Thermal noise calculations for cryogenic optics
R. Nawrodt, I. Martin, A. Cumming, W. Cunningham,
S. Rowan, J. Hough
ET-WP2 Workshop, La Sapienza - University of Rome
26th/27th February 2009
Overview

sources of thermal noise

bulk material + coatings

up scaling to necessary size

suspensions

problems, open questions
Motivation
h [1/ Hz]
10
10
-21
[Punturo, ET talk at the LSC meeting, Amsterdam 2008]
10
-22
-23
thermal noise limited
10
10
-24
-25
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
Thermal Noise

Brownian Noise




Thermoelastic Noise




Bulk
Coating
Suspension
Bulk
Coating
Suspension
Photothermal Noise

Coating  Bulk
Modelling Thermal Noise

Reference geometry = advanced detector optics
w0 = 60 mm
D = 340 mm
L = 200 mm
D/2w0 = 2.8

aim: cross check with existing calculations
Bulk Material (1)

demands:




low thermal noise
suited for coatings (surface treatment, polish...)
big sizes available (materials: Fused Silica, CaF2, Si,
Sapphire)
thermal noise contribution:


Brownian thermal noise
thermoelastic noise
Bulk Material (2)

Brownian Thermal Noise


reminder:
 origin - thermal energy
 fluctuation-dissipation-theorem
gives spectral distribution
idea:
low loss material will concentrate Brownian noise around
the internal resonances (which are above the detection
band)
Bulk Material (3)

Brownian Thermal Noise

infinite half space
S
ITM
X
2k B T 1   2
(f , T )  3 / 2 
 substrate(f , T)
 f wY
[Liu, Thorne 2000]
 … Poisson ratio, Y … Young’s modulus, T … temperature, f … frequency,
substrate ... mechanical loss of the substrate, w … beam radius (1/e 2 definition)

finite mirror material (analytical calculation)
SFTM
(f , T)  C2FTM  SITM
X
X (f , T)
[Liu, Thorne 2000, Bondu, Hello, Vinet 1998]
correction term nearly temperature independent and < 1
Thus, the infinite half space always gives an upper limit.
Bulk Material (4)

Brownian Thermal Noise (cont’ finite mirror)
C 2FTM
SFTM
(f , T )
 XITM
SX (f , T )
1  2
U0 
aY
2k B T 1   2
S (f , T )  3 / 2 
 substrate(f , T)
 f wY
4k B T
SFTM
(
f
,
T
)

 substrate(f , T)  U 0  U 
X
f
exp(  2m r02 / 2a 2 )
Um

 m J 0 ( m ) 2
m 1


[Liu, Thorne 2000]
ITM
X
1  Q 2m  4k m HQ m
Um 
(1  Q m ) 2  4k 2m HQ m
a2
2 4 2
2
2
U 

H
p

12

H

p
s

72
(
1


)
s
0
0
6H 3Y

J0 … Bessel function of order zero
m … m‘th zero of the zero order Bessel function J1(m)
Qm  exp  2k m H
k m  m / a
exp(  2m r02 / 4a 2 )
s
 2m J 0 ( m )
m 1

exp( k 2m r02 / 4)
pm 
a 2 J 02 ( m )
Bulk Material (5)

Brownian Thermal Noise

finite mirror material (FEA, ANSYS)
direct use of Levin‘s approach
S x (f , T ) 
4k BT U max
 2  substrate
f
F0
[Levin 1998]
ANSYS
Bulk Material (6a)

Brownian Thermal Noise

Young’s modulus
Material
Y [GPa]
Material
Y [GPa]
Fused Silica
72
Si(100)
130
CaF2
135
Si(110)
169
Sapphire
350
Si(111)
188
Diamond
>1000
Bulk Material (6b)
Brownian Thermal Noise


10
Q-factor
10
10
10
impurity effects
(e.g. doping, oxygen)
Mechanical loss
10
9
8
7
[McGuigan 1978]
10
6
1
3
10
30
temperature [K]
100
300
Material properties collected and summarized for ET homepage.
Bulk Material (7)

Thermoelastic Noise

reminder:
 origin – entropy production during heat flux between
compressed and expanded regions causes
thermoelastic loss


a given temperature fluctuation T is converted into a
displacement fluctuation x by means of the thermal
expansion coefficient 
dependent on material thermal properties
Bulk Material (8)
Thermoelastic Noise (Material properties)
thermal expansion coefficient [1/K]
heat capacity [J/kg K]
700
600
500
400
300
200
100
0
0
50
100
150
200
temperature [K]
250
300
x 10
-6
1400
2.5
2
1.5
1
0.5
0
-0.5
0
50
100
150
200
temperature [K]
250
300
1200
1000
800
[Hull 1999]
3
800
thermal conductivity [W/m K]

