Constraints on Yukawa-Type Deviations from Newtonian Gravity at 20 Microns

0.95 MB·English

by  S. J. Smullin

Constraints on Yukawa-Type Deviations from Newtonian Gravity at 20 Microns
S. J. Smullin,1,∗ A. A. Geraci,1 D. M. Weld,1 J. Chiaverini,2,† S. Holmes,3 and A. Kapitulnik1,4,‡
1Department of Physics, Stanford University, Stanford, CA 94305, USA
2National Institute of Standards and Technology, Boulder, Colorado 80305, USA
3Department of Statistics, Stanford University, Stanford, CA 94305, USA
4Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
(Dated: February 2, 2008)
Recenttheoriesofphysicsbeyondthestandardmodelhavepredicteddeviationsfrom Newtonian
gravity at short distances. In order to test these theories, we have a built an apparatus that can
measureattonewton-scaleforcesbetweengoldmassesseparatedbydistancesontheorderof25µm.
Amicromachinedsiliconcantileverwasusedastheforcesensor,anditsdisplacementwasmeasured
withafiberinterferometer. Wehaveusedourmeasurementstosetboundsonthemagnitudeαand
length scale λ of Yukawa-type deviations from Newtonian gravity; our results presented here yield
6 thebest experimental limit in therange of λ=6–20 µm.
0
0 PACSnumbers: 04.80.Cc
2

n
I. INTRODUCTION Lamoreaux
a Gauge
J Bosons
1 A. Motivation 10 8

Stanford 1
2 The theories ofNewton andEinsteinare powerfuland Dilaton
v
accurate in describing the observed natural world. In
4 10 4
0 spiteofthis,gravitystillpresentsseveraltheoreticalchal- a Stanford 2
2 lenges, including the gauge hierarchy problem, the cos-
8 mological constant problem, and the lack of a quantum Moduli Colorado
0 description of gravity. Many theories of physics beyond
KK gravitons
5 the standard model, in particular those theories that at- 10 0 Washington
0 tempt to unify the standard model with gravity, predict
/
h theexistenceofextradimensions,exoticparticles,ornew
p forcesthatcouldcauseamasscouplinginadditiontothe 10 -6 10 -5 10 -4 10 -3
p- Newtonian gravitationalpotential at short distances. Of l (meters)
particular interest to us have been some recent theories
e
h that predict new forces in a range measurable by table-
FIG. 1: [color online] Experimental results (solid lines) and
: top experiments [1, 2, 3, 4, 5, 6, 7, 8]. In many cases,
v theoretical predictions shown in α-λ space. The area to the
i the modification to Newtonian gravityis predicted to be upper right of the experimental lines (from Refs. [9, 14, 15,
X a Yukawa-type potential, arising either from coupling to 16, 31] and our latest results from this paper) shows where
r massiveparticlesorthe compactificationofextradimen- Yukawa-type deviations from Newtonian gravity have been
a
sions. With this addition, the gravitationalpotentialbe- excluded. The line labeled “Stanford 1” gives results from
tween two masses m and m separated by distance r is [9]; the line labeled “Stanford 2” shows the results described
1 2
predicted to be of the form: in this paper. Dashed lines show roughly the predictions for
thedilaton[8]andthefirstKaluza-Kleinmodeoftwosimply-
m m
V(r)= G 1 2 1+αe−(r/λ) . (1) compactifiedextradimensionsasdescribedinRef.[2]. Shaded
− r h i regions to the left of the experimental lines show predictions
for moduli and gauge bosons from Ref. [32].
Here,GisNewton’sconstant,αisthestrengthofthenew
potentialascomparedtotheNewtoniangravitationalpo-
tential, and λ is its range.
Withtheseideasinmind,wehaveconstructedadevice 13]. Our experiment was designed to measure Yukawa-
to measure attonewton-scale forces between masses sep- type forces in the range of λ= 5–50 µm with as small a
arated by distances on the order of 25 µm [9, 10, 11, 12, value of α as possible. This range of parameters is rel-
| |
evant to recent theoretical predictions [2, 4, 5, 8] and is
complementary to recent experimental attempts to test
these new ideas [9, 14, 15, 16]. Our results are reported
∗Present address: Physics Department, Princeton University, in terms of an upper bound on α(λ) of a Yukawa poten-
Princeton,NJ08544,USA
tial that could be consistent with our data; we present
†Presentaddress: LosAlamosNationalLaboratory,MSD454,Los
results at the 95% confidence level. Unless otherwise
Alamos,NM87545,USA
‡Electronicaddress: [email protected] stated, all measurements and techniques described ap-
2
Optical were fabricated to have small spring constants and high
Test mass
Fiber
quality factors in vacuum. The thermal noise limit in
Cantilever this experiment was approximately 2.5 10−16 N/√Hz,
Drive mass ×
at cryogenic temperatures in vacuum.
In order to be able to test and characterize the appa-
ratus by measuring a known force much larger than any
Drive Mass
expected gravitational force, a magnetic analog to the
Gold Motion
gravitational experiment was built into the apparatus.
Silicon Piezoelectric
Magnetic test masses were fabricated with a nickel layer
Bimorph
Actuator on them. An electrical current passed through the drive
z mass meander created a spatially-varying magnetic field
Shield
(cutaway) y that would couple to the magnetic moment of the test
x mass and drive the cantilever.
For both magnetic and gravitational tests, the force
FIG. 2: [color online] Schematic (not to scale) showing the wasmeasuredasafunction ofthe equilibriumy-position
drivemassbelowthecantileverthatbearsthegoldtestmass. between the drive mass and the test mass. Any cou-
A gold-coated silicon nitride shield membrane separates the pling between masses would show a distinct periodicity
masses.
in the measured force as a function of this y-position.
