Supplemental Material for “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
(LSC Instrument List)
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
2
26
University of British Columbia, Vancouver, BC V6T 1Z4, Canada
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
In the Letter, we present an analytical method to calculate nonlinear thermal effects of absorbers from first prin-
ciples. The temperature profile at the highly reflective (HR) surface is solved from iterating the nonlinear integral
equation (Eq. (6)):
T
HR
(
r
) =
H
−
1
0
[
̃
T
(
k,z
=
h
2
)]
=
2
πK
∫
r
0
dr
′
r
′
(
I
(
r
′
)
−
g
(
T
HR
(
r
′
)))
1
r
K
(
r
′
2
r
2
)
+
2
πK
∫
∞
r
dr
′
(
I
(
r
′
)
−
g
(
T
HR
(
r
′
))
K
(
r
2
r
′
2
)
(S1)
Given the temperature distribution
T
(
r,z
) solved in previous section, we can find the displacement vector field
u
of
the mirror next. We follow Hello and Vinet’s formalism, but apply it to the semi-infinite solution [1]. The elasticity
equation to solve is the static Navier-Cauchy equation:
∇·
T
=
T
ij
;
j
= 0
(S2)
where
T
ij
is the stress tensor and semicolon stands for covariant derivative. The gravity is not included because we
are interested in the net thermal effect caused by point absorber. The stress tensor has thermoelastic stress-strain
relationship:
T
ij
=
−
λ
Θ
g
ij
−
2
μE
ij
+ (3
λ
+ 2
μ
)
αTg
ij
(S3)
where
E
ij
= (
u
i
;
j
+
u
j
;
i
)
/
2 is the strain tensor, Θ =Tr(
E
) is the expansion of the body, and
g
ij
is the 3-D metric.
Other constants are
μ
(first Lam ́e coefficient),
λ
(second Lam ́e coefficient), and
α
(thermal expansion coefficient). The
stress tensor (
T
ij
) should not be confused with temperature distribution
T
(
r,z
). Under a differential temperature
T
(
r,z
) =
T
abs
−
T
∞
, the mirror undergoes an additional expansion of
αT
in all directions. Assuming axial symmetry
and no rotation, we have
u
φ
= 0 and remove one component of Eq. (S2). The other remaining two coupled partial
differential equations are broken down in terms of our most interested variable
u
i
:
u
i
;
j
=
u
i,j
+ Γ
ijl
u
l
(S4)
where comma stands for partial derivative. The Christoffel symbols are Γ
φφr
= Γ
φrφ
=
1
r
, Γ
rφφ
=
−
1
r
, and all rest
are zero. The survived components of strain tensor are
E
rr
=
u
r
;
r
=
u
r,r
+ Γ
rrl
u
l
=
u
r,r
E
φφ
=
u
φ
;
φ
=
u
φ,φ
+ Γ
φφl
u
l
=
u
r
r
E
zz
=
u
z
;
z
=
u
z,z
+ Γ
zzl
u
l
=
u
z,z
E
rz
=
1
2
(
u
r
;
z
+
u
z
;
r
) =
1
2
(
u
r,z
+ Γ
rzl
u
l
+
u
z,r
+ Γ
zrk
u
k
) =
1
2
(
u
r,z
+
u
z,r
)
(S5a)
(S5b)
(S5c)
(S5d)
∗
wenxuanj@mit.edu
3
The new stress-strain relationship is
T
rr
=
λ
(Θ
−
3
αT
(
r,z
)) + 2
μ
(
E
rr
−
αT
(
r,z
)) =
λ
Θ + 2
μE
rr
−
(3
λ
+ 2
μ
)
αT
(
r,z
)
T
φφ
=
λ
(Θ
−
3
αT
(
r,z
)) + 2
μ
(
E
φφ
−
αT
(
r,z
)) =
λ
Θ + 2
μE
φφ
−
(3
λ
+ 2
μ
)
αT
(
r,z
)
T
zz
=
λ
(Θ
−
3
αT
(
r,z
)) + 2
μ
(
E
zz
−
αT
(
r,z
)) =
λ
Θ + 2
μE
zz
−
(3
λ
+ 2
μ
)
αT
(
r,z
)
T
rz
= 2
μE
rz
(S6)
All other components of the stress tensor are zero. In the steady-state condition with no external pressure (ignoring
gravitational contribution), the total force
F
acting on the volume element is zero:
F
=
−
∫
∂V
T
·
d
A
=
−
∫
V
∇·
T
dV
= 0
(S7)
So
∇·
T
=
T
ij
;
j
=
T
ij,j
+ Γ
ijl
T
lj
+ Γ
jjl
T
il
= 0
⇒
T
rj
;
j
=
∂T
rr
∂r
+
∂T
rz
∂z
+
T
rr
−
T
φφ
r
= 0
T
φj
;
j
=
∂T
rφ
∂r
+
∂T
zφ
∂z
+
3
T
rφ
r
= 0
,
trivial because all
T
iφ
= 0
T
zj
;
j
=
∂T
rz
∂r
+
∂T
zz
∂z
+
T
rz
r
= 0
(S8)
These two coupled partial differential equations can be represented in the variable
u
i
of interest.
