/CPSdiff.tex

@@ -24,7 +24,7 @@
 \chapter{Control of seismic platforms motion and LSC offloading}
 \label{CPSdiff}
 During 2019, I spent some months working at the 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 performance at low-frequency, focussing on the reduction of seismic motion of the platforms where the optics are located.\\
-In this chapter I will demonstrate how we can modify seismic control configuration of LIGO: in particular, this study should help reducing the differential motion between the chambers, making them move in sync, and help reducing and stabilizing the rms motion of the auxiliary sensors, through an LSC offload. The final goal is to obtain different and possibly better performance for seismic motion stabilization, faster and longer locking mode and, ultimately, more gravitational waves detections.  The detailed computations included in this chapter are original and partially presented to the LIGO community and stored in LIGO DCC \cite{proposal} \cite{technote1} .\\
+In this chapter I will demonstrate how we can modify seismic control configuration of LIGO: in particular, this study should help reduce the differential motion between the chambers, making them move in sync, and help reduce and stabilize the rms motion of the auxiliary sensors, through an LSC offload. The final goal is to obtain different and possibly better performance for seismic motion stabilization, faster and longer locking mode and, ultimately, more gravitational waves detections.  The detailed computations included in this chapter are original and partially presented to the LIGO community and stored in LIGO DCC \cite{proposal} \cite{technote1} .\\
 This work has been developed in collaboration with LIGO Hanford and LIGO Livingston laboratories, Stanford University, MIT and UoB and completed at UoB during 2020.\\
 This chapter is partially including some technical notes I shared with LIGO collaboration and the contents of this study have been presented at conferences and workshops \cite{chiatalk}.\\
 Essential information about the sections of LIGO involved in this study has been exposed in detail in Chapter \ref{LIGO}.

main/GW.tex

@@ -32,7 +32,7 @@ This thesis presents a study for the enhancement of the detectors for gravitatio
 Chapter 2. In this chapter we will see that there are some gravitational-wave sources emitting at lower frequency to which the current detectors are blind: it is in this frame that the experiments proposed in this thesis have been done. The final and ambitious goal is to improve the sensitivity of the detectors at lower frequencies.\\
 
 \noindent
-Chapter 3. This chapter describes briefly how an interferometric detector for gravitational waves works. In particular, the detector LIGO, with which I collaborated, is illustrated. Specific details of the instruments to which the author has contributed are explained and referred to throughout the experimental work in the following chapters.\\
+Chapter 3. This chapter describes briefly how an interferometric detector for gravitational waves works. In particular, the detector LIGO, with which I collaborated, is illustrated. Specific details of the instruments to which I contributed are explained and referred to throughout the experimental work in the following chapters.\\
 
 \noindent
 Chapter 4. This chapter contains the first experimental work performed in the first year of my PhD study: an optical lever for the reduction of tilt motion has been designed and built at UoB, and then tested at the AEI. The details of the experiment and the results are explained in detail.\\

main/LF.tex

@@ -13,7 +13,7 @@ Fig. \ref{spec} summarizes the possible objects that can be gravitational-waves
 \end{figure}
 
 \noindent
-The best modelled sources are binary systems, orbiting each other around a common central point. The Fig. \ref{binary} shows the main phases of the evolution of these kind of systems and the emission of gravitational waves at different frequencies, depending on the phase.
+The best modelled sources are binary systems, orbiting around a common central point. The Fig. \ref{binary} shows the main phases of the evolution of these kind of systems and the emission of gravitational waves at different frequencies, depending on the phase.
 
 \begin{figure}[h!]
 \centering
@@ -29,7 +29,7 @@ Currently, the ground-based observatories are tuned to detect emission from bina
 \noindent
 The first detection of gravitational waves happened on the 14th September 2015 and confirmed the Theory of General Relativity, opening a new window on the Universe: the signal from a merger of two black holes have been observed thanks to the emission of gravitational waves, confirming the existence of these objects, still mostly unknown \cite{first}.\\
 \noindent
-The detector responsible of the new discovery is based in the USA and it is one of the terrestrial interferometers currently in use for gravitational-wave detection \footnote{The working principles of the interferometers and details about the US instrument are exposed in Chapter \ref{LIGO}}. 
+The detector responsible of the new discovery is based in the USA and it is one of the terrestrial interferometers currently in use for gravitational-wave detection \footnote{The working principles of the interferometers and details about the US instrument are explained in Chapter \ref{LIGO}}. 
 
