A widely used assumption within the gravitational-wave community has so far
been that a test mass acts like a rigid body for frequencies in the detection
band, i.e. for frequencies far below the first internal resonance. In this
article we demonstrate that localized forces, applied for example by a photon
pressure actuator, can result in a non-negligible elastic deformation of the
test masses. For a photon pressure actuator setup used in the gravitational
wave detector GEO600 we measured that this effect modifies the standard
response function by 10% at 1 kHz and about 100% at 2.5 kHz

Acemese F

B Willke

H Grote

H Lück

Hild S

I Martin

J Hough

J R Smith

K Danzmann

K Mossavi

K Strain

M Brinkmann

M Hewitson

M Weinert

N Rainer

P Willems

S Hild

S Reid

Sigg D

Smith R

Takahashi R

W Winkler

English

arXiv

A widely used assumption within the gravitational-wave community has so far been that a test mass acts like a rigid body for frequencies in the detection band, i.e. for frequencies far below the first internal resonance. In this paper, we demonstrate that localized forces, applied for example by a photon pressure actuator, can result in a non-negligible elastic deformation of the test masses. For a photon pressure actuator setup used in the gravitational-wave detector GEO 600, we measured that this effect modifies the standard response function by 10% at 1 kHz and about 100% at 2.5 kHz

Hild, S.

Brinkmann, M.

Danzmann, K.

Grote, H.

Hewitson, M.

Hough, J.

Luck, H.

Martin, I.

Mossavi, K.

Rainer, N.

Reid, S.

Smith, J.R.

Strain, K.

Weinert, M.

Willems, P.

Willke, B.

Winkler, W.

NARCIS 

Photon-pressure-induced test mass deformation in gravitational-wave detectors

Maastricht University Research Portal

Lück, H.

Smith, J. R.

