Interferometric testbed for nanometer levelstabilization of environmental motion
over long time scales
Kenji Numata1,2,* and Jordan Camp2
1Department of Astronomy, University of Maryland, College Park, Maryland, 20742, USA2NASA Goddard Space Flight Center, Code 663, Greenbelt, Maryland, 20771, USA
Precision interferometric experiments at long timescales are necessary for a variety of space sciencemissions. These include Laser Interferometer SpaceAntenna (LISA), a gravitational-wave search requir-ing measurement precision of 10pm over 1000 s [1],and the proposed Gravity Recovery and ClimateExperiment (GRACE) follow-on, an Earth-orbitingpair of satellites designed to map the Earth’s gravityfield, requiring precision of 1nm over 100 s [2]. Thedevelopment of these missions requires extensivetesting of their interferometric systems on theground before launch.Sensitive interferometric measurements require a
very stable environment. However, these measure-ments take place in a laboratory environment whererelative drift of the components may exceed 1 μmover a typical length of meters. As a result, the mainmeasurement may be obscured. In this paper, we pre-
sent the design and test of an apparatus that sensesrelative drift between two platforms by stable inter-ferometers and provides suppression of the driftwith the use of mechanical actuators. The stabilizedplatforms then provide an environment wheresensitive interferometry over long time scales maybe performed.
A further advantage of the use of stabilized plat-forms is that the mechanical actuators may be usedto modulate the position and alignment of the appa-ratus supported by the platforms, allowing detailedstudies of transfer functions, noise, and cross cou-plings. This concept is also scalable to long distanceswithout change in performance. In contrast, fixingthe apparatus to a long continuous support does notallow for test actuation and limits the stability to thethermal expansion of the support, of an order of afraction of a micrometer over long time scales for1m of stainless steel.
The concept of this stabilized platform interferom-eter is similar to the suspension-point interferometer[3], which is proposed for the next-generationground-based interferometric gravitational-wave
detectors. In the suspension-point interferometer, aninterferometer measures and controls the distancebetween intermediate stages of suspended pendula,stabilizing the system at frequencies of up to about10Hz. In contrast, our platform is designed for sta-bilization at longer time scales, greater than 1 s.We were able to suppress the thermally and seis-
mically induced relative motion between two plat-forms separated by 1m from 4 μm to 8nm, a factorof 500 improvement, over a time scale of 1000 s.An iodine-stabilized Nd:YAG laser was used as thelight source of the interferometer, providing a stableabsolute frequency (length) reference. The relativedistance between optical benches supported by theplatforms was measured by the interferometer andlocked in reference to the laser wavelength by mov-ing one of the platforms with a piezoelectric transdu-
cer (PZT) driven hexapod actuator. The resultantdisplacement noise was limited by cross couplingfrom degrees-of-freedom (DoF) that were not con-trolled by the hexapod.
2. Experimental Setup
A. Conceptual Design
Figure 1 shows the conceptual setup of our experi-ment. We have two stable optical benches with twofiducial points. The optical benches are shown assquares in the figure. One of them (right square) ison a hexapod actuator. The relative position of thetwo optical benches change at long time scales dueto thermal and/or seismic motions (Fig. 1, top). Wedetect this environmental motion by measuring
Fig. 1. Concept of our experiment. Without active stabilization (top), the two separated optical benches (indicated as squares) drift withrespect to each other over long time scales. Wemeasure this motion with stabilization interferometers bymeasuring distances between thefiducial points on the optical benches. With active stabilization (bottom), the distances are kept constant by actuating the platform sup-porting the optical bench. The “virtually-connected” optical benches provide a stable environment for interferometric testing.
the distances between the fiducial points with stabi-lization interferometers.By feeding back the detected environmental
motions to the actuator, the distances between thefiducial points are held constant (Fig. 1, bottom).In Fig. 1, length and yaw information from thestabilization interferometers are fed back to thehexapod to null out the relative motion. Since thestabilized platforms become virtually connected,measurements performed on top of the platforms willtake place in a drift-free environment. The interfe-rometer used in such measurement is named “mea-surement interferometer” in this paper. As weexplain in detail below, we paid great attention tominimize drifts between and within interferometersto get the highest possible stability.The entire stabilized platform interferometer sys-
tem was set in a typical laboratory environmentwithout special external temperature or vibrationcontrol. The hexapod platforms, optical benches,and interferometers were placed in a vacuum tankwith pressure∼10−4 Torr. A single optical table madeof stainless steel was used to support the vacuumtank. The optical benches were separated by 1m.
