Point Absorber Limits to Future Gravitational-Wave Detectors

Wenxuan Jia ,

Hiroaki Yamamoto ,

Kevin Kuns ,

Anamaria Effler ,

Matthew Evans,

Peter Fritschel,

R. Abbott,

C. Adams,

R. X. Adhikari,

A. Ananyeva,

S. Appert,

K. Arai,

J. S. Areeda,

Y. Asali,

S. M. Aston,

C. Austin,

A. M. Baer,

M. Ball,

S. W. Ballmer,

S. Banagiri,

D. Barker,

L. Barsotti,

J. Bartlett,

B. K. Berger,

J. Betzwieser,

D. Bhattacharjee,

G. Billingsley,

S. Biscans,

1,2

C. D. Blair,

R. M. Blair,

N. Bode,

14,15

P. Booker,

14,15

R. Bork,

A. Bramley,

A. F. Brooks,

D. D. Brown,

A. Buikema,

C. Cahillane,

K. C. Cannon,

X. Chen,

A. A. Ciobanu,

F. Clara,

C. M. Compton,

S. J. Cooper,

K. R. Corley,

S. T. Countryman,

P. B. Covas,

D. C. Coyne,

L. E. H. Datrier,

D. Davis,

C. Di Fronzo,

K. L. Dooley,

22,23

J. C. Driggers,

P. Dupej,

S. E. Dwyer,

T. Etzel,

T. M. Evans,

J. Feicht,

A. Fernandez-Galiana,

V. V. Frolov,

P. Fulda,

M. Fyffe,

J. A. Giaime,

6,3

K. D. Giardina,

P. Godwin,

E. Goetz,

6,13,26

S. Gras,

C. Gray,

R. Gray,

A. C. Green,

E. K. Gustafson,

R. Gustafson,

E. D. Hall,

J. Hanks,

J. Hanson,

T. Hardwick,

R. K. Hasskew,

M. C. Heintze,

A. F. Helmling-Cornell,

N. A. Holland,

J. D. Jones,

S. Kandhasamy,

S. Karki,

M. Kasprzack,

K. Kawabe,

N. Kijbunchoo,

P. J. King,

J. S. Kissel,

Rahul Kumar,

M. Landry,

B. B. Lane,

B. Lantz,

M. Laxen,

Y. K. Lecoeuche,

J. Leviton,

J. Liu,

14,15

M. Lormand,

A. P. Lundgren,

R. Macas,

M. MacInnis,

D. M. Macleod,

G. L. Mansell,

11,1

S. Márka,

Z. Márka,

D. V. Martynov,

K. Mason,

T. J. Massinger,

F. Matichard,

2,1

N. Mavalvala,

R. McCarthy,

D. E. McClelland,

S. McCormick,

L. McCuller,

J. McIver,

2,26

T. McRae,

G. Mendell,

K. Merfeld,

E. L. Merilh,

F. Meylahn,

14,15

T. Mistry,

R. Mittleman,

G. Moreno,

C. M. Mow-Lowry,

S. Mozzon,

A. Mullavey,

T. J. N. Nelson,

P. Nguyen,

L. K. Nuttall,

J. Oberling,

Richard J. Oram,

C. Osthelder,

D. J. Ottaway,

H. Overmier,

J. R. Palamos,

W. Parker,

3,32

E. Payne,

A. Pele,

R. Penhorwood,

C. J. Perez,

M. Pirello,

H. Radkins,

K. E. Ramirez,

J. W. Richardson,

K. Riles,

N. A. Robertson,

2,21

J. G. Rollins,

C. L. Romel,

J. H. Romie,

M. P. Ross,

K. Ryan,

T. Sadecki,

E. J. Sanchez,

L. E. Sanchez,

T. R. Saravanan,

R. L. Savage,

D. Schaetzl,

R. Schnabel,

R. M. S. Schofield,

E. Schwartz,

D. Sellers,

T. Shaffer,

D. Sigg,

B. J. J. Slagmolen,

J. R. Smith,

S. Soni,

B. Sorazu,

A. P. Spencer,

K. A. Strain,

L. Sun,

M. J. Szczepa

ń

czyk,

M. Thomas,

P. Thomas,

K. A. Thorne,

K. Toland,

C. I. Torrie,

G. Traylor,

M. Tse,

A. L. Urban,

G. Vajente,

G. Valdes,

D. C. Vander-Hyde,

P. J. Veitch,

K. Venkateswara,

G. Venugopalan,

A. D. Viets,

T. Vo,

C. Vorvick,

M. Wade,

R. L. Ward,

J. Warner,

B. Weaver,

R. Weiss,

C. Whittle,

B. Willke,

15,14

C. C. Wipf,

L. Xiao,

Hang Yu,

Haocun Yu,

L. Zhang,

M. E. Zucker,

1,2

and J. Zweizig

LIGO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

LIGO, California Institute of Technology, Pasadena, California 91125, USA

LIGO Livingston Observatory, Livingston, Louisiana 70754, USA

California State University Fullerton, Fullerton, California 92831, USA

Columbia University, New York, New York 10027, USA

Louisiana State University, Baton Rouge, Louisiana 70803, USA

Christopher Newport University, Newport News, Virginia 23606, USA

University of Oregon, Eugene, Oregon 97403, USA

Syracuse University, Syracuse, New York 13244, USA

University of Minnesota, Minneapolis, Minnesota 55455, USA

LIGO Hanford Observatory, Richland, Washington 99352, USA

Stanford University, Stanford, California 94305, USA

Missouri University of Science and Technology, Rolla, Missouri 65409, USA

Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany

Leibniz Universität Hannover, D-30167 Hannover, Germany

OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia

RESCEU, University of Tokyo, Tokyo 113-0033, Japan

OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia

University of Birmingham, Birmingham B15 2TT, United Kingdom

Universitat de les Illes Balears, IAC3

—

IEEC, E-07122 Palma de Mallorca, Spain

SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom

Cardiff University, Cardiff CF24 3AA, United Kingdom

The University of Mississippi, University, Mississippi 38677, USA

University of Florida, Gainesville, Florida 32611, USA

The Pennsylvania State University, University Park, Pennsylvania 16802, USA

University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

University of Michigan, Ann Arbor, Michigan 48109, USA

