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Gravitational Wave Interferometry

Gravitational Wave Interferometry. Jean-Yves Vinet ARTEMIS Observatoire de la Côte d’Azur Nice (France). Summary. Shot noise limited Michelson Resonant cavities Recycling Optics in a perturbed space-time Thermal noise Sensitivity curve. 0.0 Introduction. Virgo principle

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Gravitational Wave Interferometry

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  1. Gravitational Wave Interferometry Jean-Yves Vinet ARTEMIS Observatoire de la Côte d’Azur Nice (France) VESF School

  2. Summary • Shot noise limited Michelson • Resonant cavities • Recycling • Optics in a perturbed space-time • Thermal noise • Sensitivity curve VESF School

  3. 0.0 Introduction Virgo principle (LIGO as well) Resonant cavity Recycling cavity 20kW 20W 1kW 3km Laser recycler splitter photodetectorr VESF School

  4. 0.1 Introduction R. Weiss Electromagnetically coupled broadband gravitational antenna.Quar. Prog. Rep. in Electr., MIT (1972), 105, 54-76 The first idea MIT R.L. Forward Wideband laser-interferometer gravitational-radiation experiment.Phys. Rev. D (1978) 17 (2) 379-390 The first Experiment Hughes J.-Y. Vinet et al. Optimization of long-baseline optical interferometers for gravitational-wave detection.Phys. Rev. D (1988) 38 (2) 433-447 B.J. Meers Recycling in a laser-interferometric gravitational-wave detector.Phys. Rev. D (1988) 38 (8) 2317-2326 theory VESF School

  5. 1.1 Shot noise Detection of a light flux (power P) by a photodetector Integration time : Number of detected photons : In fact, n is a random variable, of statistical moments The photon statistics is Poissonian VESF School

  6. 1.2 Shot noise We can alternatively consider the power P(t) as a random Process, of moments May be viewed as the inverse of the bandwidth of the photodetector Is the spectral density of power noise VESF School

  7. 1.3 Shot noise In fact, the « one sided spectral density » is (white noise) The « root spectral density » is P=20W,  = 1.064 m VESF School

  8. 1.4 Michelson Mirror 2 b Splitter Light source A a Mirror 1 B photodetector VESF School

  9. 1.5 Michelson Amplitude reaching the photodetector: Detected power: Assume with Linearization in x VESF School

  10. 1.6 Michelson : tuning of the output fringe. VESF School

  11. 1.7 Michelson The signal must be larger than the shot noise fluctuations of of spectral density: Spectral density of signal: Signal to noise ratio: VESF School

  12. 1.8 Michelson VESF School

  13. 1.9 Michelson The interferometer must be tuned near a dark fringe. The optimal SNR is now The shot noise limited spectral sensitivity in x corresponds to SNR=1: For 20W incoming light power and a Nd:YAG laser : VESF School

  14. 1.10 Michelson For detecting GW: (L : arm length, 3 km) Increase ? Increase L ? VESF School

  15. 2.1 Resonant cavities Relative phase for reflection and transmission 1 T R A B VESF School

  16. 2.2 Resonant cavities The Fabry-Perot interferometer A E B L Intracavity amplitude: resonances Reflected amplitude: VESF School

  17. 2.3 Resonant cavities Finesse :  = FSR/Full width at half max Free Spectral Range linewidth f VESF School

  18. 2.4 Resonant cavities Free spectral range (FSR): Linewidth Being a resonance, Assume with Reduced detuning Total losses : Coupling coeff. VESF School

  19. 2.5 Resonant cavities A Fabry-Perot cavity B f f VESF School

  20. 2.6 Resonant cavities Instead of For a single round trip Effective number of bounces VESF School

  21. 2.7 Resonant cavities Gain factor at resonance for length 2L f VESF School

  22. 3.1 Recycling Perfectly symmetrical interferometer F splitter a F At the black fringe and cavity resonance: VESF School

  23. 3.2 Recycling Recycling mirror Mic Recycling gain Recycling cavity Recycling cavity at resonance: Optimal value of : Optimal power recycling gain: VESF School

  24. 3.3 Recycling The cavity losses are likely much larger than other losses Recycling gain limited by New sensitivity to GW 50 50 VESF School