Caltech Authors - Main

IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 24 (2007) 5681–5688 doi:10.1088/0264-9381/24/22/025
Photon-pressure-induced test mass deformation in
gravitational-wave detectors
S Hild1, M Brinkmann1, K Danzmann1, H Grote1, M Hewitson1,
J Hough1, H Lu¨ck1, I Martin2, K Mossavi1, N Rainer1, S Reid2,
J R Smith3, K Strain2, M Weinert1, P Willems4, B Willke1
and W Winkler1
1 Max-Planck-Institut fu¨r Gravitationsphysik (Albert-Einstein-Institut) and Leibniz Universita¨t
Hannover, Callinstr 38, D-30167 Hannover, Germany
2 SUPA, Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
3 Syracuse University, Department of Physics, 201 Physics Building, Syracuse, New York
13244-1130, USA
4 The LIGO project, California Institute for Technology, Mail Stop 18-34, Pasadena,
California 91125, USA
E-mail: stefan.hild@aei.mpg.de
Received 9 August 2007, in final form 28 September 2007
Published 6 November 2007
Online at stacks.iop.org/CQG/24/5681
Abstract
A widely used assumption within the gravitational-wave community has so far
been that a test mass acts like a rigid body for frequencies in the detection
band, i.e. for frequencies far below the first internal resonance. In this paper,
we demonstrate that localized forces, applied for example by a photon pressure
actuator, can result in a non-negligible elastic deformation of the test masses.
For a photon pressure actuator setup used in the gravitational-wave detector
GEO 600, we measured that this effect modifies the standard response function
by 10% at 1 kHz and about 100% at 2.5 kHz.
PACS numbers: 04.80.Nn, 95.75.Kk
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The currently operating laser-interferometric gravitational-wave detectors GEO 600 [1], Virgo
[2], LIGO [3] and TAMA300 [4] are using cylindrical test masses made of fused silica to
probe changes in the metric originating from gravitational waves. Usually, such test masses
are considered to behave as rigid bodies for any applied force.
However, in this paper, we show that a non-uniformly distributed force acting on a test
mass can cause a significant deformation. We evaluate this effect quantitatively, using the
0264-9381/07/225681+08$30.00 © 2007 IOP Publishing Ltd Printed in the UK 5681
5682 S Hild et al
Figure 1. Experimental arrangement for measurements with the photon pressure actuator. M, far
mirror in the north building of the GEO 600 detector. Illumination of this mirror with light from
the laser diode produces a differential arm-length change that is measured at the main output of
the interferometer.
Table 1. Parameters used for calculating the photon-pressure-induced test mass deformation.
Test mass diameter 180 mm
Test mass thickness 100 mm
Test mass material Fused silica (Suprasil)
Beam of photon pressure actuator 5 mm diameter (Gaussian)
Interferometer beam 50 mm diameter (Gaussian)
photon pressure actuator installed at the GEO 600 interferometer. Such a photon pressure
actuator was first demonstrated by Clubley et al [5] to be employed as an easy and independent
method for displacement calibration of a gravitational-wave detector.
Figure 1 shows a diagram of the experimental setup of the photon pressure actuator in the
GEO 600 detector. The light of a laser diode is collimated by a lens, L, and enters the vacuum
system via a viewport, before it is impinging on one of the main mirrors, M. More detailed
descriptions of the setup can be found in [6, 7].
In section 2, we calculate the expected test mass deformation induced by the photon
pressure using three different methods. Two-dimensional and three-dimensional finite element
analyses are performed as well as an analytical estimation. The corresponding displacement
sensed by the gravitational-wave detector is evaluated in section 3 using a simple model.
A prediction is made of how the response to the photon actuator changes when the test
mass deformation effect is taken into account. The presence of the test mass deformation is
experimentally confirmed by measurements presented in section 4.
2. Models used to calculate the test mass deformation
In order to evaluate the effect from any test mass deformation introduced by photon pressure,
we have to calculate the actual deformation caused by the setup of the GEO photon pressure
actuator. Table 1 indicates the parameters used for the calculations.
Altogether we compared three different ways to calculate the mirror deformation: two
finite element analyses have been performed in addition to a (quasi-)analytical calculation.
The following list gives a short description of each of the methods applied:
• ANSYS: ANSYS is a well-known finite element software package [8]. The three-
dimensional model consisted of about 42 000 elements. A non-homogeneous meshing
Photon-pressure-induced test mass deformation in gravitational-wave detectors 5683
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
di
sp
la
ce
m
en
t [
m
/N
]
radius [m]
Application of a force with 
Gaussian distribution (radius of 2.5mm)
ANSYS
Analytical
Comsol
Figure 2. Radial profile of the test mass deformation originating from a force continuously applied
with a Gaussian distribution over a radius of 2.5 mm around the center of the test mass. For the
analysis a GEO 600 standard test mass was used (18 cm diameter, 10 cm thickness, made of fused
silica (Suprasil)). Shown are the results of two finite element analysis. The result produced by
the ANSYS software is represented by the blue dashed-dotted line, while the COMSOL software
produced the orange dashed line. An analytical calculation is indicated by the solid green line.
The corresponding effective displacements (see table 2) agree within 10% for all three results.
(Explanation is given in the text.)
was chosen, employing a very dense meshing in the center of the test mass in order to
resolve the maximum of the deformation.
• Comsol: Comsol multiphysics [9] is also a widely used finite element analysis software
package. The deformation of the cross-section of the test mass was analyzed by
performing a two-dimensional simulation.
• Analytical: The analytical calculation is based on an algorithm developed by Bondu,
