Download G010415-00 - DCC

Proposal
Thermal and Thermoelastic Noise Research
for Advanced LIGO Optics
Norio Nakagawa
Center for Nondestructive Evaluation
Iowa State University
in collaboration with
E.K. Gustafson and M.M. Fejer
Ginzton Laboratory, Stanford University
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
1
Outline
• Objective and Approach
• Output of the Prior NSF Support
– Laser phase noise formulas for optical resonator and delay line
– Coating noise estimation
• Proposed Work
– Coating noise studies for mirrors of edge geometry
– Thermo-elastic noise of coated mirrors
– Thermal & thermo-elastic noises of realistic mirror designs
• Tasks & Time Lines
• Summary
• (Broader Impacts)
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
2
Objective and Approach
Objectives
• To develop laser-phase-noise
formulas
– Green’s-function-based
– Analytical and computational
– two-point laser-phase-fluctuation
correlations
– complex test mass objects.
• To extend the noise estimation
method to thermo-elastic noises.
• To estimate thermal and thermoelastic noises of coated mirrors for
advanced LIGO designs.
• To examine merits of interferometer
design options for future LIGO.
November 29-30, 2001
LIGO-G010415-00-Z
Approaches
• Calculate phase noise via
Green’s function method
– Elasticity
– Thermo-elasticity & thermal
diffusion
• Analytical mirror models
– Half space
– Quarter space
– Thin coating layer
• Numerical calculations
– Realistic mirror shapes
– Coating loss
– Delay-line vs. Fabry Perot
Talk at PAC-11, LHO
3
Output to Date
•
Publications
– N. Nakagawa, Eric Gustafson, P. Beyersdorf and M. M. Fejer
“Estimating the off resonance thermal noise in mirrors, Fabry-Perot
interferometers and delay-lines: the half infinite mirror with uniform
loss,” to appear in Phys. Rev. D
– N. Nakagawa, A. M. Gretarsson, E.K. Gustafson, and M. M. Fejer,
“Thermal noise in half infinite mirrors with non-uniform loss: a slab of
excess loss in a half infinite mirror,” submitted to Phys. Rev. D
– N. Nakagawa, E.K. Gustafson, and M. M. Fejer, “Thermal phase noise
estimations for fabry-perot and delay-line interferometers using coated
mirrors,” in preparation.
– D. Crooks, et al., “Excess mechanical loss associated with dielectric
mirror coatings on test masses in interferometric gravitational wave
detectors,” submitted to Classical and Quantum Gravity.
– Gregory M. Harry, et al., “Thermal noise in interferometric gravitational
wave detectors due to dielectric optical coatings,” submitted to Classical
and Quantum Gravity.
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
4
Phase-Noise Correlation
• Requirement
– Compute laser phase noise correlation



x1 , t1 x2 , t2   
 
S  , x1 , x2 
d i t1 t 2 

 2

x1,t1 
e

x2 ,t2 

x1

x2
E, ,
E,  , 
Ex. Coating noise model
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
5
Intrinsic Thermal Noise
• Phase Noise Formula
 
2k T
S  , r1 , r2   4k 2 B
  w  
 dS  dS  r   r1  00 r   r2 
w
00






 

  







  d x  i  nj  x , r ; c  cijkl  x   k  nl  x , r ; c 
3
V


  icijkl
  1  i  cijkl

cijkl
 cijkl
 
S  , r1 , r2 


ui r u j r 

 
r1 , r2

2

2 r


 r e
w
00
w2

November 29-30, 2001
LIGO-G010415-00-Z
the two-point laser-beam phasenoise power-spectrum correlation
w, k
the laser beam spot size (amplitude
radius), and wave number
the displacement spectral
correlation
ij
elastic Green’s function
cijkl [c’ijkl, c”ijkl]
elastic constants [dispersive and
absorptive parts]
()
loss function
kB , T
Boltzmann constant, and temperature
The laser beam reflection points
(the beam centers)
the Gaussian laser-beam profile
function
Talk at PAC-11, LHO
6
Fluctuation-Dissipation Relation
• Surface Force density  Strain


 

  
  
w 
F r   r1   uij x; r1   F  dS  i  nj x , r ; c 00 r   r1 
w
00




 
    
