arXiv:1611.08997v1 [physics.ins-det] 28 Nov 2016
First Demonstration of Electrostatic Damping of Parametri
c Instability at Advanced
LIGO
Carl Blair
1
,
∗
Slawek Gras
2
, Richard Abbott
5
, Stuart Aston
3
, Joseph Betzwieser
3
, David Blair
1
, Ryan
DeRosa
3
, Matthew Evans
2
, Valera Frolov
3
, Peter Fritschel
2
, Hartmut Grote
4
, Terra Hardwick
5
,
Jian Liu
1
, Marc Lormand
3
, John Miller
2
, Adam Mullavey,
3
, Brian O’Reilly
3
, and Chunnong Zhao
1
1
University of Western Australia, Crawley, Western Austral
ia 6009, Australia
2
Massachusetts Institute of Technology, Cambridge, Massac
husetts 02139, USA
3
LIGO Livingston Observatory, Livingston, Louisiana 70754
, USA
4
Max Planck Institute for Gravitational Physics, 30167 Hann
over, Germany
5
California Institute of Technology, Pasadena 91125, USA
and
6
Louisiana State University, Baton Rouge, Louisiana 70803,
USA
B. P. Abbott,
1
T. D. Abbott,
2
C. Adams,
3
R. X. Adhikari,
1
S. B. Anderson,
1
A. Ananyeva,
1
S. Appert,
1
K. Arai,
1
S. W. Ballmer,
4
D. Barker,
5
B. Barr,
6
L. Barsotti,
7
J. Bartlett,
5
I. Bartos,
8
J. C. Batch,
5
A. S. Bell,
6
G. Billingsley,
1
J. Birch,
3
S. Biscans,
1
,
7
C. Biwer,
4
R. Bork,
1
A. F. Brooks,
1
G. Ciani,
10
F. Clara,
5
S. T. Countryman,
8
M. J. Cowart,
3
D. C. Coyne,
1
A. Cumming,
6
L. Cunningham,
6
K. Danzmann,
11
,
12
C. F. Da Silva Costa,
10
E. J. Daw,
13
D. DeBra,
14
R. DeSalvo,
15
K. L. Dooley,
16
S. Doravari,
3
J. C. Driggers,
5
S. E. Dwyer,
5
A. Effler,
3
T. Etzel,
1
T. M. Evans,
3
M. Factourovich,
8
H. Fair,
4
A. Fern ́andez Galiana,
7
R. P. Fisher,
4
P. Fulda,
10
M. Fyffe,
3
J. A. Giaime,
2
,
3
K. D. Giardina,
3
E. Goetz,
12
R. Goetz,
10
C. Gray,
5
K. E. Gushwa,
1
E. K. Gustafson,
1
R. Gustafson,
17
E. D. Hall,
1
G. Hammond,
6
J. Hanks,
5
J. Hanson,
3
G. M. Harry,
18
M. C. Heintze,
3
A. W. Heptonstall,
1
J. Hough,
6
K. Izumi,
5
R. Jones,
6
S. Kandhasamy,
16
S. Karki,
19
M. Kasprzack,
2
S. Kaufer,
11
K. Kawabe,
5
N. Kijbunchoo,
5
E. J. King,
20
P. J. King,
5
J. S. Kissel,
5
W. Z. Korth,
1
G. Kuehn,
12
M. Landry,
5
B. Lantz,
14
N. A. Lockerbie,
21
A. P. Lundgren,
12
M. MacInnis,
7
D. M. Macleod,
2
S. M ́arka,
8
Z. M ́arka,
8
A. S. Markosyan,
14
E. Maros,
1
I. W. Martin,
6
D. V. Martynov,
7
K. Mason,
7
T. J. Massinger,
4
F. Matichard,
1
,
7
N. Mavalvala,
7
R. McCarthy,
5
D. E. McClelland,
22
S. McCormick,
3
G. McIntyre,
1
J. McIver,
1
G. Mendell,
5
E. L. Merilh,
5
P. M. Meyers,
23
R. Mittleman,
7
G. Moreno,
5
G. Mueller,
10
J. Munch,
20
L. K. Nuttall,
4
J. Oberling,
5
P. Oppermann,
12
Richard J. Oram,
3
D. J. Ottaway,
20
H. Overmier,
3
J. R. Palamos,
19
H. R. Paris,
14
W. Parker,
3
A. Pele,
3
S. Penn,
24
M. Phelps,
6
V. Pierro,
15
I. Pinto,
15
M. Principe,
15
L. G. Prokhorov,
25
O. Puncken,
12
V. Quetschke,
26
E. A. Quintero,
1
F. J. Raab,
5
H. Radkins,
5
P. Raffai,
27
S. Reid,
28
D. H. Reitze,
1
,
10
N. A. Robertson,
1
,
6
J. G. Rollins,
1
V. J. Roma,
19
J. H. Romie,
3
S. Rowan,
6
K. Ryan,
5
