Point Absorber Limits to Future Gravitational-Wave Detectors
Wenxuan Jia,
1,
∗
Hiroaki Yamamoto,
2
Kevin Kuns,
1
Anamaria Effler,
3
Matthew Evans,
1
Peter Fritschel,
1
R. Abbott,
2
C. Adams,
3
R. X. Adhikari,
2
A. Ananyeva,
2
S. Appert,
2
K. Arai,
2
J. S. Areeda,
4
Y. Asali,
5
S. M. Aston,
3
C. Austin,
6
A. M. Baer,
7
M. Ball,
8
S. W. Ballmer,
9
S. Banagiri,
10
D. Barker,
11
L. Barsotti,
1
J. Bartlett,
11
B. K. Berger,
12
J. Betzwieser,
3
D. Bhattacharjee,
13
G. Billingsley,
2
S. Biscans,
1, 2
C. D. Blair,
3
R. M. Blair,
11
N. Bode,
14, 15
P. Booker,
14, 15
R. Bork,
2
A. Bramley,
3
A. F. Brooks,
2
D. D. Brown,
16
A. Buikema,
1
C. Cahillane,
2
K. C. Cannon,
17
X. Chen,
18
A. A. Ciobanu,
16
F. Clara,
11
C. M. Compton,
11
S. J. Cooper,
19
K. R. Corley,
5
S. T. Countryman,
5
P. B. Covas,
20
D. C. Coyne,
2
L. E. H. Datrier,
21
D. Davis,
9
C. Di Fronzo,
19
K. L. Dooley,
22, 23
J. C. Driggers,
11
P. Dupej,
21
S. E. Dwyer,
11
T. Etzel,
2
T. M. Evans,
3
J. Feicht,
2
A. Fernandez-Galiana,
1
V. V. Frolov,
3
P. Fulda,
24
M. Fyffe,
3
J. A. Giaime,
6, 3
K. D. Giardina,
3
P. Godwin,
25
E. Goetz,
6, 13, 26
S. Gras,
1
C. Gray,
11
R. Gray,
21
A. C. Green,
24
E. K. Gustafson,
2
R. Gustafson,
27
E. Hall,
1
J. Hanks,
11
J. Hanson,
3
T. Hardwick,
6
R. K. Hasskew,
3
M. C. Heintze,
3
A. F. Helmling-Cornell,
8
N. A. Holland,
28
J. D. Jones,
11
S. Kandhasamy,
29
S. Karki,
8
M. Kasprzack,
2
K. Kawabe,
11
N. Kijbunchoo,
28
P. J. King,
11
J. S. Kissel,
11
Rahul Kumar,
11
M. Landry,
11
B. B. Lane,
1
B. Lantz,
12
M. Laxen,
3
Y. K. Lecoeuche,
11
J. Leviton,
27
J. Liu,
14, 15
M. Lormand,
3
A. P. Lundgren,
30
R. Macas,
22
M. MacInnis,
1
D. M. Macleod,
22
G. L. Mansell,
11, 1
S. M ́arka,
5
Z. M ́arka,
5
D. V. Martynov,
19
K. Mason,
1
T. J. Massinger,
1
F. Matichard,
2, 1
N. Mavalvala,
1
R. McCarthy,
11
D. E. McClelland,
28
S. McCormick,
3
L. McCuller,
1
J. McIver,
2, 26
T. McRae,
28
G. Mendell,
11
K. Merfeld,
8
E. L. Merilh,
11
F. Meylahn,
14, 15
T. Mistry,
31
R. Mittleman,
1
G. Moreno,
11
C. M. Mow-Lowry,
19
S. Mozzon,
30
A. Mullavey,
3
T. J. N. Nelson,
3
P. Nguyen,
8
L. K. Nuttall,
30
J. Oberling,
11
Richard J. Oram,
3
C. Osthelder,
2
D. J. Ottaway,
16
H. Overmier,
3
J. R. Palamos,
8
W. Parker,
3, 32
E. Payne,
33
A. Pele,
3
R. Penhorwood,
27
C. J. Perez,
11
M. Pirello,
11
H. Radkins,
11
K. E. Ramirez,
34
J. W. Richardson,
2
K. Riles,
27
N. A. Robertson,
2, 21
J. G. Rollins,
2
C. L. Romel,
11
J. H. Romie,
3
M. P. Ross,
35
K. Ryan,
11
T. Sadecki,
11
E. J. Sanchez,
2
L. E. Sanchez,
2
T. R. Saravanan,
29
R. L. Savage,
11
D. Schaetzl,
2
R. Schnabel,
36
R. M. S. Schofield,
8
E. Schwartz,
3
D. Sellers,
3
T. Shaffer,
11
D. Sigg,
11
B. J. J. Slagmolen,
28
J. R. Smith,
4
S. Soni,
6
B. Sorazu,
21
A. P. Spencer,
21
K. A. Strain,
21
L. Sun,
2
M. J. Szczepa ́nczyk,
24
M. Thomas,
3
P. Thomas,
11
K. A. Thorne,
3
K. Toland,
21
C. I. Torrie,
2
G. Traylor,
3
M. Tse,
1
A. L. Urban,
6
G. Vajente,
2
G. Valdes,
