Fiber-optic extrinsic Fabry–Perotvibration-isolated interferometer for use inabsolute gravity meters

Eric L. Canuteson and Mark A. Zumberge

In an absolute gravitymeter, a laser interferometermeasures the position of a test mass that is falling ina vacuum. The calculated value of gravity is the average acceleration of the mass during a set ofdrops. Since systematic accelerations of the optical system will bias the measured value of gravity,various interferometer geometries have been implemented in the past to isolate the optical system fromground motion. We have developed and tested a low-finesse fiber-optic extrinsic Fabry–Perotinterferometer that is fixed to the mass of a critically damped seismometer in which the effects ofsystematic ground motion and acoustic vibrations are reduced. r 1996 Optical Society of America

1. Introduction

The need to determine accurately the local accelera-tion that is due to the Earth’s gravity can be found inseveral important geophysical and metrologic appli-cations. Such absolutemeasurements of the Earth’sgravity usually rely on secondary length and timestandards—an iodine-stabilized laser for length anda rubidium oscillator for time. Examples of theinstances in which gravity g must be absolutelyknown include calibration of relative gravity metersused in mineral and oil exploration, measurementsof deformations in the Earth’s crust, calibration ofpressure meters, and studies of gyroscopic systems.1Potentially, absolute gravity information will be-come a critical component in a new definition of thekilogram.The methodology for absolute gravity measure-

ments has been well developed over the past twodecades.2,3 Figure 1 depicts the technique utilizedby virtually all absolute gravity meters. A massholding a retroreflector is dropped within an evacu-ated chamber while its position is tracked with alaser interferometer. As the falling retroreflectoraccelerates, it generates interference fringes at anincreasingly rapid rate 1one fringe is generated for

The authors are with the Institute of Geophysics and PlanetaryPhysics, Scripps Institution of Oceanography, University of Cali-fornia, San Diego, La Jolla, California 92093-0225.Received 3 April 1995; revised manuscript received 29 August

1995.0003-6935@96@193500-06$10.00@0r 1996 Optical Society of America

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each half-wavelength of the retroreflector’s motion2.One can obtain a record of themass’s fall through thetiming and counting of interference fringes. Fromthis record, the acceleration that is due to gravity g iscalculated. In a typical instrument, the length offall is approximately 15 cm, lasting 0.175 s andgenerating 474,000 interference fringes.Attention must be paid to sources of systematic

error in the measurement. While there are cases ofmeters nearly identical in design that produce mea-surements that agree to 1 µGal, agreement among acollection of differently designed meters is at the5–10-µGal level.4 11 µGal 5 1026 cm@s2, which isapproximately 1029g; the Gal is the widely acceptedunit of acceleration in geodesy and metrology.2There is still a need to examine possible sources ofsystematic error that may come into play at the5-µGal level. One of these is the effect of vibrationson the optical system.In Figure 1, the optical system is attached to the

Earth’s surface. The falling mass, of course, is notin contact with anything. If the ground acceleratesduring the drop 1because of seismic motion, forexample2, then themeasurement will yield the differ-ence between the falling mass’s acceleration and theground’s acceleration. To ameliorate this source ofnoise, the second retroreflector is usually mountedon a spring suspension. The spring isolates themeasurements from ground motion at frequencieshigher than the resonance frequency of the isolationmass–spring system. For simplicity, inmost instru-ments only the reference retroreflector is attached tothe spring. In that case, the system is isolated only

from vertical motions of the ground 1they change thetwo path lengths in the Michelson interferometerequally2.In some instruments, the mass–spring resonance

period is relatively short 1e.g., 1 s2. In others, theperiod of the spring is made longer electronically1e.g., 30 s2 to isolate more of the seismic spectrum.5Examination of Fig. 1 reveals that perturbations

