A Low-Frequency Torsion Pendulum with Interferometric Readout
M.P. Ross, K. Venkateswara, C.A. Hagedorn, C.J. Leupold, P.W.F. Forsyth, J.D. Wegner, E.A. Shaw, J.G. Lee, J. H. Gundlach
AA Low-Frequency Torsion Pendulum with Interferometric Readout
M.P. Ross, a) K. Venkateswara, C.A. Hagedorn, C.J. Leupold, P.W.F. Forsyth, J.D. Wegner, E.A. Shaw, J.G. Lee, and J. H. Gundlach Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington, 98195,USA University of Washington Bothell, Bothell, WA 98011, USA OzGrav-ANU, Centre for Gravitational Astrophysics, College of Science, The Australian National University,Acton, ACT 2601, Australia
We describe a torsion pendulum with a large mass-quadrupole moment and a resonant frequency of 2.8 mHz,whose angle is measured using a modified Michelson interferometer. The system achieved noise levels of ∼
200 prad / √ Hz between 0.2-30 Hz and ∼
10 prad / √ Hz above 100 Hz. Such a system can be applied to abroad range of fields from the study of rotational seismic motion and elastogravity signals to gravitationalwave observation and tests of gravity.
I. INTRODUCTION
Since the days of Coulomb , Cavendish , andE¨otv¨os , torsion balances have been used for a wide va-riety of precision measurements. Over the years theyhave allowed precise measurement of the gravitationalconstant , tests of the behavior of gravity , hunts fordark matter , and searches for novel fifth forces . Re-cently, large-moment torsion balances have begun to bebuilt with the goal of measuring gravitational waves and gravity gradients . Additionally, these instrumentshave been proposed to study elastogravity signals fromseismic event. In this paper, we describe the Michelson Interferom-eter Torsion-balance (MINT) that employs a modifiedMichelson interferometer to achieve sub-nrad / √ Hz angu-lar readout of a low resonant frequency, dumbbell-shapedtorsion balance.
II. MECHANICAL DESIGN
Parameter Value κ . × − N m/radI 9 . × − kg m f Q κ is the torsional spring constant, I isthe moment of inertia, f is the resonant frequency, and Q isthe observed quality factor. The MINT balance, shown in Figure 1, consists of a11.4-cm-wide aluminum pendulum hung from a 20- µ m-diameter and 8.9-cm-long tungsten fiber. The relativelywide pendulum allowed for an increased angular sensi- a) [email protected] tivity as well as decreased the resonant frequency of thebalance.The pendulum was suspended from an intermediatecopper mass which itself is suspended from a rigid struc-ture with a 75- µ m-thick 7.6-cm-long “pre-hanger” tung-sten fiber and a beryllium-copper leaf spring. This inter-mediate mass is place within an exterior magnetic fieldto form an eddy current damper which is detailed in Sec-tion III. This damper is attached to a translation stagewhich can move in both horizontal directions as well asrotate about vertical. Additionally, two plane electrodesare placed on the back side of the pendulum to allow foractive control. Beam SplitterPendulum AutocollimatorMotorized Steering MirrorCapacitive Actuator
FIG. 1. A photograph of the instrument showing the pen-dulum, interferometer optics, capacitive actuators, and au-tocollimator optics. Not shown is the magnetic damper andalignment stage.
A CAD rendering of the pendulum is shown in Fig-ure 2. To avoid spurious forces and to decrease lossesdue to gas damping, the pendulum is housed in a vac-uum chamber held at ∼ a r X i v : . [ phy s i c s . i n s - d e t ] J a n PendulumAdjustment ScrewInterferometer Mirrors Torsion FiberDamper Mass
FIG. 2. CAD rendering of the MINT pendulum and magneticdamper disk.
