New Design of Electrostatic Mirror Actuators for Application in High-Precision Interferometry
NNew Design of Electrostatic Mirror Actuators forApplication in High-Precision Interferometry
H Wittel , S Hild G Bergmann , K Danzmann andK A Strain E-mail:
[email protected] Max–Planck–Institute for Gravitational Physics and Leibniz University ofHannover, D-30167 Hannover, Germany SUPA, School of Physics and Astronomy, The University of Glasgow, Glasgow,G12 8QQ, UK
Abstract.
We describe a new geometry for electrostatic actuators to be usedin sensitive laser interferometers. The arrangement consists of two plates at thesides of the mirror (test mass), and therefore does not reduce its clear apertureas a conventional electrostatic drive (ESD) would do. Using the sample case ofthe AEI-10m prototype interferometer, we investigate the actuation range andinfluences of relative misalignment of the ESD plates in respect to the test mass.We find that in the case of the AEI-10 m prototype interferometer, this new kindof ESD could provide a range of 0.28 µ m when operated at a voltage of 1 kV. Inaddition, the geometry presented is shown to provide a reduction factor of about100 in the magnitude of actuator motion coupling to test mass displacement.We show that therefore in the specific case of the AEI-10m interferometer it ispossible to mount the ESD actuators directly on the optical table, without spoilingthe seismic isolation performance of the triple stage suspension of the main testmasses.PACS numbers: 04.80.Nn, 95.75.Kk
1. Motivation
Interferometric gravitational wave detectors, such as GEO 600 [1], Advanced LIGO(aLIGO) [2], Advanced Virgo [3] and KAGRA [4] are large laser interferometers withthe mirrors/test masses hung at the bottom of multi-stage pendulum chains. For theoperation of these detectors, it is necessary to have low-noise, contact-free actuatorsfor controlling the longitudinal and angular degrees of freedom of the mirrors. Thisis traditionally done with either magnet-coil actuators or electrostatic drives (ESDs).GEO 600 has operated since 2001 with ESDs as the main longitudinal actuators forcontrolling the differential arm length. Based on this experience, ESDs are nowemployed in aLIGO, using a very similar configuration to the original GEO 600 design.The GEO/aLIGO ESDs are designed in the form of a metallic comb structure that hasbeen coated onto a reaction mass and is located a few millimeters behind the mirror.Schematic drawings and a photograph of this conventional ESD setup are shown inFigure 1. The ESD on the reaction mass needs its own seismic isolation, to avoidthe coupling of ground motion to the seismically isolated test mass. Therefore, it isalso hung as the lowest stage of a multi stage pendulum, which again requires its own a r X i v : . [ phy s i c s . op ti c s ] N ov ew Electrostatic Mirror Actuators for High-Precision Interferometry LASER mirror(test mass) reaction mass ESD side view front view(reaction mass) Photograph of a reaction mass of GEO 600
Figure 1:
Left and center:
Schematic drawing of a conventional ESD setup. The shownESD setup consists of four independent ESD comb-shaped electrode pairs on a singlereaction mass. The anodes are colored in red, the cathodes are colored blue. Thecenter of the reaction mass is kept free so that a laser beam can pass without significantclipping.
Right:
Photograph of a reaction mass of GEO 600sensors and actuators for alignment and damping of the reaction mass. Furthermore,this conventional type of ESD reduces the possible free aperture in transmission. Thismay be problematic for experiments which require the largest possible free aperture,such as the planned AEI-10m prototype interferometer [6][7] or the speedmeter proofof principle experiment in Glasgow [8][9].With the AEI-10m interferometer in mind, we investigate a new ESDconfiguration, featuring a simpler geometry, that uses only two plates at the sidesor at top and bottom of the mirror, pictured in Figure 3. Since the force that may beobtained per applied voltage of this setup is smaller than for the conventional ESDconfiguration, this type of ESD is mainly suited for the case of light ( ≤
100 g) mirrorssuch as in the AEI-10m prototype interferometer or in the Glasgow speed meter proofof principle experiment.