6
D. C. Vander-Hyde,
9
P. J. Veitch,
16
K. Venkateswara,
35
G. Venugopalan,
2
A. D. Viets,
37
T. Vo,
9
C. Vorvick,
11
M. Wade,
38
R. L. Ward,
28
J. Warner,
11
B. Weaver,
11
R. Weiss,
1
C. Whittle,
1
B. Willke,
15, 14
C. C. Wipf,
2
L. Xiao,
2
Hang Yu,
1
Haocun Yu,
1
L. Zhang,
2
M. E. Zucker,
1, 2
and J. Zweizig
2
1
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
3
LIGO Livingston Observatory, Livingston, LA 70754, USA
4
California State University Fullerton, Fullerton, CA 92831, USA
5
Columbia University, New York, NY 10027, USA
6
Louisiana State University, Baton Rouge, LA 70803, USA
7
Christopher Newport University, Newport News, VA 23606, USA
8
University of Oregon, Eugene, OR 97403, USA
9
Syracuse University, Syracuse, NY 13244, USA
10
University of Minnesota, Minneapolis, MN 55455, USA
11
LIGO Hanford Observatory, Richland, WA 99352, 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
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 British Columbia, Vancouver, BC V6T 1Z4, Canada
arXiv:2109.08743v1 [physics.ins-det] 17 Sep 2021
2
27
University of Michigan, Ann Arbor, MI 48109, USA
28
OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia
29
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India
30
University of Portsmouth, Portsmouth, PO1 3FX, UK
31
The University of Sheffield, Sheffield S10 2TN, UK
32
Southern University and A&M College, Baton Rouge, LA 70813, USA
33
OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia
34
The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
35
University of Washington, Seattle, WA 98195, USA
36
Universit ̈at Hamburg, D-22761 Hamburg, Germany
37
Concordia University Wisconsin, 2800 N Lake Shore Dr, Mequon, WI 53097, USA
38
Kenyon College, Gambier, OH 43022, USA
High-quality optical resonant cavities require low optical loss, typically on the scale of parts per
million. However, unintended micron-scale contaminants on the resonator mirrors that absorb the
light circulating in the cavity can deform the surface thermoelastically, and thus increase losses by
scattering light out of the resonant mode. The point absorber effect is a limiting factor in some high-
power cavity experiments, for example, the Advanced LIGO gravitational wave detector. In this
Letter, we present a general approach to the point absorber effect from first principles and simulate
its contribution to the increased scattering. The achievable circulating power in current and future
gravitational-wave detectors is calculated statistically given different point absorber configurations.
Our formulation is further confirmed experimentally in comparison with the scattered power in the
arm cavity of Advanced LIGO measured by in-situ photodiodes. The understanding presented here
provides an important tool in the global effort to design future gravitational wave detectors that
support high optical power, and thus reduce quantum noise.