III. EDDY CURRENT DAMPER
In order to simplify the system, one would prefer tooperate an instrument with a single degree of freedom.However, any torsion fiber will also permit pendulum“swing” modes. These being the modes whose restoringforce is predominantly gravity and not the spring of thefiber. The swing modes are readily excited due to am-bient seismic motion and can become the most energeticmodes if not controlled appropriately.In the MINT apparatus, these modes are passivelydamped by suspending the pendulum from an intermedi-ate stage consisting of a copper disk suspended by both avertical spring and a thin fiber, shown in Figure 3. Thevertical spring is formed by a laser cut beryllium cop-per leaf spring which gives a vertical, “bounce”, mode of ∼ Damper MassLeaf SpringPre-hanger FiberRing MagnetTo Pendulum
FIG. 3. CAD rendering of the eddy current damper assembly.
IV. OPTICAL READOUT
Two independent, in-vacuum optical readouts areoperated to allow for both coarse and fine angular mea-surements of the pendulum. A two-dimensional autocol-limator senses the angle of a mirror attached to the centerof the pendulum with a sensitivity of ∼
20 nrad / √ Hz.The autocollimator was used for initial alignment andcross-calibration.The primary readout was formed by a modified Michel-son interferometer, shown in Figure 4, whose arms wereformed by two mirrors attached to opposite ends of thependulum. Angular motion induces a phase difference, φ , between the two arms given by: φ = 4 π rλ θ (1)where r is the distance from the fiber to the location ofthe beam on the mirror, λ is the wavelength of light, and θ is the angle of the pendulum. This phase differencecauses changes in the light intensity entering the outputport. A 1310-nm fiber coupled laser was connected to theinput port of the interferometer and the output port wasattached to a in-air photodiode. Both of these connec-tions were made via a pair of teflon optical-fiber vacuumfeedthroughs . V. CONTROLS
In order to operate the interferometer in a linearregime, the pendulum is locked in feedback using two x y PendulumAutocollimator BeamInterferometerSteering MirrorBeam Splitter Fiber couplersInput OutputCapacitive Actuator
FIG. 4. Optical layout of the MINT apparatus. The autocol-limator is comprised of the beam path at the top of the imagewhile the Michelson interferometer is formed by the bottombeam paths. parallel-plate capacitive actuators. The lock is achievedin two stages, first the autocollimator is used as the feed-back sensor to decrease the amplitude of the torsionalmode and provide a course alignment. Then the feed-back control is switched to the interferometer readoutto allow for low noise operation. The feedback employsa PID loop whose unity gain frequency was in the 10-30 mHz range. The physical voltage-to-force gain of thecapacitive actuator depends on the gap between the actu-ator and the pendulum. A relatively large gap ( ∼ VI. CROSS COUPLING MINIMIZATION
Any coupling of non-torsional motion to the angularreadout decreases the performance of the measurementsand thus must be minimized. Additionally, although themagnetic damper described in Section III decreases themotion in the swing modes, the residual motion in theswing resonances adds significant artifacts in the readoutif these couplings aren’t minimized. Below, we give asimple description of the methods used to decrease thecoupling of the readout to these swing modes. For a moredetailed study of such couplings see Shimoda et al. .If the plane of the interferometer is not perpendicular to the torsion fiber then the swing motion about the y-axis couples to the interferometer readout. This is due tothe two arms effectively sensing pendulums of differentlengths. The path length difference due to this follows:∆ δ = 2 r tan φ sin θ y (2)where ∆ δ is the difference in path length, r is the dis-tance from the center of the pendulum to the center ofthe mirrors, φ is the angle between the plane of the inter-ferometer and the pendulum, and θ y is the angle of thependulum about the y-axis with respect to vertical. Thiscoupling was minimized by tilting the optical table aboutthe x-axis, which rotates the plane of the interferometerwhile keeping the angle of the fiber fixed relative to thelocal vertical. This angle was changed to minimize theobserved motion at the y-axis swing resonant