PHYSICAL REVIEW LETTERS

127,

241102 (2021)

0031-9007

127(24)

241102(7)

241102-1

OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia

Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India

University of Portsmouth, Portsmouth PO1 3FX, United Kingdom

The University of Sheffield, Sheffield S10 2TN, United Kingdom

Southern University and A&M College, Baton Rouge, Louisiana 70813, USA

OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia

The University of Texas Rio Grande Valley, Brownsville, Texas 78520, USA

University of Washington, Seattle, Washington 98195, USA

Universität Hamburg, D-22761 Hamburg, Germany

Concordia University Wisconsin, 2800 North Lake Shore Drive, Mequon, Wisconsin 53097, USA

Kenyon College, Gambier, Ohio 43022, USA

(Received 22 September 2021; accepted 27 October 2021; published 7 December 2021)

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.

DOI:

10.1103/PhysRevLett.127.241102

Introduction.

—

Awide variety of precision optical experi-

ments rely on resonant optical cavities to enable precise

measurements of space, time, and fundamental physics.

These experiments often require high optical intensity

incidents on the mirrors of the cavity to boost the signal-

to-noise ratio. However, unintended defects may be depos-

ited 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 containing

high circulating power. The point absorber becomes a

limiting factor in various precision measurement experi-

ments 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 detectors

–

.Itis

thus necessary to develop a quantitativeunderstanding 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 the Advanced Laser

Interferometer Gravitational-Wave Observatory (aLIGO)

serves as a good example of the point absorber effect

[8]

aLIGO is a dual-recycled Fabry-Perot Michelson interfero-

meter 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 increasing the arm

power or by manipulating the quantum 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 instabilities

[12

–

15]

. The

arm power during the third observing run O3 was limited to

one-third of the designed value of 750 kW, mainly due to

point absorbers on the mirror

[8,9]

that scatter light out of

the fundamental cavity mode.

Point absorbers were known to exist since the first

observing 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 Letter, we provide a more

general approach from first principles. The traditional

formalism is extended to include arbitrary heating functions

with any nonlinear boundary conditions, such as the Stefan-

Boltzmann law. With the correction from the nonlinear

boundary condition, we can make more accurate statistical

estimations of the arm power for the next planned upgrade

of aLIGO (known as

“

”

) 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 deformation

of the mirror is derived using a thermoelasticity equation.

Next, this deformation is incorporated in a FFT-based

simulation to obtain the field in the arm cavity, which is

used to calculate its round-trip loss and achievable power.

In addition, we simulate the low-angle scattered light

PHYSICAL REVIEW LETTERS

127,

241102 (2021)

241102-2

intensity and compare this with

in situ

measurements. 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 absorbing

laser power, which thermally distorts the mirror surface 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 derived first.

Consider a cylindrical optic with radius

and thickness

. Choose cylindrical coordinates at the center of the mirror

with the

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

sub

r; z

Þ¼

∇

2

r; z

Þ¼

0

;

where

sub

r; z

Þ¼

r; z

Þþ

∞

is the temperature of the

optic substrate, and

is the temperature departure from the

constant ambient temperature

∞

. The boundary condi-

tions include the intensity of a heating source

on the

HR surface due to the point absorber,

−

∂

2

−

Þþ

Þj

2

;

where

is the thermal conductivity,

is the thermal

flux of blackbody radiation

Þ¼

εσ

½ð

∞

r; z

ÞÞ

4

−

4

∞

;

is the thermal emissivity or absorptivity (assumed to

be unity throughout this Letter), and

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

0

of the Laplace equation in cylindrical

coordinates with respect to

[20]

, we get

−

2

∂

2

̃

k; z

Þ¼

0

;

where the Hankel transform

0

is defined as

̃

k; z

Þ¼

0

r; z

Þg

≡

∞

0

drrJ

0

r; z

Þð

and

0

is the Bessel function of the first kind. The

angular dependence is dropped due to cylindrical

symmetry. The solution of Eq.