Introduction -
A wide variety of precision optical
experiments rely on resonant optical cavities to enable
precise measurements of space, time, and fundamental
physics. These experiments often require high optical in-
tensity incidents on the mirrors of the cavity to boost the
signal-to-noise ratio. However, unintended defects may
be deposited on the reflective surface of the mirror during
the coating process or exposure to a dusty environment
[1]. These localized defects, known as “point absorbers”,
absorb optical power and cause undesired thermal effects
on the optics under irradiation, especially in cavities con-
taining high circulating power. The point absorber be-
comes a limiting factor in various precision measurement
experiments that require a high-finesse cavity with low
round-trip loss, such as cavity QED [2], axion detection
[3], qubit experiments [4], and gravitational-wave detec-
tors [5–7]. It is thus necessary to develop a quantitative
understanding of the point absorber effect in high-power
optical cavities.
With a 4 km long baseline and a circulating power of
more than 200 kW, the arm cavity of Advanced Laser In-
terferometer Gravitational-Wave Observatory (aLIGO)
serves as a good example of the point absorber effect
[8]. aLIGO is a dual-recycled Fabry–P ́erot Michelson in-
terferometer designed to measure tiny perturbations of
spacetime with unprecedented precision [9]. One of the
fundamental noises that limit aLIGO’s performance is
quantum shot noise, which can be reduced either by in-
creasing the arm power or by manipulating the quan-
tum states of light through squeezing [10]. However, arm
power can be limited by point absorbers (studied here),
other thermal distortions [11], and a variety of instabil-
ities [12–15]. The arm power during the third observ-
ing run was limited to one-third of the designed value
of 750 kW, mainly due to point absorbers on the mir-
ror [8, 9] that scatter light out of the fundamental cavity
mode.
Point absorbers were known to exist since the first ob-
serving run. Many analyses have been carried out to
understand how they deform the optics and scatter light
out of the cavity [1, 16, 17]. In this paper, we provide
a more general approach from first principles. The tra-
ditional formalism is extended to include arbitrary heat-
ing functions with any nonlinear boundary conditions,
such as Stefan-Boltzmann law. With the correction from
nonlinear boundary condition, we can make more accu-
rate statistical estimations of the arm power for the next
planned upgrade of aLIGO (known as “A+”) and the
next generation of gravitational-wave detectors with a
variety of potential point absorber configurations.
We start by calculating the differential temperature
profile from single point absorber heating with proper
boundary conditions. Then the thermoelastic deforma-
tion of the mirror is derived using thermoelasticity equa-
tion. Next, this deformation is incorporated in an FFT-
based simulation to obtain the field in the arm cavity,
which is used to calculate its round-trip loss and achiev-
able power. In addition, we simulate the low-angle scat-
tered light intensity and compare this with in-situ mea-
surements. Our results reveal a good match between
these measurements and simulation, thus confirming our
understanding of point absorbers.
Theoretical modeling -
Point absorbers degrade the
performance of high-power optical resonators by absorb-
3
ing laser power, which thermally distorts the mirror sur-
face and thereby scatters light out of the resonant mode
of the cavity. The analytical solution of the differential
temperature under a general boundary condition is de-
rived first.
Consider a cylindrical optic with radius
a
and thickness
h
. Choose cylindrical coordinates at the center of the
mirror with the
z
direction pointing into the cavity. One
point absorber is put at the center of the high-reflective
(HR) side for cylindrical symmetry. When the cavity
is held on resonance, the system is static, and the heat
equation reduces to the Laplace equation
∇
2
T
= 0
(1)
where
T
(
r,z
) is the temperature departure from the am-
bient temperature
T
∞
. The boundary conditions include
the intensity of a heating source
I
(
r
) on the HR surface
due to the point absorber:
−
K
∂T
∂z
∣
∣
∣
∣
z
=
h
2
=
−
I
(
r
) +
g
(
T
)
∣
∣
∣
∣
z
=
h
2
(2)
where
K
is the thermal conductivity,
g
(
T
) is the thermal
flux of blackbody radiation
g
(
T
) =
σ
[
(
T
∞
+
T
(
r,z
))
4
−
T
4
∞
]
,
(3)
is the thermal emissivity/absorptivity (assumed to
be unity throughout this paper), and
σ
is the Stefan-
Boltzmann constant.