Hello and Vinet [10] for the calculation of thermo-elastic noise in an infinite half-plane
mirror. In addition, our calculation includes corrections and extensions from Liu and
Thorne [11].
Figure 2 shows the radial profile of the calculated displacement of the test mass surface,
D(r). The three methods agree pretty well over the full radius of the test mass.
The effective displacement actually measured by the interferometer, Dtotal, is determined
by the overlap of the mirror deformation and the main interferometer beam, which can be
described by a Gaussian beam. The radial intensity of the interferometer beam, I (r), is given
by
I (r) = exp
(−2r2
ω2
)
, (1)
where ω = 25 mm is the radius of the beam. Each point on the mirror surfaces contributes
to the total effective displacement, Dtotal, weighted by the product of the power of the
5684 S Hild et al
Table 2. Results of the three different calculations of the photon-pressure-induced test mass
deformation. The three values for the effective displacement agree to within 10% and the resulting
notch frequencies (see section 3) match within 5%.
Effective Resulting notch
displacement (m N−1) frequency (Hz)
ANSYS 3.83 × 10−10 3598
Comsol 3.49 × 10−10 3768
Analytical 3.54 × 10−10 3741
main interferometer beam, I (r), and the corresponding displacement D(r).5 For a radially
symmetric setup, as described above, the total effective displacement can be expressed by a
single integral
Dtotal =
∫ 0.09m
0
∫ 2π
0
Deff · r · dr · dϕ =
∫ 0.09m
0
2π · r · kI · I (r) · D(r) · dr. (2)
The factor kI is a normalization factor for I (r) in order to give∫ 0.09m
0
2π · r · kI · I (r) · dr = 1. (3)
Using equation (2) the total effective displacement for all three calculated test mass
deformations can be derived. The corresponding results are displayed in the second column
of table 2. For the three different calculations of the test mass deformation the effective
displacement sensed by the main interferometer, Dtotal, agrees within about 10%.
3. Predicted effect of the photon-pressure-induced test mass deformation
The previous section showed that the test masses are not completely rigid and that the photon
pressure actuator beam really causes a non-negligible deformation. Next, we have to evaluate
how strong the displacement originating from the non-rigidity of the test mass is, compared to
the displacement of the center of mass (originating from the pendulum response). Since both
responses are linear in the applied light power of the photon pressure actuator we can simply
compare their responses.
The pendulum response, αpen, follows a 1/f 2-law, has a magnitude of 5 × 10−7 m N−1
at 100 Hz and is 180◦ out of phase with the light power modulation (see blue dashed trace
figure 3)6. The response of the mirror deformation, αdef , is assumed to be flat in frequency and
in phase with the modulated light power for frequencies below the first internal resonances of
the test mass, which in the case of a GEO test mass, has a frequency of roughly 11 kHz [12].
In the previous section, effective displacements in the range 3.54–3.83 × 10−10 m N−1 were
found. The magnitude and phase of αdef from the analytical test mass deformation is shown
by the red dashed-dotted trace in figure 3. The total response, αtotal, plotted in green (solid) is
the sum of the two individual responses:
αtotal = αpen + αdef . (4)
5 The phase change of the interferometer light due to the mirror deformation is proportional to the light amplitude√
I (r). At the sensing point of the interferometer (which is a photodiode) the signal is beaten with the local oscillator,
which also has a spatial profile of
√
I (r). Taking both these effects into account we have to use I (r) for weighting
the radial test mass deformation D(r).
6 Due to the fact that we are only interested in frequencies far above the resonance of the pendulum which is around
1 Hz, excitation and pendulum motion have opposite phase.
Photon-pressure-induced test mass deformation in gravitational-wave detectors 5685
103 104
10
10
m
a
gn
itu
de
 [m
/N
]
pendulum response
effective displacement from mirror deformation
total displacement response
103 104
0
100
ph
as
e 
[d
eg
]
103 104
0
1
2
3
ra
tio
 w
ith
ou
t/
w
ith
de
fo
rm
a
tio
n
Frequency [Hz]
Figure 3. Simple model for the photon pressure calibrator taking into account the responses from
the pendulum and from the mirror deformation effect (analytical). The pendulum response follows
a 1/f 2-law and is 180◦ out of phase from the photon pressure excitation. The mirror deformation
has a flat response and is in phase with the photon pressure actuation. If both responses are added
a notch appears at the frequency where both responses have equal size. The purple trace shows
the expected discrepancy between the calibrations with and without accounting for the test mass
deformation.
The total response shows a steep notch at the frequency where the individual responses
have the same magnitude and compensate each other completely due to having opposite
phase. At the frequency of the notch the phase of αtotal jumps from −180 to 0◦. The
modeled notch frequencies for the three different methods are displayed in the third column of
table 2. The resulting discrepancy between the photon pressure actuator response without and
with test mass deformation is shown in the lowest subplot of figure 3. Already at 1 kHz the test
mass deformation causes an effect of 10%. At 2 kHz the error, when not taking the test mass
deformation into account, amounts to 40% and between 2.6 kHz and 4.6 kHz the discrepancy
is larger than 100%.
4. Measurement of the photon-pressure-induced test mass deformation
In order to verify this model, we injected signals to the photon pressure actuator over a wide
frequency range including frequencies as high as 6 kHz to be able to resolve any potential
5686 S Hild et al
103
10
10
10
R
es
po
ns
e 
of
 p
ho
to
n 
pr
es
su
re
 a
ct
ua
to
r 
[a
.u.
]
103
0
Ph
as
e 
[d
eg
]
Frequency [Hz]
without test mass deformation
model including test mass deformation
measurement
Figure 4. Result of the long duration photon pressure actuator injections over a wide frequency
range. The pink dashed line represents the photon pressure actuator response without taking test
