1

Mirror Thermal Noise: Gaussian vs Mesa beams

J.Agresti, R. DeSalvo

LIGO-Virgo Thermal Noise Workshop

2

Mirror thermal noise problem:

Advanced-Ligo sensitivity

Dominated by test-masses thermoelastic (S-TM) or coating (FS-TM) thermal noises.

Can we reduce the influence of thermal noise on the sensitivity of the interferometer?

Sapphire TM

Fused Silica TM

Without drastic design changes

3

Mirror Thermal Noise:

Fluctuating hot spots and cold spots inside the mirror

Expansion in the hot spots and contraction in the cold spots creating fluctuating bumps and valleys on the mirror’s surface

Interferometer output: proportional to the test mass average surface position, sampled by to the beam’s intensity profile.

Mirror surface

Created by stochastic flow of heat within the test mass

Thermoelastic noise Brownian noise

Due to all forms of intrinsic dissipations within a material (impurities, dislocations of atoms, etc..)

Surface fluctuations

4

Indicative thermal noise trends

Coating Brownian noise

Substrate Brownian noise

Substrate thermoelastic noise

Exact results require accurate information on material properties and finite size effects must be taken in account.

2

2

3

1

1

1

1

wS

wS

wS

wS

cBX

sBX

cTEX

sTEX

Coating thermoelastic noise

Noise spectral densities in the Gaussian beam case

(infinite semi-space mirror)

5

Diffraction prevents the creation of a beam with a rectangular power profile…but we can build a nearly optimal flat-top beam:

The mirror shapes match the phase front of the beams.

Flat-top beam

Gaussian beam

R

rki

w

r

G eru 2

2

2

2

)(

20

2

2

)1()(

2)( w

irr

DrFT erdru

0

Lw

k

6

Thermal noise for finite sized mirrors:

1. Precise comparative estimation of the various thermal noise contributions for finite test masses (design optimization).

2. Noise suppression using Mesa beam.

7

Thermal noise calculations

Interferometer is sensitive to the test mass surface displacement

)(),()( 2 rftrurdtXMirror

z

Levin’s approach to Fluctation Dissipation Theorem 2

02

8)(

F

WTkS dissB

X

dissW Is the energy dissipated by the mirror in responce to the oscillating pressure

)cos()(),( 0 trfFtrP

8

BHV+LT (accurate) approximate analyical solution of elasicity equations for a cylindrical test mass

Assumptions in our analysis

Pressure distribution

),( trPQuasistatic approximation for the oscillations of stress and strain induced by P.

GWsound

Adiabatic approximation for the substrate thermoelastic problem (negligible heat flow during elastic deformation).

beamheat rr

Coating is an isotropic and homogeneous thin film

Brakes down for coating thermoelastic problem

Perturbative approach

9

Material properties:

Parameters : (c.g.s. units) Fused Silica: Sapphire: Coating

Ta2O5 SiO2

Density ( g/cm3) 2.2 4 6.85 2.2

Young modulus ( erg/cm3) 7.2 1011 4 1012 1.4 1012 7.2 1011

Poisson ratio 0.17 0.29 0.23 0.17

Loss angle 5 10-9 3 10-9 10-4 (total)

Lin. therm. expansion coeff. (K-1) 5.5 10-7 5 10-6 3.6 10-6 5.1 10-7

Specific heat per unit mass (const. vol.) (erg/(g K) ) 6.7 106 7.9 106 3.06106 6.7 106

Thermal conductivity (erg/(cm s K)) 1.4 105 4 106 1.4 105 1.4 105

Total thickness (cm) variable variable

1419 n 2419 n

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Ideas behind calculations

• Fixed total mirror mass = 40 Kg.

• The beam radius is dynamically adjusted to maintain a fixed diffraction loss = 1ppm (clipping approximation).

• The mirror thickness is also dynamically adjusted as a function of the mirror radius in order to maintain the total 40 Kg mass fixed.

• Calculation at the frequency 100 Hz

Noise TE-s 1

f

Noise B-s / B-c 1

f

11

UW sdiss 2

masstest

ijij dVU 21

masstest

diss dVTT

W2

21

C

TYT

C

rt tbeam rr

,,,, 21

z

u

r

u

z

u

r

u

r

u rzrz

zzz

rrrr

zzrrrzrziiii ,2,2

Substrate Brownian noise

Substrate thermoelastic noise

12

ccdiss UW 2 dUU cc

dSU cij

cijSc 2

1

)0()0()0( zzz zzzzcc

rrrrc

zzcc

rrcc

rzc

crzc

iic

cc

ciic ,2,2

0crz

Coating Brownian noise

Boundary condition

13

beamt rrd

tC

TYT

zK

t )21()(

2

2

cs,

iTKi )(

dz

ss

ccdzsc

Hz

s

z

c

z

TK

z

TKTT

z

T

z

T

,,0,00

cc

V

cs

s

V

sdiss dV

z

T

TdV

z

T

TW

cs

22

Coating thermoelastic noise

at the surface

Boundary condition

14

Results for Gaussian and Mesa beam

Substrate Thermoelastic

Coating Brownian

Substrate Brownian

Coating Thermoelastic

Gaussian beam

Mesa beam

FS

CB 1.7

CT 1.7

SB 1.55

ST 1.92

MBX

GBX SS

S

CB 1.6

CT 1.5

SB 1.4

ST 1.4

MBX

GBX SS

15

Gaussian beam

Mesa beam

Comparison between Gaussian and Mesa beam

Gain factor Gain factor6.1 1.7

4.222 Ha6.22 Ha

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GB MB

NS-NS range

177 Mpc

228 Mpc

Sensitivity improvement

17

2

22242

2

02

222 )(~

1

12

)2(

4)(

qg

dqqK

qdqqdq

C

KTkS z

beffX

0 0 )()(2)(~ rqJrfdrrqg

DrforDrforD

rfFT 0,1

)(2

3)100( HzfS

SFTX

GBX

04wD

cmw 6

cmw 6.20

dT

dn

)(4

)(22

21

2121

nn

nneff

Coating Thermo-refractive noise estimation

• Infinite mirrors

• Perfect square beam

18

Finite size test mass correction for Gaussian Beam

Substrate Thermoelastic

Coating Brownian

Substrate Brownian

Coating Thermoelastic

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5ppm Diff. loss