(4)

is the sum of growth

modes

and decay modes

−

, where the latter

vanishes by the semi-infinite boundary conditions

→

∞

Þ¼

r; z

→

−∞

Þ¼

0

Let

be the heating function from a point absorber;

here we use a Gaussian profile

Þ¼

−

2

2

with

absorber radius

and irradiation intensity

at the

absorber center. Let

Þ¼

r; z

2

be the tem-

perature profile at the HR surface. The Hankel transform of

the boundary condition at the HR surface

(2)

gives

̃

k; z

Þ¼

−

2

0

−

Þþ

Þg

Therefore,

is found by taking the inverse transform

̃

k; z

2

Þ¼

2

0

−

(

)

2

2

2

∞

−

(

)

2

2

;

where

is the complete elliptic integral of the first kind.

Equation

(7)

is a nonlinear integral equation with no

closed-form solution, but an approximate solution can be

found by either linearizing the boundary function

using successive approximation.

The linearized boundary solution has been given in

[19]

The heat from a small point absorber is primarily dissipated

by conduction. This estimate breaks down for a large

absorber of radius

Þð

2

1

4

which is

∼

100

min

an aLIGO arm cavity

[17]

. The correction of nonlinear

radiation will matter if the radiative contribution becomes

024681012141617

Radius [cm]

-4

-3

-2

-1

HR Displacement [nm]

0.5

100

500

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

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.

(12)

] are also shown.

PHYSICAL REVIEW LETTERS

127,

241102 (2021)

241102-3

significant. This motivates us to find a solution to the

general boundary condition.

The nonlinear integral equation

(7)

can be solved by

successive approximation with feedback. We start the

zeroth iteration with an initial guess

0

. The real

solution is denoted as

, and the zeroth error function

0

Þ¼

0

−

. Plugging this into Eq.

(7)

and

keeping the first nontrivial order of the error,

0

≡

0

Þ ¼

Þþ

4

εσ

∞

0

3

0

2

Assuming the error has weak variation over radius

0

≈

0

,wehave

0

Þ ¼

−

0

0

Þ¼

−

0

−

0

;

where

0

Þ¼

8

εσ

0

∞

0

3

2

2

∞

0

3

2

2

We can then iterate the temperature profile

1

Þþ

1

until we reach convergence at

Þ¼

and

1

[21,22]

The displacement vector field of the optic can be found

given the temperature solution

r; z

. We follow Hello and

Vinet

’

s formalism but apply it to our solution

[23]

(see the

Supplemental Material for detailed derivations

[24]

Figure

shows the resulting displacement of the HR

surface. For small point absorbers, the differential temper-

ature is relatively low, and the radiative correction is

negligible; the linearized boundary solution is accurate.

However, the correction becomes significant for an

absorber with radius larger than

100

m, up to a factor

of 3 in the

500

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 nanometers in height and a few

centimeters in size.

An analytical fit to displacement of the linearized

solution is given by

≈

−

2

2

Þþ

2

;

where

is the first Lam ́

e coefficient,

is the second

Lam ́

e coefficient, and

is the thermal expansion coefficient.

Note that Eq.

(12)

breaks down at high absorbed power, as

shown in Fig.

. 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.

—

Advanced gravitational-wave detectors are Michelson

interferometers using Fabry-Perot cavities as arms to

increase optical power and thus the signal produced by

gravitational-wave strain. The arm power is further

increased 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 linearly

proportional to the input power with the slope set by the

round-trip loss of the cold cavity (gray lines in Fig.

However, the thermoelastic deformation from the point

absorbers contributes to the optical loss by scattering 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 the

achievable arm power in realistic situations with multiple

absorbers is important in planning future detectors

—

for

example, the next-generation gravitational-wave detector

Cosmic Explorer (CE)

[25,26]

. CE will achieve a factor of

10 increase in sensitivity relative to A

by scaling up the

design to use 40-km-long arm cavities and increasing

the arm power by a factor of 2. The key parameters of the

coupled arm cavities of both detectors are summarized in

Table

To investigate the achievable arm power in CE and

the upcoming A

observing runs, we conducted a

Monte Carlo statistical analysis of round-trip loss by

calculating fields under 1000 point absorber maps gener-

ated on the arm cavity mirrors. For each map, the absorber

locations are uniformly distributed; radii are governed by a

Rayleigh distribution, and numbers are governed by a

Poisson distribution with mean number density one per

60

2

, characteristic of coated aLIGO mirrors. We inves-

tigate the cases of mean absorber radius

i¼

5

(optimistic) and larger absorbers with

i¼

12

m (pes-

simistic). The FFT-based simulation package Stationary

Interferometer Simulation

[27]

is used to calculate the field

amplitudes in the cavity given these point absorber maps.