frequency.Additionally, if the faces of the mirrors are not parallelto the fiber, the swing motion about the x-axis couplesdue to the change in angle of the faces of the mirrors.Assuming that both mirrors are inclined by the sameamount, this coupling follows:∆ δ = 2 L tan α sin θ x (3)where L is the length of the pendulum, α is the angle ofthe mirrors with respect to vertical, and θ x is the angle ofthe pendulum about the x-axis with respect to vertical.This coupling was minimized by iteratively shifting trimscrews placed at the back of the pendulum. By shiftingthe screws the center of mass of the pendulum shifted inthe x-direction which caused the pendulum to tip aboutthe y-axis.With these two methods, the coupling of the interfer-ometer to swing motion was decreased to ∼ − rad/rad.The residual effect of the swing resonance was compara-ble to the readout noise. In addition to decreasing thenoise at the swing resonant frequency, this cross-couplingminimization decreased broadband noise due to horizon-tal tilts of the instrument driven by ambient seismic mo-tion. VII. NOISE PERFORMANCE
With the cross coupling to non-torsional modes min-imized, the instrument achieves the noise performanceshown in Figure 5. Since the pendulum is in active feed-back the readout of the instrument is the sum of theangular equivalent of capacitor feedback signal and theoutput of the interferometer. Also shown is the motionmeasured by both axes of the autocollimator.The high frequency performance is shown in Figure 6.At these frequencies, the feedback contribution to thereadout is negligible and is thus omitted. These spectrawere taken at separate but subsequent times due to filesize constraints.It is apparent that the instrument has a noise floor of ∼
200 prad / √ Hz between 0.2-30 Hz and ∼
10 prad / √ Hz -3 -2 -1 -10 -8 -6 -4 FIG. 5. Amplitude spectral density of the angle noise of bothautocollimator directions, the interferometer readout, and thecapacitor feedback. Also shown is the expected thermal noiselimit of the pendulum. -12 -11 -10 -9 -8 -7 -6 -5 FIG. 6. High frequency amplitude spectral density of theangle noise of both autocollimator directions and the inter-ferometer readout. above 100 Hz. Below 0.2 Hz, the noise rises significantlyto ∼
10 nrad / √ Hz at 30 mHz. This rise is due to bothresidual seismic noise coupling and temperature effects.In later runs, the noise in the 0.5-30 mHz range was re-duced to the level of the suspension thermal noise bysurrounding the pendulum with a thin aluminum hous-ing.Multiple structures are apparent throughout the spec-tra. The collection of lines above 5 Hz are due to me- chanical resonances of the apparatus and optics while thebroader structures between 0.1-5 Hz are due to seismicmotion. Particularly, the broad structure between 0.2-0.3 Hz is due to the oceanic microseism. We believe thatthis is true torsional seismic motion and not due to resid-ual cross-couplings as it is independent of small changesin either α or φ . However, a second, co-located torsionbalance is required to verify that the observed feature istruly the torsional microseism. VIII. POSSIBLE APPLICATIONS
The MINT apparatus, or similar instruments, have awide range of applications. Here we will explore a collec-tion of promising avenues.
A. Inertial Seismic Sensing
Seismic waves cause both translational and rotationalmotion. Traditionally, seismology has been limited tostudy only translational motion as rotation sensors didnot meet the required sensitivity. Recently, a number ofsensors have been developed to sense such motion .These observations have allowed a number of uniquemethods and studies.
The MINT apparatus senses the torsional ground mo-tion by using the pendulum as an inertial reference whilehaving the optics rigidly attached to the ground. Thesensed angle is then related to the rotation of the groundvia: ˜ θ ( ω ) = − ω ω − iQ ω − ω ˜ θ g ( ω ) (4)where ˜ θ is the observed angular motion, ˜ θ g is the angu-lar motion of the ground, ω is the angular frequency ofmotion, Q is the quality factor, and ω is the resonantangular frequency of the pendulum.The MINT apparatus joins a small class of sensors withsub-nrad / √ Hz torsional noise down to 100 mHz. The in-ertial angular performance of MINT is shown in Figure 7.This performance allows for the observation of both tele-seismic and regional Love waves.