T. Sadecki,
5
E. J. Sanchez,
1
V. Sandberg,
5
R. L. Savage,
5
R. M. S. Schofield,
19
D. Sellers,
3
D. A. Shaddock,
22
T. J. Shaffer,
5
B. Shapiro,
14
P. Shawhan,
29
D. H. Shoemaker,
7
D. Sigg,
5
B. J. J. Slagmolen,
22
B. Smith,
3
J. R. Smith,
30
B. Sorazu,
6
A. Staley,
8
K. A. Strain,
6
D. B. Tanner,
10
R. Taylor,
1
M. Thomas,
3
P. Thomas,
5
K. A. Thorne,
3
E. Thrane,
31
C. I. Torrie,
1
G. Traylor,
3
G. Vajente,
1
G. Valdes,
26
A. A. van Veggel,
6
A. Vecchio,
32
P. J. Veitch,
20
K. Venkateswara,
33
T. Vo,
4
C. Vorvick,
5
M. Walker,
2
R. L. Ward,
22
J. Warner,
5
B. Weaver,
5
R. Weiss,
7
P. Weßels,
12
B. Willke,
11
,
12
C. C. Wipf,
1
J. Worden,
5
G. Wu,
3
H. Yamamoto,
1
C. C. Yancey,
29
Hang Yu,
7
Haocun Yu,
7
L. Zhang,
1
M. E. Zucker,
1
,
7
and J. Zweizig
1
(LSC Instrument Authors )
(LSC Collaboration)
(Dated: December 7, 2016)
Interferometric gravitational wave detectors operate wit
h high optical power in their arms in order
to achieve high shot-noise limited strain sensitivity. A si
gnificant limitation to increasing the optical
power is the phenomenon of three-mode parametric instabili
ties, in which the laser field in the arm
cavities is scattered into higher order optical modes by aco
ustic modes of the cavity mirrors. The
optical modes can further drive the acoustic modes via radia
tion pressure, potentially producing an
exponential buildup. One proposed technique to stabilize p
arametric instability is active damping of
acoustic modes. We report here the first demonstration of dam
ping a parametrically unstable mode
using active feedback forces on the cavity mirror. A 15,538 H
z mode that grew exponentially with
a time constant of 182 sec was damped using electro-static ac
tuation, with a resulting decay time
constant of 23 sec. An average control force of 0.03 nN rms was
required to maintain the acoustic
mode at its minimum amplitude.
2
Introduction
Three-mode parametric instability (PI)
has been a known issue for advanced laser interferometer
gravitational wave detectors since first recognised by Bra-
ginsky et al [1], and modelled in increasing detail [2–6].
The phenomenon was first observed in 2009 in micro-
cavities [7], then in 2014 in an 80 m cavity [8] and soon
afterwards during the commissioning of Advanced LIGO
[9]. Left uncontrolled PI results in the optical cavity con-
trol systems becoming unstable on time scales of tens of
minutes to hours [9].
The first detection of gravitational waves was made by
two Advanced LIGO laser interferometer gravitational
wave detectors with about 100 kW of circulating power
in their arm cavities [10]. To achieve this power level
required suppression of PI through thermal tuning of the
higher-order mode eigen-frequency [11] explained later in
this paper. This tuning allowed the optical power to be
increased in Advanced LIGO from about 5 % to 12 % of
the design power, sufficient to attain a strain sensitivity
of 10
−
23
Hz
−
1
2
at 100 Hz.
