23/05/06VESF School1 Gravitational Wave Interferometry Jean-Yves Vinet ARTEMIS Observatoire de la Côte d’Azur Nice (France)

23/05/06VESF School2 Summary Shot noise limited Michelson Resonant cavities Recycling Optics in a perturbed space-time Thermal noise Sensitivity curve

23/05/06VESF School3 0.0 Introduction Resonant cavity splitter photodetectorr Laserrecycler 3km 20W 1kW 20kW Recycling cavity Virgo principle (LIGO as well)

23/05/06VESF School4 0.1 Introduction R. Weiss Electromagnetically coupled broadband gravitational antenna. Quar. Prog. Rep. in Electr., MIT (1972), 105, R.L. Forward Wideband laser-interferometer gravitational-radiation experiment. Phys. Rev. D (1978) 17 (2) J.-Y. Vinet et al. Optimization of long-baseline optical interferometers for gravitational-wave detection. Phys. Rev. D (1988) 38 (2) B.J. Meers Recycling in a laser-interferometric gravitational-wave detector. Phys. Rev. D (1988) 38 (8) The first idea The first Experiment theory MIT Hughes

23/05/06VESF School5 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

23/05/06VESF School6 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 1.2 Shot noise

23/05/06VESF School7 1.3 Shot noise In fact, the « one sided spectral density » is (white noise) The « root spectral density » is P=20W, =  m

23/05/06VESF School8 1.4 Michelson Light source Mirror 2 Mirror 1 Splitter photodetector a b A B

23/05/06VESF School9 1.5 Michelson Amplitude reaching the photodetector: Detected power: Assume with Linearization in x

23/05/06VESF School Michelson : tuning of the output fringe.

23/05/06VESF School Michelson The signal must be larger than the shot noise fluctuations of of spectral density: Spectral density of signal: Signal to noise ratio:

23/05/06VESF School Michelson

23/05/06VESF School 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 :

23/05/06VESF School Michelson For detecting GW: ( L : arm length, 3 km) Increase L ? Increase ?

23/05/06VESF School Resonant cavities Relative phase for reflection and transmission 1 R T AB

23/05/06VESF School Resonant cavities The Fabry-Perot interferometer L A B E Intracavity amplitude: Reflected amplitude: resonances

23/05/06VESF School Resonant cavities linewidth Free Spectral Range  = FSR/Full width at half max Finesse : f

23/05/06VESF School Resonant cavities Assume Free spectral range (FSR): Linewidth with Being a resonance, Reduced detuning Total losses :Coupling coeff.

23/05/06VESF School Resonant cavities A B Fabry-Perot cavity f f

23/05/06VESF School Resonant cavities Instead ofFor a single round trip Effective number of bounces

23/05/06VESF School Resonant cavities Gain factor at resonance for length 2L f

23/05/06VESF School22 Perfectly symmetrical interferometer F F splitter At the black fringe and cavity resonance: 3.1 Recycling a

23/05/06VESF School Recycling Mic Recycling mirror Recycling cavity at resonance: Optimal value of : Optimal power recycling gain: Recycling gain

23/05/06VESF School Recycling The cavity losses are likely much larger than other losses Recycling gain limited by New sensitivity to GW 50

23/05/06VESF School Recycling 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. The heuristic (naive) preceding theory is not sufficient.

23/05/06VESF School Optics in a perturbed Space Time t L One Fourier component of frequency :

23/05/06VESF School27 If the round trip is a light ray’s 4.2 Optics in a perturbed Space Time t if then Creation of 2 sidebands

23/05/06VESF School Optics in a perturbed Space Time Assume Round trip GW Linear transformation

23/05/06VESF School 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

23/05/06VESF School Optics in a perturbed Space Time Transmission of already existing sidebands New contribution to sidebands : GW!