mass deformation in account. The green solid line indicates the photon pressure actuator response
taking the test mass deformation into account. The actual measurement of the response is displayed
by the blue circles. The presence of the expected notch structure and by this the non-rigidity of the
test masses in the detection band of the gravitational wave is clearly confirmed by the measurement.
notch structure. The measurements presented here include long duration measurements to
achieve reasonable snr at high frequencies. Individual measurement points are made of single
discrete Fourier transforms (DFT) containing up to 10 h of data. Such amounts of data are
difficult to handle with standard computers. A heterodyne technique was employed to reduce
the volume of data. The time series of the data containing the signal of interest, Esig ·sin(ωsigt),
is multiplied by a sine wave with a slightly lower frequency, ωhet:
Esig · sin(ωsigt) · sin(ωhett) = 12Esig[cos(ωsig − ωhet)t − cos(ωsig + ωhet)t]. (5)
The second term on the right-hand side of equation (5) still contains the signal, but shifted
towards even higher frequencies. The signal component we are interested in is shifted to
a very low frequency, (ωsig − ωhet), which for our investigations was chosen to be 9 Hz.
After heterodyning, the data stream is strongly low-pass filtered and down-sampled to give
a data stream that can be handled by desktop computers. Figure 4 shows the result of
these measurements. The pink dashed line represents the photon pressure actuator response
without taking test mass deformation into account, while the green solid line indicates the
Photon-pressure-induced test mass deformation in gravitational-wave detectors 5687
photon pressure actuator response taking the test mass deformation into account. The actual
measurement of the response is displayed by the blue circles.
The presence of the expected notch structure and by this the non-rigidity of the test masses
in the detection band of the gravitational wave is clearly confirmed by the measurement. The
magnitude of the response is about a factor of 3 below the pendulum response for frequencies
between 3 and 4 kHz. At 5 kHz the measurement matches the pendulum response and finally
at 6 kHz the measured response clearly exceeds the 1/f 2 behavior of the pendulum response.
Around the notch frequency, the phase of the photon pressure actuator response also changes
significantly from about −165◦ at 2.8 kHz to about −30◦ at 4.8 kHz. A phase of nearly 0◦ at
high frequencies clearly indicates that the total response of the system is no longer dominated
by the pendulum response.
It can be seen that the notch is smeared out in the experimental data. This originates
from beam jitter of the main interferometer beam resulting in a varying overlap of the main
interferometer beam and deformed test mass. Different overlaps correspond to different
effective displacements
Dtotal = r · kI
∫ 0.09m
0
∫ 360◦
0
· I (r, ϕ) · D(r, ϕ) · dr · dϕ, (6)
which finally correspond to slightly shifted notch frequencies. Within the long measurement
intervals of up to 10 h the measurement averages over a variety of notch frequencies resulting
in a smeared out notch. A detailed description of this effect, including a quantitative analysis
can be found in [7].
5. Summary and conclusions
We evaluated the deformation of a GEO 600 test mass caused by a narrow localized force,
applied by a photon pressure actuator using finite element models as well as an analytical
calculation. Based on these results, we developed a simple model showing that for frequencies
above 1 kHz the test mass deformation has a non-negligible effect on the response of the photon
pressure actuator. Actual measurements confirmed the predictions of our model, in particular,
around 3.8 kHz the presence of a notch structure in the photon pressure actuator response is
clearly observed in the measurements.
Test masses of the currently operating laser-interferometric gravitational-wave detectors
cannot be considered to be rigid bodies for frequencies within the detection band. The effect of
a test mass deformation needs to be taken into account when using the photon pressure actuation
for high-precision calibration of any gravitational-wave detector. Furthermore, it should be
mentioned that the actuation with widely used coil-magnet actuators can potentionally lead to
test mass deformation effects as well.
Acknowledgments
The authors are grateful for the support from PPARC and the University of Glasgow in UK,
and the BMBF and the state of Lower Saxony in Germany. LIGO was constructed by the
California Institute of Technology and Massachusetts Institute of Technology with funding
from the National Science Foundation and operates under agreement PHY-0107417. In
addition, we would like to thank R Savage, P Kalmus, E Goetz, M Landry, B O’Reilly and the
ILIAS collaboration for many fruitful discussions about the photon pressure calibrators. This
paper has LIGO document number LIGO-P070074-00-Z.
5688 S Hild et al
References
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[3] Sigg D et al 2004 Commissioning of LIGO detectors Class. Quantum Grav. 21 S409–15
[4] Takahashi R (the TAMA Collaboration) 2004 Status of TAMA300 Class. Quantum Grav. 21 S403–8
[5] Clubley D A et al 2001 Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave
detector using photon pressure Phys. Lett. A 283 85
[6] Mossavi K et al 2006 A photon pressure calibrator for the GEO600 gravitational wave detector Phys. Lett.
A 353 1–3
[7] Hild S 2007 Beyond the first generation: extending the science range of the gravitational wave detector GEO
600 PhD Thesis University of Hannover
[8] www.ansys.com
[9] www.comsol.com
[10] Bondu F et al 1998 Thermal noise in mirrors of interferometric gravitational wave antennas Phys. Lett.
A 246 227–36
[11] Liu Y T and Thorne K S 2000 Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-
wave test masses Phys. Rev. D 62 122002
[12] Smith R et al 2004 Mechanical quality factor measurements of monolithically suspended fused silica test masses
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Smith, J.