B. Laser Source
The source for the interferometric sensors were low-noise nonplanar ring oscillator (NPRO) lasers. Inorder to get enhanced long-term stability, the laserfrequency is locked to an iodine absorption line usinga saturation spectroscopy technique [4]. Figure 2shows the setup of the iodine frequency stabilization.The 1064nm infrared light from the laser is con-verted into 532nm green light by a frequency-doubling crystal (PPLN:MgO) in an oven at 53 °C.The green light is split into two paths, each directedthrough an electro-optic modulator or an acousto-optic modulator, and then injected into an iodine cellas a counterpropagating beams. They behave as aphase-modulated probe beam and a frequency-shifted, chopped pump beam with modulation fre-quencies of 20MHz and 8kHz, respectively. The op-
tical path length within the cell was 30 cm. The coldfinger of the cell was kept at 5 °C. The signal fromhyperfine absorption line of iodine was fed back tothe laser through PZT and thermal tuning terminalsafter double demodulation. The infrared laser lightused for the interferometric sensor was picked offafter the frequency-doubling crystal using a dichroicmirror to filter the green light. We built two identicaliodine systems in order to evaluate actual frequencystability and to provide independent beam sourcesfor the stabilization interferometers and the mea-surement interferometer. The laser stabilization sup-presses frequency noise by a factor of over 1000; inAllan variance, the frequency stability is 10�12 in1000 s with our stabilization system, which is goodenough to allow picometer level measurements overa total path length of 1m, where we have used therelationship of ΔL=L ¼ Δν=ν. Here, ΔL is the mea-surable length variation, L is the path length, andΔν=ν is the relative frequency stability of the laser.
The laser beams are coupled into polarization-maintaining fibers and sent into the vacuum tanksthrough fiber feedthroughs. The output beams fromthe fiber couplers go into free-space interferometersinside the vacuum tank.
C. Laser Interferometers
Figure 3 shows the setup of the stabilization interfe-rometers. There are two homodyne Michelson inter-ferometers, which have nonpolarized beam splittersand corner cubes. The use of corner cubes allows usto align these interferometers easily, removing un-wanted angle–length couplings. We obtained the dis-placement signal by differentiating two fringesignals from both sides of the beam splitter and bycontrolling the hexapod at the middle (zero point)of the fringe, where the intensity change is propor-tional to the length change. This differential signalenables us to suppress direct coupling from a laserintensity fluctuation to an interferometer output.
We used two types of measurement interferom-eters. The first was a homodyne Michelson
Fig. 2. Laser frequency stabilization system using iodine. FI, Faraday isolator; CR, frequency-doubling crystal in an oven; DM, dichroicmirror; HWP, half-wave plate; EOM, electro-optic modulator; AOM, acousto-optic modulator; PBS, polarized beam splitter; M, mirror; L,lens; D, detector; MX, mixer; LPF, low-pass filter; f E, EOM modulation frequency; f A, AOM modulation frequency.
interferometer with corner cubes as end mirrors(Fig. 4). To read out the displacement informationcontinuously at any location of the fringe, we useda λ=8 wave plate and recorded two interferometeroutputs, which provide two distinct light polariza-tions. The λ=8 wave plate provides an additionalπ=2 phase shift in round trip for the polarization com-ponent parallel to the slow axis of the wave plate.When one of the polarizations is at the fringe peak,the other polarization is at the middle fringe due tothe phase shift. Thus, the length signal is obtained bycalculating the arctangent of the ratio of the two sig-nals. This type of interferometer is used in metrologysystems in which continuous length output overdistances greater than a light wavelength is re-quired [5].The second type of measurement interferometer
was a heterodyne interferometer, shown in Fig. 5.Here a third laser source is offset-phase locked tothe iodine-stabilized laser of the measurement inter-
ferometer by a digital phase-locking servo with atypical offset frequency of 4:3MHz. The heteorodyneinterferometer consists of four beam splitters. One ofthe interfering signals is used to phase lock the thirdlaser and the other signal is used as a length outputat the other optical bench. A phase change of thelength-output signal contains the length informationand is read out by a phasemeter [6]. The phasemetercompares the phase of the fringe and that of the re-ference RF signal, using a internal phase-locked loopalgorithm. The phasemeter does not measure con-stant drift; however, it allows us to measure opticalphase variation continuously as long as phase doesnot jump over 2π between the sampling points.Our concept of phase locking and phase measure-ment at each end is similar to the plan for the LISAinterspacecraft interferometer [7].
