@@ -6,18 +6,18 @@ During the first year of my PhD studies, I investigated the use of optical lever
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{poster2}.
\section{Inertial sensors affected by tilt-coupling}
There are many contributions affecting aLIGO sensitivity at low frequency. One of the most investigated is the tilt of HAM vacuum chamber of ISI benches, 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 \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.
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.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/HAMoplev.PNG}
\includegraphics[scale=0.9]{images/HAMoplev.PNG}
\caption[Example of tilt-coupling contributions at LHO]{Plot of the contributions to the Suspension point L motion at LHO HAM5. The pitch (RX) contribution dominates above 1\,Hz (Figure taken from \cite{lantz}).}
\end{figure}
\newpage
\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.\\
One of the most important problems, in order to achieve good isolation, is the sensitivity of the horizontal sensors to rotation. If we could independently measure the rotation, we could calculate the true translation motion.\\
\begin{figure}[h!]
\centering
...
...
@@ -25,8 +25,9 @@ The most important problem, in order to achieve good isolation, is the sensitivi
\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:
Referring to Fig. \ref{a}, when a rotation around the center of mass of the sensor occurs, an additional term to the equation of motion (F$_{tilt}$) appears:
\begin{equation}
\centering
...
...
@@ -43,22 +44,25 @@ So we have the following situation:
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:
where $x$ is the direction of motion, k and b constants.\\
We assume that the angle is very small, in such a way $\sin\theta$$\simeq$$\theta$. So the equations of motion are:
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 $w = x-y$:
where $x$ is the displacement of the mass and $y$ is the displacement of the support of the sensor.\\
Since we want to study the system in the frequency domain, we substitute $w = x-y$ and apply the Laplace transform:
\begin{equation}
\centering
m(W+Y)s^2 = -bWs -kW + mg\theta
m(W+Y)s^2 = -bWs -kW + mg\theta.
\end{equation}
\noindent
After some manipulations, we obtain:
\begin{equation}
\centering
W = \frac{ms^2}{ms^2 + bs +k}\left(-y + \frac{g}{s^2}\theta\right)
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...
@@ -86,7 +90,7 @@ Since we have a factor $\omega^2$ at the denominator, it has more contributions
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:
If we are dealing with vertical sensor, referring to Fig. \ref{v}, in presence of tilt we have:
\begin{equation}
\centering
...
...
@@ -114,12 +118,13 @@ When the optic is tilted by an angle $\theta$, we have the situation illustrated
\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.\\
What if we have both horizontal and vertical seismometers on the same bench, as on aLIGO? In this case, we have two instruments that are sensitive to horizontal and vertical ground motion at the same time. When the bench is tilted, they are tilted at the same time of the same angle, but they are not affected in the same way, as we have seen, but we are not able to deduce the tilt motion at low frequency because of the limitations given by the sensors noises.\\
\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.\\
If could measure both vertical and horizontal motions and decouple the contribution of the tilt for the horizontal one, we could know exactly the amount of corrections the actuators have to perform.\\
With optical lever systems we can measure the angle of the tilt, even if it is extremely small: in this way we could be able to directly measure the tilt angle $\theta$ and apply corrections to the horizontal sensor through a dedicated active system.\\
\noindent
The device described in this chapter should involve sensing and actuation for the seismic motion on aLIGO. The position device can not be set on the same bench where the other sensors are, because it would be affected by the same ground motion. So it has to be placed on another bench, at some distance L, and an actuation system is associated to it in order to adjust the tilt of the bench under exam. The 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}.
The device described in this chapter should involve sensing and actuation for the seismic motion on aLIGO. The position device can not be set on the same bench where the other sensors are, because it would be affected by the same ground motion. So it has to be placed on another bench, at some distance L, and an actuation system is associated to it in order to adjust the tilt of the bench under exam. The angular sensitivity increases with the distance L. Moreover, even the bench where the position device is set needs to be stable: another optical lever could be placed on it, with the associated actuation, mirroring the first one and keeping both platforms stable. A basic picture of the whole system is shown in Fig. \ref{z}.
\begin{figure}[h!]
\centering
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@@ -130,7 +135,7 @@ The device described in this chapter should involve sensing and actuation for th
\noindent
The purpose when thinking of interferometers is to help reducing the Rx motion on the HAM chambers that propagates into the suspensions.
\section{Noise budget}
\section{Experiment design}
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}.
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@@ -148,7 +153,7 @@ The translation coupling noise due to the motion of the platform where sensors a
\subsection{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 quadrant Q in Fig. \ref{j}.\\
When light is incident on the sensor, a photocurrent $I$ is detected by each quadrant Q in Fig. \ref{j}.\\
\begin{figure}[h!]