MPG.PuRe

Crossref

IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITYClass. Quantum Grav. 24 (2007) 5681–5688 doi:10.1088/0264-9381/24/22/025Photon-pressure-induced test mass deformation ingravitational-wave detectorsS Hild1, M Brinkmann1, K Danzmann1, H Grote1, M Hewitson1,J Hough1, H Lu¨ck1, I Martin2, K Mossavi1, N Rainer1, S Reid2,J R Smith3, K Strain2, M Weinert1, P Willems4, B Willke1and W Winkler11 Max-Planck-Institut fu¨r Gravitationsphysik (Albert-Einstein-Institut) and Leibniz Universita¨tHannover, Callinstr 38, D-30167 Hannover, Germany2 SUPA, Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, UK3 Syracuse University, Department of Physics, 201 Physics Building, Syracuse, New York13244-1130, USA4 The LIGO project, California Institute for Technology, Mail Stop 18-34, Pasadena,California 91125, USAE-mail: stefan.hild@aei.mpg.deReceived 9 August 2007, in final form 28 September 2007Published 6 November 2007Online at stacks.iop.org/CQG/24/5681AbstractA widely used assumption within the gravitational-wave community has so farbeen that a test mass acts like a rigid body for frequencies in the detectionband, i.e. for frequencies far below the first internal resonance. In this paper,we demonstrate that localized forces, applied for example by a photon pressureactuator, can result in a non-negligible elastic deformation of the test masses.For a photon pressure actuator setup used in the gravitational-wave detectorGEO 600, we measured that this effect modifies the standard response functionby 10% at 1 kHz and about 100% at 2.5 kHz.PACS numbers: 04.80.Nn, 95.75.Kk(Some figures in this article are in colour only in the electronic version)1. IntroductionThe currently operating laser-interferometric gravitational-wave detectors GEO 600 [1], Virgo[2], LIGO [3] and TAMA300 [4] are using cylindrical test masses made of fused silica toprobe changes in the metric originating from gravitational waves. Usually, such test massesare considered to behave as rigid bodies for any applied force.However, in this paper, we show that a non-uniformly distributed force acting on a testmass can cause a significant deformation. We evaluate this effect quantitatively, using the0264-9381/07/225681+08$30.00 © 2007 IOP Publishing Ltd Printed in the UK 56815682 S Hild et alFigure 1. Experimental arrangement for measurements with the photon pressure actuator. M, farmirror in the north building of the GEO 600 detector. Illumination of this mirror with light fromthe laser diode produces a differential arm-length change that is measured at the main output ofthe interferometer.Table 1. Parameters used for calculating the photon-pressure-induced test mass deformation.Test mass diameter 180 mmTest mass thickness 100 mmTest mass material Fused silica (Suprasil)Beam of photon pressure actuator 5 mm diameter (Gaussian)Interferometer beam 50 mm diameter (Gaussian)photon pressure actuator installed at the GEO 600 interferometer. Such a photon pressureactuator was first demonstrated by Clubley et al [5] to be employed as an easy and independentmethod for displacement calibration of a gravitational-wave detector.Figure 1 shows a diagram of the experimental setup of the photon pressure actuator in theGEO 600 detector. The light of a laser diode is collimated by a lens, L, and enters the vacuumsystem via a viewport, before it is impinging on one of the main mirrors, M. More detaileddescriptions of the setup can be found in [6, 7].In section 2, we calculate the expected test mass deformation induced by the photonpressure using three different methods. Two-dimensional and three-dimensional finite elementanalyses are performed as well as an analytical estimation. The corresponding displacementsensed by the gravitational-wave detector is evaluated in section 3 using a simple model.A prediction is made of how the response to the photon actuator changes when the testmass deformation effect is taken into account. The presence of the test mass deformation isexperimentally confirmed by measurements presented in section 4.2. Models used to calculate the test mass deformationIn order to evaluate the effect from any test mass deformation introduced by photon pressure,we have to calculate the actual deformation caused by the setup of the GEO photon pressureactuator. Table 1 indicates the parameters used for the calculations.Altogether we compared three different ways to calculate the mirror deformation: twofinite element analyses have been performed in addition to a (quasi-)analytical calculation.The following list gives a short description of each of the methods applied:• ANSYS: ANSYS is a well-known finite element software package [8]. The three-dimensional model consisted of about 42 000 elements. A non-homogeneous meshingPhoton-pressure-induced test mass deformation in