By comparing our measurements to predictions from fi-
nite element analysis (FEA), a bound on Yukawa-type
ply to the Cooldown04 experiment, fromwhich the new
deviations was derived.
α(λ) bound was derived. These latest results yield close
The separation of the signal frequency from the drive
to anorderofmagnitudeimprovementoverourprevious
frequency, the shield between the masses, and the use of
bounds on the Yukawa potential at distances on the or-
non-magnetic test masses for the gravitational measure-
der of 20 µm [9]. Fig. 1 summarizes our results together
mentreducedoreliminatedmanypossiblesourcesofnon-
withsomeofthe theoreticalpredictionsandotherrecent
gravitational background. The geometry of the appara-
experimental results.
tusprovidedanimportantdegreeoffreedomthatpermit-
tedadiscriminationoftrue couplingbetweenthemasses
from certain electrical or mechanical backgrounds.
B. Overview
To measurevery short-rangeforces,we usedmassesof
C. Organization of the Paper
size comparable to the distance between them. A gold
prism was attached to the end of a single-crystal silicon
This paper primarily describes the apparatus and
cantilever. Adrivemass,comprisingagoldmeanderpat-
results from the experiment labeled Cooldown 04.
tern embedded in a silicon substrate to create an alter-
Cooldown 01 is described in Refs. [9, 10], Cooldown 02
nating pattern of gold and silicon bars, was mounted on
is describedin Sec.VI ofthis paper,and Cooldown03is
apiezoelectricbimorphatadistancefromthecantilever-
described elsewhere[11, 13].
mounted test mass. The face-to-face vertical separation
In the second section of this paper, the apparatus is
between the masses was 25 µm, limited in partby the
∼ described in detail. The following sections describe our
presence of a stiff, metallized silicon nitride shield mem-
finiteelementanalysis,theexperimentalmethod,andthe
brane between the masses.
averaging used to convert the raw data to a force mea-
AsdiagrammedinFig.2,thedrivemasswasoscillated
surement. The sixth section of this paper describes the
along the y-direction underneath the test mass, main-
magnetic experiment used to test the apparatus and the
taining the vertical (z-) separation between the masses.
measurement of thermal noise. In the final sections, our
Due to the differing mass densities of the gold and sili-
experimental results, fitting techniques, error analysis,
conbars,analternatinggravitationalfieldwascreatedat
and the resulting bound on α(λ) are presented, followed
thetestmasslocation. Thedrivemasswasoscillatedata
by conclusions and a discussion of future prospects for
subharmonic of the resonantfrequency of the cantilever.
this experiment.
Due to the geometry of the drive mass and the ampli-
tude of oscillation, any gravitational coupling between
the masses would thus create a force on the cantilever
II. APPARATUS
at harmonics of this drive frequency, including the can-
tilever’sresonantfrequency. Themotionofthecantilever
on resonance was measured using a fiber interferometer; A. Cantilever
fromthismeasurement,theforcebetweenthemasseswas
deduced. Thecantileversusedinthisexperimentwerefabricated
Thermal noise provided a limit on the measurement from single-crystal silicon using standard micromachin-
of cantilever motion. To minimize this limit, cantilevers ingtechniques[10]. Thecantileverswerefabricatedfrom
3
100 Si,orientedinthe 110 direction;theywere50µm
h i h i TABLEI: Dimensions of thetest mass.
wide, 250 µm long, and 0.33 µm thick, yielding an ex-
pected spring constant k =0.005 N/m [17, 18]. Parameter Value Error Units
The resonant frequency of a mass-loaded cantilever is
Width 51 1.5 µm
determinedby the springconstantk andthe massofthe
Length 51 1.5 µm
test mass m :
t Rectangular height 32 1.5 µm
ω2 =k/m , (2) Roundedvolume 7800 3000 µm3
0 t Missing volume from side face 4000 1500 µm3
where ω is the angular frequency of the first bending Missing volume duetoporosity 2.5 2.5 %
0
mode ofthe cantilever. The spring constantofeachcan- Density of gold 19.3 0 g/cm3
tilever was deduced from the resonant frequency, which Total Mass of test mass 1.64 0.13 µg
was measured very precisely, and the mass of the test
mass, discussed in the next section. Adding the test
mass reduced the resonant frequency of the cantilever
to 300 Hz. As found from the resonant frequency, the
∼
addition of the test mass increased the spring constant A group of thirty test masses was weighed (in sets of
to 0.007 N/m; this increase was due to the shortened five or six) with a microbalance to determine the aver-
∼
effective length of the cantilever once the test mass was age mass. Because many of these test masses were im-
attached. perfect specimens, this measurement was only used as a
Cantilevers exhibit a Lorentzian transfer function be- guide. After weighing, test masses were examined under
tween driving force and amplitude; when driven at the a scanning electron microscope (SEM) and etched by a
resonantfrequencyf byforceF,themaximumdisplace- focused ion beam (FIB) to determine more precisely the
0
ment x at the center of mass of the test mass is typical dimensions and porosity of the test masses. All
the nonmagnetic test masses were fabricated together;
x(f0)=F(f0)Q/k, (3) those examined in the FIB were assumed to be repre-
sentative of the entire batch, including the one used in
where Q is the quality factor. The quality factors of Cooldown 04. The SEM showed the test masses to be
cantilevers used in this experiment were found to be as slightly larger than intended (as confirmed by the mi-
high as 80000in vacuum and at cryogenictemperatures. crobalance measurements), with a rounded face where
The energy from the thermal environment provides a the evaporated gold was polished. Etching with the FIB
constant series of random-phase impulses at all frequen- showed the evaporated gold to have some porosity near
cies tothe device. The resultingmotionofthe cantilever the side faces of the test mass.