 \section{Opening the low frequency window}
 As we will see in the next chapter, the ground-based detectors involved in the search of gravitational waves cover a wide range of frequencies, but they are affected by some noises which make them unable to detect waves from sources emitting below 30 Hz. We will later see the nature of these noises. The reason why it is important to open the lower frequency window is that it can give access to the detection of gravitational waves emitted by sources whose physical structure and astrophysical features are still unknown.\\
@@ -47,7 +47,7 @@ The emitted frequency from a source of gravitational waves depends on the masses
 
 \noindent
 where $M_c$ is the combination of the two involved masses $m_1$ and $m_2$, defined as \textit{chirp mass} $M_c$ = ($m_1$ $\cdot$ $m_2$)$^{3/5}$/($m_1$ + $m_2$)$^{1/5}$.\\
-This equation is particularly useful if we want to know information about the radiation emitted by a certain mass, at a certain frequency, at a certain time before the merger. Predictions about this time and the frequency where it is possible to detect the radiation are essential for several reasons, going from efficiency of the detector in detecting different of sources to Multimessenger astronomy, in which timing is important to assure a correct localization of the source \cite{branchesi}.\\
+This equation is particularly useful if we want to know information about the radiation emitted by a certain mass, at a certain frequency, at a certain time before the merger. Predictions about this time and the frequency where it is possible to detect the radiation are essential for several reasons, going from efficiency of the detector in detecting different sources to Multimessenger astronomy, in which timing is important to assure a correct localization of the source \cite{branchesi}.\\
 In our case of interest, if we apply the lowest range of frequency available by ground-based detectors ($\sim$ 10 Hz in order of magnitude) and consider $M_c$ = 1.21 M$_{\odot}$, it is possible to observe the radiation emitted at $\tau$ = 17 minutes to coalescence. Hence the equation says that the larger the time to coalescence is, the smaller the masses involved are \footnote{A useful exercise to prove this is by applying the Kepler's law for different emitting frequencies and masses. Some interesting examples are given in \cite{mag}.}.\\
 Recalling Fig. \ref{spec}, the range of the frequencies of emission below 10 Hz lies almost entirely in the domain of the space-based detectors. Opening this frequency window would allow the ground-based detectors to access to a frequency bandwidth which has still not been investigated and would allow the detection from sources whose physics is still unknown. 
 
@@ -61,7 +61,7 @@ f_{obs} = f_{gw}/(1+z).
 
 \noindent
 The implication of this effect lies in a factor (1+z) multiplied to the masses involved \cite{mag}.\\
-An important consequence is that if the instrument could be able to detect in a broader range of lower frequencies, it could be possible to identify objects located at higher redshifts, i.e. more ancient, or apparent high masses increased by the cosmological distance \cite{yu}. Examples of these objects are Intermediate Mass Black Holes (IMBH) or stellar-mass BHs, whose nature and physics are still unknown.
+An important consequence is that if the instrument could detect in a broader range of lower frequencies, it is possible to identify objects located at higher redshifts, i.e. more ancient, or apparent high masses increased by the cosmological distance \cite{yu}. Examples of these objects are Intermediate Mass Black Holes (IMBH) or stellar-mass BHs, whose nature and physics are still unknown.
 