A semi-infinite assumption can be made by treating
the optic as a semi-infinite solid [18, 19]. We present a
general way of solving the Laplace equation under either
linearized or nonlinear boundary conditions. Taking the
zeroth-order Hankel transform
H
0
of the Laplace equa-
tion in cylindrical coordinates with respect to
r
[20], we
get
(
−
k
2
+
∂
2
z
)
̃
T
(
k,z
) = 0
(4)
The angular dependence is dropped due to cylindrical
symmetry. The solution of Eq. (4) is the sum of growth
modes
A
(
k
)e
kz
and decay modes
B
(
k
)e
−
kz
, where the
latter vanishes by the semi-infinite boundary conditions
T
(
r
→∞
,z
) =
T
(
r,z
→−∞
) = 0.
Let
I
(
r
) be the heating function from a point absorber;
here we use a Gaussian profile
I
(
r
) =
I
b
e
−
r
2
/w
2
with
absorber radius
w
and irradiation intensity
I
b
at the ab-
sorber center. Let
T
HR
(
r
) =
T
(
r,z
=
h/
2) be the tem-
perature profile at the HR surface. The Hankel transform
of the boundary condition at the HR surface Eq. (2) gives
̃
T
(
k,z
) =
−
e
k
(
z
−
h/
2)
Kk
H
0
[
−
I
(
r
) +
g
(
T
HR
(
r
))]
(5)
0
2
4
6
8
10
12
14
16
17
Radius [cm]
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
HR Displacement [nm]
0.5
m
1
m
5
m
10
m
50
m
100
m
500
m
Linearized boundary
Linearized boundary fit
Nonlinear boundary
FIG. 1. Thermoelastic displacements on the HR surface by
various point absorber radii (labeled near each curve). The
edge of the 17-cm radius optic has zero deformation. The
incident intensity on the centered absorber is 4
.
1
×
10
7
W
/
m
2
,
equivalent to the center intensity of the 240 kW beam on the
mirror of aLIGO arm cavity. Analytic fits to the linearized
boundary solution (Eq. (11)) are also shown.
Therefore,
T
HR
(
r
) is found by taking the inverse trans-
form of
̃
T
(
k,z
=
h/
2):
T
HR
(
r
) =
2
πK
∫
r
0
d
r
′
r
′
[
I
(
r
′
)
−
g
(
T
HR
(
r
′
))]
1
r
K
(
r
′
2
r
2
)
+
2
πK
∫
∞
r
d
r
′
[
I
(
r
′
)
−
g
(
T
HR
(
r
′
))]
K
(
r
2
r
′
2
)
(6)
where
K
is the complete elliptic integral of the first kind.
Eq. (6) is a nonlinear integral equation with no closed-
form solution, but an approximate solution can be found
by either linearizing the boundary function
g
(
T
) or using
successive approximation.
The linearized boundary solution has been given in
[19]. The heat from a small point absorber is primar-
ily dissipated by conduction. This estimate breaks down
for a large absorber of radius (
K/I
b
)(
I
b
/
2
σ
)
1
/
4
which is
∼
100
μ
m in an aLIGO arm cavity [17]. The correction
of nonlinear radiation will matter if the radiative contri-
bution becomes significant. This motivates us to find a
solution to the general boundary condition.
The nonlinear integral equation Eq. (6) can be solved
by successive approximation with feedback. We start the
zeroth iteration with an initial guess
T
0
(
r
). The real so-
lution is denoted as
T
S
(
r
), and the zeroth error function
is
ε
0
(
r
) =
T
0
(
r
)
−
T
S
(
r
). Plugging this into Eq. (6) and
keeping the first non-trivial order of the error:
g
0
(
r
) =
g
(
T
0
(
r
)) =
g
S
(
r
) + 4
σ
(
T
∞
+
T
0
)
3
ε
0
+
O
(
ε
2
) (7)
Assuming the error has weak variation over radius:
4
FIG. 2. Circulating power in the arm cavity versus input power for two different detectors and mean radii of point absorbers
(optimistic 5
μ
m and pessimistic 12
μ
m). The solid line is the median with shadings corresponding to the 16th and 84th
percentile. The gray lines (no absorber case) increases linearly with the initial slopes set by the round-trip loss of the cold
cavity (Table I), and the designed power is 750 kW for A+ and 1
.