23/05/06VESF School 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)

23/05/06VESF School Optics in a perturbed Space Time Example of a Fabry-Perot cavity : reduced frequency detuning (wrt resonance) : reduced gravitational frequency

23/05/06VESF School Optics in a perturbed Space Time In particular, at resonance Showing the decreasing efficiency at high frequency

23/05/06VESF School Optics in a perturbed Space Time Computing the SNR GW carrier Carrier + 2 sidebandsS Shot noise : proportional to Signal : proportional to (root spectral density)

23/05/06VESF School Optics in a perturbed Space Time Computing the SNR Shot noise limited spectral sensitivity: General recipe: compute

23/05/06VESF School Optics in a perturbed Space Time SNR for a Michelson with 2 cavities: SNR for a recycled Michelson with 2 cavities Optimal recycling rate:

23/05/06VESF School Optics in a perturbed Space Time Spectral sensitivity : (SNR=1)

23/05/06VESF School Optics in a perturbed Space Time Spectral sensitivity of a power recycled ITF

23/05/06VESF School Optics in a perturbed Space Time 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 Optimizing the finesse But increasing the finesse is not only an algebrical game! (Thermal lensing problems)

23/05/06VESF School Optics in a perturbed Space Time With optimal recycling rate

23/05/06VESF School41 Reasons for reading The VIRGO PHYSICS BOOK (Downloadable from the VIRGO site)

23/05/06VESF School Optics in a perturbed Space Time Other types of recycling: signal recycling (Meers) Power recycler Signal recycler FP Signal extraction (Mizuno) FP Ring cavity Synchronous Recycling (Drever) Narrowing the bandwidth broadband Resonant (narrowband)

23/05/06VESF School43 All these estimations of SNR were done in the « continuous detection scheme ». In practice, one uses a modulation-demodulation scheme ITF USO modulator sidebands GW detector demodulator Low pass filter signal « video »+ « audio » sidebands 4.16 Optics in a perturbed Space Time

23/05/06VESF School 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) Pendulum motion Violin modes of wires Elastodynamic Modes of mirrors

23/05/06VESF School Thermal noise m x(t) A few Harmonic oscillator Dissipation due to viscous damping: Damping factor Resonance freq. Langevin force

23/05/06VESF School Thermal noise Fourier transform: Spectral density: A constant (white noise) is determined by the condition

23/05/06VESF School Thermal noise Quality factor : damping time Spectral density concentrated on the resonance. Increase the Q!

23/05/06VESF School Thermal noise Q = 10 Q = Viscoelastic RSD of thermal noise Const at low f

23/05/06VESF School Thermal noise Thermoelastic damping By thermal conductance Heat flux M TE oscillator Gaz spring M equibrium

23/05/06VESF School50 Thermoelastic damping In suspension wires Thermoelastic damping In mirrors 5.6 Thermal noise

23/05/06VESF School Thermal noise The Fluctuation-dissipation (Callen-Welton) theorem Let Z be the mechanical impedance of a system described by the degree of freedom x, i.e. Where F is the driving force. Then In the viscous damping case, we had

23/05/06VESF School Thermal noise The FD theorem allows to treat the case of the thermo-elastical Damping (the most likely). In a not too low frequency domain we have: : loss angle, analogous to 1/ Q We dont know much about But:

23/05/06VESF School Thermal noise Following the FD theorem:

23/05/06VESF School Thermal noise thermoelastic RSD of thermal noise

23/05/06VESF School Thermal noise Main limitation to GW interferometers: Elastodynamic modes Of mirrors. Coupling between Surface and light beam Surface equation: Reflected beam: A Equivalent displ:

23/05/06VESF School Thermal noise Direct approach: Find the resonances of the solid by solving the elastodynamical problem. No exact solution.

23/05/06VESF School Thermal noise Interaction energy: or Is analogous to a pressure of gaussian profile For finding the mechanical impedance, we can regard As resulting from the pressure p Heuristic point of view

23/05/06VESF School Thermal noise Assume an oscillating force If is much lower than the resonances of the solid, and if is the surface distortion caused by the pressure We have (One neglects the inertial forces). is the thermo elastical loss angle.

23/05/06VESF School Thermal noise : elastical energy stored in the solid under pressure. : energy for a pressure normalized to 1 N The problem amounts to compute U Result due to Levin

23/05/06VESF School Thermal noise For an infinite half space: Poisson ratio Young modulus Beam width Fused silica: Input mirror: BHV solution

23/05/06VESF School61 The sensitivity curve (Virgo) pendulum Shot noise mirrors

23/05/06VESF School62