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 rd Einstein Telescope General 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

0.Abstract 2 (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 3

1.Introduction Thermal noise of mirrors : Fundamental noise of interferometric gravitational wave detector around 100 Hz 4 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.Introduction For example … 5 Divergence ?

1.Introduction More serious problem 6 Loss angle of thermoelastic noise Fluctuation Dissipation Theorem Thermoelastic noise should be constant in low frequency region. contradiction !

1.Introduction 7 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.

8 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)

9 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) ) heat flows along optical axis (coating is thin). 999 Coating Laser beam Substrate Heat flux

10 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 Laser beam Substrate Heat flux

11 2. Thermo-optic noise Cut off (beam radius) corrected formula Constant

12 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.

13 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-node multilayer thermo-elastic element along the mirror surface. Substrate is modeled using 20-node multilayer 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

14 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) , 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

15 3. Substrate thermoelastic noise M. Cerdonio et al., Physical Review D 63 (2001) 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.

16 3. Substrate thermoelastic noise M. Cerdonio et al., Physical Review D 63 (2001) 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 ?

17 3. Substrate thermoelastic noise In the case of small beam … Cut off (beam radius) Cut off (Mirror size) f -1 f -1/4 Constant

18 3. Substrate thermoelastic noise In the case of small beam … Cut off (beam radius) Cut off (Mirror size) Beam radius dependence Mirror size dependence No size dependence

19 In the case of LCGT (Sapphire 20 K) … Cut off (beam radius) Cut off (Mirror size) 4. ET and LCGT

20 4. ET and LCGT In the case of ET-LF (Silicon 10 K) … S. Hild et al., Classical and Quantum Gravity 27 (2010)

21 4. ET and LCGT In the case of ET-LF (Silicon 20 K) …

22 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.

23 2. Thermo-optic noise Cut off (beam radius)

24 2. Thermo-optic noise Cut off (beam radius) In this region, thermo-optic noise formula breaks down.