D. Optical Bench
To get high stability between the optical componentsof the interferometers, we used monolithic glass
Fig. 3. Stabilization interferometers and control scheme. COL, collimator; CPL, coupler; PM, polarization-maintaining fiber; NBS, non-polarized beam splitter; CC, corner cube; HVAmp., high-voltage amplifier. Other notations are the same as at Fig. 2. The optical setup formeasurements interferometer is not shown in this figure.
Fig. 4. Measurement interferometer (homodyne type). HQWP, λ=8 wave plate; GT, Glan–Thompson polarizer. Other notations are thesame as at Fig. 2. Stabilization interferometers are not shown here.
optics. The bottom sides of the corner cubes andbeam splitters were cut into flats and optically po-lished. The optical bench was made of polished ultra-low expansion (ULE) glass plates, and the polishedoptical components were rigidly attached to it duringthe process of optical alignment. We used solution-assisted optical contacting [8] for optical componentsthat needed to be removed occasionally. For perma-nent bonding, we applied sodium silicate solution be-tween the two mating surfaces and silicate-bondedthe components to the ULE glass plates [9]. Thiskind of silicate-bonded monolithic interferometerhas been adopted for the LISA Pathfinder mission,showing stability at the picometer level [10].
E. Hexapod System
Figure 6 shows the hexapod actuator that we built.The hexapod provides six DoF movements with arigid structure. It is typically designed to have largedynamic range and has been used for many applica-tions that include optics and mechanics. In our case,the required dynamic range is of the order of 10 μm.Therefore, we designed our hexapod with PZT actua-tors (P-845.30 from Physik Instrumente) with 40 μm
travel, giving the same order of range for the hexa-pod. They are connected to a top platform (200mmdiameter) and a bottom plate through flexible tipjoints. Uralane was inserted between the joints todamp their resonances. Nominal load, includingthe top platform, is about 5kg.
The hexapod PZTs were driven by a matrix circuitand high-voltage amplifiers (Fig. 3). The matrix cir-cuit generates appropriate linear combinations ofsignals for each leg. The linear combination was cal-culated from our mechanical geometries and the ana-lysis found in [11]. Output signals from the matrixcircuit were connected to the high-voltage amplifiers,which provide driving voltages for the PZTs. Figure 7shows the measured transfer function from the ap-plied voltage to the horizontal displacement of thetop platform. The hexapod has resonances around200Hz with Q of 5. Below the resonances, the hexa-pod has almost flat response in the frequency do-main. Our closed-loop measurements with theinterferometer showed that the response was con-stant down to at least a 10pm peak-to-peak actua-tion level.
F. Control System
In this setup, there were two stabilization inter-ferometers to control two DoFs (length and yaw)and one measurement interferometer to read outone DoF (length). The length and yaw are obtainedby summing or differencing two length signals ofthe stabilization interferometers (Fig. 3). In orderto apply appropriate servo filters, the signals fromthe interferometers are input to a digital signal-processing system (dSPACE from dSPACE, Inc.),and its outputs are connected to the matrix circuit.The digital signal-processing system was useful forour application because the time constant of theservo filter is large. The control bandwidth was
Fig. 5. Measurement interferometer (heterodyne type). A third laser source is used to generate heterodyne signal. Notations are the sameas at Fig. 2. Stabilization interferometers are not shown.
Fig. 6. Hexapod. The top platform is supported and driven by sixPZT actuators, which are connected by flexible tip joints. The topplatform has mounting holes for optics.
10Hz for both DoFs and was limited by the mechan-ical resonances of the hexapod.