5 MW for CE. In the absence of point absorbers, the required
input power is 120 W for A+ and 140 W for CE. In the zoomed-in graph, the data points collected from LIGO Livingston
Observatory throughout Observing Run 3b are fit to obtain the radii of point absorbers. It is statistically more confident to
achieve the designed power with
〈
w
〉
= 5
μ
m.
ε
0
(
r
)
≈
ε
0
, we have
T
(
T
0
(
r
)) =
T
S
(
r
)
−
ε
0
C
0
(
r
) =
T
S
(
r
)
−
(
T
0
(
r
)
−
T
S
(
r
))
C
0
(
r
)
(8)
where
C
0
(
r
) =
8
σ
πK
[
∫
r
0
d
r
′
r
′
r
(
T
∞
+
T
0
)
3
K
(
r
′
2
r
2
)
+
∫
∞
r
d
r
′
(
T
∞
+
T
0
)
3
K
(
r
2
r
′
2
)]
(9)
We can then iterate the temperature profile
T
i
+1
=
T
(
T
i
) +
C
i
T
i
1 +
C
i
(10)
until we reach convergence at
T
(
T
i
) =
T
i
and
T
i
+1
=
T
i
[21, 22].
The displacement vector field of the optic can be found
given the temperature solution
T
(
r,z
). We follow Hello
and Vinet’s formalism but apply it to our solution [23]
(see the supplemental material for detailed derivations).
Fig. 1 shows the resulting displacement of the HR surface.
For small point absorbers, the differential temperature is
relatively low, and the radiative correction is negligible;
the linearized boundary solution is accurate. However,
the correction becomes significant for absorber with ra-
dius larger than 100
μ
m, up to a factor of three in the
500
μ
m case. A typical point absorber with a few tens of
microns in radius can cause surface deformation on the
scale of several tens of nm in height and a few cm in size.
An analytical fit to displacement of the linearized so-
lution is given by
h
(
r
)
≈
0
.
12
(
3
λ
+ 2
μ
λ
+
μ
)
I
b
w
2
α
K
ln
(
a
2
r
2
(
1
−
w
2
a
2
)
+
w
2
)
(11)
where
μ
is the first Lam ́e coefficient,
λ
is the second Lam ́e
coefficient, and
α
is the thermal expansion coefficient.
Note that Eq. (11) breaks down at high absorbed power,
as shown in Fig. 1. With the deformation known, we
can superpose it onto the mirror phase map data and
simulate fields in a static cavity.
Implications for Gravitational Wave Detectors -
Ad-
vanced gravitational-wave detectors are Michelson in-
terferometers using Fabry–P ́erot cavities as arms to in-
crease optical power and thus the signal produced by
gravitational-wave strain. The arm power is further in-
creased by the addition of a mirror at the symmetric port
of the interferometer to form a power-recycling cavity [5].
However, the power buildup can be degraded by the point
absorber effect as follows.
Without any thermoelastic deformation, the round trip
loss in the cavity is constant, and the arm power is lin-
early proportional to the input power with the slope set
by the round-trip loss of the cold cavity (gray lines in
Fig. 2). However, the thermoelastic deformation from the
point absorbers contributes to the optical loss by scatter-
ing light out of the fundamental cavity mode. Thus, an
increase in arm power leads to an increase in the optical
loss of the arm, which decreases the optical gain of the
power recycling cavity [1]. As a result, for sufficiently
high power levels the arm power saturates and becomes
largely independent of the input power.
Understanding the limitations of point absorbers on
5
the achievable arm power is important in planning future
detectors, for example, the next-generation gravitational-
wave detector Cosmic Explorer (CE) [24, 25]. CE will
achieve a factor of ten increase in sensitivity relative to
A+ by scaling up the A+ design to use 40 km long arm
cavities and increasing the arm power by a factor of two.
The key parameters of the coupled arm cavities of both
detectors are summarized in Table I.
