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\begin{document}
\title{Controlling calibration errors in gravitational-wave detectors by precise location of calibration forces.}
\author{H P Daveloza$^1$, M Afrin Badhan$^2$, M Diaz$^1$, K Kawabe$^3$, P N Konverski$^1$, M Landry$^3$ and R L Savage$^3$}
\address{$^1$ Center for Gravitational Wave Astronomy, University of Texas at Brownsville, Brownsville, Texas, USA}
\address{$^2$ Mount Holyoke College, South Hadley, Massachusetts, USA}
\address{$^3$ LIGO Hanford Observatory, Richland, Washington, USA}
\ead{hernan.daveloza@ligo.org}
\begin{abstract}
We present results of finite element analysis simulations which could lead to more accurate calibration of interferometric gravitational wave detectors. Calibration and actuation forces applied to the interferometer test masses cause elastic deformation, inducing errors in the calibration. These errors increase with actuation frequency, and can be greater than $50\%$ at frequencies above a few kilohertz. We show that they can be reduced significantly by optimizing the position at which the forces are applied. The Advanced LIGO~\cite{aLIGO} photon calibrators use a two-beam configuration to reduce the impact of {\em local} deformations of the test mass surface. The position of the beams over the test mass can be chosen such both the local and the {\it bulk} induced elastic deformation are minimized. Our finite element modeling indicates that with two beams positioned within $\pm 1$ mm of their optimal locations, calibration errors due to test mass elastic deformation can be kept below $1\%$ for frequencies up to $3.5$ kHz. We thus show that precise control of the location of calibration forces could considerably improve calibration accuracy, especially at high frequencies.
\end{abstract}
\section{Introduction}
Optimizing searches for gravitational wave sources at frequencies above 1 kHz~\cite{HF1}~\cite{HF2}, will require improvements in the calibration of ground-based interferometric gravitational wave detectors. Lindblom has shown that optimizing scientific benefit requires calibration accuracies on the order of $5\%$ for first detection and $0.5\%$ for later measurements~\cite{Lindblom}.
The Photon Calibrator (Pcal) is a calibration method that uses radiation pressure from auxiliary lasers to induce reference displacements of the interferometer's test masses. The Pcal has become one of the favored methods for determination of the interferometer's response function, due to its simplicity and ability to calibrate with forces that induce displacements that are close to what is expected from candidate Gravitational Wave (GW) sources. However, Hild et al. at GEO600~\cite{Hild}, and Goetz et al. in LIGO~\cite{Goetz} demonstrated that calibration forces exerted by the action of a centered Pcal beam produce local deformations that significantly change the sensed displacement at frequencies above about $500$ Hz due to local elastic deformation of the test mass surface. Even though this type of local elastic deformation cannot be avoided, its effects can be mitigated by changing the configuration of the Pcal from one laser beam in the middle of the face of the test mass, to two laser beams diametrically opposed and sufficiently displaced from the center of the face such that local elastic deformations are not sensed by the interferometer. This scheme was tested and implemented in LIGO as part of the Enhanced LIGO upgrade~\cite{iLIGO}~\cite{eLIGO}.
Willems~\cite{Phil} and Afrin Badhan et al.~\cite{Mahmuda} studied another possible problem, the {\em bulk} elastic deformation of the End Test Masses (ETM). Their studies showed that these deformations also significantly impact the calibration accuracy, especially at high frequencies.
Our investigations show that actuation and calibration forces induce bulk deformations with spatial distributions given by the test mass normal mode shapes. The magnitude of the induced deformation increases dramatically as the frequency of excitation approaches the frequency of a given normal mode, affecting the interferometer length calibration. We incorporated the actual initial LIGO ETM geometry and the actual physical parameters of the ETM (e.g. density, Young's modulus, Poisson's ratio, damping parameter or the inverse of the quality factor) into to our Finite Element Analysis (FEA). We also analyzed the elastic deformations of Advanced LIGO ETMs, predicting the deformations that would be produced by Pcal forces and identifying the optimal positions for the beams.