\centering
...
...
@@ -157,18 +162,18 @@ When light is incident on the sensor, a photocurrent I is detected by each quadr
\label{j}
\end{figure}
\noindent
The normalized coordinates (X, Y) for the beam's location are given by the following equations:\\
The normalized coordinates (X, Y) for the beam's location depend on the detected photocurrents and are given by the following equations:\\
\begin{equation}
\centering
X = \frac{(I2+I3)-(I1+I4)}{I1+I2+I3+I4}
X = \frac{(I_2+I_3)-(I_1+I_4)}{I_1+I_2+I_3+I_4}
\end{equation}
\begin{equation}
\centering
Y = \frac{(I1+I2)-(I3+I4)}{I1+I2+I3+I4}
Y = \frac{(I_1+I_2)-(I_3+I_4)}{I_1+I_2+I_3+I_4}
\end{equation}
\noindent
If a symmetrical beam is centred on the sensor, four equal photocurrents will be detected, resulting in null difference signals and, hence, the normalized coordinates will be (X, Y) = (0, 0). The photocurrents will change if the beam moves off center, producing difference signals that are related to the beam displacement from the center of the sensor.
If a symmetrical beam is centred on the sensor, four equal photocurrents will be detected, resulting in null difference signals and, hence, the normalized coordinates will be (X, Y) = (0, 0). The photocurrents will change if the beam moves off center, producing different signals that are related to the beam displacement from the center of the sensor.
%\subsection{Structure of the device}
%As we have seen, a QPD is formed by 4 photodiodes placed in such a way to form a circle. There are several way to build the circuit to convert the quadrant outputs into x and y position signals. In Fig. \ref{qpd} it is shown a schematic example of a circuit of the detector.
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@@ -184,7 +189,7 @@ If a symmetrical beam is centred on the sensor, four equal photocurrents will be
%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}).
\subsection{Spot position and displacement}
\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}
...
...
@@ -269,6 +274,9 @@ The same computation gives the result for the coordinate y:
In order to estimate the resolution of the device and provide an estimate of its performances, we need to account for the noises coming from the QPD and external sources.
\subsection{Photon shot noise}
\label{sn}
Because of the fact that the working principle of the QPD is based on tracking the motion of the centroid of power density, it is useful to compute the contribution of the shot noise.\\
...
...
@@ -287,7 +295,7 @@ the fractional fluctuation of the number of photons is then:
\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:
The fractional fluctuation of the input power $\sigma_p$is given by the fractional fluctuation of the number of photons:
\begin{equation}
\centering
...
...
@@ -306,7 +314,7 @@ For t=1 s:
\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:
With the laser wavelength $\lambda$ = 1064 nm and an input power P$_0$ = 1 mW that we chose for this experiment, we obtain:
\begin{equation}
\centering
...
...
@@ -342,8 +350,8 @@ To compute R, consider that the output voltage is given by:
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:
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:
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:
So, according to the block diagram in Fig. \ref{BD}, to obtain the angular measurement we have that:
@@ -380,13 +388,13 @@ This value is of the order of magnitude of the sensitivity of optical levers ant
In order to obtain a plot of the noise budget for the optical lever prototype, we need to take into account some more elements to add to the ones just computed:
\begin{itemize}
\item The motion along z of the platforms is used as noise: however, at low frequency, sensors are not sensitive to this motion so what we need is a differential motion between HAM chambers (say HAM4 and HAM5 for this derivation); the best estimation we have is the platform z motion measured by GS13s. This motion is given by channels of LIGO Livingston data;
\item The motion along z of the platforms is used as noise: however, at low frequency, most sensors are not sensitive to this motion so what we need is a differential motion between HAM chambers (say HAM4 and HAM5 for this derivation); the best estimation we have is the platform z motion measured by GS13s. This motion is given by channels of LIGO Livingston data;
\item The best performance of current tested optical levers is the one tested at the AEI and shown in Fig. 4.4 of reference \cite{sina};
\item The ground z motion of the chambers is given by the Beam Rotation Sensors (BRS) and used as noise source. This motion is taken from channels of LIGO Livingston data;
\item The ground $\theta$ motion of the chambers is given by the Beam Rotation Sensors (BRS) and used as noise source. This motion is taken from channels of LIGO Livingston data;
\item The Rx motion is given by the CPS on HAM4 and HAM5 and it is used for comparison with the optical lever performance. This motion is taken from channels of LIGO Livingston data.
\item The RX motion is given by the CPS on HAM4 and HAM5 and it is used for comparison with the optical lever performance. This motion is taken from channels of LIGO Livingston data.
