1 kazuhiro yamamoto istituto nazionale di fisica nucleare sezione di padova substrate thermoelastic...

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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:

Program tree

--> APFEM_ini.m |--> APFEM_mesh.m | |--> APFEM_solver.m --> APFEM_aux.m

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3. Substrate thermoelastic noise

M. Cerdonio et al., Physical Review D 63 (2001) 082003.

Substrate thermoelastic noise :

Relaxation of temperature gradient in substrate

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

M. Cerdonio et al., Physical Review D 63 (2001) 082003.

Substrate thermoelastic noise :

Relaxation of temperature gradient in substrate

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 noiseIn the case of small beam …

Cut off(beam radius)

Cut off(Mirror size)

f -1

f -1/4Constant

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3. Substrate thermoelastic noiseIn the case of small beam …

Cut off(beam radius)

Cut off(Mirror size)

Beam radiusdependence

Mirror sizedependence

No size dependence

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In the case of LCGT (Sapphire 20 K) …

Cut off(beam radius)

Cut off(Mirror size)

4. ET and LCGT

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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.