\section{Elastic Theory}
ETMs are made of fused silica. Although this is a stiff material, with a Young's modulus of $72.7 \,\text{GPa}$ and a Poisson's ratio of $0.16$, they can experience small deformations when acted upon by calibration or actuation forces. For example, deformations on the order of $10^{-16}$ m induced by oscillating calibration forces of $10^{-8}$ N at a frequency of $3.5$ kHz. Because the deformations are small, the response to the excitations induced by calibration forces can be represented with the appropriate linear combination of normal modes. We perform investigations to correctly identify these modes and their effects on the detector performance. As we will show, the amplitude in each mode, and then the deformation sensed by the interferometer, depends on the position of the calibration and actuation forces on the High-Reflective (HR) and Anti-Reflective (AR) surfaces of the ETMs.
The test masses of GW interferometers are right circular cylinders with one wedged flat surface. As introduced by Love~\cite{Love}, the first study of an ideal cylinder's elastic deformations was made by Pochhammer in 1876. Hutchinson, summarized this studies and presented a numerical method to calculate the normal modes of a solid cylinder~\cite{Hutchinson}.
GW interferometers measure the axial projection of the displacement of the surface of the test mass, $w=\vec{u}\cdot\vec{z}$, where $\vec{u}$ is the displacement vector and $\vec{z}$ is the direction parallel to the axis of the cylinder. Following Hutchinson, the solutions for this displacement are:
\begin{center} $w = 2 \delta J_n(\alpha r) \left\{\begin{array}{r} -sin(\delta z)\\ cos(\delta z) \end{array} \right\} \sin(n \theta)$ \end{center}
and
\begin{equation} w = \alpha^2 J_n(\alpha r) \left\{\begin{array}{r} -sin(\beta z)\\ cos(\beta z) \end{array} \right\} \cos(n \theta) \end{equation}
where $J_n$ denotes the $n$th-order Bessel function and $r, \theta$ and $z$ are the usual cylindrical coordinates. The numbers $\alpha$, $\beta$ and $\delta$ are related by $\alpha^2+\delta^2=(2\pi f)^2(1-2\nu)/2(1-\nu)$ and $\alpha^2+\beta^2=(2\pi f)^2$, where $f$ is the frequency and $\nu$ is the Poisson's ratio.
The sensing by the main interferometer laser beam can be modeled as a Gaussian-weighted integral of the displacement over the entire HR surface. Due to the periodic azimuthal dependence ($cos$ or $sin$), the integrals for a centered interferometer beam will vanish except for the case of $n=0$, in which case the azimuthal dependence vanishes. In this case we have a solution for $w$ with only a Bessel dependence on the radial direction (the integral is performed over a surface, i.e. at $z = constant$). This mode is referred to as the {\it drumhead} mode.
We can reduce or eliminate the excitation of this mode by applying forces at the nodes of this function. A rough approximation of the ratio between this optimal radial position for the applied forces and the radius of the cylinder, is given by the ratio between the first root of $J_0(x)$ ($x \sim 2.4048$), that is the first node, and the first root of its derivative ($x \sim 3.8317$), which is approximately the radius of the cylinder. Then, the $ratio = 2.4048/3.8317 = 0.6276$. Afrin Badhan found a similar solution for the initial and enhanced LIGO control system by inspection~\cite{Mahmuda}. Initial and enhanced LIGO use four magnets glue to the AR surface and four voice coil actuator to control the position of the ETMs. She concluded that for a right circular cylinder of diameter $D=250$ mm and thickness $\tau=100$ mm with four forces applied over the AR surface, the appropriate radius for each force is $r\sim 82$ mm, giving a ratio $\sim 0.654$. Our study for initial and enhanced LIGO ETMs shows that if the coil actuators, the four forces over the AR surface, were positioned at the optimal radius ($r\sim 78.7$ mm) the discrepancy between the displacement sensed by the interferometer and the displacement of an ideal rigid-body it would go from $1.8$ to $1$ at $6$ kHz as shown in Figure {\bf 1}.
Figure {\bf 1} compares the ratio between the displacement sensed by the interferometer and a free-mass rigid-body motion, for the voice coil actuator at the real position and the optimal position.
These FEA results agree reasonably well, discrepancy less than $2\%$, with LIGO calibration measurements for actuation frequencies up to $4$ kHz; above this frequency a difference of up to $20\%$ is observed between the FEA results and the measurements; this may be due to a known defect in one of the voice coil actuators~\cite{S6_budget}.
\begin{center}
\includegraphics[bb= 400 0 500 610,scale=0.3]{plot/coil_comp_color.jpg}
\end{center}
\begin{center}
{\bf Figure 1} \\
{\it Ratio between total sensed motion and rigid-body motion vs. excitation frequency, for actual (Real) and Optimal calibration force positions.}
\end{center}
\section{Finite Element Analysis}
For our FEA, we used COMSOL Multiphysics~\cite{comsol}. LIGO ETMs have a thin-film dielectric coatings applied to the flat surfaces to produce the HR surface, the one that receives the main interferometer beam, which is not modeled in our FEA. The initial (and enhanced) LIGO End Test Masses (iETM) have a diameter $D=250.75$ mm, a maximum thickness $\tau=99.63\, \text{mm}$, and a $2^{\circ}$ wedge on the AR surface. The advanced LIGO End Test Masses (aETM) have $D=340.13\,\text{mm}$, $\tau=200.2$ mm, a wedge of $0.077^{\circ}$ on the AR surface, and two flats on the sides with a separation of $326.5$ mm between them.
LIGO ETMs are suspended as pendulums to isolated them from seismic noise. In our FEA models, we treat the ETMs as free bodies, a reasonable assumption for frequencies much larger than the ETMs pendulum resonance frequency, approximately $1$ Hz. The total displacement of the HR surface is the combination of the free-mass motion and the displacement resulting from elastic deformation of the ETMs. The effective displacement measured by the interferometer is the overlap of this HR surface displacement with the main interferometer gaussian beam. The gaussian weighting for a centered interferometer beam is given by:
\begin{equation}
\frac{exp(-2(x^2+y^2)/r_{ifo}^2)}{2 \pi \times r_{ifo}^2}
\end{equation}
where the interferometer beam spot size, $r_{ifo}$, is $45$ mm for the iETM, and $62$ mm for the aETM. The denominator is a normalization factor.
Using the elastic theory described in Section 2, we analyzed the coupling from the actuation of the photon calibrator forces to the displacement of each mode. Our analysis shows that two normal modes are most relevant for calibration frequencies below $6$ kHz, the {\it butterfly} mode and the drumhead mode. The butterfly and drumhead mode resonant frequencies are $6612$ Hz and $9221$ Hz for the iETM, and $5953$ Hz and $8151$ Hz for the aETM, respectively. Figure {\bf 2} shows the deformation shape of the butterfly and drumhead mode for an aETM.
\begin{center}
\begin{tabular}{ | c | c | } \hline
\bf{Butterfly} & \bf{Drumhead}\\ \hline
\includegraphics[bb= 0 0 450 400,scale=0.3]{plot/aETM_B+_color.png} &
\includegraphics[bb= 0 0 450 400,scale=0.3]{plot/aETM_D_color.png}\\ \hline
\end{tabular}\\
\end{center}
\begin{center}
{\bf Figure 2} \\
{\it Advanced LIGO End Test Mass normal modes.}
\end{center}
The drumhead mode has no azimuthal variation. The amplitude of the mode can be minimized by precise location of the actuation or calibration forces at the optimal radius, which is close to the nodal circle for the mode. The butterfly mode has a periodic variation with azimuthal angle. We already showed that for a perfect cylinder this periodic variation results in an effective zero displacement. Nevertheless, details of the actual test mass geometry, e.g. flats and wedges, affect, as Miller points out~\cite{Miller}, the shapes and frequencies of the normal modes. This is easy to visualize with {\it isosurfaces}; they are three dimensional surfaces with a constant value of the displacement of the elements. Figure {\bf 3} shows zero axial displacement {\it isosurfaces}, that is surfaces where $w = 0$, for an ideal cylinder and for an aETM.
\begin{center}
\includegraphics[bb= -100 0 500 430,scale=0.3]{plot/isosur_cyl_aETM_color.png}
\includegraphics[bb= 0 0 500 430,scale=0.3]{plot/isosur_aETM_color.png}\\
{\bf Figure 3:}\\
\emph{Butterfly mode $w=0$ isosurfaces for an ideal cylinder (left) and an advanced LIGO End Test Mass (right). The arrows show the displacement vectors for the elements.}
\end{center}
For ideal cylinders, the $w = 0$ isosurfaces cross the cylinder through its center, leaving an almost zero effective displacement. However, the $w = 0$ isosurfaces for the ETMs do not cross through the center of the cylinder, leaving an additional displacement that sums with the free-mass motion. Our study shows that the amplitude of the displacement produced by the elastic deformation for the butterfly mode can be minimized using three or more forces.
We also calculated the transfer function from the photon calibrator forces to the displacement sensed by the interferometer to find an optimal placement of the photon calibrator beams. The discrepancy between the ideal rigid-body motion and the effective displacement measured by the interferometer is weakly dependent on the position of the interferometer beam, but strongly dependent on the positions of the driving forces. The resulting potential calibration errors are plotted versus excitation frequencies for a two-beam Pcal calibrator configuration in Figure {\bf 4}, and for a three-beam Pcal calibrator configuration in Figure {\bf 5}. For two optimally located beams the discrepancy is less than $0.1\%$ up to $3.2$ kHz and less than $1\%$ for frequencies up to $4.3$ kHz, but increases to almost $4\%$ at $5$ kHz.
\includegraphics[bb= -120 50 500 650,scale=0.35]{plot/result_poster_color.jpg} \\
\begin{center}
{\bf Figure 4: }\\
\emph{Advanced LIGO two-beams Photon calibrator configuration, ratio between total sensed motion and rigid-body motion vs. excitation frequency, for optimally positioned beams and $\pm 1$ mm and $\pm 3$ mm offsets from the optimal locations.}\\
\end{center}
The discrepancy increases dramatically for beams that are offset from their optimal positions. This increase in discrepancy for frequencies near and above $5$ kHz can be understood by considering the increased compliance in the butterfly and drumhead deformation patterns as the excitation frequency approaches their resonance frequencies. The amplitude of the deformation related to the butterfly mode is important only for frequencies close to it's resonant frequency, $6612\,\text{Hz}$ for the iETM and $5953$ Hz for the aETM, because far away from this frequencies the shape of this mode is governed by the azimuthal periodic variation that averages to zero for a centered interferometer beam. Thus, in general, the behavior of the discrepancy is governed by the amplitude of the elastic deformation related to the drumhead mode, because it directly involves the center of the ETMs with no periodic azimuthal dependence. For beams positioned in radius smaller than the optimal radius, the displacement produced by the drumhead mode is $180^{\circ}$ out of phase with the free-mass motion, so these motion are subtracted. On the other hand, if the beams are positioned in a radius larger than the optimal radius, the displacement due to the drumhead mode are in phase with the free-mass motion and they add, so the discrepancy between the ideal rigid-body and the effective displacement always increases with frequency when approaching $6\,\text{kHz}$. Figure {\bf 4} shows this behavior. Note that above about $4$ kHz, the discrepancies for Pcal beams positioned at radius smaller than the optimal radius, which initially decreased, begins to increase as the frequency approaches $5$ kHz. This is because the excitation of the butterfly deformation pattern as the excitation frequency approaches its resonance frequency, which is smaller than the drumhead mode resonance frequency.