B. Gravitational Wave Observation
If a × -polarization gravitational wave passes verticallythrough the instrument, tidal forces cause an angular de-flection of the pendulum which follows :˜ θ ( ω ) = ω q I ( ω − iQ ω − ω ) ˜ h × ( ω ) (5)where q is the dynamic quadrupole, I is the momentof inertia, and h × is the gravitational wave strain. For -2 -1 -11 -10 -9 -8 -7 FIG. 7. Amplitude spectral density of the inertial angularnoise. the MINT pendulum the ratio q /I ≈ .
94. Note thatsince this ratio is approximately one, the angular sensi-tivity shown in Figure 7 can be converted to strain bymultiplying by a factor of two.Although other instruments are much more sensitiveto gravitational waves , this apparatus can be used as aprototype for future torsion-balance based gravitationalwave detectors. Additionally, this instrument may al-low for the study of atmospheric gravity gradient noise with the addition of an identical orthogonally-orientedapparatus. C. Elasto-gravity Signals
The prompt gravitational signal caused by earthquakeshas become a promising avenue for earthquake earlywarning. Torsion balances have begun to be exploredas a method to observe these signals with high fidelity. At its current sensitivity, MINT is expected to be ca-pable of observing the elastogravity signals only fromnearby earthquakes where the warning advance is mini-mal. However, such observations could increase the preci-sion of earthquake source modeling. Additionally, withfuture upgrades we hope to reach observable distanceswith meaningful advanced warning.
D. Torque Sensing
Torsion balances are used to sense weak torques ina variety of experiments. These include short rangegravity experiments , tests of the equivalence principle ,and measurements of the gravitational constant . Thetorque sensitivity of the MINT apparatus is shown in Figure 8. The torque noise performance makes this in-strument particularly promising for searches of ultra-lightdark matter . -2 -1 -14 -13 -12 FIG. 8. Amplitude spectral density of the torque noise.
IX. CONCLUSION
We have described a simple torsion pendulum withresonant frequency of 2.8 mHz whose angle is readoutwith both an autocollimator and a modified Michelsoninterferometer. The pendulum is locked in feedback us-ing a pair of capacitive actuators in order to control thenatural motion and to linearize the interferometric read-out. This system achieves angular noise performance of ∼
200 prad / √ Hz between 0.2-30 Hz and ∼
10 prad / √ Hzabove 100 Hz.This apparatus can contribute to a variety of fields in-cluding rotational seismology, gravitational wave obser-vation, and the study of elastogravity signals. Althoughfirst conceived of in the 18th-century, modernized torsionbalances are at the forefront of today’s precision mea-surement and provide the ideal apparatus for a numberof scientific experiments.
ACKNOWLEDGMENTS
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.This work was supported by funding from the NSF un-der Awards PHY-1607385, PHY-1607391, PHY-1912380and PHY-1912514. Charles Augustin Coulomb. Recherches th´eoriques etexp´erimentales sur la force de torsion: sur l’´elasticit´e des fils dem´etal: application de cette th´eorie `a l’emploi des m´etaux dansles arts dans diff´erentes exp´eriences de physique: constructionde diff´erentes balances de torsion, pour mesurer les plus petitsdegr´es de force: observations sur les loix de l’´elasticit´e de la co-herence. 1784. Henry Cavendish. Experiments to determine the density of theearth.
Philosophical Transactions of the Royal Society of Lon-don , 88:469–526, 1798. ISSN 02610523. URL . Roland v. E¨otv¨os, Desiderius Pek´ar, and Eugen Fekete.Beitr¨age zum gesetze der proportionalit¨at von tr¨agheit undgravit¨at.