The round-trip loss for each map is calculated at several

arm powers from which the power recycling gain is

computed. The recycling gain is then converted to the

input power required to reach a given arm power.

Figure

shows the results for these two cases for both

the A

and CE arm cavities. The medians are shown as

PHYSICAL REVIEW LETTERS

127,

241102 (2021)

241102-4

solid lines and the shadings correspond to the 16th and 84th

percentile. The arm power saturation is evident in the

i¼

12

m case and, while it may be possible for A

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

i¼

5

m pose little risk of damaging the

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

observing run O3, which deviate from linear growth at high

power due to the point absorber effect. These data, shown in

the inset graph of Fig.

,arefittoyielda

12

6

mradius

absorber and 66 ppm round-trip loss of the cold cavity. The

thermal absorptivity is taken as unity to break its degeneracy

with the radius of the point absorber [Eq.

(12)

]. The data sit

in the predicted region of the pessimistic case. These results

are also consistent with measurements of the total absorbed

power of the point absorber

[28]

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 in the

interferometer increases, the absorbers cause thermoelastic

aberration of the HR surface of the test mass, which in turn

results in an increased scattering. PD1 and PD4 facing the

cavity sample the Airy patterns of scattered light, as shown

in Figs.

3(a)

and

3(b)

There were roughly a dozen point absorbers scattered

around the surface of the end mirror, including one

dominant absorber with the largest size near the center

of the mirror. After the observing run O3, 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 reveal that this large and centrally located

absorber dominates the optical scatter (Fig.

). The FFT

results are shown in Figs.

3(c)

and

3(d)

for each of the 23

spot locations. We moved the beam to the same location

repeatedly at indices 12, 17, 18, and 23. It is seen that

the measurements at these indices are equal, indicating

that our measurements are reproducible over a week.

TABLE I. Parameters of the

arm cavity of LIGO Livingston

Observatory measured in observing run O3b and the proposed

Cosmic Explorer. Note that, with the exception of the optical gain

and round-trip loss, the A

design parameters are the same as

those of aLIGO.

Parameter

aLIGO

Designed arm power

750 kW

1.5 MW

Optical gain of

Power recycling cavity

Arm cavity

270

280

Round-trip loss of

Power recycling cavity

500 ppm

Cold arm (no absorber)

66 ppm

40 ppm

Arm cavity length

3995 m

40 km

Mirror

Aperture

34 cm

70 cm

Material

Fused silica

Temperature

290 K

Beam radius on

Input mirror

5.2 cm

12 cm

End mirror

6.1 cm

12 cm

FIG. 2. Circulating power in the arm cavity versus input power for two different detectors and mean radii of point absorbers (optimistic

5

m and pessimistic

12

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

), 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 enlarged graph, the data points collected from LIGO Livingston Observatory throughout observing run O3b are fit to obtain the

radii of point absorbers. It is statistically more confident for A

to achieve the designed power with

i¼

5

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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 consistency

between data and simulation lends further credibility to

our modeling and improves our understanding of the point

absorber effect. Without the scattering due to point absorb-

ers, the simulated relative scatter magnitude is roughly a

factor of 10 lower, and the simulated variations show little

coherence with the PD measurements.

Conclusion.

—

In summary, we carried out an analytical

approach to the point absorber problem in a high-power

resonant cavity. We propose an analytical solution to the

thermoelastic deformation of the optics with arbitrary point

absorber heating function and boundary conditions. Both

temperature and displacement profiles are derived and

incorporated in the state-of-the-art FFT-based optical sim-

ulation. With a more advanced and accurate understanding

of the point absorber effect, we make a statistical prediction

of arm power in current 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

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. Finally, 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

estimate the degradation on the Michelson contrast, which

impacts the signal-to-noise ratio and thus the sensitivity of

the gravitational-wave detectors.

The author acknowledges the support of MathWorks

Science Fellowship and Sloan Foundation, and thanks

The MathWorks, Inc. for its generous computing support.

Advanced LIGO was constructed by the California Institute

of Technology and Massachusetts Institute of Technology

with funding from the NSF and operates under Cooperative

Agreement No. PHY-1764464. Advanced LIGO was built

under Grant No. PHY-0823459.

wenxuanj@mit.edu

[1] A. F. Brooks

et al.

, Point absorbers in Advanced LIGO,

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