2k T 1
3
  





 ukl x; r2 
S  , r1 , r2   4k 2 B
d
x
u
x
;
r
c
x
ij
1
ijkl
2 V
 F
• Laudau-Lifshitz
E mech
 
 2

2
  ijkl  
  T0  dV T  2  dV uijijklukl , cijkl

V
V



 
structural
thermo elastic

 
2 4 k BT 1

S  , r , r   4k

E
mech
2
2
 F
November 29-30, 2001
LIGO-G010415-00-Z
viscous
viscous
Talk at PAC-11, LHO

7
Various Optical Configurations
• Phase noise formulas
Computed explicitly for
– Single-reflection mirror
 
SSingle    S  , r , r 
1
 1  rI 2  

2rI
S    
1
cos 2  SE    rI2 SI  
2 
2

 1  rI   1  rI
FP


N n 1
S ( )   S  , rn , rn   2 cos[2( n  q) ]  SE  , rn , rq 
N
DL
E
n 1
N 1
November 29-30, 2001
LIGO-G010415-00-Z
N 1 n 1
  S  ,  n ,  n   2 cos[2( n  q) ]  SI  ,  n ,  q 
– Fabry-Perot resonator
– Optical delay line
n  2 q 1
I
n 1
n  2 q 1
rI
the input mirror reflection coefficient

the transit time
SE  , SI  
the single-reflection phase noises of the input and
end-point mirrors.

rn
the positions of the N-time reflections on the endmirror surface
p

the positions of the (N-1)-time reflections on the inputmirror surface
E, 
If Half space:
Young’s modulus, and Poisson ratio
 
8k T  k 2 1   2  r1  r2 2
S  , r1 , r2   B
e
  w E
Talk at PAC-11, LHO
2 w2

  2
I 0 r1  r2  2 w 2

8
Fabry-Perot vs. Delay Line
• Fabry-Perot vs Delay lines
15
– Analytical half-space mirror model
– Fabry-Perot interferometer vs several
delay lines
– Storage time as proposed for LIGO II.
– Delay line beam centers
R
10
2
3
• evenly spaced
• on a circle
4
5
5
• When the spots are not overlapping
appreciably, the delay line is less
noisy than the Fabry-Perot.
6
0
– Noise levels are similar if
• the spot circle radii comparable to the
beam spot size
• the spots are largely overlapping, and
above several hundred Hertz.
November 29-30, 2001
LIGO-G010415-00-Z
2w
1
50
150
100
Frequency [Hz]
200
Figure 1: Comparison of the phase noise from a delay-line and a Fabry-Perot
interferometer. The solid curve (1) is for a 4 km Fabry-Perot interferometer
with an input mirror power reflectivity of RI=0.97 and an end mirror power
reflectivity of RE=1.00. Curves 2, 3, 4, 5 and 6 correspond to 4 km delay lines
all with 130 spots on the end mirror and laser beam spots of 1/e field radius w in
a pattern with their centers on a circles of radius R=w/3 (2), R=2w/3 (3),
R=5w/2 (4), R=10w (5) and R=20w (6) where w=3.5 cm, the spot size used for
both mirrors of the Fabry-Perot interferometer. The mirror Q is assumed to be
3´108 and the material properties are those of Sapphire E=71.8 GPa and s=0.16
however we are treating sapphire as isotropic for the purpose of this illustration
and assuming a single loss function.
Talk at PAC-11, LHO
9
Coating Noise; Problem Statement
• Coating noise model
– Based on half-space mirror model
• a lossy layer (thickness d) on a lossy host material
– Requirement: Compute laser phase noise correlation
– Approach: via analytical Green’s function

x1,t1 

x2 ,t2 


x1

x2
E, ,
E,  , 
November 29-30, 2001
LIGO-G010415-00-Z

  x1 , t1   x2 , t2 
Talk at PAC-11, LHO

d i t1 t2 
  2


e


  x1   x2  
 
S  , x1  x2 
d i t1 t2 

  2

e
10
Coating Noise
• Static Green’s function for layer-on-substrate

sos


sos
z 

 ~p, z  
1 1
pE 1


sos ~
1


 21     pz  i 1 G  p,0  2 pz 3 cosh pz



sos ~
3 4
1









pz


1

2


G
p
,
0


pz
i

sinh
pz


3
2
1
2
2




 ~p, z   211  2 pz 3G sos  ~p,0  21     pz  i 1 cosh pz
 211  21  pz i 1 G sos  ~
p,0  pz 3  1  2  2 sinh
where

pz

 1    cosh 2 pd  411  324   1  2 1  2   2 3  4  sinh 2 pd



 
 