3. Experimental Result
Figure 8 shows the displacement motions with andwithout the stabilization controls in the frequencydomain. It shows results using both the homodyneand the heterodyne measurement interferometers.
Without the stabilization, the relative displace-ment motion is about 0:5 μm=
ffiffiffiffiffiffiffi
Hzp
at 1mHz, roughlyconsistent with thermal expansion of stainless steel(thermal linear expansion of 10−5=K) and ameasuredtemperature variation of 10−1 K=
ffiffiffiffiffiffiffi
Hzp
and 10−2 K=ffiffiffiffiffiffiffi
Hzp
outside and inside the vacuum tank, respec-tively, at 1mHz. With the stabilization, the relativemotion becomes about 1nm=
ffiffiffiffiffiffiffi
Hzp
and 4nm=ffiffiffiffiffiffiffi
Hzp
with the homodyne and the heterodyne
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ase [deg
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Fig. 7. Transfer function of hexapod. The transfer function from matrix input voltage to displacement is shown. The peak structuresabove 200Hz are mechanical resonances.
Fig. 8. Stabilization result in frequency domain. “Free-running” indicates displacement motion between two optical benches before sta-bilization. “Stabilized (Homodyne)” and “Stabilized (Heterodyne)” indicate measured motions with the homodyne and the heterodynemeasurement interferometers, respectively, after stabilization. The dashed–dotted curve shows LISA stability requirement needed forground testing using two optical benches.
interferometers, respectively, at 1mHz. The suppres-sion factor is about 500 and 125 and is nearly con-stant below 0:01Hz. To our knowledge, thissuppression factor is the largest ever achieved forthis kind of experiment. Above 10Hz, there is no ef-fect of stabilization due to the 10Hz controlbandwidth.Figure 9 shows the stabilization result in the time
domain. Without the stabilization, the drift motionbetween the two benches was typically 4 μm peakto peak over 1 day. This low-frequency componentis believed to be from thermal drift of the opticaltable and the hexapod. Stabilized motion was about8nm peak to peak with the homodyne measurementinterferometers.
4. Discussion
In this section, we discuss the noise sources in thestabilization experiment. In Fig. 10, we plot esti-mated noise contributions in the frequency domain.We identified the performance limiting factor to becross coupling from uncontrolled DoFs.
A. Noise Sources
“Inloop noise” in Fig. 10 indicates the noise level cal-culated from the spectrum of control error signal ofthe stabilization interferometer. At frequencies be-low 0:1Hz, the servo gain is high enough so thatthe actual stability measured by the measurementinterferometer is limited by other noises, not bythe loop gain. Between 0.1 and 10Hz, the loop gainis low and it limits the performance.Laser frequency noise is suppressed below the
measured level with the iodine frequency reference.The frequency noise level was estimated from thebeat note between the two independent but identicallaser systems.Electronics noise sources, such as control circuit
noise and detector noise, do not limit the stabilityperformance. Although we have relatively large con-trol circuit noise (input equivalent noise of μV=
ffiffiffiffiffiffiffi
Hzp
)due to the digital system, the noise level is sup-
pressed by the large loop gain and does not appearin the stabilized motion.
We have evaluated other noise sources, such ascomponent drift within a single optical bench, ther-mal path-length change in transmissive optical com-ponents, beam pointing fluctuation, laser intensitynoise, and phasemeter noise in the case of the hetor-odyne measurement interferometer. Through dedi-cated measurements, all these contributions wereevaluated to be too small to explain the measuredmotions in Fig. 8.
B. Mechanical Cross Coupling
The most likely explanation for the measured motionis mechanical cross coupling from uncontrolled DoFs.In Fig. 10, we plotted 0.2% of free-running motion as“Coupling from other DoF.” As we show below, itslevel agrees with the calculated coupling from free-running motion, indicating that noise couplings fromuncontrolled translational motions (shear and pistonin Fig. 1) limit the interferometer sensitivity atthis level.
