corrections

main/oplevs.tex

@@ -33,7 +33,7 @@ Referring to Fig. \ref{a}, when a rotation around the center of mass of the sens
 \centering
 F_{tilt} = mg\sin\theta
 \end{equation}
-\bigskip
+
 \noindent
 where m is the mass and g = 9.8 m/s$^2$ is the gravitational acceleration.\\
 \noindent
@@ -43,6 +43,7 @@ So we have the following situation:
 \centering
 m\ddot{x} = -kx - b\dot{x} + F_{tilt},
 \end{equation}
+
 \noindent
 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:
@@ -51,6 +52,7 @@ We assume that the angle is very small, in such a way $\sin \theta$ $\simeq$ $\t
 \centering
 m\ddot{x} = -b(\dot{x} - \dot{y}) - k(x-y) + mg\theta,
 \end{equation}
+
 \noindent 
 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:
@@ -75,6 +77,7 @@ Remembering that s = i$\omega$ in a steady-state situation, we have:
 \centering
 W(\omega) = \frac{-m\omega^2}{-m\omega^2 + ib\omega +k} \left(-y - \frac{g}{\omega^2}\theta\right).
 \end{equation}
+
 \noindent
 The relative sensitivity to translation and tilt are included in the second term in brackets. We expected this result, as the general one is that, for a horizontal seismometer, the ratio of the sensitivity to rotation (seismometer signal per radian of angle) to the sensitivity to horizontal motion (seismometer signal per meter of translation) at a particular frequency $\omega$ is:
 
@@ -82,6 +85,7 @@ The relative sensitivity to translation and tilt are included in the second term
 \centering
 \frac{rotation \; response}{translation \; response} = \frac{g}{\omega^2}.
 \end{equation}
+
 \noindent
 If we know the size of our system, it is possible to calculate  the angle $\theta$.\\
 \noindent
@@ -117,6 +121,7 @@ When the optic is tilted by an angle $\theta$, we have the situation illustrated
 \caption{Concept of the optical lever working principle: when the optic is tilted by a known angle, the displacement is detected by the photodiode.}
 \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, but we are not able to deduce the tilt motion at low frequency because of the limitations given by the sensors noises.\\
 \noindent
@@ -132,8 +137,9 @@ The device described in this chapter should involve sensing and actuation for th
 \caption{Basic principle of the optical lever used for sensing and actuation for seismic isolation.}
 \label{z}
 \end{figure}
+
 \noindent
-The purpose when thinking of interferometers is to help reducing the Rx motion on the HAM chambers that propagates into the suspensions.
+The purpose when thinking of interferometers is to help reducing the RX motion on the HAM chambers that propagates into the suspensions.
 
 \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.\\
@@ -146,6 +152,7 @@ Let's start from the block diagram of the system, in Fig. \ref{BD}.
 \caption{Block diagram of the optical lever system.}
 \label{BD}
 \end{figure}
+
 \noindent
 In the block diagram all the noises we have to deal with are described: the most relevant in terms of contributions are the shot and the thermal noises; then there are all the noises related to the electronics, like dark current, flicker and op-amp noises, usually given in the datasheet of the devices.\\
 Beyond them, we have to consider the relative intensity noise (RIN), due to instabilities in the laser intensity: this kind of noises reduces the signal-to-noise ratio, limiting the performances of the electronic transmission. This may be reduced by making the signal positions independent of illumination intensity.\\
@@ -161,6 +168,7 @@ When light is incident on the sensor, a photocurrent $I$ is detected by each qua
 \caption[QPD segmented details for beam position detection]{View of the segmented photodiode. Each quadrant Q receives a photocurrent which is the signal responsible for any displacement detection: depending on which quadrant is receiving more or less photocurrent, it is possible to derive the position of the beam onto the active area.}
 \label{j}
 \end{figure}
+
 \noindent
 The normalized coordinates (X, Y) for the beam's location depend on the detected photocurrents and are given by the following equations:\\
 
@@ -172,6 +180,7 @@ X = \frac{(I_2+I_3)-(I_1+I_4)}{I_1+I_2+I_3+I_4}
 \centering
 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 different signals that are related to the beam displacement from the center of the sensor.
 