  25. 3.4 Recycling The heuristic (naive) preceding theory is not sufficient. The spectral sensitivity is not flat (white). Efficiency is expected to decrease when the GW frequency is larger than the cavity linewidth. Rough argument : for high GW frequencies, the round trip duration inside the cavity may become comparable to the GW period, so that internal compensation could occur. Necessity of a thorough study of the coupling between a GW and a light beam. VESF School

  26. 4.1 Optics in a perturbed Space Time L t One Fourier component of frequency : VESF School

  27. 4.2 Optics in a perturbed Space Time If the round trip is a light ray’s t if then Creation of 2 sidebands VESF School

  28. 4.3 Optics in a perturbed Space Time GW Assume Round trip Linear transformation VESF School

  29. 4.4 Optics in a perturbed Space Time Operator « round trip » Any optical element can be given an associated operator of this type. Example: reflectance of a mirror VESF School

  30. 4.5 Optics in a perturbed Space Time Transmission of already existing sidebands New contribution to sidebands : GW! VESF School

  31. 4.6 Optics in a perturbed Space Time Evaluation of the signal-to-noise ratio for any optical setup amounts to a linear algebra calculation leading to the overall operator of the setup. The set of all operators having the structure form a non-commutative field (all properties of R except commutativity) VESF School

  32. 4.7 Optics in a perturbed Space Time Example of a Fabry-Perot cavity : reduced frequency detuning (wrt resonance) : reduced gravitational frequency VESF School

  33. 4.8 Optics in a perturbed Space Time In particular, at resonance Showing the decreasing efficiency at high frequency VESF School

  34. 4.9 Optics in a perturbed Space Time Computing the SNR GW carrier S Carrier + 2 sidebands Shot noise : proportional to (root spectral density) Signal : proportional to VESF School

  35. 4.10 Optics in a perturbed Space Time Computing the SNR Shot noise limited spectral sensitivity: General recipe: compute VESF School

  36. 4.11 Optics in a perturbed Space Time SNR for a Michelson with 2 cavities: SNR for a recycled Michelson with 2 cavities Optimal recycling rate: VESF School

  37. 4.12 Optics in a perturbed Space Time Spectral sensitivity : (SNR=1) VESF School

  38. 4.13 Optics in a perturbed Space Time Spectral sensitivity of a power recycled ITF VESF School

  39. 4.14 Optics in a perturbed Space Time Optimizing the finesse Increasing the finesse leads to A gain in the factor A narrowing of the linewidth of the cavitya lower cut-off An increase of the cavity losses There is an optimal value depending on the GW frequency But increasing the finesse is not only an algebrical game! (Thermal lensing problems) VESF School

  40. 4.15 Optics in a perturbed Space Time With optimal recycling rate VESF School

  41. Reasons for reading The VIRGO PHYSICS BOOK (Downloadable from the VIRGO site)  VESF School

  42. 4.15 Optics in a perturbed Space Time Other types of recycling: signal recycling (Meers) Narrowing the bandwidth Synchronous Recycling (Drever) FP Power recycler FP FP Ring cavity Signal recycler FP Signal extraction (Mizuno) Resonant (narrowband) broadband VESF School

  43. 4.16 Optics in a perturbed Space Time All these estimations of SNR were done in the « continuous detection scheme » . In practice, one uses a modulation-demodulation scheme GW modulator ITF detector sidebands « video »+ « audio » sidebands demodulator USO Low pass filter signal VESF School

  44. 5.0 Thermal noise Mirrors are hanging at the end of wires. The suspension system Is a series of harmonic oscillators. At room temperature, each degree of freedom is excited with Energy (k : Boltzmann constant) Violin modes of wires Elastodynamic Modes of mirrors Pendulum motion VESF School

  45. Harmonic oscillator 5.1 Thermal noise m A few x(t) Dissipation due to viscous damping: Langevin force Damping factor Resonance freq. VESF School

  46. 5.2 Thermal noise Fourier transform: Spectral density: A constant (white noise) is determined by the condition VESF School

  47. 5.3 Thermal noise Quality factor : damping time Spectral density concentrated on the resonance. Increase the Q! VESF School

  48. 5.4 Thermal noise Viscoelastic RSD of thermal noise Q=10 Const at low f Q=10000 VESF School

  49. 5.5 Thermal noise Thermoelastic damping equibrium TE oscillator Gaz spring M Heat flux By thermal conductance M VESF School

  50. 5.6 Thermal noise Thermoelastic damping In mirrors Thermoelastic damping In suspension wires VESF School

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