\chapter{Interferometry and Advanced LIGO}
\label{LIGO}
Most of the work exposed in this thesis has been physically done in laboratories and on LIGO sites. In this chapter I will briefly introduce interferometers and LIGO, and I will explain in details only the structures of LIGO that have been subject of study: 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, and I will often refer to the information contained in this chapter throughout the thesis.

\section{Interferometric detectors}
The interaction of gravitation waves with two objects moving along the x axis produces effects on their distance $d = x_2 - x_1$:

\begin{equation}
\centering
s \simeq d \left(1 + \frac{1}{2}h_+ \cos \left[\omega \left(t-\frac{z}{c}\right)\right]\right).
\end{equation}
\\
\noindent
So the effect of the gravitational waves can be observed on the distance of the masses involved. A method is to measure the time it takes for light to travel from one mass to the other: this is the basic principle of the \textit{interferometer}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/itf.png}
\caption{Basic features of an 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 recomposed beam is then deviated to a power sensor.\\
If we consider the length of arms oriented to the x and y directions to be $L_x = L_y = L$, the power measured depends on the difference of path length travelled by the two beams:

\begin{equation}
\centering
P_{out} = E^{2}_{0} \sin^2 [k(L_x - L_y)] 
\end{equation}

\noindent
where $E^{2}_{0}$ 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}:

\begin{equation}
\centering
\Delta L = \frac{1}{2}hL
\end{equation}

\noindent
and so the key feature of this detector is that the beam coming from the recombination of the two ones that travelled in the arms brings a phase difference:

\begin{equation}
\centering
\Delta \phi = \frac{4\pi h L}{\lambda}
\end{equation}

\noindent
which results in a variation of the power measured:

\begin{equation}
\centering
P_{out} = E^{2}_{0} \sin^2 [k(L_x - L_y) + \Delta \phi].
\end{equation}

\noindent
The amplitude of a gravitational wave is typically very small and corresponds to a variation of the arm length of the order of $\Delta L \sim 10^{-18}$ m. This means that, if we want to measure a considerable phase shift, the sensitivity of the instrument depends on the length of the arms.

\paragraph{Fabry-Perot cavities}
A useful method to increase the length of the arms is to make the laser beam travel back and forth into a cavity delimited by two mirrors, called \textit{Fabry-Perot cavity}: here, the optical path length will be longer, due to the multiple reflections, and the field amplitude will increase due to constructive interference. This process returns a longer arm length, proportionally to the quality factor of the cavity, which depends on the reflection coefficients of the two mirrors and it is called \textit{Finesse} (F):

\begin{equation}
\centering
L_{F} \propto L \frac{2 F}{\pi}
\end{equation}

\noindent
which gives a phase shift:

\begin{equation}
\centering
\Delta \phi_{F} = \frac{8hFL}{\lambda}.
\end{equation}

\noindent
The higher is F, the higher is the effective length of the cavity and higher is the measureble phase shift.\\

\section{Advanced LIGO}

The ambition of this work is to give a contribution to the improvement of one of the interferometric detectors in use at present time, based in the USA: the Advanced Laser Interferometric Gravitational-wave Observatory (aLIGO).\\

\noindent
The configuration of aLIGO is shown in Fig. \ref{aligo}: it is a Michelson interferometer provided with Fabry-Perot cavities, power and signal recycling cavities and 4 km-long arms. The light source is a solid-state Nd:YAG laser of wavelength $\lambda$= 1064 nm, injected at a power between 5 - 125 W.\\ The instrument design is extremely intricate in its details: this thesis will provide technical information useful for the understanding of the work made on specific sections of aLIGO.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.9]{images/aligo.png}
\caption[Advanced LIGO layout]{Advanced LIGO configuration as proposed in \cite{ligo}. As a second generation detector, it is provided with two Fabry-Perot resonant cavities in the arms, delimited by the Input Test Masses (ITM) and the End Test Masses (ETM), and two additional dual recycling cavities: the power recycling (PR) and the signal recycling (SR), which optics are all suspended. Compensation Plates (CP) take care of thermal effects occurring when high powers pass through the ITMs; mode cleaners in input and outputs keep the selected mode in resonance.}
\label{aligo}
\end{figure}

\noindent
The fused silica mirrors at the end of each arm, called End Test Masses (ETM), are 34 cm $\times$ 20 cm in size and 40 kg in weight. A photodiode (PD) detects the power at the output. The optic able to split the injected beam into two parts along the arms is called Beam Splitter (BS) and it is placed at 45$^{\circ}$ between the arms.\\
There are two LIGOs in the USA, one in Hanford (WA) and one in Livingston (LA): some of the work that will be presented in the next chapters has been physically done in Hanford, in remote collaboration with Livingston team.

