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Direct Measurement of Thermo-Optic Coefficients in Coatings by
Photothermal Spectroscopy
Greg Ogin, Eric Black, Eric Gustafson,
Ken Libbrecht
Matt Abernathy Presenting
LIGO-G1200935
LSC/VIRGO Conference, Rome, Italy, 10 September 2012
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The AdLIGO Noise Curve
Source: Evans et al, LIGO-P080071-00
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Thermo-optic Noise: TO = TE + TR
• Thermo-Elastic (TE): Mirror’s surface expands into probe beam. By convention, negative dφ/dT
t 4
Ttt
tTE 4
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Thermo-optic Noise: TO = TE + TR• Thermo-Refractive (TR): Coating layers deviate from λ/4 condition
– due to both physical expansion and change in index of refraction. To first order, this manifests as a change in the phase of the reflected beam.
EreEE i
)('' TRii reEerEE
Quarter-wave stack:
After expansion, index change:
TRieE
E+
E+
E-
E-
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NPRO
Beam Dump
PBS
λ/2
λ/2
Vacuum Chamber
Beam Dump
CO
2
AOM
Data AcquisitionElectronics
Fringe LockingElectronics
PZT
Test Mirror
Photothermal Apparatus
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Mirror Under Test
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Expected Signal: Canonical Form
TRTE
)45()90(0 4 tieff
ti ea
ea
A
P
Substrate CTE
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Expected Signal: Canonical Form
TRTE
)45()90(0 4 tieff
ti ea
ea
A
P
Substrate CTECoating properties (including coating CTE effects)
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Sapphire SubstrateResponse Magnitude
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Sapphire SubstrateResponse Phase
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Silica SubstrateResponse Magnitude
K
WP
eff /101.1
6.04
0
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Silica SubstrateResponse Magnitude
K
WP
eff /101.1
6.04
0
+/- 20%
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Recent Results: Silica Substrate
K
WP
eff /101.1
6.4
0
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Combined TE/TR Results
• QWL
• Bragg
Keff /10)03.18.1( 4
Keff /10)04.08.1( 4
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Gold coatings for pure TE measurements
Challenge: 80% CO2 absorption drops down to 0.5% CO2 absorption.
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Much lower SNRDisplacement (m)
10-12
10-11
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Gold Coated “TE alone” Results
• QWL
• Bragg
Keff /10)2.03.3( 4
Keff /10)2.01.3( 4
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Extracting Values
CrHLHLeff 6.1041.148.49.137.21
For quarter-wavelength coatings
For 1/8-3/8 coatings
For quarter-wavelength TE only
For 1/8-3/8 coatings TE only
CrHLHLeff 6.1018.187.621.64.30
CrHLeff 6.101728
CrHLeff 6.104.840
(Cr? Chromium.)
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The Measurement Matrix
Which we invert to get…
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The Parameter Estimation Matrix
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Our Results…
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Our Measurements of α
SiO2 – Low Index• 2.1x10-6 K-1
– Cetinorgu et al, Applied Optics 48, 4536 (2009)
• 5.1x10-7 K-1
– Crooks et al, CQG (2004)• 5.5x10-7 K-1
– Braginsky et al, Phys Lett A 312, 244 (2003)
Ta2O5 – High Index• + 4.4x10-6 K-1
– Cetinorgu et al, Applied Optics 48, 4536 (2009)
• + 3.6x10-6 K-1
– Crooks et al, CQG (2004)• - 4.4x10-5 K-1
– MN Inci, J Phys D 37, 3151 (2004)
• + 5x10-6 K-1
– Braginsky et al, arXiv: gr-qc/0304100v1 (2003)
(5.5 ± 1.2)x10-6 K-1 (8.9 ± 1.8)x10-6 K-1
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Our Measurements of β
SiO2 – Low Index• 8x10-6 K-1
– GWINC v2 (“Braginsky”)
Ta2O5 – High Index• 1.21x10-4 K-1
– MN Inci, J Phys D 37, 3151 (2004)
• 6x10-5 K-1 *
– Gretarsson, LIGO-G080151-00-Z (2008) *Assumes α
(1.9 ± 8.0)x10-6 K-1
(1.2 ± 0.4)x10-4 K-1
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AdLIGO Baseline (GWINC v3)
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AdLIGO with Our ParametersDisclaimer: This Is Not an AdLIGO Prediction
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Conclusions
• Measuring these parameters is non-trivial, but we have demonstrated a technique, and reported initial results
• We have the ability to measure exactly what AdLIGO needs
• Thermo-optic noise, and these parameters in particular, could be critical and need further study for future generations of gravitational wave detectors
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Future Directions
• Characterize and reduce systematic errors
• Perform measurements on AdLIGO coatings with Cr layers (or at the very least Ion Beam Sputtered coatings and Ti:Ta2O5 coatings)
• Look at measurements of other materials and geometries
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Acknowledgements
• Greg Ogin
• Ken Libbrecht, Eric Black
• Eric Gustafson
• Caltech LIGO-X, Akira Villar
• Family and friends
• LIGO and the NSF – Award PHY-0757058
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Questions?
