1. Thermal noise and thermal deformations of mirrors
Yuri Levin, Monash
Fluctuation-dissipation theorem
Thermo-mechanical noise; reciprocity relations;
general optical readouts; gratings
Thermal deformations of mirrors
4. Thermo-refractive noise; standing wave
5. Thermo-chemical noise

2. 1. Fluctuation-dissipation theorem
Callen & Welton 1951, Kubo+many others
Macroscopic
degree of
freedom
Microscopic
degrees of freedom
Coupling

3. 1. Fluctuation-dissipation theorem
Callen & Welton 1951, Kubo+many others
dissipation
Macroscopic
degree of
freedom
Microscopic
degrees of freedom
Coupling

4. 1. Fluctuation-dissipation theorem
Callen & Welton 1951, Kubo+many others
dissipation
Macroscopic
degree of
freedom
Microscopic
degrees of freedom
Coupling
Thermal motion
fluctuations

5. 1. Fluctuation-dissipation theorem
Callen & Welton 1951, Kubo+many others
dissipation
Macroscopic
degree of
freedom
Microscopic
degrees of freedom
Coupling
Thermal motion
fluctuations

6. 2. Thermo-mechanical noise
Callen and Welton 51, Levin 98
Readout variable:
Interaction
1. oscillating pressure
2. Compute/measure dissipated power 3.

7. Two questions:
1. How does one figure out the readout variable in a general opto-mechanical
set up?
Optical reciprocity
Heinert + 13
2. If the dissipation is local, how does the local random stress communicate
to macroscopic surface displacement? How do we quantify this
communication?
dissipation
Elastodynamic
reciprocity
fluctuation

8. Reciprocity relations
Normal computational scheme
Compute all consequences of the perturbation
Extract the quantity you want to know
Using reciprocity relationship
Identify the quantity you want to know
Solve the reverse problem (often easier). Construct a map between perturbations and readout quantity.
Do it once!

9. Opto-mechanical readout variables
Question: how does the mode frequency change when dielectric interface moves?
Adiabatic theorem for oscillators so
Mode
energy
Interface
displacement
Optical pressure
on the interface
Useful for thermal noise calculations from e.g. gratings
(cf. Heinert et al. 2013; correction ~25%)

10. Opto-mechanical readout variables
Linear optical readout, e.g. phase measurements
Carrier light +
Perturbation
Phase
Form-factor

11. Part 6: opto-mechanics with interfaces
Linear optical readout, e.g. phase measurements
Photo-diode
Phase
Form-factor

12. Opto-mechanical readout variable
1. Generate imaginary beam
with oscillating dipoles
Photo-diode
2. Calculate induced optical
pressure on the interface
3. The phase

13. 3. Thermal deformations of mirrors
Not an issue for
advanced KAGRA.
Major issue for LIGO
& Virgo
High-temperature region
cf. Hello & Vinet 1990
New
coordinates
Basis functions: Zernike polynomials,
Hermite-Gauss functions
Treat this as a readout variable

14. Elastodynamic reciprocity relations
form-factor
Force
density
Readout
variable
displacement
form-factor

15. Elastodynamic reciprocity relations
form-factor
Force
density
is invariant with
respect to interchange of
and
Readout
variable
displacement
form-factor

16. How to calculate Apply pressure to the mirror face
King, Levin, Ottaway, Veitch to be
submitted.
Apply pressure to the mirror face
Calculate trace of the induced deformation tensor
Have to do it only once!
Calculate the thermal deformation
Young
modulus
Thermal
expansion
Temperature
perturbation

18. Computation of scattering coefficients
imaginary
pressure
Impressive
Agreement!

19. Thermo-refractive noise
Braginsky, Gorodetsky, & Vyatchanin 2000
medium
Index of refraction:
beam 3
phaseshift
temperature
fluctuation
intensity
readout
variable

20. Thermo-refractive noise
Levin 2008
Readout variable:
1. oscillating entropy injection
2. Compute/measure dissipated power, e.g. 3.

21. Standing waves, GEO600 beamsplitter
Image credit: Prof. Nollert
effective
ineffective
Benthem & Levin 2009

22. Thermochemical and Carrier-Density Noise
Intensity profile
effective
ineffective
Thermochemical noise: due to chemical impurities
Benthem & Levin 2009
Carrier-density noise: due to semiconductor
charge carriers
Heinert et al. submitted
Both seem to not threaten GW interferometry, but one has to be vigilant, especially
for the ET

23. Conclusions
Linear systems (elastic, optomechanical) feature reciprocity relations
They give direct computation of readout variable
non-trivial geometries, and they allow fast computations of thermal distortions and scattering coefficients
Vigilance for new types of thermal noise