gravitational-wave detectors 56830 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.04500.511.522.533.54x 10displacement [m/N]radius [m]Application of a force with Gaussian distribution (radius of 2.5mm)ANSYSAnalyticalComsolFigure 2. Radial profile of the test mass deformation originating from a force continuously appliedwith a Gaussian distribution over a radius of 2.5 mm around the center of the test mass. For theanalysis a GEO 600 standard test mass was used (18 cm diameter, 10 cm thickness, made of fusedsilica (Suprasil)). Shown are the results of two finite element analysis. The result produced bythe ANSYS software is represented by the blue dashed-dotted line, while the COMSOL softwareproduced the orange dashed line. An analytical calculation is indicated by the solid green line.The corresponding effective displacements (see table 2) agree within 10% for all three results.(Explanation is given in the text.)was chosen, employing a very dense meshing in the center of the test mass in order toresolve the maximum of the deformation.• Comsol: Comsol multiphysics [9] is also a widely used finite element analysis softwarepackage. The deformation of the cross-section of the test mass was analyzed byperforming a two-dimensional simulation.• Analytical: The analytical calculation is based on an algorithm developed by Bondu,Hello and Vinet [10] for the calculation of thermo-elastic noise in an infinite half-planemirror. In addition, our calculation includes corrections and extensions from Liu andThorne [11].Figure 2 shows the radial profile of the calculated displacement of the test mass surface,D(r). The three methods agree pretty well over the full radius of the test mass.The effective displacement actually measured by the interferometer, Dtotal, is determinedby the overlap of the mirror deformation and the main interferometer beam, which can bedescribed by a Gaussian beam. The radial intensity of the interferometer beam, I (r), is givenbyI (r) = exp(−2r2ω2), (1)where ω = 25 mm is the radius of the beam. Each point on the mirror surfaces contributesto the total effective displacement, Dtotal, weighted by the product of the power of the5684 S Hild et alTable 2. Results of the three different calculations of the photon-pressure-induced test massdeformation. The three values for the effective displacement agree to within 10% and the resultingnotch frequencies (see section 3) match within 5%.Effective Resulting notchdisplacement (m N−1) frequency (Hz)ANSYS 3.83 × 10−10 3598Comsol 3.49 × 10−10 3768Analytical 3.54 × 10−10 3741main interferometer beam, I (r), and the corresponding displacement D(r).5 For a radiallysymmetric setup, as described above, the total effective displacement can be expressed by asingle integralDtotal =∫ 0.09m0∫ 2π0Deff · r · dr · dϕ =∫ 0.09m02π · r · kI · I (r) · D(r) · dr. (2)The factor kI is a normalization factor for I (r) in order to give∫ 0.09m02π · r · kI · I (r) · dr = 1. (3)Using equation (2) the total effective displacement for all three calculated test massdeformations can be derived. The corresponding results are displayed in the second columnof table 2. For the three different calculations of the test mass deformation the effectivedisplacement sensed by the main interferometer, Dtotal, agrees within about 10%.3. Predicted effect of the photon-pressure-induced test mass deformationThe previous section showed that the test masses are not completely rigid and that the photonpressure actuator beam really causes a non-negligible deformation. Next, we have to evaluatehow strong the displacement originating from the non-rigidity of the test mass is, compared tothe displacement of the center of mass (originating from the pendulum response). Since bothresponses are linear in the applied light power of the photon pressure actuator we can simplycompare their responses.The pendulum response, αpen, follows a 1/f 2-law, has a magnitude of 5 × 10−7 m N−1at 100 Hz and is 180◦ out of phase with the light power modulation (see blue dashed tracefigure 3)6. The response of the mirror deformation, αdef , is assumed to be flat in frequency andin phase with the modulated light power for frequencies below the first internal resonances ofthe test mass, which in the case of a GEO test mass, has a frequency of roughly 11 kHz [12].In the previous section, effective displacements in the range 3.54–3.83 × 10−10 m N−1 werefound. The magnitude and phase of αdef from the analytical test mass deformation is shownby the red dashed-dotted trace in figure 3. The total response, αtotal, plotted in green (solid) isthe sum of the two individual responses:αtotal = αpen + αdef . (4)5 The phase change of the interferometer light due to the mirror deformation is proportional to the light amplitude√I (r). At the sensing point of the interferometer (which is a photodiode) the signal is beaten with the local oscillator,which also has a spatial profile of√I (r). Taking both these effects into account we have to use I (r) for weightingthe radial test mass deformation D(r).6 Due to the fact that we are only interested in frequencies far above the resonance of the pendulum which is around1 Hz, excitation and pendulum motion have opposite phase.Photon-pressure-induced test mass deformation in gravitational-wave detectors 5685103 1041010magnitude [m/N]pendulum responseeffective displacement from mirror deformationtotal displacement response103 1040100phase [deg]103 1040123ratio without/withdeformationFrequency [Hz]Figure 3. Simple model for the photon pressure calibrator taking into account the responses fromthe pendulum and from the mirror deformation effect (analytical). The pendulum response followsa 1/f 2-law and is 180◦ out of phase from the photon pressure excitation. The mirror deformationhas a flat response and is in phase with the photon pressure actuation. If both responses are addeda notch appears at the frequency where both responses have equal size. The purple trace showsthe expected discrepancy between the calibrations with and without accounting for the test massdeformation.The total response shows a steep notch at the frequency where the individual responseshave the same magnitude and compensate each other completely due to having oppositephase. At the frequency of the notch the phase of αtotal jumps from −180 to 0◦. Themodeled notch frequencies for the three different methods are displayed in the third column oftable 2. The resulting discrepancy between the photon pressure actuator response without andwith test mass deformation is shown in the lowest subplot of figure 3. Already at 1 kHz the testmass deformation causes an effect of 10%. At 2 kHz the error, when not taking the test massdeformation into account, amounts to 40% and between 2.6 kHz and 4.6 kHz the discrepancyis larger than 100%.4. Measurement of the photon-pressure-induced test mass deformationIn order to verify this model, we injected signals to the photon pressure actuator over a widefrequency range including frequencies as high as 6 kHz to be able to resolve any potential5686 S Hild et al103101010Response of photon pressure actuator [a.u.]1030Phase [deg]Frequency [Hz]without test mass deformationmodel including test mass deformationmeasurementFigure 4. Result of the long duration photon pressure actuator injections over a wide frequencyrange. The pink dashed line represents the photon pressure actuator response without taking testmass deformation in account. The green solid line indicates the photon pressure actuator responsetaking the test mass deformation into account. The actual measurement of the response is displayedby the blue circles. The presence of the expected notch structure and by this the non-rigidity of thetest masses in the detection band of the gravitational wave is clearly confirmed by the measurement.notch structure. The measurements presented here include long duration measurements toachieve reasonable snr at high frequencies. Individual measurement points are made of singlediscrete Fourier transforms (DFT) containing up to 10 h of data. Such amounts of data aredifficult to handle with standard computers. A heterodyne technique was employed to reducethe volume of data. The time series of the data containing the signal of interest, Esig ·sin(ωsigt),is multiplied by a sine wave with a slightly lower frequency, ωhet:Esig · sin(ωsigt) · sin(ωhett) = 12Esig[cos(ωsig − ωhet)t − cos(ωsig + ωhet)t]. (5)The second term on the right-hand side of equation (5) still contains the signal, but shiftedtowards even higher frequencies. The signal component we are interested in is shifted toa very low frequency, (ωsig − ωhet), which for our investigations was chosen to be 9 Hz.After heterodyning, the data stream is strongly low-pass filtered and down-sampled to givea data stream that can be handled by desktop computers. Figure 4 shows the result ofthese measurements. The pink dashed line represents the photon pressure actuator responsewithout taking test mass deformation into account, while the green solid line indicates thePhoton-pressure-induced test mass deformation in gravitational-wave detectors 5687photon pressure actuator response taking the test mass deformation into account. The actualmeasurement of the response is displayed by the blue circles.The presence of the expected notch structure and by this the non-rigidity of the test massesin the detection band of the gravitational wave is clearly confirmed by the measurement. Themagnitude of the response is about a factor of 3 below the pendulum response for frequenciesbetween 3 and 4 kHz. At 5 kHz the measurement matches the pendulum response