showstheLorentziantransferfunction,withaforcespec-
After data acquisition, the test mass and cantilever
tral density S on resonance of
f used in Cooldown 04 were examined under an SEM
(Fig. 3). It was found that the test mass was mounted
S = 4kk T/Qω , (4)
f B 0 with its rounded side closer to the cantilever and with
p
the top flat side tilted at 0.35 rad with respect to the
where T is the cantilever temperature and k is the
B cantilever in the y-z plane. Estimates of the size and
Boltzmann constant. We used floppy (low spring con-
uncertainty in the dimensions of the test mass were de-
stant), high quality factor cantilevers at low tempera-
rivedfromexaminationofthistestmassandofthelarger
tures to reduce this limit. Force measurements were av-
group of masses from the same fabrication batch. SEM
eragedaslongasneededtoseeasignalabovenoiseoras
images of the Cooldown 04 test mass showed that the
long as was practical.
side face that was further away from the cantilever (due
to the tilt of the test mass) was partially recessed; an
estimate of this missing volume was included in the cal-
B. Test Mass culation of the mass. Error on these estimates was due
to error in the SEM measurements and uncertainty in
To fabricate test masses, gold was deposited (using a theexactshapeoftheroundedpartonthepolishedside.
thermal evaporator)into molds of plasma-etchedsilicon. The density ofthe goldwasassumedto be the bulk den-
After polishingofthetopsurfaceofevaporatedgold,the sityof19.3g/cm3; theobservedporositywasincludedin
siliconwas dissolvedto releasethe test masses. To make volume and mass uncertainty. The dimensions and den-
magnetictestmasses,1000˚AofNiwasevaporatedbefore sity of the test mass and the experimental uncertainties
the gold. The test masses were designed to be prisms in these parametersare shownin Tbl. I. The mass given
(50 50 30) µm3 in size; with these dimensions, a test in Tbl. I agrees well (within the given experimental un-
× ×
masscouldbeaffixedtoacantileverwithoneofitslarger certainties)with the mass deducedfroma comparisonof
facesflushagainstthecantileverandalignedwiththeend the resonant frequencies of the cantilever with the mass
of the cantilever. and neighboring cantilevers without masses.
4
FIG. 3: Scanning electron micrograph of the test mass and
cantilever used in Cooldown 04. A piece of dust is visible on
one corner of the test mass. The cantilever is 50 µm wide.
FIG.4: [coloronline]Opticalmicrographofthepolishedside
of a drive mass showing the meander pattern. In the exper-
C. Drive Mass
iment, the polished side of the drive mass was facing away
from the test mass.
The drive mass was also fabricated by evaporat-
ing gold into a mold of silicon. After polishing, the
silicon substrate was diced into dies approximately testmass. BecausetheYukawapotentialsinquestionare
1.8 mm 1.3 mm in size. short-range,this imperfection at a distance of >100 µm
×
Goldandsiliconhavedifferingelectricalconductivities, away from the test mass would have little effect on the
as well as differing mass densities. To eliminate the pos- results. In the analysis, this imperfection was taken as a
sibility of a periodic coupling between the masses due to small error on the bulk density of the gold in the drive
the Casimir force or charging of the drive mass, the pat- mass. It was assumed that the drive mass had poros-
tern of the drive mass was buried beneath a thin ground ity on the sides of the gold bars similar to the amount
plane. To bury the drive mass pattern, the polished side observed in the etched test mass.
of the die was mounted on a quartz backplane, which
became the bottom of the drive mass. The bulk of the
remainingsilicononthetopwasremoved,leavingalayer D. Shield Membrane
less than 2 µm thick. On top of this layer of silicon was
deposited a thin layer of aluminum oxide (for electrical The test mass and cantilever were isolated from elec-
insulation) and 1000 ˚A of gold on top of an adhesion trostatic and Casimir excitations by a shield membrane
layer of titanium. This gold film was continued around betweenthecantileverwaferdieandthedrivemass. The
tothesideofthedrivemass,whereelectricalcontactwas cantilever was held within a silicon wafer die approxi-
made to a ground wire glued on with silver epoxy. This mately 1 cm2 in size. This die was glued to another
thingroundplanemaskedvariationsintheCasimirforce silicon die, which was etched into a frame bearing a 3-
without notably affecting the varying gravitational field µm thick membrane of silicon nitride across an area of
of the drive mass. 5.2mm 2.8mm. Theentireshieldwaferdie,including
×
As shown in Fig. 4, the main part of the drive mass the membrane, was coatedwith gold onboth sides, with
pattern comprisedfive sets of goldand silicon bars,each a ground wire attached to one corner of the die. Due
1 mm long and 100 µm in each of the cross-sectional to the geometry and the tensile stress in the membrane
dimensions. The rest of the drive mass pattern pro- [19, 20, 21], the membrane was expected to be much
vided leads to which electrical contact could be made stiffer than the cantilever.
for grounding the meander (for the gravitational exper- If there was some force between the drive mass and
iment) or for passing electrical current through the me- the shield atf0, the shield wouldmove atf0. The shield
ander (for the magnetic experiment). motion could drive the cantilever capacitively or via the
A drive mass similar to the one used for measurement Casimir force. However, because of the stiffness of the
was etched by the FIB and examined under the SEM. shield and the separation of the drive frequency from
Onthepolishedsideofthedie,eachgoldbarhadaband the signal frequency, any interaction between the shield
of indeterminate composition on either side, where the andthedrivemasswouldlikelybetoosmalltomakethe
polishing created a wedge of silicon, gold, and polishing shielddeflectenoughtodrivethecantileverameasurable
gritmixedtogether. Thiswedgeextended10–20µminto amount on resonance.
