1

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

2

The AdLIGO Noise Curve

Source: Evans et al, LIGO-P080071-00

3

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

4

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-

5

NPRO

Beam Dump

PBS

λ/2

λ/2

Vacuum Chamber

Beam Dump

CO

2

AOM

Data AcquisitionElectronics

Fringe LockingElectronics

PZT

Test Mirror

Photothermal Apparatus

6

Mirror Under Test

7

Expected Signal: Canonical Form

TRTE

)45()90(0 4 tieff

ti ea

ea

A

P

Substrate CTE

8

Expected Signal: Canonical Form

TRTE

)45()90(0 4 tieff

ti ea

ea

A

P

Substrate CTECoating properties (including coating CTE effects)

9

Sapphire SubstrateResponse Magnitude

10

Sapphire SubstrateResponse Phase

11

Silica SubstrateResponse Magnitude

K

WP

eff /101.1

6.04

0

12

Silica SubstrateResponse Magnitude

K

WP

eff /101.1

6.04

0

+/- 20%

13

Recent Results: Silica Substrate

K

WP

eff /101.1

6.4

0

14

Combined TE/TR Results

• QWL

• Bragg

Keff /10)03.18.1( 4

Keff /10)04.08.1( 4

15

Gold coatings for pure TE measurements

Challenge: 80% CO2 absorption drops down to 0.5% CO2 absorption.

16

Much lower SNRDisplacement (m)

10-12

10-11

17

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.)

19

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

23

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

24

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

27

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

28

Acknowledgements

• Greg Ogin

• Ken Libbrecht, Eric Black

• Eric Gustafson

• Caltech LIGO-X, Akira Villar

• Family and friends

• LIGO and the NSF – Award PHY-0757058

29

Questions?

Supplimentary Slidesfollow

30

31

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

32

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

33

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.

34

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

36

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

38

(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

39

Expected Signal: Canonical Form

)45()90(0

4ti

effti e

ae

a

A

P

0eff

40

Expected Signal: Canonical Form

)45()90(0

4ti

effti e

ae

a

A

P

0eff

41

Expected Signal: Canonical Form

)45()90(0

4ti

effti e

ae

a

A

P

0eff

42

Expected Signal: Canonical Form

)45()90(0

4ti

effti e

ae

a

A

P

0eff

43

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?!

?

46

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

49

Thanks Mathematica

4/

4)( 2

0

4/0

20

irlEiExpIntegraiie

eC

Patl

irti

50

Thanks Mathematica

4/

4)( 2

0

4/0

20

irlEiExpIntegraiie

eC

Patl

irti