Annalen der Physik , 373(9):11–66, 1922. doi:10.1002/andp.19223730903. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19223730903 . Qing Li, Chao Xue, Jian-Ping Liu, Jun-Fei Wu, Shan-Qing Yang,Cheng-Gang Shao, Li-Di Quan, Wen-Hai Tan, Liang-Cheng Tu,Qi Liu, et al. Measurements of the gravitational constant usingtwo independent methods.
Nature , 560(7720):582–588, 2018. J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer, andB. R. Heckel. New test of the gravitational 1 /r law at separa-tions down to 52 µ m. Phys. Rev. Lett. , 124:101101, Mar 2020.doi:10.1103/PhysRevLett.124.101101. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.101101 . T A Wagner, S Schlamminger, J H Gundlach, and E G Adel-berger. Torsion-balance tests of the weak equivalence princi-ple.
Classical and Quantum Gravity , 29(18):184002, aug 2012.doi:10.1088/0264-9381/29/18/184002. URL https://doi.org/10.1088%2F0264-9381%2F29%2F18%2F184002 . Peter W. Graham, David E. Kaplan, Jeremy Mardon, SurjeetRajendran, and William A. Terrano. Dark matter direct de-tection with accelerometers.
Phys. Rev. D , 93:075029, Apr 2016.doi:10.1103/PhysRevD.93.075029. URL https://link.aps.org/doi/10.1103/PhysRevD.93.075029 . W. A. Terrano, E. G. Adelberger, C. A. Hagedorn, and B. R.Heckel. Constraints on axionlike dark matter with masses downto 10 − eV /c . Phys. Rev. Lett. , 122:231301, Jun 2019. doi:10.1103/PhysRevLett.122.231301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.231301 . W. A. Terrano, E. G. Adelberger, J. G. Lee, and B. R. Heckel.Short-range, spin-dependent interactions of electrons: A probefor exotic pseudo-goldstone bosons.
Phys. Rev. Lett. , 115:201801,Nov 2015. doi:10.1103/PhysRevLett.115.201801. URL https://link.aps.org/doi/10.1103/PhysRevLett.115.201801 . Ayaka Shoda, Yuya Kuwahara, Masaki Ando, Kazunari Eda,Kodai Tejima, Yoichi Aso, and Yousuke Itoh. Ground-based low-frequency gravitational-wave detector with multi-ple outputs.
Phys. Rev. D , 95:082004, Apr 2017. doi:10.1103/PhysRevD.95.082004. URL https://link.aps.org/doi/10.1103/PhysRevD.95.082004 . D J McManus, P W F Forsyth, M J Yap, R L Ward, D AShaddock, D E McClelland, and B J J Slagmolen. Mechan-ical characterisation of the TorPeDO: a low frequency gravi-tational force sensor.
Classical and Quantum Gravity , 34(13):135002, jun 2017. doi:10.1088/1361-6382/aa7103. URL https://doi.org/10.1088%2F1361-6382%2Faa7103 . J. Harms, J.-P. Ampuero, M. Barsuglia, E. Chassande-Mottin,J.-P. Montagner, S. N. Somala, and B. F. Whiting. Transientgravity perturbations induced by earthquake rupture.
Geophysi-cal Journal International , 201(3):1416–1425, 04 2015. ISSN 0956-540X. doi:10.1093/gji/ggv090. URL https://doi.org/10.1093/gji/ggv090 . Eric R.I. Abraham and Eric A. Cornell. Teflon feedthrough forcoupling optical fibers into ultrahigh vacuum systems.
Appl.Opt. , 37(10):1762–1763, Apr 1998. doi:10.1364/AO.37.001762.URL http://ao.osa.org/abstract.cfm?URI=ao-37-10-1762 . Tomofumi Shimoda, Naoki Aritomi, Ayaka Shoda, YutaMichimura, and Masaki Ando. Seismic cross-coupling noisein torsion pendulums.