the gold bar at each gold/silicon boundary, tapering off
atadepth of 10µm intothe bar;this bandofindeter-
minate compo∼sition thus comprised a small part of the E. Piezoelectric Bimorph Actuator
(100 100)µm2cross-sectionofeachgoldbar. Afterfinal
×
fabrication and mounting of the buried drive mass, this Apiezoelectricbimorphactuator(hereafterreferredto
polished side of the drive masswas facing awayfromthe as the “bimorph”) was used to move the drive mass lon-
5
gitudinally underneath the test mass. The bimorph was 1˚A(rms)onresonance;the mechanicalexcitationatthe
driven at a subharmonic of the cantilever resonant fre- baseofthecantileverwasrequiredtobereducedafactor
quency f ; the particular subharmonic (either f /3 or of Q beyond this. In order to make a sensitive measure-
0 0
f /4) was chosen so the drive frequency was below the ment, it was crucial to ensure that the bimorph was not
0
resonance of the bimorph but high enough to gain a res- simplymechanicallyexcitingthecantilever. Eventhough
onant enhancement in the amplitude and a reduction of thebimorphwasmovedatasubharmonicoftheresonant
nonlinearities in the bimorph motion. At low tempera- frequencyofthecantilever,nonlinearitiesinthebimorph
tures,self-heatingofthebimorph,resonantenhancement motion (as in any piezoelectric device) meant that some
ofthe motion, anddriving voltageslargerthanthe room component (typically a few percent) of its motion was
temperature limits for the device allowed an amplitude at f ; this necessitated vibration isolation between the
0
of 100–125 µm of motion at a drive frequency f of 90– bimorph stage and the cantilever wafer stage.
d
120 Hz. Finite element analysis showed that the magni- The bimorph and cantilever wafer were separated by
tudeofthetime-varyingforceat3f fromtheNewtonian two simple spring-massvibrationisolationstages. These
d
andanyYukawapotentialbetweenmasseswouldvaryas stages each had resonant frequencies of 2.5 Hz, cal-
∼
afunctionofthebimorphamplitude,withthe maximum culated to yield roughly six orders of magnitude of at-
occurring at a bimorph amplitude of 135 µm. tenuation at the bimorph frequency and eight orders of
∼
The bimorph was secured at its base in a brass and magnitude of attenuation at the signal frequency f .
0
Cirlex [22] clamp. Onone side of the bimorphwas glued The electrical wires and the optical fiber that ran the
acapacitiveelectrodefacingacounter-electrodemounted length of the probe could have, if not loose enough,
on the clamp. On the other side of the bimorph was shorted out the vibration isolation system. Measure-
mounted a small mirror. Prior to being mounted in the mentsofthe cantileveratlowtemperaturesprovedto be
probe, the bimorph was calibrated. Room-temperature the best test of the vibration isolation system. The me-
calibration of the bimorph was performed by measuring chanical coupling between bimorph and cantilever could
the rms capacitance between the two electrodes with a be assessed by measuring the motion of the cantilever
givendrivingvoltageonthebimorph. Thecorresponding with the bimorph moving at a large ( 1 mm) vertical
∼
amount of motion was determined by reflecting a laser separationbetween masses,with the signal frequency on
beam off the mirror onto a linear CCD array. At low and off resonance of the cantilever. Such couplings were
temperatures, the amplitude of the bimorph motion was found to vary over the course of one cooldown, with no
determined from a measurement of the rms capacitance. clear indication of failure of the vibration isolation.
Uncertainties in the bimorph motion were tabulated Externalvibrationisolationprotectedthe fragileparts
through the linear fits used to compare the capacitance of the probe from human mechanical disturbances when
measurement in situ to the calibration. The total ex- the drive mass was only microns away from the shield
perimental uncertainty in the amplitude of motion was membrane. Duringdata-acquisition,theentirecryogenic
10µm,mostlyaresultofuncertaintyintheoriginalmea- system was suspended from a thick concrete ceiling by
surement of the laser spot position from the CCD array. springs, yielding a resonant frequency for the entire sys-
A small ground cap was glued on top of the bimorph, tem of close to 2 Hz.
to help isolate the shield membrane from the large ac
voltages used to drive the bimorph. The drive mass was
glued on top of this ground plane. G. Capacitive Position and Tilt Sensors
For purposes of the calibration, the bimorph bending
shape was considered to be an arc with constant radius
Capacitive position and tilt sensors were used to indi-
alongthe lengthofthe bimorph. Forouranalysis,itwas
catetherelativepositionbetweenthebimorphandwafer
assumed that the drive mass was moving purely in the
stages, so that the masses could be aligned to maximize
horizontalplane; in fact as the bimorphbends, the drive
thegravitationalforcebetweenthem. Thecapacitivepo-
mass will tip up at the ends, changing the vertical sep-
sition sensor (CPS) was based on the design described
aration between the test mass and the drive mass below
in Ref. [23]. A quadrature pattern of gold electrodes
it. Depending on the alignment of the masses, the re-
waspatternedonaquartzsubstrateandtheseelectrodes
sulting change in the verticalseparationbetween masses
were mounted on the bimorph stage. Above this, on the
may be as much as 2 µm over the course of one period
wafer stage, was mounted a single sense electrode. An
of drive mass motion. This effect was not expected to
ac voltage was applied to the quadrature electrodes and
substantially change the signal and a full modeling of it
the current was read from the counter-electrode via a
has been left for future work.
dual-phase lockin amplifier. With a phase shift of 90
degrees between the signals applied to each of the four
quadratureelectrodes,thetwochannelsofthelockinam-
F. Vibration Isolation plifierprovidedreadingslinearintherelativexandy po-
sitions between the bimorph stage and the wafer stage.
Measurement at the thermal noise limit required me- With the signal applied in-phase to each of the quadra-
chanical excitation at the cantilever tip to be less than tureelectrodes,thesignalprovidedanindicationof1/z ,
c
6
wherez istheverticalseparationbetweenthesenseand
c
quadrature electrodes.