@@ -196,6 +205,7 @@ At the light of what we have seen about QPDs, we have to compute where the beam
 \centering
 I(x) = \frac{P_0}{\pi w_{x}^2}e^{-2\left(\frac{x}{w_{x}}\right)^2},
 \end{equation}
+
 \noindent
 for coordinate x, the same for y; w$_{x}$ is the beam size (radius) in x direction and represents the distance from the x axis to which the amplitude reduces by 1/e and the intensity by 1/e$^2$; P$_0$ is the input power.\\
 \noindent
@@ -205,6 +215,7 @@ If we want to obtain the signal in terms of power, we should integrate the Gauss
 \centering
 P_{x} = \frac{P_0}{\pi w_{x}^2} \int^R_0 e^{-2\left(\frac{x}{w_{x}}\right)^2} dx,
 \end{equation}
+
 \noindent
 where R is the radius of the detector.\\
 The integral of the Gaussian function is the Error Function, defined as:
@@ -213,6 +224,7 @@ The integral of the Gaussian function is the Error Function, defined as:
 \centering
 {erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} dt.
 \end{equation}
+
 \noindent
 So we have:
 
@@ -220,6 +232,7 @@ So we have:
 \centering
 P_{x} = \frac{P_0}{\pi w_{x}^2} \int^R_0 e^{-2\left(\frac{x}{w_{x}}\right)^2} dx = \frac{P_0}{\pi w_{x}} \frac{\sqrt{\pi}}{2} \sqrt{2} w_{x} {erf}(x).
 \end{equation}
+
 \noindent
 The first term of the Taylor expansion of the error function is erf($x$) $\approx$ $\frac{2}{\sqrt{\pi}}x$, so we have:
 
@@ -259,6 +272,7 @@ Using the linear approximation, the displacement in x is given by:
 \centering
 \Delta x  \approx  \frac{\pi w_{x}}{\sqrt{2}P_0} \Delta P_{x};
 \end{equation}
+
 \noindent
 and the ratio between the variation of the power and the displacement in the x direction is given by:
 
@@ -266,6 +280,7 @@ and the ratio between the variation of the power and the displacement in the x d
 \centering
 \frac{\Delta P_{x}}{\Delta x} \approx  \sqrt{2}\frac{P_0}{\pi w_{x}} [{W}/{m}].
 \end{equation}
+
 \noindent
 The same computation gives the result for the coordinate y:
 
@@ -287,6 +302,7 @@ If P$_{0}$ is the input power and $\omega$ is the frequency, the number of photo
 \centering
 <N> = \frac{P_{0}t}{\hbar \omega};
 \end{equation}
+
 \noindent
 the fractional fluctuation of the number of photons is then:
 
@@ -294,6 +310,7 @@ the fractional fluctuation of the number of photons is then:
 \centering
 \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 $\sigma_p$ is given by the fractional fluctuation of the number of photons:
 
@@ -327,7 +344,6 @@ With the laser wavelength $\lambda$ = 1064 nm and an input power P$_0$ = 1 mW th
 %\caption{Plot of the shot noise with respect to a given range of frequency.}
 %\end{figure}
 
-\bigskip
 
 %\noindent
 %APPROFONDISCI FACCENDA DEL QUADRANTE DIVISO!\\
@@ -341,16 +357,18 @@ The other, important noise affecting the measurements is the thermal noise due t
 \centering
 V_{th} = \sqrt{\frac{4K_{B}T}{R}} \frac{A}{\sqrt{Hz}},
 \end{equation}
+
 \noindent
-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 0.77 A/W.\\
+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:
 
 \begin{equation}
 \centering
 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 $.\\
+and because the output voltage is limited by the range imposed by the interferometer itself, i.e. [+10, -10] V, we have that R = 5 $\times$ 10$^4$ $\Omega $.\\
 So, considering T = 300 K at room temperature, we have:
 
 \begin{equation}
@@ -381,6 +399,7 @@ So, according to the block diagram in Fig. \ref{BD}, to obtain the angular measu
 \centering
 \alpha = 3 \times 10^{-12} \frac{rad}{\sqrt{Hz}}.
 \end{equation}
+
 \noindent
 This value is of the order of magnitude of the sensitivity of optical levers anticipated earlier.
 