1
  2 1  2  cosh 2 pd  1 1  2 sinh 2 pd 
 
 
sos ~
1  
2
2
1
G  p ,0     81 2 1  2 3  4   2 1  2 3  4    1  2 3  4  sinh pd  2 
 
 
2
  811 2 1   1   3  4  pd 
 
 
 
1
 41  pd 1   1   3  4  3



2


  cosh pd  2 11  1  2    3  4 sinh pd cosh pd  2 11  1  2    sinh pd 
 4 11 2 1   1   3  4  pd 
2
  11 EE , 1,2,3  Pauli matrices
November 29-30, 2001
LIGO-G010415-00-Z
D. M. Burmister, J. Appl. Phys. 16, 89-94, 1945.
Talk at PAC-11, LHO
11
Coating Noise
• Intrinsic Thermal Phase-Noise Estimation

2k T
Scoating  , r   4k 2  B

1
w
 1   2  r 2 2 w2
I 0 r 2 2 w2 
  E e

2
2

 1  2 1    1




1


1

2

E 







2


2
2
d
r2
1


E
1


E


e



w



2
1


1

2


1






1
E 


Scoating , r 
The phase noise two-point correlation for a
coated half-space mirror; double-sided
E, , 
Young’s modulus, Poisson ratio, and loss
function of the substrate material
E , ,
Those of the coating material
d
The coating thickness
I0(z)
The 0-th order modified Bessel function of the
first kind; I0(0)=1
November 29-30, 2001
LIGO-G010415-00-Z
  2f

r
w2





 O d 2 w2 



Frequency
a relative position vector between
 the two beam
centers on the coating surface; r  0 for a
single reflection.
k, w
kB , T
Talk at PAC-11, LHO
The laser beam wave number, and spot size
(amplitude radius)
The Boltzmann constant and the temperature
12
Coating Noise
• Resonator
coating
S
 ,0  4k
2

2k BT

1 
 1   2  1  2 1    1

 d 








  w 
E
1
E
 w 


• Delay lines

2k T
Scoating  , r   4k 2  B

1
w

 1   2  r 2 2 w2
 1  2 1    1
 d r 2
2
2
  
e
I 0 r 2 w   
 
e









E
1
E

 w

1
r 


r
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO


w2



13
60
Phase Noise [prad/root-Hz]
Fabry-Perot resonator
w/ coated mirrors
(Various beam radii)
40
wo/ coatings
w=3.5cm
w=3.5cm
2*w
2*w
5*w
5*w
10*w
10*w
Substrate=Fused silica
 E  72.6 GPa

σ  0.16
Q  3 107

Coating= average of
Al2O3 and Ta2O5
(Crooks et al.)
 E  260 GPa

σ  0.26
Q  1.6 10 4

20
0
50
100
150
200
Frequency [Hz]
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
14
60
Phase Noise [prad/root-Hz]
Optical Delay Lines
w/ coated mirrors
(w=3.5cm)
40
wo/ coating
R=w/3
R=w/3
R=2w/3
R=2w/3
R=5w/2
R=5w/2
R=10w
R=10w
20
0
50
100
150
200
Frequency [Hz]
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
15
Proposed Work
• Edge effects on coated mirror noise
– Analytical quarter-space model
• Extension to thermo-elastic noise
– Coated mirrors
– Delay-line interferometers
– Mirrors at low temperatures
• Thermal & thermo-elastic noises
– Mirrors of realistic shapes
• Numerical Green’s function calculation
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
16
Quarter-Space Mirror Model
• Edge effects on noise calculation
– Analytical model

November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
17
Quarter-Space Mirror Model (con’t)
• Finite-mirror size effect on thermal noise (scalar model)
15
Phase Noise [prad/sq. root Hz]
Beam (power-)radius=6cm
Number of Delay-line spots=130
Diameter of D.-L. circle=27cm
F.-P. (Mirror dia.=30cm)
D-L #1 (Mirror dia.=30cm)
10
D-L #2 (Mirror dia.=45cm)
D-L #3 (Mirror dia.=60cm)
D-L #4 (Mirror dia.=300cm)
5
0
50
November 29-30, 2001
LIGO-G010415-00-Z
100
Frequency [Hz]
Talk at PAC-11, LHO
150
200
18
Thermo-elasticity & Thermal diffusion
• From previous identification