The tiltsensorsweresimpler,eachcomprisinganelec-
trode mounted on the bimorph stage and a counter-
electrode on the wafer stage above. In each case, an
ac voltage was applied to one electrode, with the cur-
rent read from the corresponding counter-electrode via
a lockin amplifier. The tilt sensors provided an indica-
tion of the vertical separation between the stages at two
other locations in addition to the CPS. The geometry of
the three capacitive sensors is shown in Fig. 5.
Together, the capacitive sensors provided an indica-
tion of the relative position and tilt between the stages,
encompassing five degrees of freedom. The relative rota-
tionbetweenthestagesaboutthez-axiswasfixedduring
probe assembly.
Thecapacitivesensorswerecalibratedatroomtemper-
aturebyraisingthebimorphstagewithathree-axisposi-
tionerandrecordingthemicrometerreadingsfromthez-
axis of the positioner along with the capacitive readings.
FIG. 5: Drawing of the bottom half of the probe. The bi-
The CPS was also calibrated in x and y using a similar
morph stage and wafer stage are shown with the wafer stage
technique. With the probe under vacuum,the three-axis raised and rotated about the x-axis to reveal the geometry
positioningstagetiltedduetoatmosphericpressure;this of the capacitive sensors. In practice, the two electrodes of
tilt did change the correspondence between micrometer a given sensor were separated by less than 1 mm in the z-
and CPS readings. Alignment between masses was es- direction. Thethreesensorswereseparatedby∼4cminthe
tablished in room temperature at atmospheric pressure. x-yplane. Thepadsusedformakingelectricalcontacttothe
Atlowtemperatures,withthesystemundervacuum,the capacitive sensors are omitted from this drawing.
capacitivesensorsandtheroomtemperaturecalibrations
(ratherthanthe micrometerreadings)wereusedtoindi-
cate the relative position and tilt between stages. I. Interferometer
Cantilever motion was measured with a fiber interfer-
ometer, based on the design described in Ref. [24]. An
InGaAs laserdiode senthundreds ofmicrowattsof1310-
H. Cryogenic Apparatus nm light through a bidirectional fiber coupler, leaving
1% of this light for the length of fiber that went to the
cantilever. The cleavedend of the fiber in the probe was
Cooling the cantilever was crucial to achieving a high
aligned to the test mass, forming a low-visibility Fabry-
force sensitivity. The probe was sealed in a vacuum can
Perot cavity with a length of 50 µm. Interference be-
inside a 4He cryostat, with an exchange gas space sep- ∼
tween the light reflected off the test mass and the light
arating the probe from the mechanical vibration of the
reflected from the cleavedend of the fiber was measured
boiling helium. The three-axis positioner was located
withaphotodiodeviaatransimpedanceamplifierwitha
outside of the cryostat, attaching to the bimorph stage
10 MΩ feedback resistor.
via a vacuum feedthrough. The interferometer was also
Because of the low reflectivity of the cleaved end of
locatedoutsidethecryostat,withalengthofopticalfiber
the optical fiber, beam divergence, and imperfect align-
running down the probe.
mentbetweenthefiberandthetestmass,onlyonereflec-
Base temperature of the probe was typically 10 K,
tion from each surface (the fiber end and the test mass)
though this temperature was raiseda few degreesby the
was expected to contribute to the interferometer signal.
useofspring-adjustmentheaters(describedinSec.IVE).
Thus, the dc interferometer signalwas a sinusoidalfunc-
The noise temperature of the cantilever was found to be
tion of the distance between the fiber end and the test
typically 10 K more than the base temperature of the
mass, modulo half the wavelength of the laser. The sen-
∼
probe.
sitivity of the interferometer was maximized when the
Hold time for the cryostatwas typically 4–6 days. Af- distance betweenthe cantileverandthe opticalfiber was
ter each helium transfer, all preliminary tests of the sys- adjusted to be at one of the points of maximum slope of
tem were performed. Each data set presented in this this sinusoidal fringe (the center of the fringe).
paper was recorded entirely within one helium transfer Cantilever motion was typically on the order of
of the respective cooldown. angstroms at low temperatures. For motion much less
7
than the wavelength of the laser, the interferometer sig- F(dc) (with α = 1) were calculated by sums over the
Y
nalatagivenfrequency V (f)waslinearlyrelatedto the drive mass and the test mass in the 5 µm mesh. Only
i
amplitude of cantilever motion at that frequency x(f). the vertical component of the force was considered and
Theconversionbetweenthevoltagesignalfromtheinter- the attractive force was taken to have a positive sign.
ferometer and cantilever motion depended on the peak- The Newtonian force sum is given by:
to-peak amplitude V of the sinusoidal interferometer
pp
fringe and the relative position between the cantilever F(dc) = Gδmdδmt z . (6)
afrnindgteheofotphteicainltfiebrfeerrothmaettedretdecrmleivneeld. tWhehleoncathtieondiosntatnhcies N XVd XVt (cid:20)(cid:18) r2 (cid:19)(cid:16)r(cid:17)(cid:21)
between the fiber and the cantilever was adjusted to the
Here,thetwosumsareoverthevolumesofthedrivemass
center of the fringe, the amplitude of cantilever motion
and the test mass in the given mesh, δm refers to the
d
was determined by mass of the (5 µm)3 block in the drive mass, δm refers
t
λ to the mass of the (5 µm)3 block in the test mass, and
l
x(f)=Vi(f) , (5) r is the distance between the centers of mass of these
2πV
pp two blocks in the summation. The vertical separation
where λ is the wavelength of the laser. between the two mass blocks is z; the final term in the
l
Temperaturecontrolandhigh-frequency(>100MHz) equationis a projectiononto the verticalaxis. Similarly,
modulation [25] were used to stabilize the laser during the Yukawa force (for α = 1) for a given value of λ was
data acquisition. The high-frequency modulation short- calculated from the sum:
ened the coherence length of the laser, reducing the im-
Gδm δm z
portance of stray reflections from connectors within the F(dc) = d t e−r/λ(1+r/λ) .
