1. 1 Gravitational wave interferometer OPTICS François BONDU CNRS UMR 6162 ARTEMIS, Observatoire de la Côte d’Azur, Nice, France EGO, Cascina, Italy May 2006 Fabry-Perot cavity in practice Rules for optical design Optical performances

2. 2 Contents I. Fabry-Perot cavity in practice Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront Length / Frequency measurement: cavity transfer function II. Rules for gravitational wave interferometer optical design Optimum values for mirror transmissions “dark fringe”: contrast defect “Mode Cleaner” III. Optical performances Actual performances: Mirror metrology Optical simulation Accurate in-situ metrology

3. 3 Michelson configuration at dark fringe + servo loop to cancel laser frequency noise VIRGO optical design Slave laser Master laser Fabry-Perot cavity to detect gravitational wave Suspended mirrors to cancel seismic noise L=3 km Long arms to divide mirror and suspension thermal noiseRecycling mirror to reduce shot noise Input > to filter out input beam jitter and select mode L=144m Output Mode Cleaner to filter output mode

4. 4 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors 1. Fabry-Perot cavity: A. parameters REFLECTIONTRANSMISSION Can we understand these shapes?

5. 5 Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Mirror 1 Mirror 2 E in E sto E trans E ref E rt = r 1 P -1 r 2 P E sto 1. Fabry-Perot cavity: A. parameters

6. 6 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors E rt = r 1 P -1 r 2 P E sto Round trip “losses” 1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity

7. 7 1. Fabry-Perot cavity: A. parameters SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors E rt = r 1 P -1 r 2 P E sto Free spectral range Period : Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity

8. 8 1. Fabry-Perot cavity: A. parameters SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Recycling gain RESONANCE CONDITION Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity

9. 9 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors RESONANCE CONDITION Suppose now Cavity pole 1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity

10. 10 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Finesse 1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity

11. 11 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors on resonance reflectivity 1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity

12. 12 2nd order In T+P 1st order in T+P Finesse On resonance reflection transmission 1. Fabry-Perot cavity: A. parameters

13. 13 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors 1. Fabry-Perot cavity: A. parameters T1 = 12% T2 = 5% L = 0 (finesse = 35) REFLECTIONTRANSMISSION

14. 14 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer 1. Fabry-Perot cavity: B. Matching Optimal coupling Over-coupling Under-coupling

15. 15 Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Frequency/Length tuning 1. Fabry-Perot cavity: B. Matching

16. 16 Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer NON-SCALAR MODEL: 1. Fabry-Perot cavity: B. Matching Mirror 1 Mirror 2 E in E sto E trans E ref E in (x,y) ; E sto (x,y) ; r 1, P, r 2 are operators z axis E rt = r 1 P -1 r 2 P E sto

17. 17 Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer NON-SCALAR MODEL: 1. Fabry-Perot cavity: B. Matching E sto (x,y) = k E in (x,y) (k complex number) E sto E in Wavefront matching: Superpose angles and lateral drifts of incoming and resonating beam >

18. 18 NON-SCALAR MODEL: 1. Fabry-Perot cavity: B. Matching E sto (x,y) = k E in (x,y) (k complex number) E sto E in Wavefront matching: Superpose beam positions and beam widths > Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer

19. 19 NON-SCALAR MODEL: Definition of beam coupling: Round trip coupling losses: Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer  Too small mirror diameters “clipping”  imperfect surface: local defects, random figures 1. Fabry-Perot cavity: B. Matching

20. 20 NON-SCALAR MODEL: Definition of stability: Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer Definition of stability in case of perfect surface figures: 1. Fabry-Perot cavity: B. Matching

21. 21 1. Fabry-Perot cavity: B. Matching Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer Charles Fabry (1867-1945) Alfred Perot (1863-1925) Amédée Jobin (mirror manufacturer) (1861-1945) Gustave Yvon (>1911) Marseille – beginning of 20th century “Les franges des lames minces argentées”, Annales de Chimie et de Physique, 7e série, t12, 12 décembre 1897 “A taste of Fabry and Perot’s Discoveries, Physica Scripta, T86, 76-82, 2000

22. 22 Impedance matching Frequency/length tuning (“lock”) Wavefront matching alignment beam size / position surface defects - stability The Fabry-Perot interferometer 1. Fabry-Perot cavity: B. Matching

23. 23 Phase modulated laser: mphase modulation index fmmodulation frequency SB- C SB+ 1. Fabry-Perot cavity: C. measurement

24. 24 error signal: Does not provide information about frequency behavior once locked 1. Fabry-Perot cavity: C. measurement

25. 25 Modulated laser + measurement line: nphase modulation index fnmodulation frequency SB- C SB+ f << FSR, f ≠ fm This pole 1. Fabry-Perot cavity: C. measurement

26. 26 Contents I. Fabry-Perot cavity in practice Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront Length / Frequency measurement: cavity transfer function II. Rules for gravitational wave interferometer optical design Optimum values for mirror transmissions “dark fringe”: contrast defect “Mode Cleaner” III. Optical performances Actual performances: Mirror metrology Optical simulation Accurate in-situ metrology

27. 27 2. Optical design: A. mirror transmissions Fabry-Perot cavity with Rmax transmissions as end mirrors Virgo mirrors: L RT ~500 ppm, G cavity ~ 32  reflectivity defect 1.5% Was estimated 1-5 % at design Have as much as possible power on beamsplitter  Match “losses” of cavities with recycling mirror Was estimated 8 % at design (5.5 % recent refit)

