27 November 2000

Physics Letters A 277 (2000) 143–155www.elsevier.nl/locate/pla

Tilt suppression for ultra-low residual motion vibrationisolation in gravitational wave detection

J. Winterflooda,∗, Z.B. Zhoub, L. Jua, D.G. Blaira

a Department of Physics, University of Western Australia, Perth 6907, Australiab Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, PR China

Received 27 April 2000; received in revised form 13 October 2000; accepted 17 October 2000Communicated by P.R. Holland

Abstract

It is shown that a combination of ultra-low frequency vibration isolators combined with high sensitivity tilt sensing andfeedback allows rms vibration levels in laser interferometer gravitational wave detector test masses to be reduced from∼100µmto∼1 nm greatly simplifying operation and reducing control system noise injection. 2000 Elsevier Science B.V. All rightsreserved.

PACS:04.80.N; 95.55.Y; 07.10.F; 91.30

1. Introduction

Gravitational wave (GW) detection offers a newspectrum with which to explore the universe. Thetechnology of GW detection requires the reduction ofall sources of vibration towards or below limits set byquantum measurement theory. Isolation from seismicnoise is critical — not only in the GW detection band,but also at frequencies below this because even lowfrequency disturbances make locking of high finessecavities very difficult and call for large servo controlforces to maintain lock which can themselves injectnoise (from the limited dynamic range of electronicsand through seismic coupling of imperfect actuators).It is therefore now widely recognised that a reductionof residual motion is of critical importance to the

* Corresponding author. Tel.: +61-8-93801844, fax: +61-8-93801014.

E-mail address:jwinter@physics.uwa.edu.au (J. Winterflood).

achievement of the high sensitivity goals of advancedGW detectors.

For the simplest control scheme, the dynamic rangeof the electronic control system (∼1010) should bematched to the dynamic range of the forces to be ap-plied to fringe lock the test mass, where the largestforce is the force required to maintain lock in the pres-ence of low-frequency (∼0.5 Hz) residual motion, andthe smallest is the force noise that produces motion notexceeding the instrument noise level (∼10−22 m/

√Hz

— advanced LIGO) at the low end of the detectionband (∼20 Hz). If an applied force produces 10−22 mof motion at 20 Hz, then 1010 times this force canbe expected to produce only∼10−9 m of motion at0.5 Hz. Thus the residual motion at the test mass needsto be reduced to this level if the locking forces are tobe applied directly to the test mass at this sensitivity.

The use of mechanical or electrical filters to reducethe control system noise reaching the test mass allowsthis requirement to be eased somewhat. For instance,

0375-9601/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0375-9601(00)00689-7

144 J. Winterflood et al. / Physics Letters A 277 (2000) 143–155

if the test mass is suspended as a 1 Hz pendulumfrom a previous stage and the locking forces appliedto this previous stage, then the noise at 20 Hz reachingthe test mass will be reduced by a factor of 202

bringing the residual motion requirement to a moreattainable value of∼5× 10−7 m. However, addingfilter stages between the fringe locking drive andthe test mass (within the control loop) can makecontrol considerably more difficult or even impossible.These arguments suggest a target residual motion oforder 1 nm. Arguments presented later in this Letterbased on assumed seismic levels indicate that residualmotion less than 1µm rms above 0.2 Hz cannot beachieved without tilt control.

In the following sections, we point out various prob-lems to be overcome in order to reduce the residualmotion to the nanometre level. It becomes apparentthat a key element in this progression is an adequatehigh frequency (0.1–5 Hz) seismic tilt sensor. This isan area that has remained largely unexplored in seis-mology but has recently begun to be addressed by theGW community. The design requirements of this tiltsensor are the main topic of this paper and we presentsome preliminary experimental measurements.

2. Seismic isolation

The size of the control forces required to fringelock the test mass is determined by the residual hor-izontal low frequency motion which feeds throughan isolation stack below its intended cutoff frequency(a few Hz). To enable us to express our results in lin-ear displacement units, we shall assume a standardseismic noise spectrum (the dotted curve in Fig. 1) of10−6/f 2 m/

√Hz above 1 Hz levelling out below the

microseismic peak at about 0.2 Hz. This spectrum istypical of an urban environment but the isolated sitesat which GW detectors are located may be two or eventhree orders of magnitude quieter around 1 Hz (al-though much the same at microseismic frequencies).A quieter site reduces all the absolute levels (i.e., thevertical scale on the graphs) by the seismic factor butleaves the relative relationships unchanged.

