@@ -41,11 +41,11 @@ So we have the following situation:
\begin{equation}
\centering
m\ddot{x} = -kx - b\dot{x} + F_{tilt},
m\ddot{x} = -kw - b\dot{w} + F_{tilt},
\end{equation}
\noindent
where $x$ is the direction of motion, k and b constants.\\
where $x$ is the direction of motion, k is stiffness and b is damping of the spring.\\
We assume that the angle is very small, in such a way $\sin\theta$$\simeq$$\theta$. So the equations of motion are:
\begin{equation}
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@@ -59,7 +59,7 @@ Since we want to study the system in the frequency domain, we substitute $w = x-
\begin{equation}
\centering
m(W+Y)s^2 = -bWs -kW + mg\theta.
m(W+Y)s^2 = -bWs -kW + mg\Theta.
\end{equation}
\noindent
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@@ -67,15 +67,15 @@ After some manipulations, we obtain:
\begin{equation}
\centering
W = \frac{ms^2}{ms^2 + bs +k}\left(-y + \frac{g}{s^2}\theta\right)
W = \frac{ms^2}{ms^2 + bs +k}\left(-Y + \frac{g}{s^2}\Theta\right).
\end{equation}
\noindent
Remembering that s = i$\omega$ in a steady-state situation, we have:
where W is the measured quantity, Y and Theta are the quantities we are interested in. Remembering that s = i$\omega$ in a steady-state situation, we have:
@@ -91,7 +91,7 @@ If we know the size of our system, it is possible to calculate the angle $\thet
\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.
When the seismometer is tilted, its sensitivity to angles increases as $g /\omega^2$. 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, referring to Fig. \ref{v}, in presence of tilt we have:
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@@ -101,7 +101,7 @@ If we are dealing with vertical sensor, referring to Fig. \ref{v}, in presence o
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.
Since $cos\theta$$\sim$ 1 - $\theta^{2}/2$, the $\theta^{2}$ is second order and negligible for changes of angles.
\begin{figure}[h!]
\centering
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@@ -159,7 +159,7 @@ Beyond them, we have to consider the relative intensity noise (RIN), due to inst
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.
\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.\\
The Quadrant PhotoDiodes (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}.\\
\begin{figure}[h!]
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@@ -351,20 +351,20 @@ With the laser wavelength $\lambda$ = 1064 nm and an input power P$_0$ = 1 mW th
\subsection{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:
The other, important noise affecting the measurements is the thermal noise due to the transimpedance resistor of the photodiode R. It is given by:
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 0.2 A/W.\\
To compute R, consider that the output voltage is given by:
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 0.2 A/W.\\
To compute R, consider that the output voltage from the photocurrent is given by:
\begin{equation}
\centering
V = P_0 \rho R,
V_{ph} = P_0 \rho R,
\end{equation}
\noindent
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@@ -373,7 +373,7 @@ So, considering T = 300 K at room temperature, we have:
\begin{equation}
\centering
T_{h}=1.47 \times 10^{-12}\frac{W}{\sqrt{Hz}}.
T = 1.47 \times 10^{-12}\frac{W}{\sqrt{Hz}}.
\end{equation}
\subsection{Resolution}
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@@ -430,8 +430,8 @@ The differential Z motion is given by the difference between the z motion measur
The contribution from the GS13 needs to be manipulated to give rad/$\sqrt{Hz}$: this is done converting to measured velocity to displacement.
\noindent
All the noise sources are divided by the lever arm, in order to obtain an estimation in radians.\\
A low pass filter (LP) at 1 Hz is applied to the BRS motion and a high pass filter (HP) is applied to the $\Delta Z$ motion at 0.1 Hz.\\
Summing all the noise elements in quadrature, we have the total noise performance of the optical lever, which is shown in the plot in Fig. \ref{oplevnoise}:
A low pass filter (LP) at 1 Hz is applied to the BRS motion and a high pass filter (HP) is applied to the $\Delta Z$ motion at 0.1 Hz: we use the filters to estimate the motion, since we know that the measured signals are limited by noise where we are applying the filters.\\
Summing all the noise elements in quadrature, we have the total noise performance of the optical lever, which is shown in the plot in Fig. \ref{oplevnoise}: the plot shows that the improvements that optical levers can give are limited to a restricted range of frequencies and that they suffer the differential Z motion contribution below 0.1 Hz. Given the technical difficulties of developing and installing optical levers (see following sections), the effort would only be worthwhile if the Z motion was improved via better sensors.
\caption[Optical lever noise budget]{Optical lever total noise budget. This plot shows that with the noises considered the optical levers can be helpful in a restricted range of frequencies between 0.1 Hz and 1 Hz, with respect to CPS. However, at frequencies below 0.1 Hz the optical levers suffer from the contribution of the GS13s on the ISI, compared to the CPS.}
\caption[Optical lever noise budget]{Optical lever total noise budget. This plot shows that our model predicts significant signal-to-noise ratio improvement from $\sim$0.15 Hz to 1 Hz (Vs CPS), but we are limited by the differential Z motion over a wide band.}
\label{oplevnoise}
\end{figure}
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@@ -583,7 +583,7 @@ However, with UoB pre-amp the measurements do not seem consistent with what we e
\caption[Different pre-amps test: bench motion]{Motion of Central bench measured by L4Cs and accelerometers, with AEI and UoB pre-amps. These plots highlights that the UoB electronics is not performing well (probably it is saturating) below 10 Hz.}
\caption[Different pre-amps test: bench motion]{Motion of Central bench measured by L4Cs and accelerometers, with AEI and UoB pre-amps. These plots highlights that the UoB electronics is not performing well (due to loop leakage) below 10 Hz.}