and finallyat 6 kHz the measured response clearly exceeds the 1/f 2 behavior of the pendulum response.Around the notch frequency, the phase of the photon pressure actuator response also changessignificantly from about −165◦ at 2.8 kHz to about −30◦ at 4.8 kHz. A phase of nearly 0◦ athigh frequencies clearly indicates that the total response of the system is no longer dominatedby the pendulum response.It can be seen that the notch is smeared out in the experimental data. This originatesfrom beam jitter of the main interferometer beam resulting in a varying overlap of the maininterferometer beam and deformed test mass. Different overlaps correspond to differenteffective displacementsDtotal = r · kI∫ 0.09m0∫ 360◦0· I (r, ϕ) · D(r, ϕ) · dr · dϕ, (6)which finally correspond to slightly shifted notch frequencies. Within the long measurementintervals of up to 10 h the measurement averages over a variety of notch frequencies resultingin a smeared out notch. A detailed description of this effect, including a quantitative analysiscan be found in [7].5. Summary and conclusionsWe evaluated the deformation of a GEO 600 test mass caused by a narrow localized force,applied by a photon pressure actuator using finite element models as well as an analyticalcalculation. Based on these results, we developed a simple model showing that for frequenciesabove 1 kHz the test mass deformation has a non-negligible effect on the response of the photonpressure actuator. Actual measurements confirmed the predictions of our model, in particular,around 3.8 kHz the presence of a notch structure in the photon pressure actuator response isclearly observed in the measurements.Test masses of the currently operating laser-interferometric gravitational-wave detectorscannot be considered to be rigid bodies for frequencies within the detection band. The effect ofa test mass deformation needs to be taken into account when using the photon pressure actuationfor high-precision calibration of any gravitational-wave detector. Furthermore, it should bementioned that the actuation with widely used coil-magnet actuators can potentionally lead totest mass deformation effects as well.AcknowledgmentsThe authors are grateful for the support from PPARC and the University of Glasgow in UK,and the BMBF and the state of Lower Saxony in Germany. LIGO was constructed by theCalifornia Institute of Technology and Massachusetts Institute of Technology with fundingfrom the National Science Foundation and operates under agreement PHY-0107417. Inaddition, we would like to thank R Savage, P Kalmus, E Goetz, M Landry, B O’Reilly and theILIAS collaboration for many fruitful discussions about the photon pressure calibrators. Thispaper has LIGO document number LIGO-P070074-00-Z.5688 S Hild et alReferences[1] Hild S et al 2006 The status of GEO 600 Class. Quantum Grav. 23 S643–51[2] Acemese F et al 2004 Status of VIRGO Class. Quantum Grav. 21 S385–94[3] Sigg D et al 2004 Commissioning of LIGO detectors Class. Quantum Grav. 21 S409–15[4] Takahashi R (the TAMA Collaboration) 2004 Status of TAMA300 Class. Quantum Grav. 21 S403–8[5] Clubley D A et al 2001 Calibration of the Glasgow 10 m prototype laser interferometric gravitational wavedetector using photon pressure Phys. Lett. A 283 85[6] Mossavi K et al 2006 A photon pressure calibrator for the GEO600 gravitational wave detector Phys. Lett.A 353 1–3[7] Hild S 2007 Beyond the first generation: extending the science range of the gravitational wave detector GEO600 PhD Thesis University of Hannover[8] www.ansys.com[9] www.comsol.com[10] Bondu F et al 1998 Thermal noise in mirrors of interferometric gravitational wave antennas Phys. Lett.A 246 227–36[11] Liu Y T and Thorne K S 2000 Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses Phys. Rev. D 62 122002[12] Smith R et al 2004 Mechanical quality factor measurements of monolithically suspended fused silica test massesof the GEO600 gravitational-wave detector Class. Quantum Grav. 21 1091–8

Rainer, M.

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(the TAMA Collaboration)

A et al

A photon pressure calibrator for the GEO600 gravitational wave detector Phys.

Beyond the first generation: extending the science range of the gravitational wave detector GEO

Mechanical quality factor measurements of monolithically suspended fused silica test masses of the GEO600 gravitational-wave detector Class. Quantum Grav.

Thermal noise in mirrors of interferometric gravitational wave antennas Phys.

Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses Phys.

Photon pressure induced test mass deformation in gravitational-wave
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Photon pressure induced test mass deformation in gravitational-wave detectors

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