@@ -408,7 +427,7 @@ The differential Z motion is given by the difference between the z motion measur
 \end{equation}
 
 \noindent
-GS13 motion of both chambers needs to be re-calibrated, because it does not take into account the low frequency range.\\
+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.\\
@@ -422,12 +441,12 @@ OpLev_{noise} = \sqrt{(OpLev_{AEI})^2 + (\theta^{BRS}_g)^2 + (Thermal)^2 + (Shot
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.3]{images/oplevsum.png}
-\caption[Optical lever noise budget]{Optical lever total noise budget.}
+\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.}
 \label{oplevnoise}
 \end{figure}
 
 \section{Design of the prototype}
-The optical design has been simulated, taking into account some general constraints of the sensor: generally, the QPD diameter is around 10 mm, so the beam size should not exceed 1-3 mm; gaps in quadrant photodiodes are of the order of tens $\mu$m. Moreover, it is ideal for the setup to be compact.\\
+The optical design has been simulated, taking into account some general constraints of the sensor: generally, the QPD diameter is around 10 mm, so the beam size should not exceed 1-3 mm; gaps in quadrant photodiodes are of the order of tens $\mu$m. Moreover, it is ideal for the setup to be compact: the prototype has been design to lie on a [75 mm $\times$ 36 mm] platform.\\
 \noindent
 The chosen light source is a 1064 nm wavelength fiber-coupled Nd:YAG solid-state laser. Because of the fact that the beam size impinging on the photodiode has to be around 1 mm, a fiber collimator is used at the fiber output, and a plano-convex lens is used to focus the beam at the photodiode. In this way, with the chosen collimator, the beam size at its output is 1.38 mm. This is considered the starting point for the free propagation of the laser beam. The use of the collimator ensures that the beam size enlargement after a length L of propagation is minimized: according to the simulated free propagation, after 10 m the beam size is 2.8 mm.
 \noindent
@@ -439,8 +458,9 @@ The focussing lens of focal length 150 mm is inserted 10 cm before the photodiod
 \caption{Basic sketch of the lever optical design (not in scale).}
 \label{syst}
 \end{figure}
+
 \noindent
-The prototype and its own pre-amplifying electronics have been built at UoB (Fig. \ref{oplev20}) and tested in air and in vacuum at the AEI. The purpose is to calibrate the prototype with the electronics from UoB in vacuum conditions and test the sensitivity of the device to angular displacements. This first step is necessary for a good sensing system characterization.
+The prototype and its own pre-amplifying electronics have been built at UoB (Fig. \ref{oplev20}) and tested in air and in vacuum at the AEI. The purpose is to calibrate the prototype with the electronics from UoB in vacuum conditions and test the sensitivity of the device to angular displacements. This first step is necessary for a good sensing system characterization.\\
 
 \begin{figure}[h!]
 \centering
@@ -461,6 +481,7 @@ The pin configuration of the QPDs has been reset because the AEI electronics is
 \item Q4: PIN 4 to PIN 7
 \item BIAS: PIN 5 to PIN 4
 \end{itemize}
+
 \noindent
 Two adaptor cables have been built to connect the UoB boxes to the QPDs with the new pin configurations.\\
 \noindent
@@ -479,7 +500,7 @@ Summarizing, the prototype is ready for the test with the specifications listed
 \begin{tabular}{c|lcl}
 Beam size at QPD  & w = 1 mm\\
 Power at output   & P = 3.5 mW\\
-Displacement      & $\Delta$ x = 2.22 $\times$ 10$^{-3}$\\
+Displacement      & $\Delta$ x = 2.22 $\times$ 10$^{-3}$ \\
 Lens focal lenght & F = 150 mm\\
 Shot noise                & SN = 75 nV/$\sqrt{Hz}$  \\
 Responsivity Si @ 1064 nm & $\rho$ = 0.2 A/W      \\
@@ -542,30 +563,31 @@ Some peaks at lower frequencies may be due to bench motion: if the assumption is
 The movement of South bench along z axis is used as a reference to monitor the bench adjustments with temperature variations. The variable under examination is displacement tested by a Linear Variable Displacement Transformer (LVDT).
 