2. Basic Principle
The working principle of ESDs is that an inhomogeneous electric field is built up in adielectric medium (i.e. the mirror). If the mirror is not centered longitudinally betweenthe plates, it will be subject to a force that pulls it towards the center of the plates.In order to analyze our new ESD design, we first turn to a simplified analytic model,before we use finite elements (FE) simulations later. Our simplified model shall be thecase of a dielectric slab inserted into a parallel-plate capacitor (as in [10]), which ispictured in Figure 2. Calculating the force for this simple model is a standard problemin electrodynamics. The force is given by [10]: F = ( (cid:15) − (cid:15) ) · E D · U ESD / d plates , (1)where E D is the plate depth, U ESD is the voltage across the plates and d plates is theplate separation. (cid:15) and (cid:15) r are dielectric constants (see also table 1 for the values andsymbols used in this article). This formula is only valid under certain assumptions: ew Electrostatic Mirror Actuators for High-Precision Interferometry E D d plates U ESDforce
Figure 2: Illustration of the simple model. A slab of a dielectric material between twoparallel capacitor plates. The electric field is pictured as gray arrows.(i) There is no gap between the dielectric and the plates.(ii) One end of the dielectric material is in a homogeneous electric field between theplates.(iii) The other end of the dielectric is far outside of the plates, it does not ’see’ theelectric field of the plates.We can use the formula for the simplified case to approximate the order ofmagnitude of the force that the ESD will provide. The exact strength of the force mustbe determined by finite element methods, since none of the assumptions mentionedabove is exactly fulfilled.To estimate the force that the ESD can provide using the simplified formula, firstwe compute an effective dielectric constant (cid:15) r eff for a mirror between the two plates,since it does not fill the space between the ESD plates completely. We determine a’fill factor’ of A mirror /A plates = πr /E D · d plates = 0.54. Now we can multiply thefill factor with (cid:15) r and get an effective (cid:15) r eff = 2. With the values given in Table 1 andan (cid:15) r eff =2, equation 1 gives a position independent force of the order of some µ N.
3. Quantitative Analysis using FEM
Mirror diameter M D T (cid:15) r (cid:15) Mirror mass m 102 gPendulum length l 20 cmLateral distance mirror-ESD d ESD W xE H xE D d plates n nodes ew Electrostatic Mirror Actuators for High-Precision Interferometry MME W DT d E H d X YZ
Z YX m i rr o r ESD plate
Figure 3:
Left:
Basic geometrical setup of the ESD plates with respect to the main testmasses/mirror. Please note that this sketch can be either side-view or top-view,depending on the actual installation. The probing laser beam is shown in purplewhile the beam transmitted through the mirror is painted with a lighter color. Thedimensions M T , M D , E H , E W , and d , are defined in table 1. Top right:
View of the FE model that was used in this article.
Bottom right:
A mock up drawing of what a potential installation might look like.the ESD plates. The Z-axis points from the flat mirror surface to the center betweenthe plates (sometimes called ’longitudinal direction’ or ’beam direction’). The Y-axispoints towards one of the plates and the X-axis is perpendicular to the Y and Z-axes.Figure 3 shows the geometry of the new ESD setup. The force that the ESD applieson the mirror in dependence of the relative longitudinal position between mirror andESD is plotted in Figure 4. From this dependency we also find the best operatingdistance between mirror and ESD: the maximum and to first order flat force of 1.4 µ Nappears when the mirror is shifted by 3.075 cm relative to the ESD plates. We choosethis position as our potential operating point. From the force the magnitude of theESD range can be deduced. We assume that the mirror is suspended as a pendulumof the length l . The full actuation range x is reached when the pendulum back actionforce cancels the ESD-force: x = − F operating · l/mg ≈ . µ m (at 1 kV) . (2)From the equations 2 and 1 it also follows that the range is inversely proportional tomirror mass and plate separation. That is the reason why this new kind of ESDs is ew Electrostatic Mirror Actuators for High-Precision Interferometry -6 -4 -2 0 2 4 6-1.5-1-0.500.511.5 x 10 -6 Position of the mirror in cm F o r c e a c t i n g o n t h e m i rr o r i n N force in Xforce in Yforce in Z Figure 4: Longitudinal force on the mirror, versus relative longitudinal position ofmirror and ESD. The x-axis shows the relative distance between the mirror centerand the center of the capacitor plates.only suited for the case of small mirrors. x ∝ / ( m · d plates ) (3)Figure 5a shows that the force that can be obtained scales quadratically with theapplied voltage, when the mirror is in the operating position, while Figure 5b showshow the force depends on the plate separation.