optical part of the interferometer circuit. Y XVd XVt (cid:20)(cid:18) r2 (cid:19) (cid:16)r(cid:17)(cid:21)
(7)
A time-variation (accounting for drive mass motion)
J. Cantilever Position Adjustment and
was applied to this calculated set of dc forces as a func-
Characterization
tion of y-position and from this, the expected ac (finite
frequency)forcewas extractedwith a Fouriertransform,
A piezoelectric stack (hereafter referred to as “piezo yielding ac Newtonian andYukawa forces. In the case of
stack”) mounted underneath the cantilever wafer was Cooldown 04, the third harmonic of the drive frequency
used to adjust the distance between the cantilever wafer wasstudied;theFouriercomponentofthemeasuredforce
andthe opticalfiber to maintainalignmentatthe center at this frequency is referred to as the third harmonic ac
of the interferometer fringe. This piezo stack was also force.
used to excite the cantilever for the purposes of charac- There were several inputs to the FEA model: the ge-
terization. ometry and density of the masses, the (x, y, z) position
Intheprobe,thepiezostackwasmountedbetweenthe and tilts between masses, amplitude of bimorph motion,
waferframeandthewaferstage. Uponcooling,thestain- and the range λ of the Yukawa potential being modeled.
lesssteelwaferframeandstagewouldcontractwhile the The output of the model was the ac Newtonian gravita-
piezostackwouldlengthenslightly. Atlowtemperatures, tionalforcebetweenmassesF andthe ac Yukawaforce
N
it was expected that the differential thermal contraction F forα=1andagivenλ. Forarbitraryα,theYukawa
Y
of the materials would have made the wafer frame tilt in force could simply be scaled by α.
the x-z plane. A correction for this tilt was included in Both the magnitude and phase of this Fourier compo-
the room temperature alignment between masses. nent with respect to the drive frequency are important.
Clearly, the dc gravitational force between the masses
reflects the periodicity of the drive mass pattern. In
III. FINITE ELEMENT ANALYSIS AND
fact, the ac force also shows this periodicity. As shown
SPATIAL PHASE-SENSITIVE DETECTION
in Fig. 6, the magnitude of any ac gravitational force
has a clear periodicity of 100 µm as a function of the
InordertodeduceaboundonYukawa-typedeviations y-equilibrium position of the drive mass with respect to
from Newtoniangravity,the measurementmust be com- thetestmass,correspondingtothe100µmhalf-periodof
paredto the expected force between the masses for both the drive mass pattern. Eachminimum of the third har-
a Newtonian potential and a Yukawa potential. Finite monic ac force magnitude is zero and is accompanied by
element analysis (FEA) with a mesh size of 5 µm was a discontinuous phase changeof π. The fourth harmonic
used to calculate the expected dc (without time varia- ac force (not shown) has the same periodicity, though
tion) gravitational force between the masses for a range the fourth harmonic force has minima where the third
of longitudinal (along the y-axis) positions between the harmonic force has maxima, and vice-versa.
masses, with the vertical separation held constant. To exploit this geometric feature of our design, data
At a given y-point (with other alignment parameters were recorded at several values of the y-equilibrium po-
set), the dc Newtonian force F(dc) and dc Yukawa force sitionbetweenmasses,scanning overmorethan 200µm,
N
8
allowed selection of the better test masses from the fab-
−2110 N)10500 NYuekwatownai aFno Frcoerce ricAatfitoenr baatttachch.ing test masses to cantilevers, the can-
e ( tileverwafer wasgluedinto a stainless steelwafer frame.
c
For 0 The shield wafer was then glued to this cantilever wafer,
−600 −400 −200 0 200 400 600
Equilibrium y−Position of Drive Mass (m m) using a press to keep the wafers parallel. Measurements
before and after each gluing showed the degree of tilt
( 1 mrad)betweenthe twowafers. Photographsofthe
d) 2 ≪
a two wafers before and after gluing were used to deter-
ase (r−02 minetherelativepositionbetweeneachcantileveronthe
Ph wafer and the center of the shield membrane.
−600 −400 −200 0 200 400 600 By design, the shield membrane was 10 µm below the
Equilibrium y−Position of Drive Mass (m m)
surfaceoftheshieldwaferthatwasgluedtothecantilever
wafer. The glue between the wafers added typically 1–
FIG. 6: Thecalculated magnitude(top) and phase(bottom) 5 µm to this distance. The offset of the shield and the
ofthepredictedthirdharmonicacgravitationalforcebetween thicknessofthegluelayerdidlimittheminimumvertical
masses. The larger Yukawa force for α = 5 and λ = 34 µm separation between masses. However, reducing this dis-
is shown by the dashed line; the smaller Newtonian force is tance significantly would cause the cantilever to snap-in
the solid line. The phase is the same for a Newtonian or a and adhere to the shield [26].
Yukawaforce. Inthiscalculation,thedrivemasswastakento
be27µmunderneaththetestmass,withouttilt. Themiddle
gold bar of the drivemass is centered underthetest mass at
B. Probe Assembly
y-position of 0 µm.
Before mounting in the probe, the shield membrane
the period of the drive mass pattern. Comparison of wascarefullyexaminedfordust. Anypiecesofdustwith
these y-scan measurements to FEA predictions allowed a resolvable height profile ( 2 µm) were removed with
≥
us to discriminatebetweencouplingsthat couldbe grav- a microprobe. Then the wafer was mounted in the wafer
itational in origin and spurious backgroundsthat do not stage and the optical fiber was aligned to the test mass.
followthegravitationalpattern. By“locking-in”thisway Calibrations of position sensors were performed and the
to the expected spatial periodicity, we were able to set a drive mass and the bimorph were mounted in the probe.
stronger and more accurate bound on α(λ) than what a
single force measurement would have provided.
