Khalili Etalons for the Einstein GW Telescope

Transcription

1 Khalili Etalons for the Einstein GW Telescope A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Moscow State University, University of Jena, University of Glasgow and Caltech Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 1 / 35

2 Overview of this presentation 1 Motivation for Khalili Etalons (Stefan) 2 Noise contributions for a Khalili cavity, i.e. no coupling through the substrate considered (Ronny) 3 How to calculate the noise couplings inside an etalon.? Example: Coating Brownian noise for finite size etalon (Kentaro) 4 Detailed calculations of all noise sources inside a Khalili etalon with mechanical coupling of front and back surface (Sergey and Alexey) 5 Confirmation of calculations by FEM results 6 Summary and outlook (Stefan) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 2 / 35

3 Motivation: Coating Brownian Noise in ET-C ET-C is limited by Coating Brownian noise around 100Hz. ET-C-HF detector: Room temperature, maximised beam radius (12cm) and LG33.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 3 / 35

4 How can we further decrease coating noise w/o cooling? Crude estimate of Coating Brownian Reduction from Khalili cavities From talk at WP3 meeting in Glasgow, December 2009: Assuming a reflectivity of about 0.7 for the first Khalili mirror gives: Reduce the coating noise of the end mirrors by a factor 2.7 Reduce the overall coating Brownian contribution (full IFO) by a factor 1.5.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 4 / 35

5 How much can we win with Khalili cavities? From S.Hild: Khalili etalons for ET, presentation at ET-WP3 meeting in Glasgow, December A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 5 / 35

6 Motivation of using Khalili Etalons instead of Khalili cavities? One potential disadvantage of Khalili cavities for ET is that it might be hard to control them in case the Khalili cavities are short (a few meters) compared to the arm cavities (10 km). This disadvantage might potentially be overcome by using an Khalili etalon instead of of two independent mirrors. However, nothing comes for free! If we make use of an etalon, then the coating noise benefit is reduced, because there is a mechanical coupling of noise components from the back surface to the front surface and vice versa. The questions we want to answer in this presentation Which are the dominant noise contributions of a Khalili etalon? How can we actually take the coupling through the substrate into account? How much worse is a Khalili etalon compared to a Khalili cavity?.gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 6 / 35

7 Overview of this presentation 1 Motivation for Khalili Etalons (Stefan) 2 Noise contributions for a Khalili cavity, i.e. no coupling through the substrate considered (Ronny) 3 How to calculate the noise couplings inside an etalon? Example: Coating Brownian noise for finite size etalon (Kentaro) 4 Detailed calculations of all noise sources inside a Khalili etalon with mechanical coupling of front and back surface (Sergey and Alexey) 5 Confirmation of calculations by FEM results 6 Summary and outlook (Stefan) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 7 / 35

8 Dummy set of parameters used for our analysis We decided to do our analysis with a specific set of parameters: The first end mirror surface carries 3 coating doublets leading to a reflectivity of about 0.7 and optical power inside the Khalili cavity/etalon of 300 kw for ET-C HF. parameter bulk SiO 2 layer Ta 2 O 5 layer thickness 30 cm m front m front m back m back n β = dn dt /K /K /K ρ 2202 kg/m kg/m kg/m 3 Y 72 GPa 72 GPa 140 GPa ν α /K /K /K κ 1.38 W/m K 1.38 W/m K 0.46 W/m K C 746 J/kg K 746 J/kg K 306 J/kg K φ Table: Material parameters. The layer thicknesses are total thicknesses summing over all λ/4 layers. A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 8 / 35

9 Formulas for the noise contributions Brownian bulk thermal noise Thermo-elastic bulk thermal noise S x,b (f, T ) = 2k BT (1 ν 2 b ) π 3/2 Y b fw φ b, (1) S TE,b (f, T ) = 4k BT 2 α 2 b (1 + ν b) 2 κ b π 5/2 (ρ b C b ) 2 w 2 f 2 (2) Thermo-refractive bulk thermal noise S TR,b (f, T ) = Brownian coating thermal noise 4k BT 2 β 2 b hκ b π 3 (ρ b C b ) 2 w 4 f 2 (3) S x,c (f, T ) = 2k BT (1 ν 2 b ) π 3/2 Y b fw φ c, (4) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 9 / 35