to the interferometer other than vertical groundmotions can produce unequal optical path-lengthchanges in the two interferometer arms. For ex-ample, if the platform on which the optics aremounted rotates around an axis perpendicular to theplane of the figure, an optical path-length changewill be produced in an amount proportional to thelateral separation of the two retroreflectors. Addi-tionally, acoustic vibrations that affect this separa-tion will also be sensed by the interferometer. Thesought-for accuracy in the most precise measure-ments, 1029g, requires displacement sensitivity of1023l, where l is 633 nm, the wavelength of thestabilized laser. Therefore these acoustic vibra-tions can be problematic indeed. While seismicnoise can produce large scatter in the gravity valuesdetermined in individual drops of the test mass,their time signature is essentially random. Acous-tic and mechanical vibrations caused by the instru-ment’s dropping mechanism, on the other hand, arenot. It is such phase-locked vibrations that limitthe instrument’s accuracy because they cannot beaveraged away.

2. Design of the Interferometer

One solution to the reference frame accelerationproblem would be to mount the entire optical systemonto the isolation spring. While being somewhatimpractical 1portability is a key attribute in a useful

Fig. 1. Acceleration of a retroreflector falling in a vacuumchamber determined interferometrically.

absolute gravity meter2, this configuration wouldstill be susceptible to acoustic vibrations that wouldmodulate the path lengths separating the bulk opti-cal components. Low-finesse extrinsic fiber interfer-ometers have proved useful in a variety of applica-tions that involve vibration and displacementmeasurement.6–8 We have constructed a particularlow-finesse Fabry–Perot interferometer and inte-grated it into an absolute gravity meter. The resultis an absolute gravity optical system that is tiltinsensitive, compact, and relatively insensitive toground and acoustic vibrations. The interferome-ter has the additional advantage that it is possible toalign the measurement axis with the local directionof gravity quickly and remotely.Figure 2 is a schematic diagram of the new optical

system. Light from a polarization-stabilized laser1Newport Model NL-12 is launched into a single-modeoptical fiber. This laser has been chosen both be-cause of its small size and lack of frequency modula-tion 1compared with, for example, an iodine-stabi-lized laser2. Its wavelength stability is periodicallychecked against an iodine-stabilized laser. An eva-nescent-wave fiber-optic beam splitter replaces thetraditional bulk optic beam splitter. The light trav-els toward two reflectors that form an etalon; theresulting interference pattern returns back throughthe coupler and to the detector. The two reflectorsthat comprise the etalon are the cleaved fiber end-face and a falling retroreflector. Because there is nomodulation in the laser’s wavelength, the phasenoise caused by the optical path imbalance is limitedto that caused by the linewidth of the laser 1less than0.001l for a path-length difference of less than 1 m2.The fiber-optic cable terminates at the focal point

of a positive lens with a focal length of 30 mm.Approximately 4% of the incident light reflects backinto the fiber at the interface between the fiber andthe air. The lens collimates the remainder of the

Fig. 2. New optical system that is essentially a low-finesseFabry–Perot fiber-optic interferometer.

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light, and the resultant beam impinges on the centerof a falling corner cube retroreflector. Since thebeam strikes the center of the corner cube, the lighttravels back along its original path and is refocusedinto the fiber. The primary interference is betweenthe first reflection from the air–fiber interface andthe first reflection from the corner cube; these gener-ate an approximately sinusoidal fringe signal. Ofcourse, there are multiple reflections at the interfacebetween the fiber and air; these reflections, succes-sively reduced in intensity, undergo 180° phaseshifts since the incident light is in a lower impedancemedium. The detector measures the resultant lightintensity of all the reflections at the output arm ofthe coupler. We show below that the deviation froma purely sinusoidal fringe pattern is small.The ultimate goal is an accurate determination of

the acceleration of the falling retroreflector relativeto the pseudoinertial reference frame to which thefiber endface is fixed. Therefore, wemust relate theresulting interference pattern to the acceleration ofgravity and show that we can measure the resultingintensity and determine g in an efficient way.Initially, we noticed that an interferometer such asthat depicted in Fig. 2 produced a fringe signalsimilar to that of a Michelson interferometer, havinga nearly full depth of modulation. This will be thecase only if losses occur in the coupling of the lightreturned from the retroreflector back into the fiber.It has been shown that the higher-order reflectionscan have a significant impact on the resulting fringesignal from a low-finesse Fabry–Perot interferome-ter.6 For the case of our particular instrument, wecan account for the effect from the extra reflectionseasily bymodifying a unique data-processingmethod1described below2 developed for a two-beam interfer-ometer; we have found that the effect is small.Considering the first three reflections—the initial

fiber–air reflection, the retroreflector reflection, andthe second reflection from the fiber–air interface andthe retroreflector—the resulting intensity F at thedetector is