 \subsection{Multi-messenger astronomy and low frequencies}
 Multi-messenger astronomy is a branch of astronomy born with the discovery of the first gravitational wave. It has been seen that the signal of a gravitational wave can be followed up by observatories operating in other frequency bands (say, the electromagnetic bandwidth), to localize and study the source under several other points of view \footnote{A general overview about multi-messenger astronomy can be found in \cite{branchesi}. An interesting paper about a multi-messenger GW-source detection and its implications is \cite{multi}.}.\\

main/LIGO.tex

 \chapter{Interferometry and Advanced LIGO}
 \label{LIGO}
-Most of the work exposed in this thesis has been carried on in laboratories and on LIGO sites. In this chapter, I will briefly introduce interferometers and LIGO, and I will explain in detail only the structures of LIGO that have been the subject of study in this thesis work: this is essential to fully embrace the work exposed in Chapter \ref{CPSdiff} in particular, and in general for the devices described in the whole thesis. The information contained in this chapter will often be referred to throughout this thesis.
+Most of the work reported in this thesis has been carried out in laboratories and on LIGO sites. In this chapter, I will briefly introduce interferometers and LIGO, and I will explain in detail only the structures at LIGO that have been the subject of study in this thesis work: this is essential to fully embrace the work reported in Chapter \ref{CPSdiff} in particular, and in general for the devices described in the whole thesis. The information contained in this chapter will often be referred throughout this thesis.
 
 \section{Interferometric detectors}
 The interaction of gravitation waves with two objects moving along the x axis produces effects on their distance $d = x_2 - x_1$ and hence the effect of the gravitational waves can be measured by looking at the variation of the distance of the masses involved.\\
@@ -9,12 +9,12 @@ A method to do it is to measure the time light takes to travel from one mass to
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.5]{images/itf.png}
-\caption{Basic features of an interferometer.}
+\caption{Basic features of a Michelson interferometer.}
 \label{itf}
 \end{figure}
 
 \noindent
-As shown in Fig. \ref{itf}, an interferometer is an instrument where a laser beam of wavelength $\lambda$ is split into two beams which propagate in two perpendicular arms of the same length. At the end of each arm, a mirror reflects the beam back to be recombined with the other one. The recombined beam is then deviated to a photodiode.\\
+As shown in Fig. \ref{itf}, a laser interferometer is an instrument where a laser beam of wavelength $\lambda$ is split into two beams which propagate in two perpendicular arms of the same length. At the end of each arm, a mirror reflects the beam back to be recombined with the other one. The recombined beam is then diverted to a photodiode.\\
 If we consider the length of arms oriented to the x and y directions to be $L_x = L_y = L$, the power measured by the photodiode depends on the difference of path length between the two beams \cite{mag}:
 
 \begin{equation}
@@ -24,7 +24,7 @@ P_{out} = E^{2}_{0} \sin^2 [k(L_x - L_y)]
 
 \noindent
 where $E_{0}$ is the amplitude of the electric field generated by the laser source and k = $2\pi / \lambda$.\\
-We know that the effect of a gravitational wave is to modify the distance of two masses: in the case of the interferometer the path length difference in the arms is proportional to the gravitational wave amplitude $h$ \cite{mag}:
+We know that the effect of a gravitational wave is to modify the distance between two masses: in the case of the interferometer the path length difference in the arms is proportional to the gravitational wave amplitude $h$ \cite{mag}:
 
 \begin{equation}
 \centering
@@ -117,7 +117,7 @@ Noises can be of fundamental, technical and environmental origin. Fundamental no
 Technical noises arise from electronics, control loops, charging noise and other effects; environmental noises include seismic motion, acoustic and magnetic noises: these noises can be reduced once identified and carefully studied.\\
 
 \noindent
-This thesis focuses on the improvement of the seismic isolation system. Seismic motion is measured using inertial sensors which are placed on the suspension benches. The residual motion effects the stability of the resonant cavities and limits the sensitivity of the detector in the low frequency band. The goal is to provide solutions to reduce seismic motion and improve the detector sensitivity.
+This thesis focuses on the improvement of the seismic isolation system. Seismic motion is measured using inertial sensors which are placed on the suspension benches. The residual motion affects the stability of the resonant cavities and limits the sensitivity of the detector in the low frequency band. The goal is to provide solutions to reduce seismic motion and improve the detector sensitivity.
 