To investigate the achievable arm power in CE and
the upcoming A+ observing runs, we conducted a sta-
tistical analysis of round-trip loss by calculating fields
under a thousand point absorber maps generated on the
arm cavity mirrors. For each map, the absorber loca-
tions are uniformly distributed; radii are governed by
a Rayleigh distribution, and number are governed by a
Poisson distribution with mean number density one per
60 cm
2
, characteristic of coated aLIGO mirrors. We in-
vestigate the cases of mean absorber radius
〈
w
〉
= 5
μ
m
(optimistic) and larger absorbers with
〈
w
〉
= 12
μ
m (pes-
simistic). The FFT-based simulation package Stationary
Interferometer Simulation (SIS) [26] is used to calculate
the field amplitudes in the cavity given these point ab-
sorber maps. The round-trip loss for each map is calcu-
lated at several arm powers from which the power recy-
cling gain is computed. The recycling gain is then con-
verted to the input power required to reach a given arm
power.
Fig. 2 shows the results for these two cases for both the
A+ and CE arm cavities. The medians are shown as solid
lines and the shadings correspond to the 16th and 84th
percentile. The arm power saturation is evident in the
〈
w
〉
= 12
μ
m case and, while it may be possible for A+
to reach its 750 kW design arm power, it is unlikely that
CE would ever reach its design of 1
.
5 MW with absorbers
of this size. On the other hand, our analysis suggests
that point absorbers with
〈
w
〉
= 5
μ
m pose little risk of
TABLE I. Parameters of Y-arm cavity of LIGO Livingston
Observatory and the proposed Cosmic Explorer.
Parameter
aLIGO
CE
Designed arm power
750 kW
1
.
5 MW
Optical gain of:
Power recycling cavity
40
76
Arm cavity
270
280
Round trip loss of:
Power recycling cavity
500 ppm
500 ppm
Cold arm (no absorber)
66 ppm
40 ppm
Cavity length
3995 m
40 km
Mirror
Aperture
34 cm
70 cm
Material
Fused Silica
Fused Silica
Temperature
290 K
290 K
Beam radius on:
Input mirror
5
.
2 cm
12 cm
End mirror
6
.
1 cm
12 cm
(a)
(b)
Input
Mirror
End Mirror
4000 m
0.34 m
PD 1
PD 4
(c)
(d)
PD 4
PD 1
FIG. 3. (a) Schematics of the Y arm cavity of LIGO Liv-
ingston Observatory with photodiodes (PD 1 and 4) marked.
(b) Intensity distribution of the field incident on the end mir-
ror baffle with a through hole at origin. (c-d) Experimental
measurement (with 5
×
error bar) of nomalized scatter power
landing on PD 1 and 4 versus FFT simulation with point ab-
sorber formulation incorporated. The error bar of simulation
is due to the 3 mm uncertainty of beam position. The data
is taken at 23 beam spot locations on the end mirror at four
different days. The relative scatter of clean optics without
any absorber is roughly an order of magnitude lower than the
plotted simulation curve (not shown).
damaging the A+ arm power, but it requires on average
30% more input power for CE to achieve the designed
goal. In both cases, the point absorbers limit the arm
power of CE more significantly than that of A+.
This statistical model is consistent with measured arm
powers in the LIGO Livingston observatory during Ob-
serving Run 3, which deviate from linear growth at high
power due to the point absorber effect. This data, shown
in the inset graph of Fig. 2, is fit to yield a 12
.
6
μ
m radius
absorber and 66 ppm round trip loss of the cold cavity.
The thermal absorptivity is taken as unity to break its de-
generacy with the radius of the point absorber (Eq. (11)).
The data sits in the predicted region of the pessimistic
case. These results are also consistent with measure-
ments of the total absorbed power of the point absorber
[27].
Scatter Magnitude -
Knowing the absorber radii, beam
position, and cavity parameters, we can calculate the
scattered fields through FFT simulation and compare the
theoretical modeling with measurements. Inside the arm
cavity, there are four silicon photodiodes (PDs) mounted
on each of the baffles installed in front of the test mass
optics to block and monitor scattered light. As the power
6
in the interferometer increases, the absorbers cause ther-
moelastic aberration of the HR surface of the test mass,
which in turn results in an increased scattering. PD 1 and
4 facing the cavity sample the Airy patterns of scattered
light, as shown in Fig. 3(a-b).
There were roughly a dozen point absorbers scattered
around the surface of the end mirror, including one dom-
inant absorber with the largest size near the center of
the mirror. After the Observing Run 3, we moved the
beam spot at 23 locations on the end mirror to change
the intensity incident on the absorbers while fixing the
beam spot on the input mirror. Simulations of each of
these 23 alignments reveals that this large and centrally
located absorber dominates the optical scatter (Fig. 2).