C. Drive Mass Gluing
IV. EXPERIMENTAL METHODS The drive mass was glued to the bimorph with the
bimorph positioned so that the drive mass was pushed
In order to accurately compare measurements to the againstthesiliconshoulderofashieldwafer,mountedin
FEA, alignment between masses had to be known as ac- theprobe. Thisprocess,onlyrepeatedwhenthebimorph
curately and precisely as possible. In order to maximize orthedrivemassneededtobereplaced,achievedapprox-
the gravitational force between masses, the goal was to imate parallelism between the drive mass and any can-
centerthemasseswithrespecttoeachotherwithnotilts tileverwafer. Before eachcooldown,parallelismbetween
about the y-axis (θ ) or the x-axis (θ ). The align- the shield wafer and the drive mass was again examined
xz yz
ment coordinates are diagrammed in Fig. 7. With no andadjusted. Uncertainties in the tilt betweenthe drive
tilts in the system, the third harmonic ac gravitational massandthe cantileverweredominatedbyuncertainties
forcewouldbemaximizedwiththetestmassatx=0µm in this process.
and y = 50 µm with respect to the center of the drive
mass at equilibrium.
Alignment between the masses was fixed by room- D. Alignment
temperature preparations and low-temperature adjust-
ments. Thisprocessbeganwiththe assemblyofthe can- After the drive mass was glued to the bimorph, the
tilever and shield wafers. parallelism between the shield wafer and the drive mass
was examined optically, using images recorded from a
telescopeviaaCCDcamera. Parallelismwasassessedin
A. Wafer Assembly the x-z andy-z planes by lookingatthe reflectionof the
drivemassinthegold-coatedshoulderoftheshieldwafer.
The test mass was attached to the cantilever using CPS readings and optical images were compared as the
epoxy applied with a microprobe under an optical mi- bimorph stage was moved to change the vertical separa-
croscope. The optical microscope used for this process tion between the drive mass and the shield wafer. Tilts
9
Top View Side View alignment between the wafer stage and bimorph stage,
Gold
Test Mass Tilt Optical Shield and hence the alignment between masses. At base tem-
Silicon Drive Mass Fiber Membrane perature,thetiltofthewaferstagewasadjustedbyheat-
Motion
ing two of the three springs on the upper vibration iso-
Y
* (0,0) lation stage. To heat a spring, a current on the order of
Z
X Q xz X 20mAwaspassedthroughamanganinwire(ofresistance
Piezoelectric
50 Ω) wrapped around the coils of the BeCu spring;
Bimorph Z-separation
= 100 microns ~ 25 microns ∼such heating could increase the length of the spring on
the order of 100 µm. Temperature and position of the
probe stabilized after severalhours.
FIG. 7: [color online] Schematic showing the geometry and
alignmentparametersbetweenthetestanddrivemasses. On The long time scale of the spring heating prohibited
theleft,thetopviewshowsthex-y plane;theorigin,marked fine adjustment of the tilt. However, the large separa-
withanasterisk, isthecenterofthedrivemassinthisplane. tionbetweenthe tilt sensors( 38mm) in the x-y plane
With the drive mass at equilibrium, the x-y position of the in comparison to the size of t∼he drive mass ( 1.5 mm)
test mass was defined with respect to this origin. This fig- allowed for a coarse readjustment of tilt at lo∼w temper-
ure shows the masses to scale; the shield membrane and
atures; typically, the vertical separation between the tilt
the ground plane over the drive mass are omitted. On the
sensorsfor a givenCPSz -readingwaswithin 100µmof
right, the side view shows (not to scale) the z-x plane. The c
the room temperature alignment points.
z-separation between masses is the face-to-face vertical dis-
Withthetiltadjusted,thex-y positionofthebimorph
tance. The test mass tilt, exaggerated in this figure for the
purposes of illustration, is the tilt of the test mass with re- stagewasadjustedtoregaintheroomtemperaturealign-
specttothecantilever. Theanglebetweenthedrivemassand ment points. The manual operationof the x-micrometer
thecantilever in this planewas defined as θxz. necessitatedcoarseralignmentinthisdirection,sincethe
micrometer could not safely be adjusted with the drive
mass positioned close to the shield wafer. However, the
measured in two vertical planes (θ and θ ) were ad- alignment between the masses in the x-direction only
xz yz
justed using turnbuckles on the lower vibrationisolation needed to be accurate to within a couple of hundred mi-
stage(thewaferstage),withadditionalcompensationfor crons, since the Yukawa forces being studied were short
the expected piezo-induced tilt of the wafer frame upon rangeincomparisontothe1-mmlengthofthedrivemass
cooling. Uncertainty in this part of the alignment was bars. The motor control of the y and z micrometers al-
dominated by optical limitations. lowedadjustmentsinthesedirectionstowithinamicron,
The large three-axis positioner was used to adjust the withuncertaintybeingdominatedbynoise(electricaland
alignment of the drive mass with respect to the test mechanical) on the capacitive readings.
massbothatroomtemperatureandatlowtemperatures. Additional information about the alignment between
Micrometers on this stage were accurate to 2.54 µm masses was provided by moving the bimorph frame in
(0.0001 in), with the y and z micrometers controlled by the y-direction until the drive mass contacted the side
motors. of the silicon frame bearing the shield membrane. This
After optimization of the tilt, the bimorph frame was test was performed with the bimorph static and lowered
moved so that the drive mass was centered in the x-y so that the drive mass was 200 µm away from the
∼
plane with respect to the shield membrane; a telescope shield. Thecontactbetweenthedrivemassandthewafer
with a reticle aided this centering process. The bimorph gave a mechanical impulse to the cantilever, clearly vis-
stage was then slowly raised until the drive mass con- ible on the interferometer signal. The distance between
tacted the shield. This contact gave an impulse to the the alignment point and this contact point provided ad-
shieldandthecantilever,creatingaclearsignalonthein- ditional confirmation of the drive mass alignment along
terferometer. CPSandtiltsensorreadingswererecorded the y-axis.
severaltimes during this process; these alignmentpoints After realignment, the bimorph frame could then be
were the targets for initial alignment when the system movedto the y-position,determined by the photographs
was cold. Uncertainties in this process included optical of the wafer, at which the drive mass was centered un-
limitationsandmotionofthethree-axispositioningstage derneath the test mass. The experimental values and
in the x-y plane as it was raised along the z-axis. uncertainties in the alignment parameters are given in
The drive mass and shield were examined through a Tbl. II.
telescope for any indication of dust on either surface be-
fore the vacuum can was closed.