 \section{LIGO seismic isolation system}
 \label{ligosei}
@@ -135,7 +135,7 @@ The HAMs provide five levels of isolation, among which there is the Internal Sei
 \begin{figure}[h!]
 \centering
 \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 provides isolation from ground motion.}
+\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 provide isolation from ground motion.}
 \label{ham}
 \end{figure}
 
@@ -164,7 +164,7 @@ Part of the work presented in this thesis focussed on the enhancement of the per
 Active isolation implies a sensing system of the noise to reduce and a control system to compensate the disturbance. Each platform includes relative position sensors, inertial sensors and actuators, working in all degrees of freedom.\\
 
 \noindent
-The control loop of a generic ISI stage on the X degree of freedom is  simplified in the block diagram in Fig. \ref{control}. The platform motion is the sum of the input disturbance and the contribution from the control signal and it is measured by relative position and inertial sensors. This motion is then low- and high-passed via filters suitably built to fit the requirements and tuned to obtain the best performance combining the best results of both filters. This technique is called \textit{blending}, and the frequency where the relative and the inertial sensors contribute at their best is called the \textit{blend frequency}. The result of this blend is called the \textit{super sensor}. The output of the super sensor feeds the feedback loop, where the actuators close the loop \footnote{A general overview of control loops theory is exposed in Appendix B}.\\
+The control loop of a generic ISI stage on the X degree of freedom is  simplified in the block diagram in Fig. \ref{control}. The platform motion is the sum of the input disturbance and the contribution from the control signal and it is measured by relative position and inertial sensors. This motion is then low- and high-passed via filters suitably built to fit the requirements and tuned to obtain the best performance combining the best results of both filters. This technique is called \textit{blending}, and the frequency where the relative and the inertial sensors contribute at their best is called the \textit{blend frequency}. The result of this blend is called the \textit{super sensor}. The output of the super sensor feeds the feedback loop, where the actuators close the loop \footnote{A general overview of control loops theory is given in Appendix B}.\\
 The sensor correction loop takes the ground motion signal from an inertial instrument, filtering it before adding it to the relative sensor signal. This filter is needed because the sum of the motions from the ground inertial and the relative sensors can in principle provide a measurement of the absolute motion of the platform. However, the ground sensors are affected by low frequency noise and need to be suitably filtered. 
 
 \begin{figure}[H]

main/laserstab.tex

@@ -27,7 +27,7 @@ As the name reminds, the 6D investigates the motion of a reference mass in all 6
 \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 compact interferometers, detecting the relative motion between the mass and the platform; actuators apply corrections to the platform and the whole apparatus is in vacuum.}
+\caption[6D design.]{Sketch of the 6D device (Figure taken from \cite{6d}). The working principle is based on an isolated, suspended reference mass which is monitored by compact interferometers, detecting the relative motion between the mass and the platform; actuators apply corrections to the platform and the whole apparatus is in vacuum.}
 \label{6d}
 \end{figure}
 \noindent
@@ -321,7 +321,7 @@ The results of this experiment showed that it is possible to stabilize the frequ
 interferometers of the same type that are used inside the 6D sensor. With this technology, we managed to reach a
 frequency stabilization of 3.6 $\times$ 10$^3$ Hz/$\sqrt{Hz}$ at 1 Hz, without the need of installing the
 prototype in vacuum.\\
-This is already a promising result, but not yet sufficient for the requirements of 6D, especially below 1 Hz. The results showed that we are not limited by loop gain, that acoustic noise and vibration are importnt noise sources, that intensity noise and frequency-to-intensity coupling limit performance, and that both HoQIs show different coupling to these effects. The alternative test showed an indirect measurement of laser stabilization, because the out-of-loop HoQI improved its output signal of about one order of magnitude when the laser was modulated by the controller filter. This test highlighted that the frequency stabilization through the heterodyne detection depends on the stability and robustness of the HoQIs. A concrete plan for next tests is to place the setup in vacuum: this will suppress all the external noises and will possibly highlight the intrinsic issues of HoQIs.\\
+This is already a promising result, but not yet sufficient for the requirements of 6D, especially below 1 Hz. The results showed that we are not limited by loop gain, that acoustic noise and vibrations are important noise sources, that intensity noise and frequency-to-intensity coupling limit performance, and that both HoQIs show different coupling to these effects. The alternative test showed an indirect measurement of laser stabilization, because the out-of-loop HoQI improved its output signal of about one order of magnitude when the laser was modulated by the controller filter. This test highlighted that the frequency stabilization through the heterodyne detection depends on the stability and robustness of the HoQIs. A concrete plan for next tests is to place the setup in vacuum: this will suppress all the external noises and will possibly highlight the intrinsic issues of HoQIs.\\
 This test has already been considered as a possibility during the experiment design: the HoQI setups can be placed inside the 6D vacuum tank independently from the rest of the components of the stabilization breadboard and electronics.
 