Fig. 1 indicates the residual spectrum and rms mo-tion of a typical test mass as a result of applying var-ious isolation structures. The curves on the left showthe spectral densities in m/

√Hz, while the ones on

the right have been integrated to give the rms motionin metres for all frequencies above the value shown

Fig. 1. Successive reduction of residual motion: (a) high Q-factor internal modes resonantly amplify the seismic motion at the normal modefrequencies; (b) these modes can be passively damped for a factor of 10 reduction in residual motion; (c) ULF pre-isolation can provide anadditional factor of 100 reduction; (d) dual ULF pre-isolation and active tilt suppression can gain another factor of 100.

J. Winterflood et al. / Physics Letters A 277 (2000) 143–155 145

on thex-axis. All of these curves consist of the isola-tor transfer function multiplied by the standard seis-mic spectrum. (The isolator transfer functions werecalculated by constructing a transfer matrix for eachcomponent such as a stiff pivot, pendulum link, mass,etc. and multiplying them together to obtain the com-plete system response. Where necessary matrices with4 variables were used: force, displacement, moment,angle. Subsystems constructed with 4 variables wereconverted to 2 variables — typically force and dis-placement — at connections with simplifying assump-tions such as perfectly flexible pendulum joints.)

Excellent vibration isolation in the tens of hertz tokilohertz range has been demonstrated using all-metalisolator stacks [1,2] consisting of cascaded pendu-lum stages for horizontal isolation, and incorporatingsprings for vertical isolation. With reasonable engi-neering the horizontal component remains dominant.Curve (a) shows the response of a typical 5 stage,2.3 m high, all-metal isolation stack (consisting of 4cascaded pendulums 0.5 m and 100 kg each, Q-factors∼1000, finishing with 20 kg test mass on 0.25 m,Q∼ 106 pendulum). It cuts off steeply above a fewHz, but it amplifies the seismic noise at low frequencydue to the high Q-factor normal mode resonances. Theresidual motion spectrum is dominated by these inter-nal resonant modes and the rms residual motion is al-most two orders of magnitude above the seismic level.

This amplified motion may be reduced by dampingthe Q-factors of these modes. Direct active dampingusing sensors and actuators mounted to a seismicallyaffected frame is limited in performance due to seis-mic noise pickup and injection. However, using sus-pended reaction masses, or the new passive techniquewe have called “self-damping” [3] can allow normalmode Q-factors to be reduced below 10. With suchdamping the residual motion is reduced to the levelof curve (b) — an order of magnitude improvement,but still above seismic and far above the nanometrelevel.

3. Ultra-low frequency pre-isolation

A very effective method to further reduce residualmotion is to add an ultra-low frequency (ULF) pre-isolator stage in front of the isolator stack. Such adevice can reduce the seismic drive to the normal

mode resonances and thus the residual motion, by twoorders of magnitude. It also provides a structure beforethe isolation stack to which very low frequency driftcorrection forces may be applied (suspension pointservoing) — avoiding the noise injection problemsassociated with applying large dc forces near to the testmass. For this reason only the residual motion above∼0.2 Hz is considered in this discussion as motionbelow this frequency can be easily counteracted abovethe stack at the pre-isolator level. (This scheme alsoinvalidates any attempt to damp the stack resonancesby frame mounted sensing because the frame hastwo orders higher motion than the resonances to bedamped.)

With this incentive, various new ULF pre-isolators(1-D, 2-D and vertical) have been introduced andtested, which have passive isolation performanceequivalent to springs or pendulums up to 6 km inlength [4]. These include the inverse pendulum [5], theX-pendulum [6], the folded pendulum [7], the Scott–Russel linkage [4,8], and the torsion Crank suspen-sion [9]. Adding a pre-isolation stage (of 150 kg, 25 speriod and Q-factor∼2) to the previously describedisolation stack gives the black curve (c) when onlyhorizontal translational motion (i.e., no tilting) is con-sidered. This reduces the residual rms motion (inte-grated above 0.2 Hz) down to the 1µm level. It willbe seen that this is about the best that can be achievedwithout resorting to an active tilt sensing and controlscheme.