600
400
200
0
0
50
100
150
200
temperature [K]
250
300
Collection of extracted data as txt-files and Origin-files for ET homepage.
Bulk Material (9)

Thermoelastic Noise

finite/infinite test mass
4k BT 2 2 1   
(f , T ) 
5 / 2  2 C 2 f 2 w 3
2
ITM
TE
S
FTM
ITM
STE
(f , T)  C'2FTM  STE
(f , T)

[Liu, Thorne 2000]
problem:
Most of the calculations use the adiabatic assumption.
Bulk Material (10)

Thermoelastic Noise
adiabatic limit:
a temperature
fluctuation stays in time
1/f within the beam
diameter 2w
2w
laser
w = 2 r0
valid if:
substrate

C 
 Cr02
(adiabatic limit)
Bulk Material (11)
Adiabatic limit

10
adiabatic limit [Hz]
10
10
10
10
10
4
Si(111)
Fused Silica
Sapphire
2
crystalline materials
violate adiabatic
assumption
0
-2
-4
amorphous materials
still fulfil assumption
-6
0
T<80 K:
50
200
150
100
temperature [K]
250
300
Bulk Material (12)

Thermoelastic Noise

beyond the adiabatic limit
2
8
2 k BT r0
2
STE (f , T) 
 1  
 J()

2

3 u 2 / 2
2
ue
du
dv
3 0  (u 2  v 2 )[( u 2  v 2 ) 2  2 ]
J()
J() 

10
10


C
0
10
-10
10
-5
10
[Rowan et al. 2000, Aspen Meeting]
[Cerdonio et al. 2001]
0
10

5
10
Bulk Material (13)

Bulk Material Comparison (300 K)
thermal noise [m/Hz]
10
10
10
10
-16
Fused Silica
Si(111)
Sapphire
-18
SiO2 = 4×10-10
Si = 3×10-9
Sapphire = 3×10-9
-20
TE
-22
Brownian
10
-24
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
Bulk Material (14)

thermal noise [m/Hz]
10
10
Bulk Material Si(111) (20 K)
-19
-20
Si = 5×10-10
10
10
10
-21
-22
TE
-23
Brownian
10
-24
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
Bulk Material (15)

thermal noise [m/Hz]
10
10
10
10
10
Bulk Material Comparison (20 K)
-16
Fused Silica
Si(111)
Sapphire
-18
SiO2 = 1×10-3
Si = 5×10-10
Sapphire = 3×10-9
-20
-22
-24
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
Bulk Material (16)

thermal noise [m/Hz]
10
10
10
10
Bulk Material - e.g. Si(111), f=100 Hz, real measured values
-18
properties extracted from:
McGuigan 1978
Touloukian 1972
Hull 1999
-20
-22
-24
Brownian
Thermoelastic
total
10
-26
0
50
100
150
200
temperature [K]
250
300
Coating Material (1)

Demands:




low thermal noise
low optical absorption (thermal load)
conventional stack – high difference in refraction
index
large coatings needed with properties at the limit
what is currently available
Coating Material (2)

Brownian Thermal Noise



Thermoelastic Noise



infinite/finite
important parameters mostly unknown for coating
materials (Y, , …)
infinite/finite
important thermal parameters unknown
Photothermal Noise

absorption measurement needed for all new coatings
Coating Material (3)

Brownian Thermal Noise (infinite)
2k B T 1   2
d
S x (f , T )  2
 f w 2 Y YY ' (1  '2 )(1   2 )
[Harry et al. 2002]

 Y '2 (1  ) 2 (1  2) 2 ||  YY ' ' (1  )(1  ' )(1  2)(||    )  Y 2 (1  ' ) 2 (1  2' ) 
, '  0
S x (f , T ) 
2k B T d  Y '
Y 


 

||
2f w 2 Y  Y
Y' 
If two different mechanical loss values exist then the Brownian thermal
noise of a coating is dependent on the ratio of Young’s moduli at the
interface. Lowest loss occurs if Y=Y’.

Coating Material (4)

Brownian Thermal Noise (finite)
ANSYS as an alternative
[Cunningham, Torrie]
multilayer stack is treated as a two-layer-system
S x (f , T ) 

4k B T 1
max
max
 2  U layer
1   layer1  U layer2   layer2
f
F0
analytical approaches
[Somiya, Yamamoto, LIGO-P080121-00-Z]

Coating Material (5)

Brownian Thermal Noise
Sx (f , T) 
2k BT d  Y'
Y 


 

||
2
2
 f w Y Y
Y' 
unknown parameters: Y’,  and ||
often: approximations 
= ||
measurements: see next two talks
Coating Material (6)

Thermoelastic Noise (multilayer stack)
~2
8k BT 2 L SC F
2
1  s   g()
STE (f , T)  2
2
 f w CS
[Braginsky, Fejer et al. 2004]
 C    1  