F1t2 5 C0 1 C1 sin3f0 1 2k1v0t 1 ‰gt224

1 C2 sin3f0 1 4k1v0t 1 ‰gt224, 112

where t is the time since the object began to fall, l isthe wavelength of laser light, f0 is the initial phase,v0 is the initial velocity, g is gravity, and k 5 2p@l.C0, C1, and C2 are the various intensities of theinterfering components and depend on several fac-tors, including the reflection coefficient of the air–fiber interface, the efficiency with which light re-turned from the corner cube is coupled into the fiber,and other losses. We observe that C0 is about thesame size as C1 but C2 is much smaller 1C2 < 0.01C12.This is consistent with a sizable loss in coupling lightback into the fiber, as would be caused by lensimperfections or a nonnormal cleave angle. Since

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the loss causes C2 to be small, we can ignore higher-order reflections whose losses grow geometrically.Another possible source of error in this interferom-

eter is the diffraction caused by the edges of thecorner cube. In past absolute gravity meters, thebeam has been offset a distance from the center.In our design, the beam strikes the center and so thediffraction caused by the edges is potentially asource of error. However, this error term has beenpreviously calculated9:

crack error 5 19

2p1@2

a0w02 3 diffraction error, 122

where a0 is the width of the cracks and w0 is thebeam radius. With a beam radius of 3 mm 1theoptical fiber’s numerical aperture, 0.1, times thecollimating lens focal length, 30 mm2 and a cornercube whose cracks are less than 0.125 mm, thissource of error is smaller than the diffraction error3l212pw0222 5 1.1 3 10294 by a factor of 10.

3. Fringe Processing

The fringe-processing technique for this optical sys-tem is based on a new method developed for atwo-beamMichelson interferometer.10 Rather thanthe traditional method of fringe processing in whichthe zero crossings of the ac-coupled fringe signal arecounted, the new method consists of sampling thefringe signal at constant time intervals and fittingthe resulting samples using an iterative nonlinearleast-squares approach. We have adapted this newmethod to the low-finesse Fabry–Perot interferome-ter. The original method consisted of fitting a Mi-chelson-type interferometer with the following func-tion:

F1t2 5 F0 1 A sin3f0 1 2k1v0t 1 ‰gt224. 132

In practice, the following transformation allows oneto reduce the number of nonlinear parameters:

F1t2 5 F 1 A1 sin32k1v0t 1 ‰gt224

1 A2 cos32k1v0t 1 ‰gt224, 142

where A1 5 A cos1f02 and A2 5 A sin1f02. With theadditional reflection and a similar transformation,the fitted function in our new interferometer is

F1t2 5 F0 1 A1 sin32k1v0t 1 ‰gt224

1 A2 cos32k1v0t 1 ‰gt224

1 A3 sin34k1v0t 1 ‰gt224

1 A4 cos34k1v0t 1 ‰gt224. 152

The function is then fit by the Gauss–Newtonmethod of nonlinear least squares. Any nonlinearleast-squares fit requires a certain quality of initialguess for each nonlinear parameter to ensure thatthe solution converges. One can find the initial

guess in this case by first fitting the fringe signalwith the original method designed for a Michelsoninterferometer and using the values for gravity andthe initial velocity found by this method as thestarting values with the extra terms added. Inpractice, the extra term has a small effect and onlyone or two additional iterations are required toconverge with the new terms.The extra reflection has a small but nonnegligible