At the design power it will not be possible to avoid in-
stabilities using thermal tuning alone for two reasons.
First the parametric gain scales linearly with optical
power and second the acoustic mode density is so high
that thermal detuning for one acoustic mode brings other
modes into resonance [9, 11].
Several methods are likely to be useful for controlling
PI. Active thermal tuning will minimize the effects of
thermal transients [12, 13] and maintain operation near
the parametric gain minimum. In the future, acoustic
mode dampers attached to the test masses [14] could
damp acoustic modes. Active damping [15] of acoustic
modes can also suppress instabilities, by applying feed-
back forces to the test masses.
In this letter we report on the control of a PI by
actively damping a 15.54 kHz acoustic mode of an Ad-
vanced LIGO test mass using electro-static force actua-
tors. First we review the physics of PI and the status
of PI control in LIGO. Then we discuss the electrostatic
drive system at LIGO and how it interacts with the test
mass modes. Then we summarise the experimental con-
figuration, report successful damping observations, and
discuss the implications for high power operation of Ad-
vanced LIGO.
Parametric Instability
The parametric gain
R
m
, as de-
rived by Evans et al [4] is given by;
R
m
=
8
πQ
m
P
Mω
2
m
cλ
0
∞
∑
n
=1
R
e
[
G
n
]
B
2
m,n
(1)
Here
Q
m
is the quality factor (Q) of the mechanical mode
m
,
P
is the power in the fundamental optical mode of the
cavity,
M
is the mass of the test mass,
c
is the speed of
light,
λ
0
is the wavelength of light,
ω
m
is the mechanical
mode angular frequency,
G
n
is the transfer function for
an optical field leaving the test mass surface to the field
incident on that same surface and
B
m,n
is the spatial
overlap between the optical beat note pressure distribu-
tion and the mechanical mode surface deformation.
It is instructive to consider the simplified case of a sin-
gle cavity and a single optical mode to understand the
phenomena. For a simulation analysis including arms
and recycling cavities see [4, 5] and for an explanation
of dynamic effects that may make high parametric gains
from the recycling cavities less likely see [8]. In the sim-
plified case we consider the
T EM
03
mode as it dominates
the optical interaction with the acoustic mode investi-
gated here;
R
e
[
G
03
] =
c
Lπγ
(1 + ∆
ω
2
/γ
2
)
(2)
Here
γ
is the half-width at half maximum of the
T EM
03
optical mode frequency distribution, L is the length of
the cavity, ∆
ω
is the spacing in frequency between the
mechanical mode
ω
m
and the beat note of the fundamen-
tal and TEM
03
optical modes. In general the parametric
gain changes the time constant of the mechanical mode
as in Equation 3. If the parametric gain exceeds unity
the mode becomes unstable.
τ
pi
=
τ
m
/
(1
−
R
m
)
(3)
Where
τ
m
is the natural time constant of the mechan-
ical mode and
τ
pi
is the time constant of the mode influ-
enced by the opto-mechanical interaction. Thermal tun-
ing was used to control PI in Advanced LIGO’s Obser-
vation run 1 and was integral to this experiment, so will
be examined in some detail. Thermal tuning is achieved
using radiative ring heaters that surround the barrel of
each test mass without physical contact as in Figure 2.
Applying power to the ring heater decreases the radius of
curvature (RoC) of the mirrors. This changes the cavity
g-factor and tunes the mode spacing between the fun-
damental (
T EM
00
) and higher order transverse electro-
magnetic (
T EM
mn
) modes in the cavity, thereby tuning
the parametric gain by changing ∆
ω
in Equation 2.