These couplings occur through imperfections inparallelity and perpendicularity of the interferom-eter beams. For example, when two stabilizationbeams are in parallel along the length DoF in Fig. 1and the measurement beam is tilted by θ horizon-tally, shear motion, Δy, is detected by the measure-ment interferometer as ∼θΔy when a corner cube isused as an end mirror. In our case, the location of theoptical components and beam pointing were set atmillimeter accuracy over the 1m separation betweenthe benches. Thus, the offset angle θ is of the order of0.1%. Since we have these imperfections for everybeam in both vertical and horizontal directions, theymost likely explain the 0.2% coupling observed in thehomodyne measurement interferometer. In the caseof the heterodyne interferometer, in which cornercubes were not used, the motions to uncontrolledDoFs have larger effect, since the two measurementbeams exchanged between the two benches are notperfectly parallel. Also, such an interferometer with-out corner cubes converts angle fluctuations into
-4
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[nm
]
50403020100
Time [hour]
-4
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0
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4 Disp
lacemen
t [µm
]Free-running (right axis)
Stabilized (Homodyne, left axis)
Fig. 9. Stabilization result in time domain. Note that stabilized motion is at the nanometer level (left axis), and free-running motion is atthe micrometer level (right axis). They were measured at the same time.
length fluctuations at the second order, which wouldpredict slightly worse stability with the heterodynemeasurement interferometer than with thehomodyne.We evaluated this effect experimentally using the
heterodyne measurement interferometer. Whilelength and yaw control loops were active, we drove
each of the six hexapod DoFs and measured thetransfer function from that DoF motion to the mea-surement interferometer output. The effect of eachcross coupling was estimated by multiplying thehexapod transfer function and the free-running mo-tion. We assumed that these free-running motionshad the same spectral shape and amplitude as those
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tHz]
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Frequency[Hz]
Circuit noise
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Inloop noise
Stabilized (Homodyne)
Coupling from other DoF
Fig. 10. Noise sources of interferometer (homodyne). “Stabilized (Homodyne)” indicates the stabilization results by homodyne interfe-rometer (same as in Fig. 8). 0.2% of free-running noise in Fig. 8 is plotted as “Coupling from other DoF.” The peaks around 30Hz are fromseismic motion seen in our laboratory environment.
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pla
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ise
[m/r
tHz]
0.0001 0.001 0.01 0.1 1 10 100
Frequency[Hz]
Stabilized (Heterodyne)
Coupling from shear
Coupling from pitch
Fig. 11. Noise sources of interferometer (heterodyne). “Stabilized (Heterodyne)” indicates stabilization results with the heterodyne mea-surement interferometer (same as in Fig. 8). The two noise levels due to coupling were estimated from direct measurement of transferfunction and free-running motions.
of length and yaw. Figure 11 shows the estimates ofcross coupling from pitch and shear and also thenoise of the heterodyne measurement interferom-eter. It shows that the coupling from shear had thelargest contributions and can explain the noiseperformance.
5. Applications
In this section, we discuss applications of our testbed,showing an experimental example.
A. Testing of Interferometer
As an application example of our system, wemounted a commercial interferometer on top of thestabilized platform and measured its stability.The interferometer used here is the ZMI-510 dis-
placement measuring interferometer (DMI) fromZygo Corporation. DMI is a polarized heterodyne in-terferometer in which and internally stabilizedHe–Ne laser is used as a light source. We used anoptically contacted mirror, mounted on one of theULE optical benches, as an end mirror of the DMI.The other reference mirror and a beam splitter arehoused in a single package and it was mounted onthe other ULE optical bench through a mountingblock and a silicate-bonded invar base plate. Therelative motion of the two optical benches was stabi-lized using our system.Figure 12 shows the measured result in the time
domain. We observed 100nm peak-to-peak drift over2 days. It is a factor of 25 larger than our own homo-dyne measurement interferometer. We also observeda factor of ∼25 larger displacement noise spectrum.The motion was clearly correlated to room tempera-ture. We believe this noise is mainly from the DMIlaser frequency drift, whose specification is Δν=ν ¼�0:01ppm=hour. It could give ΔL ¼ �10nm=hourdrift over 1m path length, L, and is consistent withwhat we measured. This shows our concept can beused to evaluate a stable interferometer over longtime scales.
B. Ground Testing of Space Interferometer Missions
There are several ambitious interferometric spacemissions planned to be launched within two decades.In a manner similar to our DMI test discussedin Subsection 5.A, our concept can be applied forground testing of such missions.