\subsection{LIGO sensitivity and noise sources}
The performance of LIGO in terms of how far in the Universe it can detect gravitational waves and from which sources depends on the sensitivity: this in turn depends on the quality of the technologies involved and on the limitation given by nature.
Fig. \ref{sens} shows the sensitivity of LIGO during the first observation run with the main noises shown.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/ligosens.png}
\caption[Advanced LIGO sensitivity curve]{Advanced LIGO sensitivity during the first observation run (O1) \cite{abb}. The sensitivity curve tells us that we can observe an event emitting gravitational waves of a given amplitude at a given frequency in an average observation time of 1 s. Since every source emits waves at a certain frequency and amplitude, lowering the curve means opening the viewing on currently hidden sources.}
\label{sens}
\end{figure}

\noindent
Advanced LIGO can be tuned to adjust the frequency band of detection: for each operational mode and detection frequency band there is a gravitational wave source candidate, typically mergers of neutron stars (NS-NS) and black holes (BH-BH).\\
Noise sources make LIGO blind in some frequency windows: technological limitations can be in principle overcome thanks to improvements in science, and this is what this present work is aiming to offer. The most important noise sources for LIGO are shown in the noise budget for LIGO Hanford (LHO) in Fig. \ref{lho}.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/LHO.png}
\caption[Advanced LIGO noise budget]{Noise budget of LIGO Hanford Obsevatory \cite{mar}.}
\label{lho}
\end{figure}

\noindent
Noises can be of fundamental, technical and environmental origin. Fundamental noises come from first principles, and they determine the ultimate design sensitivity of the instrument. They include thermal and quantum noise, and cannot be reduced without a major instrument upgrade. Quantum noises include shot noise of the sensors, causing power fluctuations, and radiation pressure forces, causing a physical displacement of the test masses. Thermal noise arises from the suspensions and the optical coatings and dominates in the 5-100 Hz frequency range.\\
Technical noises arise from electronics, control loops, charging noise and other effects that can be reduced once identified and carefully studied.\\
Environmental noises include seismic motion, acoustic and magnetic noises.\\

\noindent
This thesis focuses on the improvement of the seismic isolation system, which noises affect the inertial sensors placed on the suspension benches and the stabilization of the resonant cavities, which in turn limit the sensitivity of the detector in the low frequency bandwidth.

\section{LIGO seismic isolation system}
Every optic needs to be stable with respect to seismic motion, because movements in the mirrors will cause unwanted displacement of the laser beam on the optical surface, resulting in noise during the laser journey into the cavities and then at the output. The main mirrors (test masses and beam splitter) are suspended from a stabilized bench and every suspension chain is placed in vacuum chambers called \textit{Basic Symmetric Chamber} (BSC). The auxiliary optics are placed on optical benches enclosed in the \textit{Horizontal Access Module} (HAM) chambers.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/chambers.png}
\caption[Advanced LIGO vacuum system]{Schematic view of the vacuum chambers enclosing the optics \cite{mat}. There are 5 BSCs and 6 HAMs, for a total of 11 vacuum chambers for each LIGO. Each chamber provides a mixture of passive-active isolation from seismic motion, using pendulums, inertial sensors and hydraulic systems.}
\end{figure}

\noindent
The HAMs provide five levels of isolation, among which there is the Internal Seismic Isolation platform (HAM-ISI), where the auxiliary optics are placed, giving both passive and active isolation. A detailed drawing in Fig. \ref{ham} shows the design of a HAM chamber.