There are a couple of reasons for the distinctionbetween a pre-isolator and normal isolation stage.The main one being that structures that are designedto have very low resonant frequencies tend to belarger and more massive and consequently have lowerinternal resonances (which bypass their isolation) thannormal isolator stack stages. As a result they generallydo not provide useful isolation in the 10 Hz–1 kHzdetection band but are included mainly to reduceresidual motion below the detection band. In thehorizontal regime there is an additional differencebetween the two isolator types. Normal isolator stagesare generally pendulums of an appropriate height (tensof centimeter) which sets their resonant frequency(g/h)1/2 and are attached to a single (ideally) perfectpivot on the stage above. However, in a pre-isolatorstage the connection to the previous stage is multi-point allowing it to transfer a tilt moment thereby

146 J. Winterflood et al. / Physics Letters A 277 (2000) 143–155

obtaining a resonant frequency much lower than thatexpected for a pendulum of comparable height (e.g.,150 s [4]).

4. Effect of tilt on pre-isolation

Horizontal ULF pre-isolation structures work byproviding a rigid constraint against the very large grav-itational force of the total suspended mass, while alsoproviding extremely soft horizontal compliance. Thisrequires the structures to be maintained in accuratealignment with the gravitational field — i.e., rigidlyfixed against tilting. However, seismic tilts perturbthe alignment and produce horizontal accelerations di-rectly.

For a suspension system without horizontal pre-isolation the effect of seismic tilt is far smaller thanseismic translation and can almost always be ne-glected. However, when a ULF horizontal pre-isolatoris employed, tilt induces significant translational hori-zontal motion. As the period of the pre-isolator is in-creased, the residualtranslationalmotion is reducedbut the tilt coupling remains at the base level for anobject sliding on a flat friction-less surface. Once thetranslational motion is smaller than this base level,there is no improvement to be gained by further hori-zontal isolation.

The tilt coupled translation for the above ULFpre-isolated system is shown by the grey curve (c).It is of the same approximate magnitude as thehorizontal translational signal (black curve) in the0.5–1 Hz region. If the pre-isolation is improved (e.g.,by increasing its resonant period or adding a secondstage), then the horizontal signal (black curve) reducesbut the tilt coupling (grey curve) remains at the samelevel and becomes the dominant source of residualmotion. (The total residual motion is the rms sum ofthese two curves.)

In order to further improve isolation performanceand reduce residual motion, it is necessary to counter-act the tilt effect. The only effective means of achiev-ing this is by active tilt control. A tilt sensor to sensetilt in two degrees of freedom (θx andθy ) is required,and a servo system and actuators to counteract themeasured tilt. The control loop is required to providegain over a range starting near the pre-isolator resonantfrequency (∼50 mHz), peaking at∼1 Hz and falling

off as quickly as possible above that. We discuss thedesign requirements of this tilt sensor and associatedcontrol system in the following sections.

With tilt stabilisation present, it becomes advanta-geous to improve the ULF pre-isolation performanceagain. The best approach here seems to be to add asecond stage of ULF pre-isolation (rather than try forimpossibly low resonant frequencies). With tilt stabil-isation in place and with a second ULF pre-isolationstage included, it should be possible to achieve theresidual motion indicated by the lowest curves (d) inFig. 1. In order to cascade a second ULF pre-isolatorfrom a first, the first must be made rigid to tilt. Ourcascaded ULF pre-isolation design will be the topic ofa future letter.

5. Tilt suppression requirement

A tilt suppression requirement could be derivedfrom a residual motion or servo force specificationat the test mass. However, a more practical approachis to require sufficient tilt suppression to allow fullbenefit to be obtained from a second ULF pre-isolationstage. This specification has the advantage that thelevel required is independent of the actual seismiclevel at a particular site. We will use the properties ofour existing pre-isolator in deriving this specification.That is, the tilt contribution is required to be at thelevel of the translational horizontal coupling of a dualpre-isolator.