One example is LISA, in which laser interferome-try must be performed at a 20pm=
ffiffiffiffiffiffiffi
Hzp
level at1mHz between two separated optical benches. Theoptical benchesmust be tested on the ground in orderto verify the system displacement sensitivity. How-ever, μm level ground motions will disturb suchmeasurements. Therefore, our system to suppressenvironmental motions can be used to enable thetesting. It will also enable us to measure the systemresponse to disturbances at the picometer and nanor-adian levels. At this moment, our system is not satis-fying the requirements of the LISA optical bench(13pm=
ffiffiffiffiffiffiffi
Hzp
is allocated, Fig. 8) due to the crosscoupling of noise from uncontrolled DoFs (discussedin Section 4). However, we expect that this noise willbe suppressed when the testbed is scaled up to largersystem that allows us to use all-DoF detection andcontrol.
C. Active Control of Secondary Mirror of Telescope
In reflective telescopes, a hexapod has been usedas an actuator to correct images of stars (see, forexample, [12]).
In advanced telescopes such as Terrestrial PlanetFinder—Coronagraph (TPF-C), thermal motion ofsecondary relative to primary must be of the orderof 25nm over 1 day in order to get extremely highcontrast ratio in its coronagraph. This level of posi-tion control demands active length correction usinglaser metrology [13]. In the system, reference cornercubes are attached to the mirrors and several beamswill be exchanged between them to perform such sta-bilization. Our testbed results satisfy the TPF-C re-quirement and, thus, is working proof of such aconcept of active stabilization.
Fig. 12. DMI result in time domain. The curve labeled “DMI result (left axis)” represents displacement measured by DMI while ourstabilization system is working. The curve labeled “Room temperature (right axis)” represents the temperature change around theDMI laser source during the measurement.
D. Reduction of Low-Frequency Relative Drift in Ground-Based Interferometer
A final application is a reduction of relative motionfor stable operation of sensitive interferometers.The Laser Interferometer Gravitational-Wave
Observatory (LIGO) uses stabilized lasers and highlysensitive interferometers in its goal of measuringgravitational waves in the 100Hz to 1kHz range.It monitors motion at the 10−18 mrms level with sus-pended 4km Fabry–Perot cavities on resonance [14].Relative drift between the suspension platform isabout �50μm over 12h [15]. The seismic motionbelow 40Hz is suppressed by controlling the sus-pended mirrors and the platforms in order to keepthe cavity on resonance. However, this system worksonly after the cavity achieves the stable operatinglength, allowing it to stay in resonance with the laserlight (also known as lock acquisition). Advanced de-signs of LIGOmay use higher cavity finesse, with theresult that lock acquisition becomes more difficultdue to the presence of relative seismic motion thatdisturbs suspended mirrors. Our stabilization con-cept, applied to the suspension platforms, will easethe lock acquisition difficulties and eases the dy-namic range requirements of the cavity lockingservo. Therefore, it has been discussed as a possibleoption for the advanced LIGO interferometer that isin preparation [16]. It can be installed without affect-ing suspension design.
6. Summary and Conclusion
We developed an interferometric testbed to stabilizeenvironmental motion at themillihertz range using afrequency-stabilized laser, laser metrology, and ahexapod actuator system. The environmental motionwas suppressed down to the nanometer level, com-pared to an uncontrolled noise of a few μm due tothermal and seismic drifts. To our knowledge, thesuppression gain of 500 is the highest value amongthis kind of “suspension-point interferometer” ex-periment. We have identified the importance of crosscoupling from uncontrolled DoFs and we expect thatthe performance will be improved by installing moreinterferometric sensors to control such DoFs in fu-ture work. The problem of managing thermal/seismicdrifts is common for many sensitive measurements.We showed the practicality of active control of drift,which will have many applications that require ex-tremely stable relative environments, such asground testing of the LISA optical bench.
The authors would like to thank to Tupper Hyde,Gary Brown, and Armando Morell at NASA/GoddardSpace Flight Center for their contribution to the de-sign of our hexapod. The authors also would like tothank Bill Klipstein, Brent Ware, Robert Spero, andDaniel Shaddock at NASA/Jet Propulsion Labora-tory for their kindness to let us use their phasemeterin our testbed and for useful discussions. The authorswould like to acknowledge useful comments fromGerhard Heinzel at AEI Hannover.