 

main/main.tex

@@ -49,9 +49,9 @@ Chiara Di Fronzo
 \chapter{Abstract}
 
 The discovery of gravitational waves opened a new way to look at the Universe and offered new opportunities to shed light on the still unknown aspects of physical sciences. The work presented in this thesis wants to give a contribution to the development of this new type of research: the author chose to focus on the improvement of the instruments able to detect the gravitational waves. This field is important to make the detectors more sensitive, in order to see more gravitational-wave sources and help to complete the mosaic of the astrophysical science. In particular, the detectors currently in use are interferometers, which are especially blind in a range of frequency below 30 Hz: this affects the chance to detect sources emitting in this frequency band.\\ 
-This lack of sensitivity is mainly due to seismic motion, and the work exposed in this thesis focussed on new techniques to lower this noise source and allow the instruments to be sensitive below 30 Hz.\\
-During the studies, the development and test of devices able to to potentially reduce the seismic motion have been performed, such as optical levers for tilt motion reduction and laser stabilization for low frequency readout; a new concept of the seismic system on one of the interferometers (LIGO) has also been proposed.\\
-The optical levers can in principle reduce tilt motion below 1 Hz; the use of capacitive position sensors in a new software configuration for LIGO can help to suppress ground motion by a factor of 3 in order of magnitude below 0.1 Hz. A competitive frequency stabilization to 3.6 $\times$ 10$^3$ Hz/$\sqrt{Hz}$ at 1 Hz for readout at low frequency is possible with a compact and easy to handle setup. These results are promising to provide suppression of the seismic motion in the bandwidth of interest and show that it is possible for a ground-based instrument to be seismically more stable and able to detect gravitational waves where it is now forbidden. 
+This lack of sensitivity is mainly due to seismic motion, and the work presented in this thesis focussed on new techniques to lower the noise sources and allow the instruments to be sensitive below 30 Hz.\\
+During the studies, the development and test of devices capable of potentially reducing the seismic motion have been performed, such as optical levers for tilt motion reduction and laser stabilization for low frequency readout; a new concept of the seismic system on one of the interferometers (LIGO) has also been proposed.\\
+The optical levers can in principle reduce tilt motion below 1 Hz; the use of capacitive position sensors in a new software configuration for LIGO can help to suppress ground motion by a factor of 3 in order of magnitude below 0.1 Hz. A competitive frequency stabilization to 3.6 $\times$ 10$^3$ Hz/$\sqrt{Hz}$ at 1 Hz for readout at low frequency is possible with a compact and easy to handle setup. These results are promising to provide suppression of the seismic motion in the bandwidth of interest and show that it is possible for a ground-based instrument to be seismically more stable and capable of detecting gravitational waves where it is now forbidden. 
 
 %\clearpage
 \chapter*{}
@@ -64,7 +64,7 @@ The optical levers can in principle reduce tilt motion below 1 Hz; the use of ca
 \clearpage
 
 \chapter{Acknowledgements}
-I am particularly grateful to Dr. Conor Mow-Lowry and the University of Birmingham, for giving me the opportunity and the funding to join the Gravitational waves group and contribute to the development of exciting science. This was also possible thanks to the support of the Royal Astronomical Society and the Institute of Physics, which allowed me to take part to conferences and workshops abroad.\\
+I am particularly grateful to Dr. Conor Mow-Lowry and the University of Birmingham, for giving me the opportunity and the funding to join the Gravitational waves group and contribute to the development of exciting science. This was also possible thanks to the support of the Royal Astronomical Society and the Institute of Physics, which allowed me to take part in conferences and workshops abroad.\\
 During my stay at LIGO Hanford site, I need to warmly thank Caltech for providing me accommodation and travel: this experience was very important for my studies. Thanks to the Albert Einstein Institute (Hannover) for providing their facilities for my tests.\\
 The completion of the work presented in this thesis would not have been possible without the action of the UoB, which accepted my application for an extension of my studies: the lockdown in 2020 stopped my lab work and the support of the UoB has been crucial to accomplish my project in the best way.\\
 