Figure 1 shows the optical gain curve (Equation 2) for
the TEM
03
mode, with the ring heater tuning used dur-
ing Advanced LIGOs first observing run [17]. With no
thermal tuning, the optical gain curve in Figure 1 moves
to higher frequency, decreasing the frequency spacing ∆
ω
with mode group E. This leads to the instability of this
group of modes. (Note that the mirror acoustic mode
frequencies are only weakly tuned by heater power, due
to the small value of the fused silica temperature depen-
dence of Young’s modulus). If the ring heater power is
increased inducing approximately 5 m change in radius of
curvature, the beat note gain curve in Figure 1 moves left
about 400 Hz, decreasing the value ∆
ω
for mode group
A, resulting in their instability. The mode groups C and
D are stable as the second and fourth order optical modes
that might be excited from these modes are far from res-
onance. Mode Group B is also stable at the circulating
3
A
B
C
D
E
FIG. 1. The relative location of the optical and mechanical m
odes during Advanced LIGO Observation run 1. Mechanical
modes measured in transmission of the Output mode cleaner sh
own in blue with mode surface deformation generated from
FEM modeling overlay-ed. These modes appear in groups of fou
r, one for each test mass. They have line-width
∼
1
mHz
. The
beat note between the fundamental
T EM
00
and
T EM
03
optical cavity modes for a simplified single cavity is shown i
n bold red
and with the ring heater turned off, in dashed red. The shape of
the
T EM
03
mode simulated with OSCAR [16] is inset below
the peak.
optical power used in this experiment presumably due
to either lower quality factor
Q
m
or lower optical gain
G
30
of the TEM
30
mode as investigated in [18]. If the
power in the interferometer is increased by a factor of 3
there will no longer be a stable region. Mode group A
at 15.00kHz and group E at 15.54 kHz will be unstable
simultaneously.
Electrostatic Control
Electrostatic control of PI was
proposed [19] and studied in the context of the LIGO
electrostatic control combs by Miller et al [15]. Here we
report studies of electrostatic feedback damping for the
group E modes at 15.54kHz.
The main purpose of the electrostatic drive (ESD) is
to provide longitudinal actuation on the test masses for
lock acquisition [20] and holding the arm cavities on res-
onance. It creates a force between the test masses and
their counterpart reaction masses, through the interac-
tion of the fused silica test masses with the electric fields
generated by a comb of gold conductors that are de-
posited on the reaction mass. The physical locations of
these components are depicted in Figure 2. Detail of
the gold comb is shown in Figure 3 along with the force
density on the test mass.
The force applied to the test mass
F
ESD
is dominated
by the dipole attraction of the test mass dielectric to the
electric field between the electrodes of the gold comb.
Some portion
b
m
of this force that couples to the acoustic
mode as;
F
app
,
m
=
b
m
F
ESD
,
Q
=
b
m
α
Q
×
1
2
(
V
bias
−
V
Q
)
2
(4)
Here
α
Q
is the force coefficient for a single quadrant,
while
V
bias
and
V
Q
are the voltages of the ESD elec-
trodes defined in Figure 3. The overlap
b
m
between the
ESD force distribution
~
f
ESD
,
Q
and the displacement
~u
m
of the surface for a particular acoustic mode
m
can be
FIG. 2. Schematic of the gold ESD comb on the reaction mass
(RM), the ring heater (RH) and the end test mass (ETM) with
exaggerated deformation due to the 15,538 Hz mode. The
colour represents the magnitude of the displacement (red is
large, blue is small). The laser power in the arm cavity is
depicted in red (ARM). Suspension structures are not shown
and while the scale is marked to the left the distance between
RM and ETM is exaggerated by a factor of 10
4
FIG. 3. The ESD comb pattern printed on the reaction mass
(left) and the force distribution on the test mass (right) wi
th
the same voltage on all quadrants
approximated as a surface integral derived by Miller [15]:
b
m
≈
∣
∣
∣
∫∫
S
~
f
ESD
,
Q
·
(
~u
m
·
ˆ
z
) d
S
∣
∣
∣
(5)
If a feedback system is created that senses the mode
amplitude and provides a viscous damping force using
the ESD, the resulting time constant of the mode
τ
esd
is
given by;
τ
esd
=
(
1
τ
m
+
K
m
2
μ
m
)
−
1
(6)
Here
K
m
is the gain applied between the velocity mea-
surement and the ESD actuation force on a mode with
time constant
τ
m
and effective mass
μ
m
. Reducing the ef-
fective time constant lowers the effective parametric gain.