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Supplimentary Slidesfollow
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Measuring α:Cavity Assisted Photothermal Spectroscopy
2)(FinesseSignal
Black et al, J Appl Phys 95, 7655 (2004)
•Probe locked to cavity
•Pump derived from probe laser chopped to cyclically heat cavity end mirror
•Sensitivity to mirror expansion proportional to Finesse
•Heating power in cavity proportional to Finesse
•Sample coated with gold to enhance absorption
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Details of the two terms:
• Thermo-Elastic:
• Thermo-Refractive:
tT eff
TE
4
22
2 1)/(2
LH
LHLLL
HHH
TR nn
nnnTn
nTn
T
Evans et al, Physical Review D 78, 102003 (2008)
Negative phase
Positive phase
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Theory: Assumptions• The scale of periodic thermal
disturbances (a “thermal wavelength”) is much smaller than our heating spot
• The coating thickness is smaller than a thermal wavelength
thermalprobepump ww
coatingthermal t
Together, these give us a 1-D problem where the thermal dynamics are all determined by the properties of the
substrate.
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Theory: Heat Equation Solutions
• The heat equation becomes
• With solutions
2
2
z
ua
t
u
pCa
)(),( kztiCeztu
aeaik i //)1(2
2 45
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Theory: Boundary Condition
• Our boundary condition gives C(ω)
ti
z
eA
P
z
u 0
0
a
e
A
PC
i
/
450
)(45
0
/),( kzti
i
ea
e
A
Pxtu
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Expected Signal -A Coherent Sum of…
)90(0
)/(
4)(
4 tiTE e
aA
Pdzzu
)45(0 1)0,()(
tieff
cTR e
A
aPtu
ut
)90(0)270(0 1414)(
titiTE e
A
aPe
A
aPt
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Expected Signal: Canonical Form
TRTE
)45()90(0 4 tieff
ti ea
ea
A
P
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(Reminder)
• Thermo-Elastic:
• Thermo-Refractive:
tT eff
TE
4
22
2 1)/(2
LH
LHLLL
HHH
TR nn
nnnTn
nTn
T
Evans et al, Physical Review D 78, 102003 (2008)
Negative phase
Positive phase
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Expected Signal: Canonical Form
)45()90(0
4ti
effti e
ae
a
A
P
0eff
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Expected Signal: Canonical Form
)45()90(0
4ti
effti e
ae
a
A
P
0eff
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Expected Signal: Canonical Form
)45()90(0
4ti
effti e
ae
a
A
P
0eff
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Expected Signal: Canonical Form
)45()90(0
4ti
effti e
ae
a
A
P
0eff
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Recent Results: Sapphire SubstrateResponse Magnitude
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Recent Results: Sapphire SubstrateResponse Phase
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Recent Results: Sapphire SubstrateResponse Phase
Wait, what?!
?
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Sapphire: Long Thermal Wavelength
thermalpumpw really means we have a 3-D problem (axially symmetric), “plane thermal waves” don’t work
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“Cerdonio”-type solution
• Green’s function on the surface of a half-space
• Forced sinusoidally with a Gaussian profiled beam
)(4
'2/3 2
)(4
2)',,,(
ta
xx
eta
xxtG
tir
yx
eezrC
Ptzyxf
2
0
22
02
0
0),,,(00 zz
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Then all you have to do is…
• Integrate
• and again.
ir
yx
t
xxzt
eezrC
Pe
tdxdydzdzyxtu
2
0
222
0''
02
0
0)(4
'2/3
0 )(4
12'''),,,(
0
),,0,0()( dztzuatl
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Thanks Mathematica
4/
4)( 2
0
4/0
20
irlEiExpIntegraiie
eC
Patl
irti
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Thanks Mathematica
4/
4)( 2
0
4/0
20
irlEiExpIntegraiie
eC
Patl
irti