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Effect of Laguerre Gauss
modes
on thermal noise
Janyce Franc, Raffaele Flaminio, Nazario Morgado, Simon
Chelkowski, Andreas Freise, Stefan Hild
WP3 meeting-Paris
Janyce Franc-09.06.09
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Contents
1. Motivations
2. Objectives and parameters values
3. Simulations :
1. Coating Brownian Noise
2. Substrate Brownian Noise
3. Substrate Thermo-elastic Noise
4. Conclusion on Laguerre-Gauss modes
Janyce Franc-09.06.09
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Motivations and objectives
Janyce Franc-09.06.09
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Thermal noise in future GW detectors
All these noise sources are at least at some frequencies above the ET target
Frequencies of interest : 10-1000 Hz
The ideas to reduce thermal noise :
-Arm lengths
-New coating materials
-Cooling mirror @ cryogenic T°
-Change beam mode and beam size
Janyce Franc-09.06.09
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Motivation for using LG modes
Advantages :
Best power distribution on the mirror surface
-Lower Thermal Noise
-Lower Thermal Lensing
LG00
LG33
The following simulations have been done in the framework of xylophone
configuration (cf. R. Flaminio’s presentation).
The idea is to prove the efficiency of the combination of cryogenic
temperature and the use of Laguerre-Gauss modes to decrease the
thermal noise for future GW detector.
Janyce Franc-09.06.09
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References
J.-Y.Vinet : « Transverse Gaussian optical modes and thermal issues in
advanced Gravitational Wave Interferometric detectors » to be published in
« Living Reviews in General Relativity »
M. L. Gorodetsky « thermal noises and noise compensation in highreflection multilayer coating », Phys. Lett. A 372 (2008) 6813-6822
Cerdonio et al. « Thermoelastic effects at low temperatures and quantum
limits in displacement measurements » Phys. Review D, 63, 082003
Janyce Franc-09.06.09
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Introduction
Coatings
Standard : SiO2-Ta2O5
(HL)19HLL : Transmission 4 ppm
Substrate
Silicon
Temperature
10K
Mirror dimension
Diameter : 62 cm
Thickness : 30 cm
Beam dimensions
Ratio insuring 1 ppm diffraction losses
versus order of the LG mode
LG33 : w = a/4.3 = 7.2 cm
LG00 : w=a/2.6 = 11.9 cm
2 cases considered
Finite and infinite mirror
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Parameters values @ 10K
SUBSTRATE
COATING
L (K-1)
-0.25 10-6
H (K-1)
3.6 10-6
σL
0.159
σH
0.23
L (W.m-1.K-1)
0.13
2330
H (W.m-1.K-1)
0.4
s
0.5 10-9
CL (J.K-1.Kg-1)
4
Ys (Pa)
162.4 10-9
CH (J.K-1.Kg-1)
3.17
L (Kg.m-3)
2200
H (Kg.m-3)
6850
nL
1.44876
nH
2.06
YL (Pa)
60 10-9
YH (Pa)
140 10-9
s (K-1)
4.85 10-10
σs
0.2205
s (W.m-1.K-1)
2325
Cs (J.K-1.Kg-1)
2.76 10-1
s
(Kg.m-3)
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Simulations
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Coating Brownian Noise
No significant difference between finite
and infinite mirror.
The Coat. Brown. noise decreases
with LG33.
The Gorodetsky formula leads to a
higher total thermal noise due to the
more detailed and complicated
formula.
SiO2= 5.10-5
Ta2O5=2 10-4
REDUCTION FACTORS (Vinet)
Fini(LG00)/Fini(LG33)
1.71
Infini(LG00)/Infini(LG33)
1.62
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Substrate Brownian Noise
REDUCTION FACTORS (Vinet)
Fini(LG00)/Fini(LG33)
1.97
Infini(LG00)/Infini(LG33)
1.40
Substrate Brown. Noise is reduced in a finite mirror
AND decreases with a LG33 beam.
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Substrate Thermoelastic Noise
with adiabatic assumption
These curves take into account the
adiabatic assumption.
With this mirror dimension, there is no
significant difference whatever the
calculation of substrate thermoelastic
noises :
-LG33 or LG00 larger beam
-Finite or infinite mirror
REDUCTION FACTORS (Vinet)
Fini(LG00)/Fini(LG33)
1.02
Infini(LG00)/Infini(LG33)
0.96
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The adiabatic limit at cryogenic temperature
The thermo-elastic noise takes into account the adiabatic assumption.
Temperature fluctuation stays homogeneous in time within the beam diameter 2w
For T<80K (R.Nawrodt-WP2 meeting-February 2009)
Silicon violates the adiabatic assumption
2
8
2 k BT r0
2
STE (f , T) 
 1  
 J()

2
For LG00 mode :

J() 


C

3 u 2 / 2
2
ue
du
dv
3 0  (u 2  v 2 )[( u 2  v 2 ) 2  2 ]
[Cerdonio et al. 2001]
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Substrate Thermoelastic Noise
without adiabatic assumption
-Perfect superposition of the curves of finite and
infinite mirrors
-At low frequencies, a LG33 beam
decreases the substrate thermoelastic
noise.
-At high frequencies, the thermal noise is the same
with and without adiabatic assumption.
REDUCTION FACTORS (From low to high frequencies)
Fini(LG00)/Fini(LG33)
2.10 (TBC) -1.00
Infini(LG00)/Infini(LG33)
1.99 (TBC) -1.00
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Summarize for finite mirror
Laguerre-Gauss modes give promising results.
All the thermal noises are reduced.
For a 62 cm diameter and 30 cm thick mirror dimension, we obtain :
Ratio LG00/LG33 for
finite mirror
Coat. Brown. Noise
1.7
Sub. Brownian Noise
1.97
Sub. TE Noise
2.1 @ high
frequencies (TBC)
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Conclusion
T : 300K
Arm length : 3 km
Beam : 6 cm
Thermal Noise for 3rd generation GWD
• Long interferometer arms : 10 km
• Temperature : 10K
• Large beams : 12 cm (gaussian beam) and 7,2 cm
(LG33 mode)

2 Sx input  Sx end
h
104

@ 10K, with LG33 modes compare
to gaussian beam the total thermal
is reduced.
Reduction factor : 1.71
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Thank you for your attention
Janyce Franc-09.06.09
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