10 Formulas for the noise contributions (continued) Thermo-elastic coating thermal noise S TE,c (f, T ) = 8k BT 2 π 2 f t l + t h w 2 Thermo-refractive coating thermal noise S TR,coating (f, T ) = Summation of the different noise contributions α b ρ c C c (ρ b C b ) 2 (1 + ν b) 2 2 g(ω) (5) 2k B T 2 β 2 eff λ2 π 3/2 κ b ρ b C b w 2 f, (6) with S Khalili (f, T ) = S front (f, T ) + TF 2 S back (f, T ). (7) S front = S x,b + S TE,b + S x,c,front + S TE,c,front + S TR,c,front (8) S back = S x,b + S TE,b + S x,c,back + S TE,c,back + S TR,c,back + S TR,b. (9) and TF being the optical coupling transferfunction (=0.102) for the second mirror of the Khalili cavity, determined by a numerical simulation (Finesse)..Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 10 / 35

11 Summary of the different noise sources in the Khalili-etalon under investigation. thermal noise (m/!hz) Brownian bulk TE bulk TR bulk Brownian coating front Brownian coating back TE coating front TE coating back TR coating front TR coating back std mirror frequency (Hz) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 11 / 35

12 noise standard mirror standard mirror Khalili-etalon Khalili-etalon (absolut) (relativ) (absolut) (normalized) bulk Brownian m/ Hz m/ Hz 0.52 TE m/ Hz m/ Hz 0.04 TR m/ Hz front coating Brownian m/ Hz m/ Hz 1.00 TE m/ Hz m/ Hz 0.07 TR m/ Hz m/ Hz 0.14 back coating Brownian m/ Hz 0.23 TE m/ Hz 0.04 TR m/ Hz 0.01 Table: Numerical summary of the thermal noise contributions. All contributions are extracted from the total thermal noise. The effect of transfer function is included. The relative numbers are normalised to the largest noise source which is in both cases the Brownian thermal noise of the front surface. A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 12 / 35

13 Overview of this presentation 1 Motivation for Khalili Etalons (Stefan) 2 Noise contributions for a Khalili cavity, i.e. no coupling through the substrate considered (Ronny) 3 How to calculate the noise couplings inside an etalon? Example: Coating Brownian noise for finite size etalon (Kentaro) 4 Detailed calculations of all noise sources inside a Khalili etalon with mechanical coupling of front and back surface (Sergey and Alexey) 5 Confirmation of calculations by FEM results 6 Summary and outlook (Stefan) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 13 / 35

14 Mechanical separation Khalili-Cavity Mechanical separation Mechanical separation Khalili-Etalon K-cavity K-cavity K-etalon K-etalon n1 n2 n1 n2 n1 n2 n1 n2 TN = 1 x (n1 + TFBtoF x n2 ) TN = 1 x n1 + 2 x n2 TN = 1 x (n1 + TFBtoF x n2 ) TN = 1 x n1 + 2 x + 2 x (n2 + TFFtoB x n1 ) n2 complete mechanical separation + 2 x (n2 + TFFtoB x n1 ) complete mechanical separation TN partial = εmechanical 1 (n 1 + TFseparation BtoF n 2 ) (11) TN = ε 1 n 1 + ε 2 n 2 (10) partial mechanical separation Terms with same nj should be coherently +ε 2 (n 2 + added TF FtoB n 1 )(12) complete Terms mechanical with same separation nj should be coherently added Derivation of TN is different from that with a simple mirror. partial mechanical separation Derivation of TN is different from that with a simple mirror. Terms with same n j should be coherently added. Derivation of TN is different from that with a simple mirror..gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 14 / 35

15 Review of TN calculation 1. Fluctuation-dissipation Theorem thermal energy kinetic energy = work by external force dissipated energy Left-hand side = real world, right-hand side = imaginary world thermal energy k B T kinetic energy mx 2 TN Ω2 /2 work by external force F 0 x 0 (with x 0 = F 0 /(mω 2 )) dissipated energy W /ω. = S x = 8k BTW Ω 2 F 2 0 (13) (14) 2. Levin s method Gaussian force at z = 0 (boundary condition) = Elastic equation gives dissipation W..Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 15 / 35

16 Levin s method with double pump Levin s method with double pump front surface back surface Finite-size-mirror analysis Finite-size-mirror analysis New New boundary condition a at z = h Integrate W in each surface Integrate W in each surfac Motion of a surface appearing on each surface is coherently added Motion of a surface appearing on each surfa is coherently added. Khalili Etalon 4 ET / 35.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin (Moscow State University, University of Jena, University of Glasgow and Caltech)