E  
S

 
 (1  2S )  
 1


2

C
1


1


E
 S F 

S
S   AVG

~2

sinh i F
1
g()  Im 
 iF cosh iF  R sinh
2


iF 
L2
 FC2F
F 
and R 

SCS2
The adiabatic limit for amorphous materials (silica, tantala) is low
even at cryogenic temperatures  no limit/correction.
Coating Material (7)

Thermoelastic Noise (material properties)


most parameters unknown for coatings
some measurements available (e.g. densitiy, absorption)
[Morgado, 1st ET Meeting, Cascina 2008]

coatings often approximated by bulk material values
Measurements of thermal and mechanical properties
needed.
Coating Material (8)
10
Photothermal Noise
SPT (f , T) 

2 2
2 0 WabsC



1


K ( )
2
2



1
u 2e  u / 2
K ()   du  dv 2
 0  (u  v 2 )( u 2  v 2  i)
K()

10
0
10
2
-10
10
-5
10
0
10

[Cerdonio et al. 2001]

absorption of coatings governs photothermal noise
Ta2O5:TiO2 absorption, ~ 1 ppm
[Harry 2007]
5
10
Coating Material (9)

example (15 double layers Ta2O5:TiO2 / SiO2)
Si(111)
15 double layers
advanced geometry
20 K sample temperature
l = 1064 nm
Ta2O5 = 9×10-4
SiO2 = 6×10-4
Si = 5×10-10
Coating Material (10)
thermal noise [m/Hz]
10
10
10
10
10
10
-19
bulk Brownian
bulk TE
coating Brownian
coating TE
-20
coating dominates
-21
-22
-23
-24
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
Comparison to Goal
strain noise h [1/Hz]
10
10
10
10
10
-18
cryogenic mirror
ET sensitivity
advDetector sensitivity
-20
-22
-24
-26
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
4 mirrors contributing, L = 10 km
How to achieve the ET sensitivity?
increasing beam size:

10
-18
strain noise [1/Hz]
ET
adDet
10
10
10
10
assuming advDet.
aspect ratio
-20
-22
-24
w [mm]
m [kg]
60
42
100
196
140
537
-26
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
m ~ w3
ROC
How to achieve the ET sensitivity?

thermal noise [m/Hz]
10
10
10
10
10
10
increasing beam size:
-19

bulk Brownian
bulk TE
coating Brownian
coating TE
-20
smaller influence of the
coatings (e.g. waveguide
mirrors, beam-profile)
-21
-22
60 mm beam radius
would be sufficient
-23
-24
10
0
10
1
2
10
frequency [Hz]
10
3
10
4
How to achieve the ET sensitivity?

increasing beam size:

results agrees with estimates by S. Hild
for upscaling existing techniques in GWINC
[Hild et al. arXiv: 0810.0604v2]

big substrate, coating with dia. 800 mm needed

big mass will cause problems in the suspension
Suspension Material (1)

Demands:

low thermal noise
high thermal conductivity (extraction of thermal load
of residual absorption, ~ 1 ppm, ~ 1 W)
high breaking strength

available ?


Suspension Material (2)

Brownian Thermal Noise

Thermoelastic Noise

fibres, ribbons

Thermal Aspects

Connecting Bulk and Suspension?
Suspension Material (3)
breaking strength of Si is dependent
on treatment (200 MPa – 6 GPa)
1 circular fibre diameter: 25 mm2 … 1 mm2
500 kg
30% of maximum value:
4 fibres:
~ 2.5 mm2
~ 0.7 mm2 per fibre
~ 1 mm diameter
Suspension Material (4)
Thermoelastic Noise

10
thermoelastic loss
10
10
10
10
10
-2
dia.
-4

NL
TE

-6
-8
1 Y
T T
[Cagnoli, Willems 2002]
-10
 for Si unsuited for
compensation
-12
10
 
YT 


  0 
C 
Y  1  () 2
2
300 K
20 K
0
10
1
2
10
frequency [Hz]
10
3
10
4
AdvLIGO fibre thermal noise reduction

techniques developed for advDet. -> ET suspension


FE assisted analysis of refined fibre
models, fibre shapes, fibre neck shapes,
tapered fibres
additional aspects of heat extraction
needs to be taken into account
[Cumming 2008]
Challenges

high coating thermal noise

compensation: large beam diameter, large substrate

large substrate mass causes problems in the suspension


possible reduction:
finite size correction, reconsidering aspect ratio of substrat,
better/no dielectric coatings, lower temperatures
unknown parameters of coatings (thermal, mechanical) cause
uncertainties which might change the result significantly
Suggestions



database for material properties (paper + data files)
common reference curve for thermal properties
 comparison between different calculations
implementation of temperature dependent material
properties in a GWINC code (cryoGWINC?)