The translational horizontal motion also containsa component which is mechanically cross-coupledfrom vertical motion — making it also dependenton vertical pre-isolation performance. This cross-coupling is usually assumed to be at the 1% levelset by fabrication accuracy of mechanical structures,but provided it can be measured and tuned, it can bemade as low as 10−4–10−5 at a particular frequency.This has been demonstrated in centre of percussiontuning of high performance isolators achieving 100 dBisolation [4,7].

We have recently built and tested a 3-D ULF pre-isolator which is rigid to tilt [3] so that a second pre-isolator can be cascaded after it. It is constructed ascascaded horizontal and vertical sections. For the hor-izontal isolation we used a platform on 1 m inversependulum legs. This tilt-rigid structure supports a La-

J. Winterflood et al. / Physics Letters A 277 (2000) 143–155 147

Coste straight-line suspension [10] for vertical isola-tion. We obtained vertical resonant periods in excessof 25 s with Q-factor>1, and horizontal resonant pe-riods of∼40 s. For a second stage, we assume per-formance based on our previous work — a full sizedScott–Russel pendulum, in which we obtained a pe-riod of∼150 s.

For the purposes of this analysis we will aim atconservative vertical and horizontal periods of 15and 20 s, respectively, for a first tilt-rigid pre-iso-lation stage, and 20 and 100 s for a second higherperformance stage. These parameters imply that at thetop of the stack the horizontal seismic noise will beabout 2% of the vertical. This ratio is sufficient to beable to ignore vertical seismic noise in the followingdiscussion. Hence we assume two cascaded horizontalpre-isolators with the following parameters:p1= 20 s,m1 = 150 kg,p2 = 100 s,m2 = 500 kg. Cascadingthese give the mode periods apparent in Fig. 1(d) of100 and 8 s. This pair produces a seismic attenuationin the 0.5 Hz frequency region of∼2× 10−5. Thissets the tilt suppression requirement in this frequencyregion to this value or below.

6. Seismic tilt expected

The seismic tilt spectrum in the frequency range ofinterest is rarely measured and to our knowledge hasnot been correlated with the vertical and horizontalspectrum amplitudes. However, making the reasonableassumption that the seismic motion travels as waves inthe surface of the earth, suggests that the tilt spectrumamplitude will be∼2π/λ times the vertical spectrumamplitude whereλ is the wavelength of the waves. Thephase velocity, which determinesλ varies greatly withfrequency and terrain composition (which is rarelyhomogeneous). For instance, at microseismic peakfrequencies (50–200 mHz) in eastern Montana thisvelocity was measured at∼3 km/s [11], but for shorterwavelengths where the bulk of the strain exists ina much softer surface layer such as sand we expectthe velocity to be much less. A velocity given forshear waves in sand below the water table [12] is500 m/s and we have taken this value as a likelyand conservative figure for the sand plains of WesternAustralia in the frequency region of most concernwhich is around 1 Hz. (The velocity above the water

table is given as 400 m/s and Raleigh waves travelsomewhat slower than pure shear waves.)

This figure finds some support in a measurementmade of the ground tilt seismic spectrum [13] at theVIRGO laboratory near Pisa. Unfortunately, the verti-cal and tilt seismic spectra were not measured simul-taneously, but assuming the usual 10−6/f 2 m/

√Hz

above 3 Hz, this measurement suggested a phase ve-locity of 700–800 m/s. We have assumed that the hor-izontal and vertical seismic noise have similar spectra(with the level shown by the dotted curve in Fig. 1),and are related to tilt using a constant seismic wave ve-locity of 500 m/s. This relationship directly producesthe expected seismic tilt spectrum shown by the dot-ted curve in Fig. 3. The expected higher velocities atlong wavelengths will make the low frequency end ofthe spectrum fall off much quicker and higher veloci-ties in general will reduce the tilt sensor sensitivity andloop gain requirement (by reducing the area betweengrey curves (c) and (d) in Fig. 1).

