Mirror Thermal Noise: Gaussian vs Mesa beams
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Transcript of Mirror Thermal Noise: Gaussian vs Mesa beams
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Mirror Thermal Noise: Gaussian vs Mesa beams
J.Agresti, R. DeSalvo
LIGO-Virgo Thermal Noise Workshop
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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
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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
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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.
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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)
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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
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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.
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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
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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
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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
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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
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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
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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
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Coating thermoelastic noise
at the surface
Boundary condition
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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
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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
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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
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Finite size test mass correction for Gaussian Beam
Substrate Thermoelastic
Coating Brownian
Substrate Brownian
Coating Thermoelastic
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5ppm Diff. loss