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Substrate thermoelastic noise
and thermo-optic noise
at low temperature
in low frequency region
Kazuhiro Yamamoto
Istituto Nazionale di Fisica Nucleare Sezione di Padova
Kenji Numata
University of Maryland
NASA Goddard Space Flight Center
Enrico Serra
Interdisciplinary Laboratory for Computational Science (LISC),
FBK-CMM and University of Trento
24 November 2010 3rd Einstein Telescope General Workshop
@Hungarian Academy of Sciences, Budapest, Hungary
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0.Abstract
(1) All formulae for substrate themoelastic noise and
thermo-optic noise in previous papers break down
in low frequency region.
(2) Substrate thermoelastic noise and thermo-optic noise
of cryogenic interferometer (ET-LF and LCGT) are
evaluated using corrected formulae.
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Contents
1. Introduction
2. Thermo-optic noise
3. Substrate thermoelastic noise
4. ET and LCGT
5. Summary
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1.Introduction
Thermal noise of mirrors : Fundamental noise
of interferometric gravitational wave detector around 100 Hz
There are some kinds of thermal noise (dissipation).
Substrate thermoelastic noise :
Relaxation of temperature gradient in substrate
Thermo-optic noise :
Relaxation of temperature difference
between substrate and coating
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1.Introduction
For example …
Divergence ?
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1.Introduction
More serious problem
Loss angle of thermoelastic noise
Fluctuation Dissipation Theorem
Thermoelastic noise should be constant
in low frequency region.
contradiction !
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1.Introduction
Our conclusion is that
(1) All formulae for substrate themoelastic noise and
thermo-optic noise in previous papers break down
in low frequency region.
(2) This result could be important
for cryogenic interferometer.
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1.Introduction
How can we calculate thermal noise ?
Y. Levin, Physical Review D 57 (1998) 659.
Pressure whose profile is the same as laser beam
is applied on the mirror.
Time development of pressure is sinusoidal.
Frequency is the same as that of power spectrum of
thermal noise.
Dissipation caused by this pressure is related with power
spectrum of thermal noise.
(Fluctuation Dissipation Theorem)
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2. Thermo-optic noise
Relaxation of temperature difference
between substrate and coating
Heat flux : Origin of loss
In all previous papers (For example, M. Evans et al.
Physical Review D 78 (2008) 102003.)
heat flows along optical axis (coating is thin).
Substrate
Coating
Laser beam
Heat flux
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2. Thermo-optic noise
However, if frequency is extremely low
(time development of pressure is slow)
heat can flow along radius direction.
We take heat flow along radius direction into account
although it is neglected in all previous papers.
Coating
Substrate
Heat flux
Laser beam
Heat flux
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2. Thermo-optic noise
Cut off
(beam radius)
Constant
corrected formula
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2. Thermo-optic noise
Michael J. Martin (JILA, University of Colorado) also derived
formula of thermo-optic noise in low frequency region.
His consideration is perfectly independent from ours and his
result agrees with ours.
Calculations of Martin and ours are analytical. We are
proceeding with calculation using finite element method.
E. Serra and M. Bonaldi, International Journal for Numerical
Methods in Engineering 78 (2009) 691.
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Fully – coupled Finite Element formulation for evaluating thermo-optic noise
(the work in progress)
The idea is to decouple
coating + substrate FEM
Substrate is modeled using 20-node
domain
multilayer thermo-elastic elements in
the volume.
Coating is modeled with 8-node
multilayer thermo-elastic element
along the mirror surface.
This procedure reduce the
computational cost and problems
from aspect-ratio mirror - coating
The thermo-elastic dissipation is calculated by solving this algebraic system
of equation:
and using :
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Preliminary validation for the 8-node thermoelastic and 20-node
elements for modeling thin and thick structures - Ref. E. Serra M.
Bonaldi International Journal of Numerical Methods in Engineering
volume 78 (6) 671-691, 2009
Arbitrary Precision Finite Element Method (APFEM) code is in now
under development for modeling multilayer coatings:
Program tree
--> APFEM_ini.m
|--> APFEM_mesh.m
|
|--> APFEM_solver.m
--> APFEM_aux.m
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3. Substrate thermoelastic noise
Substrate thermoelastic noise :
Relaxation of temperature gradient in substrate
M. Cerdonio et al., Physical Review D 63 (2001) 082003.
Spatial Fourier transform
If frequency is lower, contribution of smaller wave number
(longer wavelength) Fourier component is larger.
Half infinite substrate : Wavelength can be longer infinitely.
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3. Substrate thermoelastic noise
Substrate thermoelastic noise :
Relaxation of temperature gradient in substrate
M. Cerdonio et al., Physical Review D 63 (2001) 082003.
In actual case, mirror has finite size.
Wavelength must be smaller than size of mirror.
We must take size of mirror into account
(not Fourier transform, but Fourier series).
Divergence is removed ?
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3. Substrate thermoelastic noise
In the case of small beam …
Constant
Cut off
(Mirror size)
f
-1/4
Cut off
(beam radius)
f -1
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3. Substrate thermoelastic noise
In the case of small beam …
Cut off
(Mirror size)
No size
dependence
Cut off
(beam radius)
Mirror size
dependence
Beam radius
dependence
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4. ET and LCGT
In the case of LCGT (Sapphire 20 K) …
Cut off
(beam radius)
Cut off
(Mirror size)
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4. ET and LCGT
In the case of ET-LF (Silicon 10 K) …
S. Hild et al., Classical and Quantum Gravity 27 (2010) 015003.
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4. ET and LCGT
In the case of ET-LF (Silicon 20 K) …
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5. Summary
(1) All formulae for substrate themoelastic noise and
thermo-optic noise in previous papers break down
in low frequency region.
(2) Substrate thermoelastic noise : Finite size mirror
Thermo-optic noise : Heat flow along radius direction
(3) Evaluation for ET and LCGT using corrected formulae
A few times smaller power spectrum
(than that derived from old formulae)
Total noise does not so change.
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2. Thermo-optic noise
Cut off
(beam radius)
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2. Thermo-optic noise
Cut off
(beam radius)
In this region,
thermo-optic noise formula
breaks down.
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