7. Tilt cancellation servo control loop

With these assumptions and tilt suppression require-ment, a control system may be designed to counter-act tilt (e.g., by means of piezo stack actuators) some-where prior to the second pre-isolation stage. The tiltmust be sensed after the actuation (through as rigid aconnection as is possible) and a high loop gain pro-vided to null the tilt as well as possible. The loop gainmay be tailored to only provide high suppression (sup-pression≈ 1/loop gain) at the necessary frequenciesand a suitable loop gain is shown in Fig. 2. It uses6 complex poles and 2 complex zeros and cuts offsharply above 20 Hz. It was manually tailored to min-imise the residual motion under free-mass condition(prior to fringe lock), and also to minimise the highfrequency performance required of the servo system.This loop gain was used to obtain the residual motioncurves shown in Fig. 1(d).

8. Tilt sensitivity and noise requirements

Clearly the sensitivity (noise floor) required from atilt sensor applied in this system must be such that tiltreadout noise does not contribute, through the servo

148 J. Winterflood et al. / Physics Letters A 277 (2000) 143–155

Fig. 2. Loop gain requirement for the tilt suppression in Fig. 1(d).

system, significant additional translation noise to thetest mass (over and above residual seismic feedthroughand final pendulum thermal motion). This sensitivityrequirement is the black curve shown in Fig. 3. At highfrequencies (>5 Hz) the thermal noise in the final pen-dulum dominates. At medium frequencies the seismictilt and translation feeding through the isolation dom-inates. At low frequencies (<0.2 Hz) a drift control

system which is not considered here will be used tocentre the entire stack using a low frequency positionerror signal and applying its control through the tilt ac-tuators. Its effect is not shown in any of the figures.

In Fig. 3 the required sensitivity is compared withthe Brownian motion thermal noise predicted froma 15 kg mass with 8 cm radius of gyration pivotedwith a resonant period of 100 s and various Q-factors.

Fig. 3. (a) Tilt sensor sensitivity requirement. (b) Thermal noise floor. (c) Shot noise floor achieved.

J. Winterflood et al. / Physics Letters A 277 (2000) 143–155 149

It is evident that for the frequency range of interest,thermal noise should not be a problem.

Also shown in Fig. 3 is the readout noise level thatwe have achieved with a relatively simple shadow sen-sor (actually, LED and 3 mm aperture with balancedphoto-diodes). This is clearly inadequate but we ex-pect that the “walk-off” angle sensor described in thefollowing sections will achieve the 10−11 rad/

√Hz re-

quired.Other sources of tilt sensing noise are cross-coupled

translational motion appearing as a tilt signal, andtemperature fluctuation gradients. Using a conductivemetal test mass in a vacuum with an isothermal shield,we do not expect temperature fluctuation noise tocontribute to the noise floor. Cross-coupled noise iscovered in the following section.

9. Recent angular accelerometer/tiltmeterresearch

Seismic tilt in this high frequency range of 0.1–5 Hzhas seen very little investigation until quite recently.In 1989 Speake and Newell [14] reported on a highfrequency (0.1–10 Hz) tiltmeter and a measurementof the seismic tilt in their laboratory. Their tiltmeteremployed a dumbbell shaped test mass of∼4 kg andlength∼20 cm, suspended very lightly and as close aspossible to its centre of mass using a commercial (Ben-dix) flex-pivot. The resonant period used was∼10 s,Q-factor∼100, and the relative position between theframe and test mass was measured with a balancedshadow sensor arrangement. A damping torque wasapplied by differentiating this position signal and ap-plying a small force using magnet and coil actuation.Low frequency stability was also obtained by integrat-ing this signal and applying it to drive the dumbbell to-wards the electrical zero for times longer than∼50 s.The seismic tilt measurement showed a level of the or-der of 1 nrad/

√Hz and was approximately flat with

frequency. However, this level was only a factor of 2above the noise floor of the instrument.

In 1992 Usher and Mat Isa [15] reported on a sim-ilar but smaller device using a test mass of∼0.3 kgand length∼10 cm, again suspended with flex-pivots.The free resonant period obtained was∼1.4 s but theyapplied force feedback to obtain an angular accelera-tion signal and the closed loop resonant frequency was

about 10 Hz (with a loop gain of∼100). They used adifferential capacitive transducer for sensing the offset(error signal) but obtained a very poor signal to noiseratio of approximately unity. This was apparently dueto the thermal noise (Brownian motion) of their testmass — exacerbated by the low frequency of interestwhich was a narrow band centred on 0.05 Hz. A factorof 4 improvement would be expected at 0.1 Hz.