4. Requirements & Noise
As mentioned above, the force on the mirror (provided by the new ESD configuration)is independent of the mirror position, as long as mirror and ESD are positionedcorrectly. Together with the performance of the seismically isolated tables in the AEI-10m prototype interferometer [11] it is possible to mount the new ESD directly to thetable without additional seismic isolation. Figure 6 shows the expected coupling ofseismic noise through the new ESDs for non-ideal positioning of the mirror in respectto the ESD plates. Even with the ESDs 3.75 mm off from the ideal longitudinalposition, the noise contribution in the AEI-10m interferometer would still be lowerthan the sum of all other classical noise sources at all frequencies.In order allow a direct comparison of the seismic coupling of a conventional ESDdesign and our new ESD design, we included the expected noise coupling from aconventional ESD design in Figure 6 (see grey traces). For this analysis we assumedthe force on the mirror for a conventional ESD design, which is given by [14]: F = a · (cid:15) · (cid:15) r · U ESD · d . , (4) ew Electrostatic Mirror Actuators for High-Precision Interferometry −6 Voltage U
ESD in V F o r c e a c t i ng on t he m i rr o r i n N data from FEM modelquadratic fit (a) Voltage change −1 −6 Separation of the plates (d plates ) in cm F o r c e a c t i ng on t he m i rr o r i n N force in Xforce in Yforce in Z (b) Plate separation Figure 5: Figure (a) shows the force on the mirror for different voltages. The relativeposition of mirror and ESD is the position with the optimal force. (b) shows howthe force depends on the plate separation (assuming that the mirror is at the idealoperating position)where d conv is the separation between the mirror and the ESD and a is a geometryfactor that depends on the actual shape of the electrode pattern, and the dimensionsof the mirror and the ESD. The strong scaling of the force with d conv , explains thelarge coupling of seismic from the ESD to the test mass. This can be seen in Figure 6,where we compare a conventional ESD to the new ESD design (both actuators providethe same longitudinal range). The new ESD design provides a reduction in seismicnoise coupling of about a factor of 100 compared to the conventional design.Moreover, we also investigated how precisely the ESD plates (of the new design)have to be positioned in the direction along their surface normal. If the mirror is closerto one of the ESD plates, it will see a force towards the closer plate. This situationhas been simulated using the FE model. The results can be seen in Figure 7. If wearbitrarily set a limit for the force towards the closer plate to 1/10 of the force inbeam direction, then the mirror must be centered between the plates within 1 mm.
5. The Effect of Asymmetries in Materials in the Vicinity of the ESDs
As mentioned earlier, the main optics of the AEI-10m interferometer will be hung asa multi-stage pendulum, which is usually the case for gravitational wave detectors. Inorder to keep the mechanical losses low, the pendulum wires are typically dimensionedsuch that they are loaded to a significant amount of their breaking stress [12]. In thecase of the AEI-10m prototype, it is planned to use fused silica fibers with a diameterof 20 µ m for the lowest pendulum stage. To protect the main optics if a fiber breaks,a ’catcher’ will be placed underneath each suspended mirror. One could argue thatsuch catchers may alter the performance of the ESDs proposed in this article, asthey break the symmetry of the electrostatic environment close to the ESDs. Suchasymmetries may introduce a (position-dependent) torque on the mirror, which wouldlead to coupling of longitudinal mirror motion to mirror alignment. We do not expectthe same effect from symmetrically positioned parts around the ESDs.We simulated the effect of a sample configuration with catcher, which is modeled ew Electrostatic Mirror Actuators for High-Precision Interferometry -21 -20 -19 -18 -17 -16 -15 Frequency in Hz D i s p l a c e m e n t i n m / s q r t ( H z ) quantum noiseSQLsum of classical noisesnew ESD, 3.75mm offnew ESD, 2.5mm offnew ESD, 1.25mm offconventional ESD, d conv = 1mmconventional ESD, d conv = 3mm Figure 6: Comparison of seismic noise coupling in the AEI-10m prototypeinterferometer for ESDs with the conventional design and our novel ESD configuration.The reddish traces indicate projections of how much table movement