1 Kazuhiro Yamamoto Istituto Nazionale di Fisica Nucleare Sezione di Padova Substrate thermoelastic noise and thermo-optic noise at low temperature in low frequency region 24 November 2010 3rd Einstein Telescope General Workshop @Hungarian Academy of Sciences, Budapest, Hungary Kenji Numata University of Maryland NASA Goddard Space Flight Center Enrico Serra Interdisciplinary Laboratory for Computational Science (LISC), FBK-CMM and University of Trento
Transcript
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Kazuhiro YamamotoIstituto Nazionale di Fisica Nucleare Sezione di Padova
Substrate thermoelastic noise and thermo-optic noise
at low temperature in low frequency region
24 November 2010 3rd Einstein Telescope General Workshop @Hungarian Academy of Sciences, Budapest, Hungary
Kenji NumataUniversity of Maryland
NASA Goddard Space Flight Center
Enrico SerraInterdisciplinary Laboratory for Computational Science (LISC),
FBK-CMM and University of Trento
0.Abstract
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(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.
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
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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
1.IntroductionFor example …
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Divergence ?
1.IntroductionMore serious problem
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Loss angle of thermoelastic noise
Fluctuation Dissipation Theorem
Thermoelastic noise should be constant
in low frequency region. contradiction !
1.Introduction
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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 noiseRelaxation 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).
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Coating
Laser beam
Substrate
Heat flux
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2. Thermo-optic noiseHowever, 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
Laser beam
SubstrateHeat flux
Heat flux
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2. Thermo-optic noise
Cut off(beam radius)
corrected formula Constant
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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 thermo-elastic dissipation is calculated by solving this algebraic system of equation:
Coating is modeled with 8-nodemultilayer thermo-elastic elementalong the mirror surface.
Substrate is modeled using 20-nodemultilayer thermo-elastic elements in the volume.
and using :
This procedure reduce the computational cost and problems from aspect-ratio mirror - coating
The idea is to decouple coating + substrate FEM domain
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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:
