Point absorbers in Advanced LIGO
A. F. Brooks,
1
G. Vajente,
1
H. Yamamoto,
1
R. Abbott,
1
C. Adams,
2
R. X. Adhikari,
1
A. Ananyeva,
1
S. Appert,
1
K. Arai,
1
J. S. Areeda,
3
Y. Asali,
4
S. M. Aston,
2
C. Austin,
5
A. M. Baer,
6
M. Ball,
7
S. W. Ballmer,
8
S. Banagiri,
9
D. Barker,
10
L. Barsotti,
11
J. Bartlett,
10
B. K. Berger,
12
J. Betzwieser,
2
D. Bhattacharjee,
13
G. Billingsley,
1
S. Biscans,
11, 1
C. D. Blair,
2
R. M. Blair,
10
N. Bode,
14, 15
P. Booker,
14, 15
R. Bork,
1
A. Bramley,
2
D. D. Brown,
16
A. Buikema,
11
C. Cahillane,
1
K. C. Cannon,
17
H. Cao,
16
X. Chen,
18
A. A. Ciobanu,
16
F. Clara,
10
C. M. Compton,
10
S. J. Cooper,
19
K. R. Corley,
4
S. T. Countryman,
4
P. B. Covas,
20
D. C. Coyne,
1
L. E. H. Datrier,
21
D. Davis,
8
C. Di Fronzo,
19
K. L. Dooley,
22, 23
J. C. Driggers,
10
P. Dupej,
21
S. E. Dwyer,
10
A. Effler,
2
T. Etzel,
1
M. Evans,
11
T. M. Evans,
2
J. Feicht,
1
A. Fernandez-Galiana,
11
P. Fritschel,
11
V. V. Frolov,
2
P. Fulda,
24
M. Fyffe,
2
J. A. Giaime,
5, 2
K. D. Giardina,
2
P. Godwin,
25
E. Goetz,
5, 13
S. Gras,
11
C. Gray,
10
R. Gray,
21
A. C. Green,
24
A. Gupta,
1
E. K. Gustafson,
1
R. Gustafson,
26
E. Hall,
11
J. Hanks,
10
J. Hanson,
2
T. Hardwick,
5
R. K. Hasskew,
2
M. C. Heintze,
2
A. F. Helmling-Cornell,
7
N. A. Holland,
27
W. Jia,
11
J. D. Jones,
10
S. Kandhasamy,
28
S. Karki,
7
M. Kasprzack,
1
K. Kawabe,
10
N. Kijbunchoo,
27
P. J. King,
10
J. S. Kissel,
10
Rahul Kumar,
10
M. Landry,
10
B. B. Lane,
11
B. Lantz,
12
M. Laxen,
2
Y. K. Lecoeuche,
10
J. Leviton,
26
J. Liu,
14, 15
M. Lormand,
2
A. P. Lundgren,
29
R. Macas,
22
M. MacInnis,
11
D. M. Macleod,
22
G. L. Mansell,
10, 11
S. M ́arka,
4
Z. M ́arka,
4
D. V. Martynov,
19
K. Mason,
11
T. J. Massinger,
11
F. Matichard,
1, 11
N. Mavalvala,
11
R. McCarthy,
10
D. E. McClelland,
27
S. McCormick,
2
L. McCuller,
11
J. McIver,
1
T. McRae,
27
G. Mendell,
10
K. Merfeld,
7
E. L. Merilh,
10
F. Meylahn,
14, 15
T. Mistry,
30
R. Mittleman,
11
G. Moreno,
10
C. M. Mow-Lowry,
19
S. Mozzon,
29
A. Mullavey,
2
T. J. N. Nelson,
2
P. Nguyen,
7
L. K. Nuttall,
29
J. Oberling,
10
Richard J. Oram,
2
C. Osthelder,
1
D. J. Ottaway,
16
H. Overmier,
2
J. R. Palamos,
7
W. Parker,
2, 31
E. Payne,
32
A. Pele,
2
C. J. Perez,
10
M. Pirello,
10
H. Radkins,
10
K. E. Ramirez,
33
J. W. Richardson,
1
K. Riles,
26
N. A. Robertson,
1, 21
J. G. Rollins,
1
C. L. Romel,
10
J. H. Romie,
2
M. P. Ross,
34
K. Ryan,
10
T. Sadecki,
10
E. J. Sanchez,
1
L. E. Sanchez,
1
T. R. Saravanan,
28
R. L. Savage,
10
D. Schaetzl,
1
R. Schnabel,
35
R. M. S. Schofield,
7
E. Schwartz,
2
D. Sellers,
2
T. Shaffer,
10
D. Sigg,
10
B. J. J. Slagmolen,
27
J. R. Smith,
3
S. Soni,
5
B. Sorazu,
21
1
arXiv:2101.05828v2 [physics.ins-det] 25 Mar 2021
A. P. Spencer,
21
K. A. Strain,
21
L. Sun,
1
M. J. Szczepa ́nczyk,
24
M. Thomas,
2
P. Thomas,
10
K. A. Thorne,
2
K. Toland,
21
C. I. Torrie,
1
G. Traylor,
2
M. Tse,
11
A. L. Urban,
5
G. Valdes,
5
D. C. Vander-Hyde,
8
P. J. Veitch,
16
K. Venkateswara,
34
G. Venugopalan,
1
A. D. Viets,
36
T. Vo,
8
C. Vorvick,
10
M. Wade,
37
R. L. Ward,
27
J. Warner,
10
B. Weaver,
10
R. Weiss,
11
C. Whittle,
11
B. Willke,
15, 14
C. C. Wipf,
1
L. Xiao,
1
Hang Yu,
11
Haocun Yu,
11
L. Zhang,
1
M. E. Zucker,
11, 1
and J. Zweizig
1
(The LIGO Scientific Collaboration)
1
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
2
LIGO Livingston Observatory, Livingston, LA 70754, USA
3
California State University Fullerton, Fullerton, CA 92831, USA
4
Columbia University, New York, NY 10027, USA
5
Louisiana State University, Baton Rouge, LA 70803, USA
6
Christopher Newport University, Newport News, VA 23606, USA
7
University of Oregon, Eugene, OR 97403, USA
8
Syracuse University, Syracuse, NY 13244, USA
9
University of Minnesota, Minneapolis, MN 55455, USA
10
LIGO Hanford Observatory, Richland, WA 99352, USA
11
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
12
Stanford University, Stanford, CA 94305, USA
13
Missouri University of Science and Technology, Rolla, MO 65409, USA
14
Max Planck Institute for Gravitational Physics (Albert
Einstein Institute), D-30167 Hannover, Germany
15
Leibniz Universit ̈at Hannover, D-30167 Hannover, Germany
16
OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia
17
RESCEU, University of Tokyo, Tokyo, 113-0033, Japan.
18
OzGrav, University of Western Australia,
Crawley, Western Australia 6009, Australia
19
University of Birmingham, Birmingham B15 2TT, UK
20
Universitat de les Illes Balears, IAC3—IEEC,
E-07122 Palma de Mallorca, Spain
21
SUPA, University of Glasgow, Glasgow G12 8QQ, UK
2
22
Cardiff University, Cardiff CF24 3AA, UK
23
The University of Mississippi, University, MS 38677, USA
24
University of Florida, Gainesville, FL 32611, USA
25
The Pennsylvania State University, University Park, PA 16802, USA
26
University of Michigan, Ann Arbor, MI 48109, USA
27
OzGrav, Australian National University, Canberra,
Australian Capital Territory 0200, Australia
28
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India
29
University of Portsmouth, Portsmouth, PO1 3FX, UK
30
The University of Sheffield, Sheffield S10 2TN, UK
31
Southern University and A&M College, Baton Rouge, LA 70813, USA
32
OzGrav, School of Physics & Astronomy,
Monash University, Clayton 3800, Victoria, Australia
33
The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
34
University of Washington, Seattle, WA 98195, USA
35
Universit ̈at Hamburg, D-22761 Hamburg, Germany
36
Concordia University Wisconsin, 2800 N Lake Shore Dr, Mequon, WI 53097, USA
37
Kenyon College, Gambier, OH 43022, USA
3
Abstract
Small, highly absorbing points are randomly present on the surfaces of the main interferom-
eter optics in Advanced LIGO. The resulting nano-meter scale thermo-elastic deformations and
substrate lenses from these micron-scale absorbers significantly reduces the sensitivity of the inter-
ferometer directly though a reduction in the power-recycling gain and indirect interactions with the
feedback control system. We review the expected surface deformation from point absorbers and pro-
vide a pedagogical description of the impact on power build-up in second generation gravitational
wave detectors (dual-recycled Fabry-Perot Michelson interferometers). This analysis predicts that
the power-dependent reduction in interferometer performance will significantly degrade maximum
stored power by up to 50% and hence, limit GW sensitivity, but suggests system wide corrections
that can be implemented in current and future GW detectors. This is particularly pressing given
that future GW detectors call for an order of magnitude more stored power than currently used in
Advanced LIGO in Observing Run 3. We briefly review strategies to mitigate the effects of point
absorbers in current and future GW wave detectors to maximize the success of these enterprises.
4
I. INTRODUCTION
The Advanced LIGO (aLIGO) gravitational wave (GW) detectors, in conjunction with
the Virgo GW detector, having completed their third observing run (O3)[1], have reported
detection of multiple gravitational wave events [2, 3] (10 binary black-hole mergers and 1
binary neutron star mergers) and issued 56 public alerts for detection candidates [4]. The
aLIGO detectors, illustrated in Figure 1, are high laser power, dual-recycled, 4 km Fabry-
Perot, Michelson interferometers operating at 1064 nm [5]. Passing gravitational waves cause
a strain in space-time, resulting in a differential length variation of the Fabry-Perot arms
that yields a detectable intensity variation at the output port of the interferometer.
The sensitivity of the interferometers is limited by a variety of technical and fundamental
sources. At frequencies above approximately 100 Hz quantum noise, in the form of shot
noise on the photodetector, is the limiting noise source [6]. Quantum noise can be decreased
by increasing the input laser power injected into the interferometer, thereby increasing the
amount of stored power in the interferometer arms. The design level for aLIGO is 125 W
input power with a stored arm power of 750 kW per arm as illustrated in Figure 1. During
O3, Advanced LIGO routinely operated at input power levels of 35 W
−
40 W [1].
Optical power is absorbed at the sub-ppm level in the surfaces of the main LIGO optics,
referred to as the Test Masses (ITMX, ITMY, ETMX and ETMY in Figure 1). As stored
power is increased, these optics are exposed to several hundred kW of resonating power,
they absorb several tens of mW causing thermo-elastic deformation of the optical surfaces
and thermo-refractive lenses in the substrates [7–9].
Absorption is classified as uniform and non-uniform.
Uniform absorption
is characterized
by a spatially invariant (or nearly invariant) absorption coefficient across the high-reflectivity
(HR) surface of the optic. In the Advanced LIGO test masses, the unambiguously measured
uniform absorption values ranged over 500
±
250 ppb. In the case of uniform absorption,
non-uniform, low-spatial frequency thermal lenses [9] and surface deformations [8] are well
approximated by spherical wavefront errors. Advanced LIGO contains a thermal compen-
sation system (TCS) [10] - not shown in Figure 1 - with actuators to provide low-spatial
correction for (a) thermal lenses in the substrates and (b) curvature errors on the surfaces
of the test masses.
Non-uniform absorption
is any form of absorption with high-spatial frequency dependence
5
125W
5.2kW
750kW
750kW
~100mW
r
i
,t
i
r
i
,t
i
r
e
,t
e
r
e
,t
e
r
p
,t
p
loss, L
A
loss, L
A
T
e
(= t
e
2
)
= 5E-6
T
i
(= t
i
2
)
= 0.014
T
p
(= t
p
2
) = 0.03
PSL
PRM
ITMX
ETMY
ETMX
ITMY
BS
SRM
input/symmetric
side
output/antisymmetric
side
carrier
9MHz sideband
45MHz sideband
OMC
PD
FIG. 1. Schematic diagram of Advanced LIGO showing the main optics, nominal power levels and
different optical frequencies present in the interferometer. The detector is a Fabry-Perot Michelson,
with power-recycling and signal-recycling cavities. The power recycling cavity (PRC) and Fabry-
Perot arm cavities hold large amounts of stored power. At 125 W of input laser power, the nominal
design yields 5
.
2 kW stored in the PRC and 750 kW in the arms. The output mode cleaner (OMC)
is designed to transmit only the fundamental spatial mode of the arms, which carries the GW
signal, and reject other spatial modes. Radio-frequency sidebands are injected into the detector to
measure and control the multiple degrees of freedom created by the coupled resonant cavities.
(where ”high spatial frequency” refers to features that are significantly smaller than the
Gaussian beam diameter of the illuminating laser beam). A salient example is a point-like
absorber (a ”point absorber”): a sub-millimeter scale, highly absorbing region on the surface
of the test mass.
6
60
40
20
0
20
40
60
X coordinate - HWS (mm)
60
40
20
0
20
40
60
Y coordinate - HWS (mm)
H1-ITMX. Contour spacing = 10nm.
210
180
150
120
90
60
30
0
OPD (nm)
FIG. 2.
Hartmann sensor measurement of optical path distortion (thermo-refractive plus
thermo-elastic) from a single point absorber on H1-ITMX. Cold reference taken at GPS time:
1 180 229 513 s, hot measurement taken 3322 s later at GPS time: 1 180 232 835 s. This measure-
ment corresponds to approximately 27
±
2
.
5 mW power absorbed in the point.
Within aLIGO, Hartmann wavefront sensors (HWS) [10, 11] measure the spatial dis-
tribution of the integrated thermo-refractive and thermo-elastic deformations in the main
LIGO optics induced by operation at high power. Measurements performed in-situ have
detected unambiguous evidence of point absorbers on at least 5 of 8 observed test masses
[12]. An example of these measurements is shown in Figure 2. Additional forensics of the
aLIGO optics (performed off-site on uninstalled optics) revealed point absorbers on multiple
optics, including several optics never exposed to high laser power in the vacuum system. A
7
FIG. 3. Dark field microscope image of point absorber measured on an Advanced LIGO optic
(corresponding to the thermal lens measurement shown in Figure 2). Also shown in Buikema et.
al. [1].
microscope image of a point absorber is shown in Figure 3.
The spatial resolution of the Hartmann wavefront sensor measurements is approximately
one sample every 7
.
5 mm in both transverse dimensions. Hence we cannot resolve the features
in the wavefront smaller than this. However, due to thermal diffusion to scales larger than
the spatial resolution, we can infer the total power absorbed by fitting thermal lens models
to this data. Observed absorption values lie within the range 100
−
350 ppb of the total
incident power, however, the number of measurements is too small to reliably describe the
8
distribution of absorption values. The thermal lens and surface deformations induced by
non-uniform absorption and point absorbers are characterized by features smaller than the
incident laser beam size (by high spatial frequencies) and, currently, aLIGO contains no
high spatial frequency thermal compensation system to correct for these effects.
The origin of the point absorbers in Advanced LIGO is currently under intense investi-
gation. The absorbers cannot be removed with standard cleaning techniques and appear
to be embedded in the coating. Initial spectroscopic analysis of absorbing points on wit-
ness samples (coated in same coating runs as aLIGO optics) show high concentrations of
aluminum. Additionally, following a recent inspection of an aLIGO optic, more point ab-
sorbers appeared on that optic - apparently introduced during the inspection process. The
nature of these new contaminants is not clear. The full scope of point absorber forensics is
beyond the scope of this paper and will be addressed in a future manuscript. Additionally,
it is not clear if the presence of point absorbers is unique to the large scale optics used in
gravitational wave detectors or is common in all dielectric layer precision optics.
As Vajente [13] described, non-uniform surface deformation of the optics in a gravitational
wave detector will scatter power into higher-order spatial modes (HOM) in the interferom-
eter. The Fabry-Perot arms cavities will enhance and suppress different HOM and may,
depending on cavity geometry and other factors discussed here, resonantly extract power
from the fundamental mode, increasing the observed loss. We briefly review this in Section
II.
Within this article, we explore the thermo-optical interaction of point absorbers with
a high-power, high-finesse, Michelson interferometer. Observations of the interferometer
response are reported in Section III and projections for future impacts on Advanced LIGO
are presented in Section IV.
To provide more context, extended background material is included in the Appendices.
The basic operation of aLIGO is described in Appendix A. The effects of point absorbers
on (a) just the thermal state of the optic are described in Appendix B, (b) the simple
laser/optic interaction in Appendix C and (c) the resonant-cavity/laser/optic interaction in
Appendix D. The impact of substrate thermal lensing on the interferometer control systems
(and subsequent noise couplings) is examined in Section E.
9
II. MODELING FORMALISM
In a dual-recycled, Fabry-Perot Michelson interferometer, the power-recycling gain (the
ratio of stored laser power in the power-recycling cavity to input laser power) is a good proxy
for monitoring the average loss in the arms of the interferometer. In the appendices of this
manuscript, we describe a formalism for modeling the effective change in power-recycling
gain due to point absorbers on the test masses. We summarize the main findings here and
interested readers are encouraged to review the appendices for further detail.
As shown in Equation B8, the surface deformation from a point absorber is approximated
by:
∆
s
≈
αP
abs
2
π κ
f
(
r
)
(1)
where
f
(
r
), the functional form of the radial spatial distortion from a point absorber, is
defined in Equation B9. The surface deformation is proportional to the coefficient of thermal
expansion,
α
, the absorbed power,
P
abs
in a point and inversely proportional to the thermal
conductivity,
κ
. When the fundamental mode in a Fabry-Perot cavity reflects off a mirror
distorted by the above surface deformation, it scatters some of that field into higher order
modes (HOM). The cavity resonantly enhances/suppresses those modes as a function of the
round-trip phase they accumulate in the cavity. The loss of power from the fundamental
mode to a HOM,
L
mn
, is approximated by Equation D1, which is reproduced here:
L
mn
=
a
2
00
|
mn
g
mn
.
(2)
We emphasize here that losses are a function of two main elements. The first element,
a
00
|
mn
, defined in Equation C5, is
single-bounce amplitude scattering
from the fundamental
mode into the
mn
−
th HOM when reflected off the mirror with the surface deformation
∆
s
located at position
r
c
. The second element is the
resonant enhancement/suppression
factor
of the HOM for the Fabry-Perot cavity geometry and accounting for additional phase
delays experienced by HOM due to surface polish errors at the edges of the mirrors, defined
in Equation D2.
The average arm loss experienced by the fundamental mode is the sum of losses across
all modes other than the fundamental mode:
10
L
A
=
L
nom
+
1
2
∑
m,n
L
mn,X
+
L
mn,Y
(3)
=
L
nom
+
b
(
P
A
100kW
)
2
(4)
From this model, the power-recycling gain of the interferometer is summarized in Equation
D11, reproduced here:
G
P
=
t
p
1
−
r
p
(
1
−
G
A
2
[
L
nom
+
b
(
P
A
100kW
)
2
])
2
(5)
In Appendix D 3, we highlight all of the physical elements that contribute to the total
loss, either through single-bounce amplitude scattering,
a
00
|
mn
, (specifically total absorbed
power, material properties and location of position absorber) or through the resonant en-
hancement/suppression factor,
g
mn
, (namely clipping losses, cavity geometry and mirror
polish errors).
As shown in Appendix C 1, the loss into HOM is dependent only on the absorbed power,
and not the size or distribution of the point absorber. Additionally, it is shown in Appendix
C 2 that time-scale for scattering into higher order modes decreases as mode-order increases
- allowing us to use the time-evolution of power-recycling gain as a proxy for the contribution
of HOM to loss in the arm.
III. OBSERVATION OF POINT ABSORBER EFFECTS IN ADVANCED LIGO
A. Optical gain vs power and position
In aLIGO, we observed the power recycling gain decay as a function of input laser power,
P
in
and arm power. The latter is determined from the product of the power-recycling gain,
the input laser power, the arm cavity optical gain and the beam-splitter transmission:
P
A
= 0
.
5
G
P
G
A
P
in
(6)
For O3, the average arm cavity optical gains,
G
A
, are 268 and 265 for Hanford (LHO)
and Livingston (LLO), respectively (for full details, see Section VB in Buikema et al.[1]).
11
0
50
100
150
200
250
300
Arm Power (kW)
35
40
45
50
55
Power recycling gain
LHO: data
LHO: fit
LLO (pre-realign): data
LLO (pre-realign): fit
LLO (post-realign): data
LLO (post-realign): fit
Model:
=30
m at x=0mm
Model:
=40
m at x=0mm
FIG. 4. Average LIGO power recycling gain versus arm power for the first 120 days of 2019. The
Livingston data has been split into two epochs: pre- and post-interferometer realignment. Arm
power is the product of the measured input laser power, the measured power recycling gain, the
arm cavity optical gain and the beam-splitter transmission. Models of the PRG are shown by
dotted yellow and green curves (assuming a 30
μ
m diameter point absorber at
x
= 0mm and a
40
μ
m diameter point absorber at
x
= 0mm, respectively). The large range covered by the models
indicates a strong variation in the PRG behavior is expected depending on the exact configuration
of point absorbers.
Figure 4 shows the power-recycling gain (averaged into 1 W bins) versus the input laser
power over a 120-day period in early 2019 for the two LIGO interferometers, LLO and LHO,
with two epochs plotted for the former. After an initial commissioning period the Livingston
interferometer was realigned to move the arm mode by approximately 30 mm on ETMY [14].
The Livingston data reflects this and is split into two epochs: pre- and post-interferometer
12
realignment, red and blue data, respectively. The optimum Hanford alignment is also shown
by green data.
Using the data sets of the measured power recycling gain vs arm power,
P
A
, we fit for
L
nom
and
b
in Equation D11, yielding the following results:
Interferometer
L
nom
(ppm)
b
(ppm)
LHO
68.0
1.0
LLO (pre-realign)
60.2
6.4
LLO (post-realign)
60.7
1.3
The new alignment, blue data in Figure 4, shows a 5
×
reduction of power-dependent loss,
strongly illustrating the position-dependent arm loss described in Section C.
Lastly, we have modeled maxima and minima for expected power recycling gain, dotted
green and yellow, respectively, calculated using the model represented by Equations 3, 4 and
5, assuming absorption ranges of 100-350ppb, varying positions on the test masses and the
same nominal loss as LLO. We assume that these curves represent the likely extremum of
PRG behavior for a single point absorber on an optic. From the range, it is reasonable to
expect large differences between hypothetical individual interferometer configurations.
B. Time evolution of PRG reduction
We measured a rapid drop in the power recycling gain with a time scale on the order of
200s. This is illustrated in Figure 5.
Using a combination of a COMSOL finite-element thermal model of point or uniform
absorption induced surface deformation on a test mass and a numerical FFT model of the
full interferometer (the Static Interferometer Simulation (SIS) [15]), we simulated the time
evolution of PRG as the interferometer was powered up for two cases. In one case we assumed
uniform absorption and in the other, we assumed a point absorber located approximately
20 mm from the center of the optic.
The LLO arm power,
P
L
(
t
)
,
is shown in Figure 5 (dashed red curve). In two time-
dependent finite-element analyses, we modeled surface deformations, ∆
s
i
(
x,y,t
), from uni-
form absorption and from a point absorber as a function of time, scaling the absorbed power
by
P
L
(
t
) in the two cases.
13
0
500
1000
1500
2000
Time (s)
35
40
45
50
55
Power recycling gain
0
10
20
30
40
50
Power (W)
LLO power up: t0 = 1230514443
PRG SIS (unif. = 371 mW)
PRG SIS (pt. = 33 mW)
Measured PRG
Laser power
Arm power/5000
FIG. 5. Time evolution of power recycling gain during a power-up at LLO. Laser power and (scaled)
arm power are indicated by the right-hand axis. Power recycling gain is shown on the left-hand
axis. Also shown are two SIS models of the power-recycling gain assuming surface deformation on
a single ETM from, uniform (unif.) and a point absorber (pt), respectively. The time-evolution of
the surface deformation was modeled in COMSOL and scaled in amplitude to yield a PRG of 40
at
t
=
∞
.
The magnitude of these surface deformations were scaled such that, when added to an
ETM in SIS, the steady-state power recycling gain was equal to 40, the same value measured
in the interferometer (dotted blue curve in Figure 5). Finally, the SIS model was run on all
times from
t
= 0 to steady-state (7200 s) and predicted the power-recycling gain.
As the results show, the rapid drop in PRG is predicted by the point-absorber SIS model.
This is particularly striking when compared to the uniform absorption SIS model that shows
a time scale more than an order of magnitude larger. The smaller drop in PRG around 1300 s
14
is thought to be due to slight misalignment.
Consistent with the time-scale analysis in Appendix C 2, this result is highly suggestive
that the majority of the power-recycling gain reduction that we see can be attributed to
point absorbers interacting with higher order modes rather than uniform absorption.
C. Noise couplings from ITM thermal lens effects
We measured the coupling of the input laser relative intensity noise (RIN) to differential
arm motion (DARM) in aLIGO. At frequencies above 500 Hz, the coupling was approxi-
mately two orders of magnitude higher than expected from simulations of an ideal interfer-
ometer, as shown in Figure 6. However, simulations including ITM thermal lenses from point
absorbers modeled at a variety of different radii showed significant increase in high frequency
RIN coupling, sufficient to explain the excess coupling observed in the interferometer.
Uncertainty in point absorber position creates a large uncertainty in the exact coupling
coefficient. Hence, the model can only reproduce the qualitative behavior of the observed
RIN coupling. Noise budget measurements reported in Buikema et al.[1] show the absolute
laser intensity contribution to overall noise is not yet a limiting noise source. Below 2 kHz, it
is at least 10
×
lower than the differential arm motion in both LIGO interferometers (Figure
[2] in Buikema et al.[1] explicitly shows this). However, the coupling, and hence the noise
contribution, is expected to increase as arm cavity power increases in future.
IV. CONSIDERATIONS FOR WORKING WITH POINT ABSORBERS IN THE
FUTURE
It is clear that point absorbers adversely affect the performance of the aLIGO interfer-
ometers and that this effect increases as input power is increased. In Figure 7 we plot the
projected arm power versus input power assuming the most optimistic scenario of solely
a power-dependent reduction in power-recycling gain. This projection corresponds to the
three different scenarios in Figure 4 (O3 LHO (green), O3 LLO pre-realignment (red) and
O3 LLO post-realignment (blue)). Briefly, all scenarios predict a significant reduction in the
maximum stored power (and, hence, performance) for aLIGO.
Despite the improvement in PRG attained by repositioning the beam on the ETM, the
15
10
1
10
2
10
3
10
4
Frequency (Hz)
10
-15
10
-14
10
-13
10
-12
10
-11
10
-10
10
-9
DARM / RIN (m / [W/W])
Measurement
Ideal IFO (sim)
Point absorber @ 0cm
Point absorber @ 1cm
Point absorber @ 2cm
Point absorber @ 3cm
Point absorber @ 4cm
Point absorber @ 5cm
FIG. 6. RIN coupling models versus measurements from the LHO interferometer. The ”ideal IFO”
trace is for a perfectly symmetric IFO, therefore the coupling is due to radiation pressure at low
frequency and sidebands RIN at high frequency. The other traces have the expected spherical
lenses on the ITM substrates, plus one point absorber map at different distances from the center.
Simulations were created using the MIST simulation tool [16].
SIS model predicts a minimum 33% deficit in arm power build-up at nominal operating
power of 125 W. Assuming no other adverse effects to interferometer operation, we would
expect a corresponding increase in the shot noise floor of 15%
−
20% above the nominal
Advanced LIGO noise floor.
Mitigating the adverse effects of point absorbers can be accomplished by addressing all
the features identified in Section D 3 and expressed by the amplitude and gain terms in
Equation 2.
Amplitude reduction strategies seek to limit or eliminate the single bounce scattering
16
0
25
50
75
100
125
Input Power (W)
0
100
200
300
400
500
600
700
800
Arm Power (kW)
LHO
LLO (pre-realign)
LLO (post-realign)
Ideal case
SIS model range
FIG. 7. Arm power versus input laser power. The dashed line shows the case of constant optical
gain. The data points show the measurements from the LIGO sites. The yellow range shows
predictions from the SIS model assuming uniform absorption and a variety of point absorbers
on the optic consistent. As the effect of the point absorber is decreased, the uniform absorption
becomes the limiting factor preventing ideal build-up of arm power.
term,
a
00
|
mn
, in Equation 2. The first, and ideal, scenario is to eliminate absorbers at
the source when they are introduced into the coating: either through modifications to the
coating process or active elimination after coating. This is being actively researched within
the LIGO Scientific Collaboration. Beyond that, the simplest method for partial reduction
of amplitude scattering in-situ is re-positioning the interferometer beam, as demonstrated at
LLO. Additionally, scattering to specific HOM can be reduced by actuating on those modes
using, for example, a high-spatial frequency corrector [17]. Finally, surface deformation (and
therefore amplitude scattering) are minimized for different optic materials; for example, the
17
proposed use of cryogenic silicon in future interferometers [18, 19] has the benefit of an,
effectively, zero coefficient of thermal expansion.
Resonant enhancement/suppression techniques seek to reduce the resonant gain term,
g
mn
, in Equation 2. Future interferometers are free modify the overall cavity design and
g-factor with respect to HOM spacing to minimize the effect from point absorbers. A more
targeted option, actively being explored for Advanced LIGO, is to deliberately polish surface
errors into the edges of the optic, such as those illustrated in Figure 14, creating a wider
HOM free region around the fundamental mode resonance [20]. In addition, conceptual
work has begun on active front surface thermal actuators that could produce such surface
errors dynamically - thus allowing for in-situ fine-tuning of the HOM resonances around the
fundamental mode.
V. CONCLUSION
We have shown that point-like absorbers present a serious impediment to obtaining full
operating power in Advanced LIGO, primarily due to increased arm losses as absorbed
power increases. We reiterate that that errors in the mirror surface profile, mirror aperture
effects and higher order mode behavior matter are crucial to understanding considering
these losses. Future gravitational wave detectors call for stored power approximately an
order of magnitude or more higher than O3 [21–25], with similar laser intensities to full-
power Advanced LIGO, so the existence of point absorbers and their interaction with the
optical fields of those detectors is of significant importance.
We have explored the physics of cavity-optic-deformation to explain the observed reduc-
tion in interferometer performance whilst, simultaneously, highlighting those elements that
are most crucial to mitigating the effects of point absorbers in Advanced LIGO and future
gravitational wave interferometers.
VI. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of the United States National Science
Foundation (NSF) for the construction and operation of the LIGO Laboratory and Ad-
vanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United
18
Kingdom, and the Max-Planck-Society (MPS) for support of the construction of Advanced
LIGO. Additional support for Advanced LIGO was provided by the Australian Research
Council. The authors acknowledge the LIGO Scientific Collaboration Fellows program for
additional support.
LIGO was constructed by the California Institute of Technology and Massachusetts In-
stitute of Technology with funding from the National Science Foundation, and operates
under cooperative agreement PHY-1764464. Advanced LIGO was built under award PHY-
0823459. This paper carries LIGO Document Number LIGO-P1900287.
Appendix A: Advanced LIGO interferometer operation
To understand the observed interferometer behavior, it is helpful to provide an overview
of how the aLIGO interferometer works. This section contains a description of the ideal
operation of the Advanced LIGO interferometer. Interested readers can find more details in
other references [5, 26–30].
1. Detector sensitivity
As shown in Figure 1, Advanced LIGO is a suspended Michelson interferometer, modified
to maximize sensitivity to GW with a bandwidth of approximately 20 Hz - 2 kHz. All
modifications are designed to either amplify the signal strength, reduce the contribution of
different noise sources or improve robustness and reliability.
A passing gravitational waves of amplitude,
h
, induces a strain in spacetime, lengthening
one interferometer arm and shortening the other by an amount, ∆
L
, where
∆
L
=
hL
(A1)
and
L
is the interferometer arm length. This imposes phase fluctuations on the laser beams
present in those arms (GW audio sidebands) that are converted, by the beamsplitter, into
intensity fluctuations on the output side of the Michelson interferometer.
As Equation A1 shows, the resulting change in arm length (and, hence, the signal
strength) is proportional to the arm length. To exploit this, the aLIGO arms are 4 km
long, substantially amplifying the signal for a given strain relative to a short interferometer.
19
The arms contain resonant Fabry-Perot cavities that increase the average number of round-
trips photons experience in the arms, and the phase they accumulate, by the optical gain of
the arms,
G
A
. For the Fabry-Perot arm cavity, since the input and end mirror transmissions
are very low (
T
i
= 1.4% and
T
e
= 5ppm, respectively), the following approximation holds:
G
A
≈
4
T
i
(A2)
resulting in a nominal phase amplification factor of approximately 280 for aLIGO. (The
optical gain is also approximated by 2
F/π
, where
F
is the cavity finesse, approximately
440).
Ideally, the Fabry-Perot arms resonate the fundamental Gaussian spatial mode (some-
times referred to as TEM00) and suppress higher order spatial modes (HOM). Higher order
modes can be described in terms of different bases, for example, Hermite-Gauss (HG) or
Laguerre-Gauss (LG) [31].
