%INTERAZIONE COL DETECTOR E LAVORO A POTENZA ZERO (DARK FRINGE)\\
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:
The amplitude of a gravitational wave is typically very small and corresponds to a variation of the arm length of the order of $\Delta L \sim10^{-18}$ 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
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@@ -34,7 +95,7 @@ The configuration of aLIGO is shown in Fig. \ref{aligo}: it is a Michelson inter
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.
\section{LIGO sensitivity and noise sources}
\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.
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@@ -154,6 +215,3 @@ During the time at LIGO Hanford, some of the work has been focussed on the optim
\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.