28. 28 Michelson simple : BS laser Pin Pout P max, P min = P out On black and white fringes 2. Optical design: B. dark fringe

29. 29 Slave laser Master laser L=3 km Input > to filter out input beam jitter and select mode L=144m Output Mode Cleaner to filter output mode 2. Optical design: C. Mode Cleaners

30. 30 Detection Output Mode-Cleaner Output Mode Cleaner on Suspended Bench Photodiodes on Detection Bench Beam

31. 31 Contents I. Fabry-Perot cavity in practice Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront Length / Frequency measurement: cavity transfer function II. Rules for gravitational wave interferometer optical design Optimum values for mirror transmissions “dark fringe”: contrast defect “Mode Cleaner” III. Optical performances Actual performances: Mirror metrology Optical simulation Accurate in-situ metrology

32. 32 Measured optical parameters 16.7 W 7.1 W F = 49±0.5 F = 51 ±1 G carrier = 30-35 (exp. 50) G SB ~ 20 (exp. 36) 1 – C < 10 -4 1 – C = 3.10 -3 (mean) Slave laser Master laser 1 W T=10% III. Optical performances Losses in input Mode Cleaner? Recycling gain? Arm finesses?

33. 33 Mirror metrology reproducibility 0.4 nm; step 0.35 mm resolution 30 ppb/cm // 20 ppb resolution of a few ppm transmission map Before and/or after the coating process, maps are measured: - Mirror surface map (modified profilometer) - bulk and coating absorption map (“mirage” bench) - scatter map (commercial instrument) - transmission map (commercial instrument) - local defects measurements - birefringency The VIRGO large mirrors: a challenge for low loss coatings, CQG 2004, 21 Instruments: ESPCI, Paris Coating, 140 m 2 room class 1: LMA, Lyon Scatterometer CASI 400x400mm Micromap 400x400 mm (local defects) Absorption Photothermal Deflection System Phase shift interferometer

34. 34 Diam35 cm Rms 2.3 nm p-p 11.5 nm Surface maps Ex: a large flat mirror -Good quality silica - Good polishing - Control of coating deposition (DIBS) with no pollutants - Surface correction III. Optical performances

35. 35 TWO optical programs: - One that propagates wavefront with FFT - One that decomposes beams on TEM HG(m,n) base - Check out cavity visibility (total losses) - Check out expected recycling gain, for varying radii of curvature - Check out expected contrast defect - Check out modulation frequency - Improve interferometer parameters… Optical simulation III. Optical performances

36. 36 Scalar defectsMapsMaps+thermal Opt mod index0.0680.172±0.0010.215 ±0.001 Opt demod phase02 ±017 ±1 Finesse N49.2649.1 ±0.249.3 ±0.2 Finesse W49.7949.6 ±0.249.7 ±0.2 dF/F [%]0.270.23 ±0.120.24 ±0.12 Asymmetry [%]1.051.00 ±0.32.78 ±0.5 Stored power N [kW]15.3810.82 ±011.15 ±0 Lost power N [W]0.234.11 ±13.70 ±1 Surtension N31.3731.18 ±0.0231.15 ±0.02 Stored power W [kW]15.5510.91 ±011.27 ±0.3 Lost power W [W]0.196.05 ±0.025.85 ±0.04 Surtension W31.7031.42 ±0.0131.48 ±0.1 Carrier power on BS [W]978.5684.5 ±0.5725.1 ±2 Sideband power on BS [W]1.708.56 ±0.110.9 ±0.2 Reflected carrier [W]17.848.42 ±0.019.82 ±0.08 Reflected sb [W]0.0270.24 ±00.26 ±0.01 CITF surtension Carrier49.0434.74 ±0.0337.10 ±0.08 CITF surtension SB36.4929.01 ±0.0224.0 ±0.1 Transmitted (detected) carrier [mW]0.064 (0.064)359 ±6 (1.6 ±0)324 ±40 (3.5 ±0.1) Transmitted (detected) sb [mW]18.7 (17.9)93.0 ±0.8 (70.0 ±1)125 ±2 (100 ±2) Sensitivity [*1E-23]2.482.87 ±0.012.96 ±0.02 Optical program: typical results (Modal simulation)

37. 37 Example: Virgo simulation with surface maps and with an incoming field of 20W Contrast defect= 0.94% North arm amplification = 31.65 West arm amplification = 32.06 Recycling gain = 34.56 III. Optical performances

38. 38 Details at F FSR Fabry-Perot cavity transfer function measurements Fit values with 95% confidence interval: fp = 479 +/- 3.3 Hz fz = -177 +/- 2.2 Hz FSR = 1044039 +/- 2.2 Hz L = 143.573326 +/- 30  m Error bars: from measurement errors, Not for constant biases. (fit both real and imaginary parts simultaneously) III. Optical performances

39. 39 Input Mode Cleaner Losses T=2427 ppm T=2457 ppm T = 5.7 ppm Roud-trip losses: Computed from mirror maps: 115 ppm From measurements: 846 +/- 5 ppm Mirror transmission measurements + transfer function details measurements => Mode mismatching 17% => Cavity transmissitivity for TEM00 83% III. Optical performances (september 2005)

40. 40 Contents I. Fabry-Perot cavity in practice Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront Length / Frequency measurement: cavity transfer function II. Rules for gravitational wave interferometer optical design Optimum values for mirror transmissions “dark fringe”: contrast defect “Mode Cleaner” III. Optical performances Actual performances: Mirror metrology Optical simulation Accurate in-situ metrology