 \subsection{Final vacuum set up}
-The pressure has been set at 5 $\times$ 10$^{-3}$ mbar. What we expect is to find no variations in terms of the peaks we think are due to power fluctuations. Variations in LVDT trend can be due to temperature stabilization and related variations of pitch and yaw are then due to the more stable bench conditions.\\
+The pressure has been set at 5 $\times$ 10$^{-3}$ mbar. What we expect is to find no variations in terms of the peaks we think are due to power fluctuations. Variations in LVDT trend can be due to temperature stabilization and related variations of pitch and yaw are then due to the more stable bench conditions (Fig. \ref{LVDT_FIN}.\\
 
-\begin{figure}[H]
+\noindent
+In this conditions, also the signals from the L4C seismometers and accelerometers (Watt's Leakage) placed on the Central bench have been measured (Fig. \ref{central}). The plots with the UoB electronics show that there is some leakage below 10 Hz, probably due to saturation, in the measurement of the accelerometers.\\
+
+\noindent
+QPD performances are shown in the plots \ref{qpd_fin}. With AEI boxes we had expected results: no variations in the power fluctuation peaks and expected behaviour of pitch and yaw.\\
+However, with UoB pre-amp the measurements do not seem consistent with what we expected: we think that some non-linearities in UoB pre-amp could be the cause of the problem. This is still under investigation at UoB.
+
+\begin{figure}[h!]
 \centering
-\includegraphics[scale=0.3]{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 improved the temperature gradient by two times faster.}
+\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.}
 \label{LVDT_FIN}
 \end{figure}
 
-\noindent
-In this conditions, also the signals from the L4C seismometers and accelerometers (Watt's Leakage) placed on the Central bench have been measured. The plots with the UoB electronics show that there is some leakage below 10 Hz, probably due to saturation, in the measurement of the accelerometers.
-
-\begin{figure}[H]
+\begin{figure}[h!]
 \centering
 \includegraphics[scale=0.3]{images/AEI_SOUTH.PNG}\\
 \includegraphics[scale=0.3]{images/UOB_SOUTH.PNG}
 \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.}
 \label{central}
 \end{figure}
-\noindent
-QPD performances are shown in the plots \ref{qpd_fin}. With AEI boxes we had expected results: no variations in the power fluctuation peaks and expected behaviour of pitch and yaw.\\
-However, with UoB pre-amp the measurements do not seem consistent with what we expected: we think that some non-linearities in UoB pre-amp could be the cause of the problem. This is still under investigation at UoB.
 
-\begin{figure}[H]
+\begin{figure}[h!]
 \centering
 \includegraphics[scale=0.3]{images/AEI_QPD_TEST.PNG}\\
 \includegraphics[scale=0.3]{images/UOB_QPD_TEST.PNG}
@@ -573,6 +595,7 @@ However, with UoB pre-amp the measurements do not seem consistent with what we e
 \label{qpd_fin}
 \end{figure}
 
+\newpage
 \paragraph*{Electronic noise}
 Noise measurements of CDS with unplugged electronics have been taken, to check if there could be issues related to it. However, they do not show any unexpected behaviour:  CDS dominates nearly everywhere and the CDS noise is lower than any of our optical measurements everywhere, typically by at least a factor of 10.
 
@@ -599,7 +622,9 @@ Noise measurements of CDS with unplugged electronics have been taken, to check i
 %SN = 8,1 \time 10^{-11} \frac{W}{\sqrt{Hz}}.
 %\end{equation}
 
+\newpage
 \section*{Conclusions}
-The measurements have shown that we had issues when calibrating the device due to problems highly related to electronics from UoB, since the tests with the AEI electronics showed that the optical setup was well built and aligned. The very short time of the visit did not allow to take more in-depth tests.\\
+The analysis of feasibility of this experiment showed that the optical lever can be in principle a good device to sense tilt motion over long lever arms. However, the noise budget indicated a small frequency window of good operation, while below 0.1 Hz the levers are limited by the ground motion along the z axis. It is anyway a good device to be tested.\\
+During the test of the prototype, the measurements have shown that we had issues when calibrating the device due to problems highly related to electronics from UoB, since the tests with the AEI electronics showed that the optical setup was well built and aligned. The very short time of the visit did not allow to take more in-depth tests.\\
 Other possible reasons to investigate for better performances might lie in the structure of the prototype: further tests might be useful to understand if the device can be improved by changing the position of the lens with respect to the QPD, and let the diode sit at the focus on the lens. This solution will concentrate the power and decrease the size of the beam.\\
 The device is currently not suitable for the purposes we tested for, but it opened the way to further tests to improve the technology: since the pitch and yaw tests have shown that the optical lever might be sensitive to the vertical motion of the bench, a reduction of this motion might be of great impact to improve the sensitivity of the levers \cite{luise}. With a good sensing system of tilt motion, the addition of an actuation system able to reduce this motion will be crucially helpful to stabilize the suspension points of the optical chains and then of the whole cavity.