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

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

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

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

1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Ein Esto Etrans Eref Mirror 1 Mirror 2 Ert = r1 P-1 r2 P Esto

1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Ert = r1 P-1 r2 P Esto Round trip “losses”

1. Fabry-Perot cavity: A. parameters Round Trip Losses Free Spectral Range Recycling gain Cavity Pole Finesse Cavity reflectivity SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Ert = r1 P-1 r2 P Esto Period: Free spectral range

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

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

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

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

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 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Optimal coupling Over-coupling Under-coupling

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 SCALAR MODEL: “plane waves” scalar transmissions, scalar losses of mirrors Frequency/Length tuning

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 NON-SCALAR MODEL: Ein Esto Etrans Eref z axis Mirror 1 Mirror 2 Ert = r1 P-1 r2 P Esto Ein(x,y) ; Esto(x,y) ; r1, P, r2 are operators

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 NON-SCALAR MODEL: Wavefront matching: Esto(x,y) = k Ein(x,y) (k complex number) Esto Ein Superpose angles and lateral drifts of incoming and resonating beam <<ALIGNMENT ACTIVITY>>

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 NON-SCALAR MODEL: Wavefront matching: Esto(x,y) = k Ein(x,y) (k complex number) Ein Esto Superpose beam positions and beam widths <<MATCHING ACTIVITY>>

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 NON-SCALAR MODEL: Definition of beam coupling: Round trip coupling losses: Too small mirror diameters “clipping” imperfect surface: local defects, random figures

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 NON-SCALAR MODEL: Definition of stability: Definition of stability in case of perfect surface figures:

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

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

1. Fabry-Perot cavity: C. measurement Phase modulated laser: SB- C SB+ m phase modulation index fm modulation frequency

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

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

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

2. Optical design: A. mirror transmissions Fabry-Perot cavity with Rmax transmissions as end mirrors Virgo mirrors: LRT ~500 ppm, Gcavity ~ 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)

2. Optical design: B. dark fringe Michelson simple : laser Pin BS Pmax, Pmin = Pout On black and white fringes Pout

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

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

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

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

Absorption Photothermal Deflection System Scatterometer CASI 400x400mm Mirror metrology Micromap 400x400 mm (local defects) 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 reproducibility 0.4 nm; step 0.35 mm resolution 30 ppb/cm // 20 ppb resolution of a few ppm transmission map Phase shift interferometer Instruments: ESPCI, Paris Coating, 140 m2 room class 1: LMA, Lyon The VIRGO large mirrors: a challenge for low loss coatings, CQG 2004, 21

Surface maps Good quality silica Good polishing Ex: a large flat mirror Good quality silica Good polishing Control of coating deposition (DIBS) with no pollutants - Surface correction Diam 35 cm Rms 2.3 nm p-p 11.5 nm III. Optical performances

Optical simulation 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… TWO optical programs: One that propagates wavefront with FFT One that decomposes beams on TEM HG(m,n) base III. Optical performances

Optical program: typical results (Modal simulation) Scalar defects Maps Maps+thermal Opt mod index 0.068 0.172±0.001 0.215 ±0.001 Opt demod phase 2 ±0 17 ±1 Finesse N 49.26 49.1 ±0.2 49.3 ±0.2 Finesse W 49.79 49.6 ±0.2 49.7 ±0.2 dF/F [%] 0.27 0.23 ±0.12 0.24 ±0.12 Asymmetry [%] 1.05 1.00 ±0.3 2.78 ±0.5 Stored power N [kW] 15.38 10.82 ±0 11.15 ±0 Lost power N [W] 0.23 4.11 ±1 3.70 ±1 Surtension N 31.37 31.18 ±0.02 31.15 ±0.02 Stored power W [kW] 15.55 10.91 ±0 11.27 ±0.3 Lost power W [W] 0.19 6.05 ±0.02 5.85 ±0.04 Surtension W 31.70 31.42 ±0.01 31.48 ±0.1 Carrier power on BS [W] 978.5 684.5 ±0.5 725.1 ±2 Sideband power on BS [W] 1.70 8.56 ±0.1 10.9 ±0.2 Reflected carrier [W] 17.84 8.42 ±0.01 9.82 ±0.08 Reflected sb [W] 0.027 0.24 ±0 0.26 ±0.01 CITF surtension Carrier 49.04 34.74 ±0.03 37.10 ±0.08 CITF surtension SB 36.49 29.01 ±0.02 24.0 ±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] 2.48 2.87 ±0.01 2.96 ±0.02

Virgo simulation with surface maps and with an incoming field of 20W 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

Fabry-Perot cavity transfer function measurements Details at FFSR 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 mm Error bars: from measurement errors, Not for constant biases. (fit both real and imaginary parts simultaneously) III. Optical performances

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

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