R
eff
=
R
m
×
τ
esd
τ
m
(7)
The force required to reduce a parametric gain
R
m
to
an effective parametric gain
R
eff
when the mode am-
plitude is the thermally excited amplitude was used by
Miller [15] to predict the forces required from the ESD
for damping PI,
F
req
=
x
m
μ
m
ω
2
m
b
m
(
R
m
−
R
eff
Q
m
R
eff
)
(8)
at the thermally excited amplitude
x
m
=
√
k
B
T/μ
m
ω
2
0
,
m
, where
k
B
is the Boltzmann constant
and
T
temperature.
Feedback Loop
Figure 4 shows the damping feedback
loop implemented on the end test mass of the Y-arm
(ETMY). The error signal used for mode damping is
constructed from a quadrant photodiode (QPD) that re-
ceives light transmitted by ETMY. By suitably combin-
ing QPD elements, we measure the beat signal between
the cavity
T EM
00
mode and the
T EM
03
mode that is
being excited by the 15,538 Hz ETMY acoustic mode.
This signal is band-pass filtered at 15,538 Hz, then phase
shifted to produce a control signal that is 90 degrees out
of phase with the mode amplitude (velocity damping).
The damping force is applied, with adjustable gain, to
two quadrants of the ETMY electro-static actuator.
SRM
OMC
ITMY
ETMY
QPDX
QPDY
OMC-PD
BS
PRM
ITMX
ETMX
ERMY
K
m
+
Q1
Q3
ESD
LS
FIG. 4. A simplified schematic of advanced LIGO showing key
components for damping PI in ETMY. Components shown in-
clude input and end test masses (ITM/ETM), beam-splitter
(BS), power and signal recycling mirrors (PRM/SRM), the
laser source (LS), quadrant photo-detectors, the output
mode cleaner (OMC), the OMC transmission photo-detector
(OMC-PD). While 4 reaction masses exist, only the Y end
reaction mass is shown (ERMY) with key components of the
damping loop. These components generate a differential sig-
nal from the vertical orientation of QPDY, filter the signal
with a 10 Hz wide band pass filter centered on the 15,538 Hz
mode, apply gain
K
m
and phase
φ
set in the digital control
system and then differentially drive of the upper right
Q
1 and
lower left
Q
3 quadrants of the ESD.
Results
PI stabilization via active damping was demon-
strated by first causing the ETMY 15,538 Hz to become
parametrically unstable; this was done by turning off the
ring heater tuning, so that the
T EM
03
mode optical gain
curve better overlapped this acoustic mode, as shown in
Figure 1. When the mode became significantly elevated
in the QPD signal, the damping loop was closed with a
control gain to achieve a clear damping of the mode am-
plitude and a control phase optimised to
±
15 degrees of
viscous damping. The mode amplitude was monitored
using the photodetector at the main output of the in-
terferometer (labelled OMC-PD in Figure 4), as it was
found to provide a higher signal-to-noise ratio than the
QPD.
The results are shown in Figure 5, which plots the
mode amplitude during the unstable ring-up phase, fol-
lowed by the ring-down when the damping loop is en-
gaged. From the ring-up phase, we estimate the para-
metric gain to be 2
.
4
±
0
.
8 from Equation 3. With the
5
FIG. 5. Damping of parametric instability. Upper panel, the
15,538 Hz ETMY mode is unstable ringing up with a time constan
t
of 182
±
9 sec and estimated parametric gain of
R
m
= 2
.
4. Then at 0 sec control gain is applied resulting in an expone
ntial
decay with a time constant of 23
±
1 sec and effective parametric gain
R
eff
,
m
= 0
.
18. Lower panel, the control force over the
same period.
damping applied,
R
eff
=
R
m
τ
eff
τ
m
+
R
m
τ
eff
(9)
the effective parametric gain is reduced to a stable value
of
R
eff
=0
.
18
±
0
.
06. The uncertainty is primarily due to
the uncertainty in the estimate of
τ
m
which was obtained
by the method described in [9].
At the onset of active damping (time t = 0 in Figure 5),
the feedback control signal produces an estimated force
of
F
esd
= 0.62 nN rms (at 15,538Hz). As the mode am-
plitude decreased the control force dropped to a steady
state value of 0.03 nN rms. Over a 20 minute period in
this damped state, the peak control force was 0.11 nN
peak.