effect on the measured value of g. Over a set of 100drops, the average change in g amounted to 5 µGal.Note that this value is an upper bound on the effectof the extra reflection. Fitting this higher-orderterm also accounts for any possible effects fromnonlinear terms in the photodetector proportional totwice the frequency of the signal. It is possible thatthe amplitude of the second-order reflection is morestrongly dependent on the distance between theretroreflector and the fiber face but we have previ-ously found the value of g to be insensitive toenvelope functions on the first-order fringe ampli-tude—most of the information is contained in thephase of the signal rather than the amplitude.We sample the intensity of the fringe signal at a

typical frequency of 0.5MHz using an 8-bit analog-to-digital converter with a 128 K memory buffer 1GageApplied Science CS2502. This results in an aliasedsignal, and while it violates the Nyquist criterion, weare able to converge on a value of g because we havesignificant a priori information about the fringesignal.10 It takes less than 10 s to fit the first tworeflections using an optimized code on a 50-MHz 486computer. The third reflection fit is done as apostprocessing step in MATLAB on a Sun workstationusing the rsults of the initial fit as the startingvalues in the nonlinear fit. It takes approximately1 min for this final step.

4. Vibration Isolation

Any interferometric measurement of the accelera-tion of gravity requires a reference that rests inpseudoinertial space. The simplest such referenceis simply the Earth; one could directly measure theacceleration of the dropped object relative to theground. And, while the ground is not inertial, theaccelerations are either small enough or well under-stood enough 1e.g., the Earth’s rotation2 that onecould obtain a reasonable estimate of gravity usingthe ground as a reference. However, in order toachieve the most precise and accurate measure, it isessential to isolate the reference arm as much aspossible from the accelerations undergone by theground. In particular, it is essential to isolate thecomponent of ground noise that is caused by themeasurement itself since this is a potential source ofa systematic error in g.The vibration-isolated interferometer we have de-

veloped uses the mass of a critically damped high-frequency seismometer as a reference frame 1Fig. 32.The apparatus is simply a mass on a spring withvelocity-dependent damping and a free period of

0.5 s. This arrangement will not reduce the noise inthe microseismic band that has a peak in its powerspectrum near a 6-s period. We only ensure thatthe high-frequency disturbances that originatewithinthe instrument itself are attenuated to the optical-path length.Initial tests with the new interferometer show an

overall reduction in the average residuals in compari-sonwith the old bulk opticsMichelson interferometer.Figure 4 shows the comparative residuals betweenthese two systems at a site where there is goodcoupling between the dropping mechanism and theoptics. This coupling is the result of both the drop-ping mechanism and the optics resting on the samerelatively soft floor. We compare stacked residualsin order to examine possible systematic errors inher-ent in the act of dropping the mass. At a sitewithout good coupling between the optics and thedropping mechanism, the stacked residuals are evenlower 1Fig. 52.Gravity data collected with the new interferome-

ter show a scatter consistent with that expected frommicroseismic noise. However, since the precision inthe estimate of the mean grows only as the squareroot of the number of drops, achieving a precision of 1µGal would require approximately 3000 drops.There are schemes to reduce the microseismic noise,most notably the superspring.5 However, there issufficient environmental noise at the 1-µGal levelthat in practice a site must be occupied for at least6–12 h in order to achieve a 1-µGal accuracy. Inprinciple, there are no barriers to mounting theinterferometer directly onto a longer-period device.

5. Practical Details of Construction

Several aspects of the construction of the systemwarrant discussion. The ferrule–lens system mustproduce a beam near the diffraction limit that isaligned with the local acceleration of gravity tobetter than 1024 rad. The ferrule is simply a glasstube in which the fiber has been epoxied. In orderto obtain the highest quality beam, we cleaved the

Fig. 3. Interferometer rests on a critically damped short periodseismometer to achieve vibration isolation.

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fiber rather than polishing the ferrule. We repeat-edly cleaved the fiber and then collimated it with a30-mm focal-length lens. To evaluate the cleavequality, we examined the far-field beam pattern andinterfered the beam in a Michelson interferometer

Fig. 4. Comparative residuals between 1a2 bulk optics and 1b2fiber-optic systems at a site with good coupling between thedropping chamber and the optics platform.