\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 gives isolation from ground motion.}
\label{ham}
\end{figure}

\noindent
The BSCs have a similar design as the HAMs, but they have two stages of ISI to support the suspensions isolating the test masses (Fig. \ref{bsc}).

\begin{figure}
\centering
\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}

\paragraph{Stabilizing the ISI}
Part of the work presented in this thesis focussed on the improvement of the performances of the active isolation system of the ISIs of both BSC and HAM chambers.\\
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; then this motion is low- and high-passed via filters suitably built to fit the requirements and tuned to obtain the best performances 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 \textit{blend frequency}. The result of this blend is called \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 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!]
\centering
\includegraphics[scale=0.7]{images/control.png}
\caption[Control loop for a generic HAM-ISI]{Control loop of a generic HAM-ISI platform. Similar block diagrams can be applied for BSC-ISI platforms, including relative position sensors between the two stages of ISIs. \textbf{Green:} there is an inertial sensor measuring the ground motion along the x axis (GNDx), a Capacitive Position Sensor (CPS) measuring relative motions between the platform and the ground. Rotational sensors take care of tilt motion and GS13 are seismometers measuring seismic motion. Tilt and GS13 sensors are both placed on the platform. \textbf{Blue:} the Sensor Correction (SC) filter is typically a Finite Impulse Response (FIR) designed to provide required magnitude and phase match at 100 mHz (where isolation is needed). High- and low-pass filters (LP and HP) manipulate the signals from the low and high frequency sensors and are blended to form the super sensor, which output is sent to the control loop in \textbf{pink}. The overall corrected signal is then sent to the plant (\textbf{yellow}), which represents the processing phase for platform motion actuation.}
\label{control}
\end{figure}

\section{LIGO Length Sensing and Control}
Length sensing and control (LSC) is a crucial feature of LIGO because the cavities need to be stable in presence of resonance as long as possible. This requires feedback controls between optical resonators and low-noise sensing systems to avoid noise coupling into the gravitational-wave readout.\\
There are several resonant cavities involved in this scheme, all important to guarantee the best performances on the sensitivity of the instrument. As we saw in Fig. \ref{aligo}, the resonators are the two Fabry-Perot cavities in the arms, the power recycling and the signal recycling cavities. The Fabry-Perot ones assure a higher sensitivity thanks to the beam bouncing into the cavity multiple times, increasing the time spent by the light into the arms and then the interaction time with a gravitational wave. The power recycling cavity is used to recover losses from power in the injection bench due to light reflected back to the laser source: this helps to increase the power travelling in each arm. The signal recycling cavity is placed at the output of the detector and is used to tune the detector to a specific bandwidth of observation.\\

\noindent
The main disturbance affecting the stabilization of the resonators is the ground motion, acting on the position of the optics and that can not be reduced by the passive isolation systems below 1 Hz. This means that an active isolation and a feedback control are required.\\

\noindent
The most important cavity lengths to keep stable are highlighted in Fig. \ref{lsc}; each length path between optics contributes to a specific signal monitored to keep the cavities in resonance. The signals are the Signal Recycling Cavity Length (SRCL), the Power Recycling Cavity Length (PRCL), the MICHelson (MICH), the Common Arm length (CARM) and the Differential Arm Length (DARM) and are described by the following relations between lengths:

\begin{equation*}
DARM = \frac{L_x - L_y}{2}
\end{equation*}

\begin{equation*}
CARM = \frac{L_x + L_y}{2}
\end{equation*}

\begin{equation*}
MICH = \frac{(l_x - l_y)}{2}
\end{equation*}

\begin{equation*}
PRCL = l_p + \frac{l_x + l_y}{2}
\end{equation*}

\begin{equation*}
SRCL = l_s + \frac{(l_x - l_y)}{2} = l_s + MICH
\end{equation*}
\\
\noindent
In particular, DARM is exactly the gravitational wave signal and thus the most important to monitor and to keep stable.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.55]{images/lsc.png}
\caption[LIGO LSC scheme.]{Cavity lengths involved in the sensing and feedback control for stabilization of resonators. The PRM is the Power Recycling Mirror, which summarize the power reclycling setup including three mirrors inside two chambers. The SRM is the same summary for the Signal Recycling.}
\label{lsc}
\end{figure}

\noindent
During the time at LIGO Hanford, some of the work has been focussed on the optimization of the time spent by cavities in resonance, using a new concept based on the communication between the optics and the platforms where they are placed.\\

\noindent
As we will see, time in stable mode is crucial to assure higher chances of detection of gravitational-wave candidates and small disturbances during the operational mode can compromise the detector while observing, losing stabilization (locking). This means that operators need to spend time to lock the instrument again and reset it in observing mode, time that is precious and that could instead be spent detecting events.\\
This work in particular intends to give a contribution to the improvement of the sensitivity and stabilization of LIGO at low frequencies.