Phys. Rev. D , 97:104003, May 2018.doi:10.1103/PhysRevD.97.104003. URL https://link.aps.org/doi/10.1103/PhysRevD.97.104003 . Stefano Maran`o and Donat F¨ah. Processing of translationaland rotational motions of surface waves: performance anal-ysis and applications to single sensor and to array measure-ments.
Geophysical Journal International , 196(1):317–339, 112013. ISSN 0956-540X. doi:10.1093/gji/ggt187. URL https://doi.org/10.1093/gji/ggt187 . Jacopo Belfi, Nicol`o Beverini, Filippo Bosi, Giorgio Carelli, Da-vide Cuccato, Gaetano De Luca, Angela Di Virgilio, Andr´eGebauer, Enrico Maccioni, Antonello Ortolan, Alberto Porzio,Gilberto Saccorotti, Andreino Simonelli, and Giuseppe Terreni.Deep underground rotation measurements: Gingerino ring lasergyroscope in gran sasso.
Review of Scientific Instruments , 88(3):034502, 2017. doi:10.1063/1.4977051. URL https://doi.org/10.1063/1.4977051 . Krishna Venkateswara, Charles A. Hagedorn, Matthew D.Turner, Trevor Arp, and Jens H. Gundlach. A high-precisionmechanical absolute-rotation sensor.
Review of Scientific In-struments , 85(1):015005, 2014. doi:10.1063/1.4862816. URL https://doi.org/10.1063/1.4862816 . M. P. Ross, K. Venkateswara, C. A. Hagedorn, J. H. Gundlach,J. S. Kissel, J. Warner, H. Radkins, T. J. Shaffer, M. W. Cough-lin, and P. Bodin. Low-Frequency Tilt Seismology with a Pre-cision Ground-Rotation Sensor.
Seismological Research Letters ,89(1):67–76, 11 2017. ISSN 0895-0695. doi:10.1785/0220170148.URL https://doi.org/10.1785/0220170148 . A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod,and K. U. Schreiber. Ring laser detection of rotations fromteleseismic waves.
Geophysical Research Letters , 27(21):3553–3556, 2000. doi:10.1029/2000GL011734. URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2000GL011734 . W. H. K Lee, M. C¸ elebi, M. I. Todorovska, and H. Igel.Introduction to the Special Issue on Rotational Seismologyand Engineering Applications.
Bulletin of the SeismologicalSociety of America , 99(2B):945–957, 05 2009. ISSN 0037-1106. doi:10.1785/0120080344. URL https://doi.org/10.1785/0120080344 . M. P. Ross, C. A. Hagedorn, E. A. Shaw, A. L. Lockwood, B. M.Iritani, J. G. Lee, K. Venkateswara, and J. H. Gundlach. Limitson the stochastic gravitational wave background and prospectsfor single-source detection with grace follow-on.
Phys. Rev. D ,101:102004, May 2020. doi:10.1103/PhysRevD.101.102004. URL https://link.aps.org/doi/10.1103/PhysRevD.101.102004 . Donatella Fiorucci, Jan Harms, Matteo Barsuglia, Irene Fiori,and Federico Paoletti. Impact of infrasound atmosphericnoise on gravity detectors used for astrophysical and geophys-ical applications.
Phys. Rev. D , 97:062003, Mar 2018. doi:10.1103/PhysRevD.97.062003. URL https://link.aps.org/doi/10.1103/PhysRevD.97.062003 . Martin Vall´ee, Jean Paul Ampuero, K´evin Juhel, Pascal Bernard,Jean-Paul Montagner, and Matteo Barsuglia. Observations andmodeling of the elastogravity signals preceding direct seismicwaves.
Science , 358(6367):1164–1168, 2017. ISSN 0036-8075. doi:10.1126/science.aao0746. URL https://science.sciencemag.org/content/358/6367/1164https://science.sciencemag.org/content/358/6367/1164