@@ -168,11 +168,14 @@ UoB = University of Birmingham\\
 \include{CPSdiff}
 \include{laserstab}
 
+\chapter*{Conclusions}
+
 \appendix
 \include{A}
 \include{B}
 \include{C}
 
+
 \backmatter
 
 \begin{thebibliography}{}

main/oplevs.tex

-\chapter{Optical Levers for tilt motion reduction}
+\chapter{Optical levers for tilt motion reduction}
 \label{oplevs}
 
 The sensors dedicated to measure the seismic motion need to account for horizontal, vertical and tilt displacements in all degrees of freedom in order to be efficient: the technology for their improvement is currently pushing and competing on sensing as low seismic motion as possible. On an interferometric detector, seismic motion affects the stabilization of the supports where the optics lie. This produces unwanted noise at low frequencies (< 30 Hz), which reduces the sensitivity of the detector.\\
@@ -8,7 +8,7 @@ The content of this chapter has been re-adapted from my MCA report \cite{mca}. A
 \section{Inertial sensors affected by tilt-coupling}
 There are many contributions affecting aLIGO sensitivity at low frequency. One of the most investigated is the tilt of HAM vacuum chamber of ISI platforms, which dominates above 1 Hz \cite{lantz}.\\
 For the rotational degrees of freedom, getting a good estimate of ground motion is not trivial because no rotational sensors capable of measuring the ground motion in rotation at low frequencies have been installed yet on aLIGO ISI platforms \cite{cooper}.\\
-However, there could be a possible way to measure angular displacements of the benches very precisely (10$^{-12}$ rad/$\sqrt{Hz}$) and to actively control them: this could be done by optical levers. In the following, we will analyse the main contributions to tilt motion and we will see how optical levers could be useful to suppress this motion.
+However, there is a possible way to measure angular displacements of the benches very precisely (10$^{-12}$ rad/$\sqrt{Hz}$) and to actively control them: this could be done by optical levers. In the following, we will analyse the main contributions to tilt motion and we will see how optical levers could be useful to suppress this motion.
 
 \begin{figure}[h!]
 \centering
@@ -465,7 +465,7 @@ The prototype and its own pre-amplifying electronics have been built at UoB (Fig
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.55]{images/OpLev20.jpg}
-\caption[Photo of the optical lever prototype]{Photo of the optical lever prototypes as built at UoB. In this picture, the devices are not connected to electronics. Each platform hosts a laser source and a sensor. Each sensor is covered by a tube to avoid spurious light on the active area, and the focusing lens in placed at the suitable distance from it.}
+\caption[Photo of the optical lever prototype]{Photo of the optical lever prototypes as built at UoB. In this picture, the devices are not connected to electronics. Each platform hosts a laser source and a sensor. Each sensor is covered by a tube to avoid spurious light on the active area, and the focusing lens is placed at the suitable distance from it.}
 \label{oplev20}
 \end{figure}
 
@@ -575,7 +575,7 @@ However, with UoB pre-amp the measurements do not seem consistent with what we e
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.25]{images/LVDT_T.PNG}
-\caption[Different pre-amps test: bench LVDT motion]{Bench motion long z axis during the vacuum pump from 30 mbar to at 5 $\times$ 10$^{-3}$ mbar pressure conditions. Pressure has been set at 30 mbar at first stage to let temperature to stabilize faster. The two-step vacuum procedure was a good idea: it accelerated the lowering of temperature by two times.}
+\caption[Different pre-amps test: bench LVDT motion]{Bench motion long z axis during the vacuum pump from 30 mbar to 5 $\times$ 10$^{-3}$ mbar pressure conditions. Pressure has been set at 30 mbar at first stage to let temperature to stabilize faster. The two-step vacuum procedure was a good idea: it accelerated the lowering of temperature by two times.}
 \label{LVDT_FIN}
 \end{figure}