17 Calculation results for Coating Brownian noise Back-surface coating Brownian noise Calculation results Total coating Brownian noise of Khalili Etalon Calculation results back-surface thermal noise total KE thermal noise back-surface The thermal ticker The the noise thicker substrate, the the mirror, more mechanical the more separation mechanical separation Default h Default = 30 cmh=30cm Etalon>> should Etalon better should be 40better cm+ be 40cm+ total KE thermal noise.gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 17 / 35

18 Overview of this presentation 1 Motivation for Khalili Etalons (Stefan) 2 Noise contributions for a Khalili cavity, i.e. no coupling through the substrate considered (Ronny) 3 How to calculate the noise couplings inside an etalon? Example: Coating Brownian noise for finite size etalon (Kentaro) 4 Detailed calculations of all noise sources inside a Khalili etalon with mechanical coupling of front and back surface (Sergey and Alexey) 5 Confirmation of calculations by FEM results 6 Summary and outlook (Stefan) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 18 / 35

19 Numerical noise spectra calculation Energy dissipated through Brownian and TE mechanism: xi+1 U B,i = πφ i dz x i R ( ) 2 Y 0 α xi+1 0 U TE,i = 2πκ 0 T dz (1 2ν 0 )C 0 ρ 0 Brownian, TE and TR noise spectra 0 (E rr T rr + E φφ T φφ + E zz T zz + 2E rz T rz )r dr S B,i = 8kT ω U B,i x i R 0 S TE,i = 8kT ω 2 ( ) 2r (Err + E φφ + E zz ) dr U TE,i S TR,s = ɛ 2 16kT 2 β0 2hκ 0 2 πρ 2 0 C 0 2w S 4 ω 2 TR,c,CM = 2 2kT 2 βeff 2 λ2 π κ 0 ρ 0 C 0 w 2 ω S TR,c,KE = (1 + ɛ 2 2) 2 2kT 2 β 2 eff λ2 π κ 0 ρ 0 C 0 w 2 ω β eff = 1 β 1 n2 2 + β 2n1 2 4 n1 2 + β2 2 n2 2 4n2 2.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 19 / 35

20 A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 20 / 35

21 Noises at 100 Hz Noise Source CM KE substrate Brownian m/ Hz m/ Hz TE m/ Hz m/ Hz TR m/ Hz total coating Brownian m/ Hz m/ Hz TE m/ Hz m/ Hz TR m/ Hz m/ Hz front coating Brownian m/ Hz m/ Hz TE m/ Hz m/ Hz TR m/ Hz m/ Hz rear coating Brownian m/ Hz TE m/ Hz TR m/ Hz.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 21 / 35

22 Influence of coupling and mirror finiteness Noises at 100 Hz Here we provide comparison of numerical results with account of coupling and and without it. with coupling no coupling noise source CM KE CM KE substrate Brownian TE TR total coating Brownian TE TR The biggest difference is in coating TE noise: coupling lessens it 150 times for CM and 71 times for KE..Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 22 / 35

23 Total KE noise gain comapred to CM Total noise gain at 100 Hz S tot,cm S tot,ke = It comes mainly from coating Brownian noise which is the main part of noise budget. Brownian coating noise gain at 100 Hz S B,c,CM S B,c,KE = Other noises are reduced more slightly. Maximum total noise gain Interesting that maximum total noise reduction is: Stot,CM /S tot,ke = at 201 Hz..Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 23 / 35

24 Quick gain estimation Omit SiO 2 layers taking into account only Ta 2 O 5 layers. CM: number of Ta 2 O 5 layers is N 0 = 20 S B,c,CM N 0 = 20 KE: numbers of Ta 2 O 5 layers are N 1 = 3 and N 2 = 17 in front and rear coatings Energy coupling of rear and front surface is evaluated to be 0.2, i.e. energy from rear coating layer comes with coefficient 0.2 while from front with 1: S B,c,KE N N 2 = = 6.4 Ratio CM to KE means gain S B,c,CM 20 = S B,c,KE 6.4 = That is a good coincidence with exact ratio Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 24 / 35

25 Overview of this presentation 1 Motivation for Khalili Etalons (Stefan) 2 Noise contributions for a Khalili cavity, i.e. no coupling through the substrate considered (Ronny) 3 How to calculate the noise couplings inside an etalon? Example: Coating Brownian noise for finite size etalon (Kentaro) 4 Detailed calculations of all noise sources inside a Khalili etalon with mechanical coupling of front and back surface (Sergey and Alexey) 5 Confirmation of calculations by FEM results 6 Summary and outlook (Stefan) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 25 / 35