would couple intothe interferometer if the novel ESD configuration is used, but the plates are not locatedideally. The dashed lines indicate the sum of all classical noise, quantum noise andthe Standard Quantum Limit (SQL) of the planned AEI-10m sub-SQL interferometer.We have also included a projection of the seismic noise coupling if conventional ESDsmounted on the tables would be used, where d indicates the distance between themirror and the ESDs (greyish traces). As one can see, the seismic noise coupling ofour novel ESD design is lower by about a factor 100 compared to a conventional ESDdesign.as a cuboid, the geometry of which is shown in Figure 8. In this simulation the ESDplates are at the sides of the mirror. The catcher is positioned in such a way that it iscentered below the mirror at the ideal operating point. It sits 1 cm below the mirrorand is 4 cm tall. In the beam direction, the mirror projects from the ESD by 1 cm oneach side. the catcher is 6.5 cm wide and is assumed to be at ground potential, whilethe ESD plates were set to +500 V and -500 V. The inclusion of a metal catcher inthe simulation caused no significant change in the longitudinal force of the ESD. Itdid however produce a lateral force of about 20% of the longitudinal force towardsthe catcher (at the operating position). Also there will be a torque in the ’tilt’ degreeof freedom. Fortunately, the torque in Y-direction, which is the strongest, is flat tofirst order when the catcher is centered with respect to the mirror. It is important tonote that this crude catcher geometry is used as a worst case scenario to set an upperlimit. To reduce the influence of mechanics around the mirror, one may, for example,use a different catcher geometry or material, such as Macor or polyether ether ketone(PEEK), a vacuum compatible polymer. The AEI-10m interferometer suspension will ew Electrostatic Mirror Actuators for High-Precision Interferometry -6 Lateral displacement of the mirror towards one of the plates (d ) in mm F o r c e i n N lateral directionlongitudinal direction Figure 7: Lateral force on the mirror, versus relative lateral position of mirror andESD. Figure 8: Geometric setup with catcher.feature ’fibre guards’, an aluminium semi-enclosure around the fibres that suspend themirror. Their presence will necessitate the ESDs to be installed at the top and bottomof the mirrors. A detailed description of the design of the 10m prototype suspensionscan be found in [13]. ew Electrostatic Mirror Actuators for High-Precision Interferometry -6 -4 -2 0 2 4 6-1.5-1-0.500.511.5 x 10 -6 Position of the mirror in cm F o r c e a c t i n g o n t h e m i rr o r i n N force in Xforce in Yforce in Z (a) Force -6 -4 -2 0 2 4 6-10-8-6-4-2024 x 10 -9 Position of the mirror in cm T o r q u e a c t i n g o n t h e m i rr o r i n N m torque around Xtorque around Ytorque around Z (b) Torque Figure 9: Force and torque with catcher. The catcher’s position was kept constant(with the z-coordinate of its center of mass at -3.075 cm), while the mirror was moved.
6. Summary and Outlook
We presented a novel type of electrostatic drives for longitudinal test mass actuationin gravitational wave detectors. The ESDs would be plates installed at the sides of thetest masses. For the AEI-10m prototype, this type of ESDs could be mounted directlyto the seismically isolated table. We find that the ESDs would have to be positionedwith an accuracy of better than 3.75 mm longitudinally and 1 mm in lateral direction.They could move the mirror by more than 0.28 µ m with 1 kV and by more than 1 µ mwhen operated at 2 kV. One aspect that has not been investigated so far refers to thenoise terms originating from electrical charges on the test masses. While it would bestraightforward to include these in our FE analysis, it is not clear at all what amountof charge and geometrical charge distribution would be sensible to assume. Thereforethis aspect needs to be tested experimentally. This work is now underway. Acknowledgments
We thank Peter Fritschel for useful comments on the manuscript. The authors wouldlike to acknowledge the support from the Max Planck Society, the European ResearchCouncil (ERC-2012-StG: 307245) and the Science and Technology Facilities Council(STFC, ST/L000946/1). This work was performed as part of our International MaxPlanck Partnership (IMPP) with the Max Planck Society, supported by SFC, EPSRCand STFC.
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