If we now focus our attention to a discrepancy of $1\%$, we observe in Figure {\bf 4} that for an optimally positioned two-beam configuration this discrepancy is reached at $4.3$ kHz. Meanwhile for an offset of $1$ mm this discrepancy is reached at $3.5$ kHz. In the case of three or four forces at the optimal radius, the discrepancy could be reduced to $0.2\%$ for frequencies up to $5$ kHz. Nevertheless, if the three or four forces are offset from the optimal locations by $1$ mm, either intentionally or unintentionally due to poor positioning accuracy, the discrepancy is again $1\%$ at $3.5$ kHz, as can be seen in Figure {\bf 5}. Thus, to realize the calibration accuracies called for in Lindblom's assessment, calibration forces must be localized within $\pm 1$ mm of their optimal locations and multiple beam configurations will be required.
\includegraphics[bb= -120 50 500 650,scale=0.35]{plot/result_poster_2_3_color.jpg} \\
\begin{center}
{\bf Figure 5: }\\
\emph{Advanced LIGO three-beams Photon calibrator configuration, ratio between total sensed motion and rigid-body motion vs. excitation frequency, for optimally positioned beams and $\pm 1$ mm offsets from the optimal locations.}\\
\end{center}
We also carried out our analysis for a sapphire ETM, a stiffer material than fused silica and a candidate material for future GW detectors test masses due to its increased thermal conductivity. For sapphire, normal modes moves to higher frequencies, butterfly $\sim9$ kHz and drumhead $\sim14\,\text{kHz}$, resulting in, compared with fused silica, smaller deformation amplitudes for frequencies below $6$ kHz which is desirable for GW detection and for high-frequency calibration accuracy.
We have also carried out a preliminary investigation of the electrostatic drives that will be used to control the aETMs~\cite{Miller}. Advanced LIGO's strict noise requirements demand that no magnets be glued to the ETMs, as was the case for initial and enhanced LIGO. The advanced LIGO suspension system, is a quadruple pendulum, in which the bottom masses are the ETMs and reaction masses are used to exert electrostatic forces on the ETMs via electrodes patterned on the reaction masses. Preliminary studies show that the electrostatic drives will produce calibration discrepancies of about $10\%$ at $2.5$ kHz and as much as $60\%$ at $5$ kHz due to the location of the forces.
\section{Conclusions}
Our analysis shows that actuation and calibration forces induce bulk deformations of test masses producing a discrepancy between the ideal free-mass, rigid-body motion and the effective displacement sensed by gravitational wave interferometers. The spatial distribution of these deformations are given by a superposition of the normal modes of the test mass with amplitudes that increase dramatically as the calibration or actuation excitation frequency approaches that of the normal modes. Even when the amplitude of the butterfly spatial deformation could be minimized using three or more forces, our study shows that, unless an accuracy of $\pm 1$ mm in locating the calibration or actuation forces is achieved, there will be no advantage in using three or more forces for frequencies below $3.5$ kHz. For an aETM with two Pcal beams positioned at a radius $111.6\pm 1.0\,\text{mm}$ it is possible to keep the discrepancy between the ideal motion and the motion sensed by the interferometer below $1\%$ up to $3.5$ kHz. For frequencies above $3.5\,kHz$, calibration accuracy will be improved by adopting a Pcal configuration with three or more beams in order to minimize the amplitude of the butterfly bulk deformation pattern.
\ack{We thank P. Willems and the staff of the LIGO Hanford Observatory for enlightening discussions. We also thank the National Science Foundation for support under award HRD0734800. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0757058. This paper has LIGO Document Number LIGO-P1100166.\\}
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\end{document}