The FFT results are shown in Fig. 3(c-d) for each of the
23 spot locations. We moved the beam to the same lo-
cation repeatedly at indices 12, 17, 18, and 23. It is seen
that the measurements at these indices are equal, indicat-
ing that our measurements are reproducible over a week.
The simulation is capable of predicting the magnitude
and variation of the low-angle scatter, even though the
field amplitude shows a great amount of structure along
the radial distance from the beam center. The consis-
tency between data and simulation lends further credi-
bility to our modeling and improves our understanding of
the point absorber effect. Without the scattering due to
point absorbers, the simulated relative scatter magnitude
is roughly a factor of ten lower, and the simulated varia-
tions show little coherence with the PD measurements.
Conclusion -
In summary, we carried out an analyt-
ical approach to the point absorber problem in a high-
power resonant cavity. We propose an analytical solu-
tion to the thermoelastic deformation of the optics with
arbitrary point absorber heating function and bound-
ary conditions. Both temperature and displacement pro-
files are derived and incorporated in the state-of-the-art
FFT-based optical simulation. With a more advanced
and accurate understanding of the point absorber effect,
we make a statistical prediction of arm power in cur-
rent and future gravitational-wave detectors for different
mean radii of point absorbers. Our analysis of resonant
field power in the cavity suggests that point absorbers of
mean 5
μ
m radii will not prevent future gravitational-
wave detectors from achieving their design sensitivity.
Active research is being carried out to mitigate both the
size and number of point absorbers on future optics. Fi-
nally, our formulation shows a strong coherence with data
when compared with in-situ measurements of scattered
light, thus confirming our model.
Future analyses on the distortion of phase and mode-
shape of the fields from point absorbers are needed to es-
timate the degradation on the Michelson contrast, which
impacts the signal-to-noise ratio and thus the sensitivity
of the gravitational wave detectors.
Acknowledgements -
The author acknowledges the
support of MathWorks Science Fellowship and Sloan
Foundation, and thanks The MathWorks Inc. for its gen-
erous computing support. aLIGO was constructed by the
California Institute of Technology and Massachusetts In-
stitute of Technology with funding from the NSF and op-
erates under Cooperative Agreement No. PHY-1764464.
aLIGO was built under Award No. PHY-0823459.
∗
wenxuanj@mit.edu
[1] A. F. Brooks
et al.
, Point absorbers in advanced ligo,
Appl. Opt.
60
, 4047 (2021).
[2] C. J. Hood, H. J. Kimble, and J. Ye, Characterization of
high-finesse mirrors: Loss, phase shifts, and mode struc-
ture in an optical cavity, Phys. Rev. A
64
, 033804 (2001).
[3] H. Liu, B. D. Elwood, M. Evans, and J. Thaler, Searching
for axion dark matter with birefringent cavities, Phys.
Rev. D
100
, 023548 (2019).
[4] J. Majer, J. Chow, J. Gambetta, J. Koch, B. Johnson,
J. Schreier, L. Frunzio, D. Schuster, A. Houck, A. Wall-
raff, A. Blais, M. Devoret, S. Girvin, and R. Schoelkopf,
Coupling superconducting qubits via a cavity bus, Na-
ture
449
, 443 (2007).
[5] R. X. Adhikari, Gravitational radiation detection with
laser interferometry, Rev. Mod. Phys.
86
, 121 (2014).
[6] F. Acernese
et al.
, Advanced virgo: a second-generation
interferometric gravitational wave detector, Classical and
Quantum Gravity
32
, 024001 (2014).
[7] K. Somiya, Detector configuration of KAGRA–the
japanese cryogenic gravitational-wave detector, Classical
and Quantum Gravity
29
, 124007 (2012).
[8] A. Buikema
et al.
, Sensitivity and performance of the
advanced ligo detectors in the third observing run, Phys.
Rev. D
102
, 062003 (2020).
[9] The LIGO Scientific Collaboration, Advanced LIGO,
Classical and Quantum Gravity
32
, 074001 (2015).
[10] M. Tse
et al.
, Quantum-enhanced advanced ligo detectors
in the era of gravitational-wave astronomy, Phys. Rev.
Lett.
123
, 231107 (2019).
[11] C. Zhao, J. Degallaix, L. Ju, Y. Fan, D. G. Blair, B. J. J.
Slagmolen, M. B. Gray, C. M. M. Lowry, D. E. McClel-
land, D. J. Hosken, D. Mudge, A. Brooks, J. Munch, P. J.
Veitch, M. A. Barton, and G. Billingsley, Compensation
of strong thermal lensing in high-optical-power cavities,
Phys. Rev. Lett.
96
, 231101 (2006).
[12] C. Blair
et al.
(LSC Instrument Authors), First demon-
stration of electrostatic damping of parametric instability
at advanced ligo, Phys. Rev. Lett.
118
, 151102 (2017).
[13] M. Evans
et al.
, Observation of parametric instability in
advanced ligo, Phys. Rev. Lett.
114
, 161102 (2015).
[14] J. A. Sidles and D. Sigg, Optical torques in suspended
Fabry Perot interferometers, Physics Letters A
354
, 167
(2006).
[15] L. Barsotti, M. Evans, and P. Fritschel, Alignment sens-
ing and control in advanced LIGO, Classical and Quan-
tum Gravity
27
, 084026 (2010).
[16] W. Winkler, K. Danzmann, A. R ̈udiger, and R. Schilling,
Heating by optical absorption and the performance of
interferometric gravitational-wave detectors, Phys. Rev.
A
44
, 7022 (1991).
7
[17] E. Hall, The effect of a point absorber in an arm cavity,
LIGO Document, LIGO-T1900038 (2019).
[18] P. Hello and J.-Y. Vinet, Analytical models of thermal
aberrations in massive mirrors heated by high power laser
beams, Journal de Physique
51
, 1267 (1990).
[19] P. P. Lu, A. L. Bullington, P. Beyersdorf, S. Traeger,
J. Mansell, R. Beausoleil, E. K. Gustafson, R. L. Byer,
and M. M. Fejer, Wavefront distortion of the reflected
and diffracted beams produced by the thermoelastic de-
formation of a diffraction grating heated by a gaussian
laser beam, J. Opt. Soc. Am. A
24
, 659 (2007).
[20] K. F. Riley, M. P. Hobson, and S. J. Bence,
Mathematical
Methods for Physics and Engineering: A Comprehensive
Guide
, 3rd ed. (Cambridge University Press, 2006) Chap.
13.3.
[21] W. R. Mann and F. Wolf, Heat transfer between solids
and gasses under nonlinear boundary conditions, Quar-
terly of Applied Mathematics
9
, 163 (1951).
[22] W. R. Mann and J. H. Roberts, On a certain nonlinear
integral equation of the Volterra type., Pacific Journal of
Mathematics
1
, 431 (1951).
[23] P. Hello and J.-Y. Vinet, Analytical models of transient
thermoelastic deformations of mirrors heated by high
power cw laser beams, Journal De Physique
51
, 2243
(1990).
[24] M. Evans
et al.
,
Cosmic Explorer: Science, Observato-
ries, Community
, Tech. Rep. CE–P2100003 (Cosmic Ex-
plorer, 2021).
[25] E. D. Hall, K. Kuns, J. R. Smith, Y. Bai, C. Wipf, S. Bis-
cans, R. X. Adhikari, K. Arai, S. Ballmer, L. Barsotti,
Y. Chen, M. Evans, P. Fritschel, J. Harms, B. Kamai,
J. G. Rollins, D. Shoemaker, B. J. J. Slagmolen, R. Weiss,
and H. Yamamoto, Gravitational-wave physics with Cos-
mic Explorer: Limits to low-frequency sensitivity, Phys.
Rev. D
103
, 122004 (2021), arXiv:2012.03608 [gr-qc].
[26] H. Yamamoto, SIS (Stationary Interferometer Simu-
lation) manual, LIGO Document, LIGO-T2000311-v2
(2020).
[27] A. Brooks
et al.
,
https://alog.ligo-la.caltech.edu/
aLOG/index.php?callRep=52400
(2020).