In 1997 the VIRGO group [13] reported on a similardevice and used it to obtain a measurement of the tiltacceleration in the laboratory above 1 Hz. They useda test mass of∼2 kg and∼24 cm long suspendedinitially with beryllium copper crossed blade springs.Their resonant frequency could be tuned between 0.3and 3 Hz and also used the force feedback techniqueto obtain angular acceleration. They used a differentialinductor transducer (LVDT) for sensing the offset andthe closed loop resonant frequency was of the orderof 10 Hz. This allowed measuring the seismic tiltacceleration spectrum up to 10 Hz and above thatthe tilt displacement spectrum was obtained with thefeedback loop open. The signal to noise ratio obtainedwas more than an order of magnitude above 2 Hz, butthere was unexplained high noise below 1 Hz.

10. New tilt sensor design

The tilt sensor we propose in Fig. 4 incorporatesseveral novel features which allow wide band sensi-tivity extending to dc, low sensitivity to translationalmotion in the band of interest, and the ability to senseboth θx andθy tilts with a single test mass (allowingsimple central mounting of a single sensor unit).

Precision electric discharge machining (EDM) al-lows very thin and wide (<50 µm thick× cm wide)flexures to be made from a monolithic block of lowloss metallic material. With an arrangement of cutssimilar to the inset in Fig. 4 it is possible to makex–y flexures with intersecting axes from a monolithicblock. Suspending a disk shaped test mass very nearto its centre of mass with such a flexure allows sens-ing both θx and θy tilts. Utilising two of thesex–yflexures with a pendulum link between provides con-tinued sensitivity down to dc (thanks to the hang of thependulum) and can be made surprisingly insensitive totranslational motion. Sensitivity to dc is not a necessityfor this application but it provides a means to measure

150 J. Winterflood et al. / Physics Letters A 277 (2000) 143–155

Fig. 4. Dual axis tilt sensor.

earth tides and other low frequency tilts for correlationin noise studies or for calibration.

11. Translational motion rejection

Knowing both the translational seismic motion ex-pected and the tilt sensitivity required, allows us to plotthe level of translational motion rejection required of

the tilt sensor (sensitivity required/motion expected).This is shown in Fig. 5. The lower broken curve is therejection required if the tilt sensor is mounted beforethe first stage of pre-isolation and the higher curve isfor mounting after it. The latter mounting has a muchmore relaxed rejection requirement because its motionnoise is reduced as a result of the pre-isolation stage.

It follows from the equivalence principle, that if adevice is made sensitive to the gravitational field in

Fig. 5. Required translational motion rejection if sensor is mounted (a) after or (b) before first stage of pre-isolation. Solid curve is a typicalsensor performance available for particular design parameters.

J. Winterflood et al. / Physics Letters A 277 (2000) 143–155 151

order to provide a signal down to dc, then it must alsobe sensitive to horizontal acceleration. For this reasonother workers have not attempted to obtain dc tiltsensitivity in order to obtain the best immunity fromtranslational cross-coupling. However, we find fromanalysis that we can obtain very good translationalmotion rejection over the main frequency band ofinterest while retaining dc tilt sensitivity below it.This can be done by tuning the spring-rates of thepivots at the top and bottom of the pendulum so thatthe angle of the test mass mirror under translationalexcitation remains constant with respect to the fixedmirror. The typical performance of such tuning isthe solid curve in Fig. 5. Between the low frequencytilting resonance and the high frequency pendulumresonance, the cross-coupling can be made very lowand almost constant. It seems that this is quite awide tolerance adjustment (of order 1% in springconstant) and the level of rejection then obtaineddepends primarily on the Q-factor (1/loss tangent) ofthe flexure material.

The best material Q-factor we have measured for athin metal flexure membrane (50µm thick by 2 cmwide) is∼105 and was obtained from niobium [16].However, for this application a value of 103 seemsto be plenty and is readily obtainable from mostmetals (e.g., steel) if cut from a monolithic block.The particular curve shown was calculated for a 15 kgdisk with 8 cm radius of gyration, suspended by a

pendulum 5 cm long with flexures made from materialwith a (hysteretic damping) Q-factor of 103. Thislength was chosen to place the pendulum resonanceabove 2 Hz where its effect is readily kept below therequired level. Also being so short, both flexures maybe cut in a single EDM operation.