F. Vertical Positioning of Drive Mass
E. Re-Alignment Typically, gravitational measurements were recorded
with the drive mass positioned 10–15 µm away from the
Under vacuum, the tilt of the positioning stage con- shield, with this safety factor allowing for the small ver-
nected to the bimorph frame significantly changed the tical motion of the drive mass over the course of the bi-
10
H. Bimorph Actuation
TABLE II: Alignment between drivemass and test mass
Parameter Value Error Units After alignment between the masses was established,
x-position -91 110 µm the bimorph was lowered to a safe distance and an ac
y-position (-12)–(283) 119a µm voltage was applied to the bimorph via a high voltage
amplifier. The capacitive measurement of the bimorph
z-separation 24–28 3.4 µm
motion indicated that the bimorph motion for a given
|θxz|× widthb 1 6 µm driving voltage increased over time, typically coming to
|θyz|× lengthc 6 9 µm equilibrium within 30 min.
aThisistheerroronthelocationoftheentirerange. Withinthis
range,therewasanuncertaintyofabout5µmoneachpoint.
bHere,widthisofthedrivemassdie: 1.3mm. I. Experimental Degrees of Freedom
cHere,lengthisofthedrivemassdie: 1.8mm.
Tomakethemostsensitivegravitationalmeasurement,
data were recorded as a function of y-position between
morph swing, backlash in the motor used to operate the masses, with a small vertical separation between masses
z-axis of the three-axis positioner, and the possibility of andthedrivefrequencyf tunedtobef /3. Diagnostics
d 0
drift during a data run. included data runs with f tuned slightly off this reso-
d
To determine the vertical (z-) separation between the nance, with the drive mass far away from the test mass,
masses, the bimorph frame was slowly raised until the with the bimorph not moving, or with f =f /4.
d 0
drive mass contacted the shield, giving an impulse to
the cantilever clearly visible on the interferometer. The
bimorph was then lowered to the desired distance from V. DATA ACQUISITION AND AVERAGING
theshieldwiththeCPSindicatingtheamountofmotion.
The largest sources of uncertainty in the z-separation Two streams of time-series data were recorded via an
between masses were bouncing during contact and data analog-to-digital converter (ADC) on a personal com-
acquisition and the possibility of dust on the shield or puter: the voltage from the interferometer and the
drive mass. Due to lack of precision in setting a given voltage from the function generator that was driving
z-separation(due to limitations of the motordriving the (through a high-voltage amplifier) the bimorph. The in-
z-micrometer), the z-separation during data acquisition terferometer signal showed the cantilever motion. The
varied over the course of a day; this is the reasonfor the function generator signal (at frequency f close to
d
range given in Tbl. II. 100Hz) providedanimportanttiming signal. Data were
recordedat a frequency of 10 kHz. Before the ADC, the
interferometer signal was ac-coupled to a pre-amplifier
G. Cantilever Characterization with a high-frequency rolloff at 3 kHz to avoid aliasing.
Anymotionofthe cantileverthatwasdue tothemov-
After reaching base temperature, the piezo stack was ing drive mass would be at a definite phase with respect
usedtoverifytheinterferometerfringe. Theshapeofthe to the drive mass motion and at a harmonic of the drive
fringe was an important indicator of alignment between frequency. The function generator signal (the drive sig-
the optical fiber and the test mass. In earlier designs nal) provided a proxy for the drive mass motion and the
of this apparatus,the fiber alignment often drifted upon analysis of the cantilever motion then included a phase
cooling so that more light was reflecting from the shield defined with respect to this drive signal.
than from the test mass. Other times, light reflected Eachdata runincluded data recordedovera periodof
from the cantilever as well as from the test mass. The timet ,withtime-seriesdataexaminedinshorterrecords
t
former situation created a large background level on the of time t . The time t , typically on the order of Q/f ,
0 0 0
interferometer; the latter would make it impossible to waschosensothateachrecordoflengtht couldbe con-
0
record data. In Cooldown 04, the shape of the fringe sidered statistically independent of other records, while
indicated no misalignment between the optical fiber and maximizing the total number of records N =t /t .
t 0
the test mass at low temperatures. For each recordof length t , the time-series data were
0
The resonantfrequencyf wasdeterminedby exciting truncated to include an integer number of periods of the
0
the cantilever at large amplitudes with the piezo stack function generatorsignal. This truncationshortenedthe
and comparing the amplitude and phase of the resulting length of each record a negligible amount and increased
signal to the drive signal. The quality factor was de- theaccuracyoftheamplitudeandphaseoftheharmonics
termined by timing the ringdown of the cantilever from of f reported by a Fourier-transformof the record.
d
excitation on resonance. The fringe height, the resonant The drive signal, resulting from a very clean function
frequency, the quality factor, and the dc level of the in- generator signal, showed an unambiguous peak on the
terferometeronthefringewerecheckedbetweendatacol- Fourier transform at the drive frequency f . The har-
d
lection runs. monicofinterestofthisdrivefrequencywasthenselected