 
4k T 1
S  , r , r   4k 2 B2
 E mech
2
 F

 
 2

Emech   T0  dV T   dV uijijklukl
V
 
V
 2




 ukl
Emech   T0  dV T  2  dV uij cijkl
V
V
• If the adiabatic condition holds
2
2

T  T0   TC0 
v

v
l
t ull
p
 
2k T
Sthel  , r1 , r2   4k 2 B

November 29-30, 2001
LIGO-G010415-00-Z

T


 w  
w 





d
S
d
S

r

r
  00 1  00 r   r2 
 v

Cv
2
l
  d x  
 vt2 
2
3
V
i l
Talk at PAC-11, LHO

 
  












x
,
r
;
c



x
,
r
;
c
nl
i k nk

19
Numerical elasticity
•
Noise study:
Cylindrical mirrors
  2ui   jTij  0
 
  2 ik   j  ijk   ik   x  x P 
– Numerical Green’s function
computation
• Betti-Rayleigh-Somigliana
formula
• Nodal discretization by the
boundary element method.
– Cylindrical mirror model
• Single reflection
• Fabry-Perot
• Delay-line
– Demonstrate the delay-line vs.
resonator assertion
– Effects of mirror aspect ratios.
November 29-30, 2001
LIGO-G010415-00-Z



  V  xP uk  xP    dS j  ui  ijk  Tij ik
S

ui  x 
Displacement

Tij  x 
stress tensor Tij  cijkl k ul 

ij  x 
the fundamental solution (Green’s function)

 ijk  x 
 cijlm l mk
cijkl
elastic constants

Density
V, S

 V x 
volume of a region and its boundary surface
Talk at PAC-11, LHO
the characteristic function of V (=1 inside V, =0
outside)
20

Static Elasticity

f
• Needs to avoid rigid-body motions

n
– Mass and moment of inertia
M   dV  , I ij   dV  x 2 ij  xi x j 
V
V
– Given surface forces f can be counterbalanced by volume forces


1
     a   V F
g x   a  b  x ,  

M 1
b   V I  K
S

g
V

K
where
 


  
F   dS f x , K   dS x  f x 
S
S


 
  
Fg   dV g x , K g   dV x  g x 
V
November 29-30, 2001
LIGO-G010415-00-Z
 
 
F  Fg  0, K  K g  0
V
Talk at PAC-11, LHO
O

b

x
 
bx
21
Tasks & Time Lines
T1. Coating noise estimation
– spatial de-correlation
T2. Numerical noise estimations
– coating
T6. Anisotropy effects
T7. Dielectric loss study
T8. Off-beam-axis scatterings
T3. Quarter space
– Analytical model
– Finite size effect
November 29-30, 2001
LIGO-G010415-00-Z
– delay-line vs. resonator
– coating
T5. Quarter space II
– Realistic mirror shapes
– Validation
Task
T1
T2
T3
T4
T5
T6,T7,T8
T4. Thermo-elastic noise estimation
3/31/2003
Talk at PAC-11, LHO
3/31/2004
3/31/2005
22
Summary
Accomplishments
• Obtained two-point phase noise
correlation formulas
– Resonator
– Delay line
– Delay lines can be quieter
• Given coating noise formulas
Proposed Activities
• Studies of intrinsic noises of
coated mirrors
– Thermal noise
– Thermo-elastic noise
– Relative significance of
coating noise
– Realistic mirror shapes
– Relative magnitudes between
coating and substrate noises.
– Their behaviors against the
beam size and other optical
parameters
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
23
Broader Impacts
• Education
– Student involvement through the collaboration with Stanford
Group
• Contribution to LSC
– The thermal noise estimation methodology
– Impact on sorting out advanced optics designs
– Impact on mirror material selection
• Impacts on other federally funded programs
– Through advancement of computational physics methodology
• NSF Industry/University Cooperative Research Center program
• DOE/NERI project; “On-line NDE for advanced reactors”
November 29-30, 2001
LIGO-G010415-00-Z
Talk at PAC-11, LHO
24