The flexure parameters for this design were notparticularly challenging although analysis shows thatreducing the effective Q-factor of the lower flexure(the low frequency resonance) gives a much flattersensitivity function at low frequency (shown in Fig. 6).This is achieved by making the lower flexure muchstiffer than need be and then offsetting this withsome inverse pendulum effect by attaching it at apoint significantly below the centre of mass of thedisk. The upper flexure was nominated to be 50µmthick and 2 mm wide to support the load at 50%yield (Rockwell 49 tool steel with extrapolated yieldstrength 2.8 GPa) giving an angular spring-rate of0.026 N m/rad, while the lower flexure was nominatedto have ten times this spring-rate but attached 1.68 mmbelow the centre of mass of the disk to give a96% spring-rate cancellation. With these values theresonant rocking period of the disk with pendulumheld rigid would be 20 s, but with the pendulum freeto tilt, the interaction gives the mode period of 65 sappearing in the graphs. The actual period of the lowfrequency rocking resonance is not important and itis intended that adjusting screws providing a small

Fig. 6. Transfer function of angle sensed from seismic tilt.

152 J. Winterflood et al. / Physics Letters A 277 (2000) 143–155

centre of mass height adjustment be used to adjustthe effective spring-rate of the lower pivot for the bestcross-coupling rejection.

A remarkable feature of most of these pendulumsuspended arrangements is that the resonances donot appear in the sensitivity function at all. Fig. 6shows the sensitivity function for the arrangementdescribed and it is apparent that there is no visibleeffect from the high Q-factor resonance just above2 Hz. The poles and zeroes are present in the transferfunction but they cancel almost perfectly (the plottingroutine used solves for all roots and plots a pointat each root). In fact, it was found that for manyareas of parameter space the low frequency resonancealso disappears entirely from the sensitivity function.Unfortunately, it does not seem possible to obtain thislow frequency resonance disappearance at the sametime as the spring-rate tuning required to minimisethe translational cross-coupling. However, in the caseof low frequencies, we intend to servo the disk tothe electrical zero, the force output becoming thetilt signal at these frequencies. This method has theuseful effect of flattening out any sensitivity variationsby the gain in the feedback. The disappearance ofthe 2 Hz resonance is convenient in that its effectdoes not need to be considered when determining thestability of the main tilt cancellation servo controlloop.

It is apparent from the cross-coupling curve inFig. 5, that both of the resonances are coupled stronglyto tilt. This means that they can be sensed and actuatedon by the control loop and magnet-coil actuators andcan be actively damped if required. This aspect has notbeen investigated however.

12. New angle sensor

The readout of angle may be achieved by variousmeans but we propose a laser beam walk-off sensor.A laser beam is bounced from a mirrored surfaceonto a quadrant photo-diode thereby obtaining bothangles with a single readout device. The laser beamdisplacement produced from a given change in test-mass angle can be greatly magnified by mountinganother mirror on the frame almost parallel to the testmass mirror surface (as shown in Fig. 4), and bouncingthe laser beam back and forth between them multipletimes before landing on the detector. We have termedthis angle sensing technique a “walk-off” sensor. Thespot displacement produced for a given mirror anglechange increases as the number of bounces squared,but due to finite beam divergence the sensitivity gainobtained will normally only increase as the numberof bounces. A report [17] on our initial experimentalinvestigation of the performance of such a device willappear shortly.

The number of bounces obtainable from a givenmirror size can be increased fourfold over the simpleparallel arrangement in Fig. 4 by presetting the fixedmirror to a small offset angle so that the laser traversesits width twice as illustrated in Fig. 7(a). If in additionone mirror is made slightly concave so that thediverging beam is re-focussed back at the detector asin Fig. 7(b), then sensitivity can be greatly increased(to go as the number of bounces squared) and thesignal becomes insensitive to laser beam pointingfluctuations.

The sensitivity (and inversely dynamic range) ofthe device is adjustable in steps by changing the

Fig. 7. (a) Walk-off sensor laser path. (b) Laser focussing profile.