Discussion
The force required to damp the 15,538Hz
mode when advanced LIGO reaches design power can be
determined from the ESD force used to achieve the ob-
served parametric gain suppression presented here, com-
bined with the expected parametric gain when operated
at high power.
F
req
F
esd
=
R
eff
R
req
R
max
−
R
req
R
m
−
R
eff
(10)
The maximum parametric gain of the 15,538Hz mode
TABLE I. List of parameters for analysis with values and
descriptions
Symbol Value
Description
Q
m
12
×
10
6
Q factor of 15,538 Hz mode
P
100 kW
Power contained in arm cavity
ω
m
/
2
π
15,538 Hz Frequency of unstable mode
M
40kg
mass of test mass
b
m
0.17
effective mass scaled ESD overlap
factor for 15,538 Hz mode
λ
0
1064 nm laser wavelength
α
Q
4
.
8
×
10
−
11
N/V
2
ESD quadrant force coefficient
L
4km
Arm cavity length
V
bias
400V
Bias voltage on ESD
V
Q
[-20,20]V ESD control voltage range
(where ∆
ω
= 0) at the power level of these experi-
ments is estimated
≈
7 given an estimated de-tuning of
∆
ω
≈
50
Hz
with zero ring heater power. At full design
power the maximum gain will be
R
max
≈
56. To obtain
a quantitative result, we set a requirement for damping
such that the effective parametric gain of unstable acous-
tic modes after damping be
R
req
= 0
.
1.
Using Equation 10, the measurements of
R
m
and
R
eff
,
the maximum force required to maintain the damped
state at high power is
F
ESD
= 1.5 nN rms. Prior to this
6
investigation Miller predicted [15] that a control force of
approximately 10 nN rms would be required to maintain
this mode at the thermally excited level.
The PI control system must cope with elevated mode
amplitudes as the PI mode may build up before PI con-
trol can be engaged. There is therefore a requirement
for some control range or safety factor such that the con-
trol system will not saturate if the mode amplitude is
a multiple of the safety factor times the damped state
amplitude. The average ESD drive voltage
V
Q
over the
duration the mode was in the damped state was 0.42 mV
rms, however during this time it peaked at
±
1.4 mV peak
out of a
±
20 V control range, leading to a safety factor
of more than 10,000. At high power the safety factor will
be reduced by the required force ratio of Equation 10
resulting in an expected safety factor of 310.
As the laser power is increased, other modes are likely
to become unstable. The parametric gain of these modes
should be less than the gain of mode group E provided
the optical beat note frequency used in these experiments
is maintained. However these modes may also have lower
spatial overlap
b
m
with the ESD. Miller’s simulation [15]
show some modes in the 30-90kHz range will require up
to 30 times the control force
F
ESD
required to damp
the group E modes. Even in this situation the PI safety
factor is approximately 10.
Conclusion
We have shown for the first time elec-
trostatic control of parametric instability. An unstable
acoustic mode at 15,538 Hz with a parametric gain of
2
.
4
±
0
.
8 was successfully damped to a gain of 0
.
18
±
0
.
06,
using electrostatic control forces. The damping force re-
quired to keep the mode in the damped state was 0.03 nN
rms. The prediction through FEM simulation was that
the ESD would need to apply approximately six times
this control force to maintain the mode amplitude at the
thermally excited level. At high power it is estimated
that damping the 15.54 kHz mode group to an effective
parametric gain of 0.1 will result in a safety factor
≈
310. It is predicted that unstable modes that are most
problematic to damp will still have a safety factor of 10.
Acknowledgments
The authors would like to acknowl-
edge the entire LIGO Scientific Collaboration for the
wide ranging expertise that has contributed to these in-
vestigations. LIGO was constructed by the California
Institute of Technology and Massachusetts Institute of
Technology with funding from the National Science Foun-
dation, and operates under Cooperative Agreement No.
PHY-0757058. Advanced LIGO was built under Grant
No. PHY-0823459. This paper has LIGO Document
Number LIGO-P1600090. The corresponding author was
supported by the Australian Research Council and the
LSC fellows program.
∗
carl.blair@uwa.edu.au
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