26 FE calculation of Brownian noise in a Khalili-cavity idea: finite element calculation as check for analytical model Levin s approach virtual pressure on surface p(r) = F 0 f (r) FE computation of total elastic energy E tot structural damping E diss = φe tot spectral noise energy density S x (ω) = kt E diss π 2 f F0 2 comparison of elastic energy is enough Symmetry radialsymmetric virtual pressure radialsymmetric geometry isotropic materials simplification from 3D to radialsymmetric 2D-analysis A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 26 / 35

27 test on substrate Brownian noise crucial for FE calculations: test problems with known analytical solutions substrate Brownian noise of finite sized test masses analytical solution by Liu and Thorne 2000 E ITM tot = 2 π 1 σ 2 Yw 0 E FTM tot results Liu and Thorne E tot = J = C 2 FTME ITM tot agreement of better than 1% diameter 50 cm height 30 cm laser beam radius w 0 90 mm frequency 100 Hz silicon (111) Young s modulus Y 188 GPa Poisson s ratio σ 0.23 density 2331 kg/m 3 Table: properties of the test substrate COMSOL E tot = J A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 27 / 35

28 test on coating Brownian noise etalon geometry cylindrical substrate with height of some 10 cm two coating stacks with height of some µm coating is negligible for elastic response numerical receipt simulate elastic response of the substrate to virtual pressure estimate coating response via boundary conditions σ xz = σ xz, σ yz = σ yz, σ zz = σ zz u xx = u xx, u yy = u yy, u xy = u xy compute coating s elastic energy w = 1 2 σ iju ij.gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 28 / 35

29 test on coating Brownian noise comparison of FE simulation of coatings with finite thickness model of infinitely small coating finite coating FE results coating thickness [m] elastic energy [J] infinite small coating w inf = J/m deviations of under 1% for coating thickness <1 mm suited for Khalili-etalon.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 29 / 35

30 elastic energies in a Khalili-etalon calculation for a given mirror geometry parameters of the cavity diameter 62 cm height 30 cm laser beam radius 120 mm frequency 100 Hz material fused silica Young s modulus 72 GPa Poisson s ratio 0.17 density 2202 kg/m 3 silica coating Young s modulus 72 GPa Poisson s ratio 0.17 tantala coating Young s modulus 140 GPa Poisson s ratio 0.23 Figure: sketch of simulated geometry, ɛ 1=0.91, ɛ 2=0.09.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 30 / 35

31 elastic energies in a Khalili-etalon FE results for the Khalili cavity Figure: z-displacement Figure: elastic strain energy A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 31 / 35

32 results for FE calculation comparison with analytical calculation yield front coatings material w analytical [J/m] w FE [J/m] silica tantala back coatings material w analytical [J/m] w FE [J/m] silica tantala successful confirmation of the analytical model A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 32 / 35

33 Overview of this presentation 1 Motivation for Khalili Etalons (Stefan) 2 Noise contributions for a Khalili cavity, i.e. no coupling through the substrate considered (Ronny) 3 How to calculate the noise couplings inside an etalon? Example: Coating Brownian noise for finite size etalon (Kentaro) 4 Detailed calculations of all noise sources inside a Khalili etalon with mechanical coupling of front and back surface (Sergey and Alexey) 5 Confirmation of calculations by FEM results 6 Summary and outlook (Stefan) A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 33 / 35

34 Summary Khalili-etalons might be an alternative to Khalili-cavities! A Khalili-etalon has more noise due to the mechanical coupling of front and back surface. A method based on Levin was developed to calculate the coating Brownian noise contributions of the front and back surface, taking the mechanical coupling into account. Finally we extended this method all relevant thermal noise contributions of the etalon. We compared the analytical derivation with FEM and found good agreement. For our dummy example the Khalili cavity gives an overall noise reduction of a factor 2.52, while the Khalili etalon would give a reduction factor of A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 34 / 35

35 Outlook Our next steps are: continue our team effort... to properly write this up... :) to perform an optimisation of the etalon parameters (coating distribution and substrate thickness), i.e. trade-off between thermal lensing and thermal noise reduction. to start investigating control issues of Khalili cavities and Khalili etalons A.Gurkovski, D.Heinert, S.Hild, R.Nawrodt, K.Somiya and S.Vyatchanin Khalili (Moscow Etalon State 4 ET University, University of Jena, University of Glasgow and Caltech) 35 / 35