1. Janyce Franc-09.06.091 Effect of Laguerre Gauss modes on thermal noise Janyce Franc, Raffaele Flaminio, Nazario Morgado, Simon Chelkowski, Andreas Freise, Stefan Hild WP3 meeting-Paris

2. Janyce Franc-09.06.092 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

3. Janyce Franc-09.06.093 Motivations and objectives

4. Janyce Franc-09.06.094 Thermal noise in future GW detectors The ideas to reduce thermal noise : -Arm lengths -New coating materials -Cooling mirror @ cryogenic T° -Change beam mode and beam size All these noise sources are at least at some frequencies above the ET target Frequencies of interest : 10-1000 Hz

5. Janyce Franc-09.06.095 LG00LG33 Advantages : Best power distribution on the mirror surface -Lower Thermal Noise -Lower Thermal Lensing Motivation for using LG modes 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.

6. Janyce Franc-09.06.096 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 high- reflection 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

7. Janyce Franc-09.06.097 Introduction CoatingsStandard : SiO 2 -Ta 2 O 5 (HL) 19 HLL : Transmission 4 ppm SubstrateSilicon Temperature10K Mirror dimensionDiameter : 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 consideredFinite and infinite mirror

8. Janyce Franc-09.06.098 Parameters values @ 10K SUBSTRATECOATING  s (K -1 ) 4.85 10 -10 σsσs0.2205  s (W.m -1.K -1 ) 2325 Cs (J.K -1.Kg -1 )2.76 10 -1  s (Kg.m -3 ) 2330 ss 0.5 10 -9 Ys (Pa)162.4 10 -9  L (K -1 ) -0.25 10 -6  H (K -1 ) 3.6 10 -6 σLσL0.159 σHσH0.23  L (W.m -1.K -1 ) 0.13  H (W.m -1.K -1 ) 0.4 CL (J.K -1.Kg -1 )4 CH (J.K -1.Kg -1 )3.17  L (Kg.m -3 ) 2200  H (Kg.m -3 ) 6850 nL1.44876 nH2.06 YL (Pa)60 10 -9 YH (Pa)140 10 -9

9. Janyce Franc-09.06.099 Simulations

10. Janyce Franc-09.06.0910 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. REDUCTION FACTORS (Vinet) Fini(LG00)/Fini(LG33)1.71 Infini(LG00)/Infini(LG33)1.62  SiO 2 = 5.10 -5  Ta 2 O 5 =2 10 -4

11. Janyce Franc-09.06.0911 Substrate Brownian Noise Substrate Brown. Noise is reduced in a finite mirror AND decreases with a LG33 beam. REDUCTION FACTORS (Vinet) Fini(LG00)/Fini(LG33)1.97 Infini(LG00)/Infini(LG33)1.40

12. Janyce Franc-09.06.0912 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

13. Janyce Franc-09.06.0913 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 [Cerdonio et al. 2001] For LG00 mode :

14. Janyce Franc-09.06.0914 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

15. Janyce Franc-09.06.0915 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. Noise1.7 Sub. Brownian Noise1.97 Sub. TE Noise2.1 @ high frequencies (TBC)

16. Janyce Franc-09.06.0916 Conclusion Reduction factor : 1.71 @ 10K, with LG33 modes compare to gaussian beam the total thermal is reduced. 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) T : 300K Arm length : 3 km Beam : 6 cm

17. Janyce Franc-09.06.0917 Thank you for your attention