(
λ
+ 2
μ
)
∂
2
r
u
r
+ (
λ
+
μ
)
∂
r
∂
z
u
z
+
μ∂
2
z
u
r
+
λ
+ 2
μ
r
∂
r
u
r
−
λ
+ 2
μ
r
2
u
r
=
α
(3
λ
+ 2
μ
)
∂
r
T
(
r,z
)
(
λ
+ 2
μ
)
∂
2
z
u
z
+ (
λ
+
μ
)
∂
r
∂
z
u
r
+
μ∂
2
r
u
z
+
λ
+
μ
r
∂
z
u
r
+
μ
r
∂
r
u
z
=
α
(3
λ
+ 2
μ
)
∂
z
T
(
r,z
)
(S9)
One solution of
u
i
(
r,z
) that satisfies Eq. (S9) is [1]
u
r
(
r,z
) =
α
(3
λ
+ 2
μ
)
2(
λ
+
μ
)
1
r
∫
r
0
dr
′
r
′
T
(
r
′
,z
) +
λ
+ 2
μ
2
μ
(3
λ
+ 2
μ
)
(
Ar
+
Brz
)
u
z
(
r,z
) =
α
(3
λ
+ 2
μ
)
2(
λ
+
μ
)
[
∫
z
h/
2
T
(
r,z
′
)
dz
′
−
∫
r
0
dr
′
r
′
∫
r
′
0
∂
z
T
(
r
′′
,h/
2)
r
′′
dr
′′
+
C
]
−
λ
μ
(3
λ
+ 2
μ
)
(
Az
+
Bz
2
2
)
−
λ
+ 2
μ
4
μ
(3
λ
+ 2
μ
)
Br
2
(S10)
where
A,B,C
are constants constrained by realistic boundary conditions.
T
rr
(
a,z
) = 0
and
u
z
(
a,h/
2) = 0
(S11)
Expand the first one:
T
rr
(
a,z
) = (
λ
+ 2
μ
)
u
r,r
+
λ
r
u
r
+
λu
z,z
−
α
(3
λ
+ 2
μ
)
T
(
a,z
)
=
−
P
abs
2
πK
μα
(3
λ
+ 2
μ
)
λ
+
μ
1
a
∫
∞
0
dke
k
(
z
−
h/
2)
e
−
k
2
w
2
/
8
1 + 4
σT
3
∞
/
(
Kk
)
J
1
(
ka
)
k
+
A
+
Bz
= 0
(S12)
These constants
A,B
are used to approximately compensate the stress
T
rr
variation at the edge along the thickness.
Since
A,B
are relatively small compared to the other terms, they are calculated with linearized
T
(
r,z
). We are most
interested in the HR surface deformation in axial direction.
h
(
r
) =
u
z
(
r,h/
2) =
α
(3
λ
+ 2
μ
)
2(
λ
+
μ
)
K
∫
r
0
dr
′
r
′
∫
r
′
0
dr
′′
r
′′
(
−
I
(
r
′′
) +
g
(
r
′′
))
−
λ
+ 2
μ
4
μ
(3
λ
+ 2
μ
)
Br
2
+
C
′
(S13)
4
where
C
′
is enforced by
h
(
a
) = 0. This concludes the solution of displacement function
h
(
r
) given the input temper-
ature
T
(
r
).
[1] 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).