Fig. 5. Stacked residuals at a site with the optics resting on anisolated pier.

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with unequal arms to estimate beam quality. Oncethe cleave produces a high-quality beam as deter-mined by both methods, the fiber is drawn back asfar as possible into the ferrule, which is filled withepoxy. The epoxy is then cured with ultravioletlight; the ferrule serves as a rigid clamp for the fiber.Once the fiber is effectively bonded into the fer-

rule, the divergent beam direction is a function of themechanical alignment of the ferrule. The next stepis to collimate the beam. An aluminum piece holdsthe collimating lens to the ferrule; its position isadjusted until the end of the fiber coincides with thefocal point of the lens 1this is noted by the constancyof the beam diameter in the far field2. The finalobstacle is to facilitate repeatable alignment of thebeam with respect to the local direction of gravity.Because the angle of the beam is now fixed relative tothe lens holder, we bond the lens holder into thecentral aluminum platform on the seismometermasswith an epoxy that has approximately the samethermal expansion coefficient as aluminum. Theorientation of the platform can now be used todetermine the direction of the optical beam. Tiltsensors 1Spectron Model TG-33T-5542 are attached tothe same structure platform. We aligned the beamwith the vertical in the laboratory using a mercuryreference mirror, and the outputs of the tiltmetersare recorded. For field and remote operation, then,the beam can be returned to the vertical by simplyadjusting the orientation of the seismometer untilthe tiltmeters produce their original outputs. Dur-ing an 11-month period that included various fieldtests and an ocean trial, the platform had a net driftof 0.23 mrad 1primarily because of one of the tiltsensors whose short-term repeatability is also suspi-ciously poor2. The gravity error corresponding to atilt is proportional to the square of the angle. Thus,while the measured drift rate corresponds to only 0.2µGal in a month, the error would be 31 µGal in ayear. While monthly calibration of the vertical isnot unreasonable 1it takes less than an hour2, wehope to reduce the drift rate by replacing the tiltme-ter we suspect is damaged and improving the cou-pling of the tiltmeters to the optics platform.We require a highly stable input stage to launch

laser light into the single-mode fiber so that ourinstrument is not limited by a constantly degradingsignal when operated remotely. We have found itpossible to achieve a stable launching configurationby mounting a quarter-pitch gradient-index lensdirectly onto a polished ferrule. The lens is epoxiedto the ferrule with optical epoxy that matches theindex of refraction of the fiber. We then eliminatedunwanted reflections from the interferometric sys-tem by orienting the lens slightly off center.

6. Conclusions

We have designed and tested a low-finesse Fabry–Perot optical fiber vibration-isolated interferometerfor use in absolute gravity meters. Our motivationin building the interferometer was to create a more

compact instrument for use in an ocean bottomabsolute gravity meter.11 While succeeding in beingsignificantly smaller than the optical systems usedpreviously in absolute gravity meters, the opticalfiber interferometer has other advantages.Because of the configuration of the interferometer,

the key elements for reducing vibration sensitivity—the fiber endface and the collimating lens—can bemounted on a small vibration-isolated platform.This new system has been shown to improve theaverage drop residuals. Under conditions in whichthe only change to the gravitymeter was the interfer-ometer, errors in the optical system caused by vibra-tion decreased by a factor of 2.A second advantage to the instrument is that it

facilitates electronic monitoring of the beam’s orien-tation with respect to the vertical. Because theoptical axis is determined by a well-definedmechani-cal axis 1that of the fiber2, it is possible to reference itto a tiltmeter. Efforts are under way to improve thetilt sensing and hence reduce the frequency withwhich drift checking must be done.Finally, in addition to being compact, the instru-

ment has proved to be fairly rugged, having beenoperated successfully on the sea floor.

We are indebted to Paul Parker who wrote much ofthe software we used in examining the data from thenew system. This research was supported by theNational Science Foundation under award numberOCE89-20058.

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