In contrast to the limited ways to amplify the signal, there are a large number of indepen-
dent noise sources in the interferometer. The majority of noise sources limit the sensitivity
below approximately 100 Hz [30, 32]. However, at high frequencies, an ideal detector is lim-
ited only by fundamental quantum sensing noise (shot noise). The Signal-to-Noise Ratio
(SNR) from shot noise is proportional the square root of the power incident on the input
(symmetric) side beamsplitter. Thus, one of the the simplest ways to improve the detector
sensitivity is to increase the input laser power (up to 125 W for aLIGO).
The incident power on the beamsplitter is further increased by adding a power recycling
mirror (PRM), see Figure 1, to return power coming out of the interferometer back into it.
This mirror forms another resonant cavity, named the power recycling cavity. The coupled
power-recycling and common arm cavities are referred to as the common coupled cavity.
For the power recycling to maintain a high optical gain, the interferometer is held close to
a dark fringe on the output (anti-symmetric) side of the beamsplitter, sending nearly 100%
of the optical power on the beamsplitter back to the PRM. The differential arm length is
held slightly off minimum (close to dark fringe) such the static amount of power at the AS
port is held constant to serve as a local oscillator for DC (homodyne) readout [33].
The coupled power-recycling and arm cavities increase the optical gain such that the
power stored in the PRC and a single arm are approximately 50
×
and 6000
×
the input
power, respectively (in fact, these numbers vary in the real interferometer depending on
20
losses in the arms and power-recycling cavities).
The storage time of photons the arm cavity is similarly increased. Intensity noise present
on the light resonant in the cavity is filtered above 1
/τ
CC
, where
τ
CC
is the coupled cavity
storage time, corresponding to a pole at a frequency of approximately 0
.
6 Hz [32]. This
filtering is essential to suppress the coupling of relative intensity noise (RIN) from the input
laser to the GW channel.
As shown in Figure 1, on the output side of the interferometer a recycling mirror (SRM)
is present, forming a resonant cavity with the test masses called the signal recycling cavity
(SRC). The SRC is held on anti-resonance, lowering optical gain and storage time of dif-
ference signals in order to increase the signal bandwidth of the interferometer (known as
resonant signal extraction [34]). The combination of the signal-recycling and arm cavities is
called the differential coupled cavity.
Finally, the output of the interferometer is spatially filtered through an output mode
cleaner (OMC) to remove higher order spatial modes that carry no GW signal, and strip off
any residual radio-frequency (RF) control sidebands from the laser, described below. At the
output of the OMC the GW signal sidebands beat with the static offset induced by the arm
differential offset to provide a power fluctuation that is detected by a pair of photodiodes.
2. Power-recycling gain
The power recycling gain,
G
P
, defined as the ratio of the stored power inside the power
recycling cavity to the input laser power incident on the PRM, is a function of the reflectivi-
ties of the power-recycling mirror, input test masses and end test masses and also a function
of the losses in the arms and the PRC. The amplitude reflectivities are denoted in lower-case,
r
a
, and corresponding power reflectivities are denoted in upper-case,
R
a
=
r
2
a
. The values
are given in Table I in Appendix F.
For this analysis, it is convenient to consider the two arm cavities as an aggregate (com-
mon) arm cavity and the power-recycling cavity as a three mirror cavity made of the PRM,
an average Fabry-Perot arm. The power recycling gain is given by:
G
P
=
(
t
p
1
−
r
p
r
FP
(1
−L
P
/
2)
)
2
(A3)
when the PRC is on resonance [35]. The amplitude reflectivity of the average arm cavity,
21
r
FP
[35], is given by:
r
FP
=
−
r
i
+
r
∗
e
1
−
r
i
r
∗
e
(A4)
≈
1
−
2
T
e
T
i
−
G
A
L
A
2
(A5)
when the arm cavities are on resonance and
T
e
,T
i
<<
1. Combining equations A3 and A5
yields an expanded form of the power-recycling gain,
G
P
≈
(
t
p
1
−
r
p
(1
−
(
G
A
T
e
−
G
A
L
A
−L
P
)
/
2)
)
2
(A6)
in which the arm loss contribution is greater than the recycling cavity loss by a factor of
G
A
, approximately 280 for aLIGO. Thus, the power-recycling gain can be used as a proxy
for losses, particularly arm losses, in the interferometer.
3. Feedback control
In order for the interferometer to function in low noise, all of the longitudinal degrees
of freedom (DOF) in the interferometer must be sensed and locked onto resonance for the
main laser field (the GW carrier). This is achieved by using the Pound-Drever Hall locking
technique that adds pairs of radio frequency sidebands onto the main laser [36] at
±
9 MHz
and
±
45 MHz. As illustrated in Figure 1, both pairs of sidebands are anti-resonant in the
Fabry-Perot arms, resonant in the PRC. The 45 MHz sidebands are resonant in the SRC.
Full details can be found in [32].
Since sidebands and carrier fields have different frequencies, their propagation inside the
meters- or km-long interferometer cavities produce different phase shifts. The net result is
that carrier and sideband fields have different resonant conditions in the various interfer-
ometer cavities. In particular, while the carrier is tuned to be resonant simultaneously in
the power recycling cavity and in the Fabry-Perot arm cavities, the sideband frequency is
chosen so that the corresponding fields are close to anti-resonance in the arms and resonant
in the power recycling cavity. In this way the sidebands provide a phase reference which
is independent of the arm cavity motions, and suitable error signals can be extracted by
demodulating the signals produced by fast photodiodes at the same modulation frequency
[32].
22
Thus linear combinations of this beating of these sidebands and carrier, measured on
RF photodiodes at different locations and in different modulation quadrature yield enough
control signals to uniquely measure all the desired degrees of freedom and, when fed back
into a control system, keep the residual longitudinal motions less than the required levels.