References
1. P. Bender, K. Danzmann, and the LISA Study Team, “Laserinterferometer space antenna for the detection of gravita-tional waves, pre-Phase A report,” 2nd ed. Tech. Rep.MPQ233 (Max-Planck-Institut für Quantenoptik, 1998).
2. M. Stephens, R. Craig, J. Leitch, R. Pierce, R. Nerem, P.Bender, and B. Loomis, “Demonstration of an interferometriclaser ranging system for a follow-on gravity mission toGRACE,” in Proceedings of IEEE International Conferenceon Geoscience and Remote Sensing Symposium (IEEE, 2006),pp. 1115–1118.
3. Y. Aso, M. Ando, K. Kawabe, S. Otsuka, and K. Tsubono,“Stabilization of a Fabry–Perot interferometer using a suspen-sion-point interferometer,” Phys. Lett. A 327, 1–8 (2004).
4. J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Opticalheterodyne saturation spectroscopy,” Appl. Phys. Lett. 39,680–682 (1981).
5. A. Araya, T. Kunugi, Y. Fukao, I. Yamada, N. Suda, S.Maruyama, N. Mio, and S. Moriwaki, “Iodine-stabilizedNd:YAG laser applied to a long-baseline interferometer forwideband Earth strain observations,” Rev. Sci. Instrum. 73,2434–2439 (2002).
6. D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B.Klipstein, “Overview of the LISA phasemeter,” in Proceedingsof Laser Interferometer Space Antenna: 6th International LISASymposium, S. M. Merkowitz and J. C. Livas, eds. (AmericanInstitute of Physics, 2006), pp. 654–660.
7. D.Weise, C. Braxmaier, M. Kersten,W. Holota, and U. Johann,“Optical design of the LISA interferometric metrologysystem,” in Proceedings of Laser Interferometer Space Anten-na: 6th International LISA Symposium, S. M. Merkowitzand J. C. Livas, eds. (American Institute of Physics, 2006),pp. 389–394.
8. D. Shaddock andA. Abromovici, “Solution-assisted optical con-tacting: components in optical contact canbeadjusted for abouta minute,”NASATech. Brief NPO30731 (NASA, 2004), http://www.nasatech.com/Briefs/Mar04/NPO30731.html.
9. E. J. Elliffe, J. Bogenstahl, A. Deshpande, J. Hough, C. Killow,S. Reid, D. Robertson, S. Rowan, H. Ward, and G. Cagnoli,“Hydroxide-catalysis bonding for stable optical systems forspace,” Class. Quantum Grav. 22, S257–S267 (2005).
10. G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann,F. Draaisma, A. Garci’a, J. Hough, O. Jennrich, U. Johann,C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Rüdiger,R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful test-ing of the LISA Technology Package (LTP) interferometer en-gineeringmodel,”Class.QuantumGrav.22, S149–S154 (2005).
11. K. Liu, J. M. Fitzgerald, and F. L. Lewis, “Kinematic analysisof a Stewart platform manipulator,” IEEE Trans. Ind. Elec-tron. 40, 282–293 (1993).
12. C. Pernechele, F. Bortoletto, and K. Reif, “Hexapod control foran active secondary mirror: general concept and test results,”Appl. Opt. 37, 6816–6821 (1998).
13. J. A. Dooley and P. R. Lawson, “Technology plan for theTerrestrial Planet Finder Coronagraph,” JPL publication05-8 (Jet Propulsion Laboratory, 2005).
14. B. Barish and R. Weiss, “LIGO and the detection of gravita-tional waves,” Phys. Today 52, 44–50 (1999).
15. F. Raab and M. Fine, “The effect of Earth tides on LIGO inter-ferometers,” LIGO document LIGO-T970059-01-D (LaserInterferometer Gravitation Wave Observatory, 1997), http://www.ligo.caltech.edu/docs/T/T970059‑01.pdf.
16. LSC Advanced Detector Committee, “LSC instrumentscience white paper 2007,” LIGO document LIGO-T070137-00-R (Laser Interferometer Gravitational Wave Observa-tory, 2007), http://www.ligo.caltech.edu/docs/T/T070137‑05/T070137‑05.pdf.