J. Winterflood et al. / Physics Letters A 277 (2000) 143–155 153

preset angle of the fixed mirror (and aim of laser) sothat greater or fewer beam bounces occur. A smoothvariation is also possible by changing the spacingbetween mirrors to obtain a better or worse focussedspot at the photo-diode.

If a flat mirror is used on the test mass, then it isclear that the device is insensitive to test mass mo-tion parallel to this flat surface (i.e.,x, y andθz). Ifthe beam is detected at the samex–y position thatit is sourced from the laser (e.g., with a beam split-ter or by sourcing and detecting at the same positionon opposite mirrors) then the sensor is also insensitive(when centred) to spacing between the mirrors (z mo-tion). Since thez dimension is quite rigid, it is prob-ably sufficient to simply mount laser and photo-diodeoffset slightly from each other. However, if the mir-ror surface is not perpendicular to the pendulum it issuspended by, then horizontal motion in the directionof this offset will appear as variation in the spacingbetween the mirrors and will be sensed as cross cou-pling. This perpendicularity can be adjusted coarselyby mass redistribution and finely by a servo force ap-plied to the test mass.

13. Experimental test

A one-dimensional tilt balance was constructedas shown in Fig. 8. A brass bar 38 cm long and8 cm diameter, weighing 13.6 kg, was suspended nearits centre by a metallic glass flexure (GoodfellowCo66/Si16/B12/Fe4/Mo2) 5.5 cm long. The thick-ness of the flexure was 25µm, and its width was20 mm giving an angular spring rate under tension of0.023 N m/rad. Suspending this bar with this flexurewith the centre of flexing at the centre of mass wouldgive a period of 17 s. It was balanced and tuned with

Fig. 8. Experimental single axis tilt sensor using shadow sensor.

the adjusting screws to a give a period of∼20 s and theQ-factor was measured at∼190. (This suggests thatthe maximum period obtainable by further raising ofthe centre of mass would be∼270 s before becomingcritically damped.)

A simple 3 mm illuminated slot and split photo-diode shadow sensor with a sensitivity of 4× 10−11

m/√

Hz was used to detect the motion of the balanceat one end as shown in Fig. 8. The whole systemwas put in a container but without vacuum. The baris zero balanced with the end screws and its resonantperiod is adjusted coarsely by adding small lead plateson the top of the bar and finely with the top screws.The tilt noise measured by the shadow sensor is thesolid curve shown in Fig. 9 (also indicated in Fig. 3),and it showed that the tilt sensor’s sensitivity was∼2× 10−10 rad/

√Hz limited by the shadow sensor

noise. The peaks at about 2.1 and 6.4 Hz were thehorizontal mode of the tilt sensor and that of thesupport frame, respectively. Even though this sensorwas twice as sensitive as Speake and Newell’s [14], itseems that the seismic tilt in our basement laboratoryis not measurable at this level.

14. Conclusion

We have shown that the use of double stage ULFpre-isolation can result in the substantial reduction inresidual motion of test masses provided it is combinedwith effective tilt control. This greatly reduces thesize of the actuation forces required to control aninterferometer. This can be quantified for our assumedseismic level as shown in Fig. 10. Forces are reducedfrom ∼0.1 N for a simple all metal pendulum stack,to ∼1 µN for a damped, double pre-isolated and tiltstabilised system.

This system also reduces residual motion to thenanometre level at which it should take of order 100 sto drift through a 1µm laser fringe prior to acquiringlock. Depending on the low frequency sensitivityrequired of the overall instrument, this system mayallow direct actuation on the test mass by framemounted actuators. We believe this system to be aminimally complex and highly practical solution toachieve the high sensitivity goals of advanced GWdetectors.

154 J. Winterflood et al. / Physics Letters A 277 (2000) 143–155

Fig. 9. Seismic tilt measurement and predicted noise levels.

Fig. 10. rms force required to hold test mass fixed: (a) no pre-isolation or damping; (b) normal modes damped; (c) single pre-isolation(black= translation, grey= tilt); (d) dual pre-isolation with active tilt suppression (black= translation, grey= tilt).

Acknowledgement

This work was supported by the Australian Re-search Council.

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