The noise in the control system must be kept very low lest the feedback loop inject it back
into the interferometer at a level which then limits the detector sensitivity.
In the interests of brevity, the above description of the feedback control system is delib-
erately terse. A full description of the control is beyond the scope of this text, references
are provided [37, 38]. However, one element key to the following discussion concerns the
differential arm motion (DARM): the degree of freedom that encodes the gravitational wave
signal. For the subsequent discussion, it is important to know that the feedback control
system is set up to maintain a constant DC power (20 mW) on the output photodiodes by
feeding the DARM error signal back to the differential arm length.
In summary, Advanced LIGO achieves good sensitivity at high frequencies by having
large amounts of stored power in those cavities and incident on the the test masses while
relying on a low-noise control system to keep the optical cavities on resonance within minimal
residual longitudinal motions.
Appendix B: Thermal-elastic surface deformation from point absorbers
Consider a point-like absorber, absorbing within a diameter 2
ω
, on the surface of an
optic that is exposed to optical power, as illustrated in Figure 8. We solve for the surface
deformation inside and outside the absorbing region.
In the specific case that the absorbing region is on the surface of an optic within a LIGO
arm cavity, the absorbed power,
P
abs
, is determined by the local intensity of the illuminating
beam multiplied by the absorbing area:
P
abs
=
(
π ω
2
)
2
P
A
π
w
2
exp
[
−
(
r
c
w
)
2
]
(B1)
where
P
A
is the laser power stored in the arm, w is the Gaussian beam radius and r
c
is the
position of the absorbing region relative to the center of the Gaussian beam. For simplicity,
we have assumed that the region is 100% absorbing.
23
P
P
P
!
P
!
P
fused silica
A
B
2w
FIG. 8. Power flowing into a point-like region. The radiated power is negligible compared to the
conduction. Hence, all power is effectively transferred by conduction. The far-field gradient is only
determine by the absorbed power and thermal conductivity.
1. Small and large scale surface deformation and time scales
a. Inside the absorbing region
Winkler et al. solved for the approximate surface deformation, ∆
s
ω
, on the surface of an
optic, within the absorbing region [7], finding
∆
s
ω
=
−
α
4
π κ
P
abs
(B2)
where
α
is the coefficient of thermal expansion,
κ
is the thermal conductivity and
P
abs
is the
power absorbed within the absorbing region. The negative sign is purely convention based
upon the deformation expanding
away
from a mirror. Expressed as a quadratic function of
position:
∆
s
=
−
α
4
π κ
P
abs
(
r
ω
)
2
.
(B3)
24
b. Outside the absorbing region
We solve for the approximate surface deformation for a half-infinite cylinder (i.e. assuming
radial and longitudinal boundaries are far away). We assume only conduction and ignore
power radiated from the front surface of the optic - given that the power radiated from one
beam radii is of the order 2%
−
3% of the total power radiated by the optic.
Ignoring boundary conditions and considering the region outside the radius of the point
absorber, the temperature distribution is solely governed by the equation of thermal con-
ductivity:
P
abs
=
−
κA
−→
∇
T
(B4)
where
A
= 2
π r
2
S
, the area of a hemispherical shell in the optic, where
r
S
is the spherical
radial coordinate (
√
x
2
+
y
2
+
z
2
). Solving in spherical coordinates, the temperature profile
is given by
T
(
r
) =
P
abs
2
π κr
S
(B5)
=
P
abs
2
π κ
√
r
2
+
z
2
(B6)
where
r
=
√
x
2
+
y
2
and
z
are the polar radial and longitudinal coordinates, respectively.
The surface deformation is approximated by the coefficient of thermal expansion,
α
, multi-
plied by the integral of temperature field:
∆
s
=
α
∫
h
0
T
(
r
) d
z
(B7)
where
h
is the thickness of the optic. Combining this with equation B3 and solving yields a
generalized approximation for the surface deformation on the optic:
∆
s
≈
αP
abs
2
π κ
f
(
r
)
(B8)
where
25
f
(
r
) =
−
1
2
(
r
ω
)
2
r
≤
ω
c
0
+ log
[
h
+
|
(
r,h
)
|
r
]
r > ω
(B9)
where
c
0
is a constant offset such that the two parts have the same value at
r
=
ω
:
c
0
=
−
1
2
−
log
[
h
+
|
(
ω,h
)
|
ω
]
(B10)
and where
|
(
r,h
)
|
=
√
r
2
+
h
2
.
(B11)
Note that the surface deformation has been referenced to the center of the point absorber
such that the surface deformation is zero at that point. Equation B8 shows that outside
of the absorbing region, the large scale surface deformation depends only on the absorbed
power and not on the size or distribution of the absorbing region. This is illustrated by
a finite-element model of surface deformation from point absorbers of different size but
fixed absolute absorbed power, shown in Figure 9. The surface deformation from uniform
absorption is shown for comparison.
c. Time scale
Finally, the time scale for a temperature distribution to form is governed by the heat
equation:
∂T
∂t
=
D
(
∂
2
T
∂x
2
+
∂
2
T
∂y
2
+
∂
2
T
∂z
2
)
(B12)
where
D
is the thermal diffusivity of the mirror, given by:
D
=
κ
ρc
(B13)
where
ρ
is the density,
c
is the specific heat capacity and
κ
is the thermal conductivity.
A cursory examination of the heat equation shows that the time-scale for the temperature
distribution to form over a given spatial scale will be proportional to that spatial scale
squared. For example, for a given spatial scale,
r
0
, the time constant for the temperature
distribution (and thus surface deformation) to form is:
26