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Reciprocity relationships for gravitational-wave interferometers

Reciprocity relationships for gravitational-wave interferometers. Yuri Levin (Monash University). 1. Example: creep noise 2. Formalism 3. Creep noise again 4. Thermal deformations of mirrors 5. Thermal noise 6. Opto -mechanical displacements 7. Discussion. Ageev et al. 97

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Reciprocity relationships for gravitational-wave interferometers

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  1. Reciprocity relationships for gravitational-wave interferometers Yuri Levin (Monash University) 1. Example: creep noise 2. Formalism 3. Creep noise again 4. Thermal deformations of mirrors 5. Thermal noise 6. Opto-mechanical displacements 7. Discussion

  2. Ageev et al. 97 Cagnoli et al.97 De Salvo et al. 97, 98, 05, 08 Part 1: creep noise

  3. Quakes in suspension fibers Levin 2012 • Sudden localized stress release: • non-Gaussian (probably), statistics not • well-understood, intensity and frequency • not well-measured. • No guarantee that it is unimportant in • LIGO II or III • Standard lore: couples through random • fiber extension and Earth curvature. • KAGRA very different b/c of inclined floor • Much larger direct coupling exists for LIGO. • Top and bottom defects much more important. defects

  4. Part 2: Reciprocity relations If you flick the cow’s nose it will wag its tail. the coupling in both directions is the same If someone then wags the cow’s tail it will ram you with its nose. Provided that the cow is non-dissipative and follows laws of elastodynamics

  5. Reciprocity relations form-factor Force density Readout variable displacement form-factor

  6. Reciprocity relations form-factor Force density is invariant with respect to interchange of and Readout variable displacement form-factor

  7. Part 3: the creep noise again stress

  8. The response to a single event: Location of the creep event Violin mode Pendulum mode

  9. Random superposition of creep events parameters, e.g. location, volume, strength of the defect. Fourier transform Probability distribution function Caveat: in many “crackle noise” system the events are not independent

  10. Conclusions for creep: • Simple method to calculate elasto-dynamic response to creep events • Direct coupling to transverse motion • Response the strongest for creep events near fibers’ ends • => Bonding!

  11. Part 4: thermal deformations of mirrors Not an issue for advanced KAGRA. Major issue for LIGO & Virgo High-temperature region cf. Hello & Vinet 1990 New coordinates Zernike polynomials Treat this as a readout variable

  12. How to calculate King, Levin, Ottaway, Veitch in prep. • Apply pressure to the mirror face • Calculate trace of the induced deformation tensor Have to do it only once! • Calculate the thermal deformation Young modulus Thermal expansion Temperature perturbation

  13. Check: axisymmetric case (prelim) Eleanor King, U. of Adelaide

  14. Off-axis case (prelim) Eleanor King, U. of Adelaide

  15. Part 5: thermal noise from local dissipation Readout variable Conjugate pressure Uniform temperature Local dissipation Non-uniform temperature. Cf. KAGRA suspension fibers See talk by Kazunori Shibata this afternoon

  16. Part 6: opto-mechanics with interfaces Question: how does the mode frequency change when dielectric interface moves? Theorem: Mode energy Interface displacement Optical pressure on the interface Useful for thermal noise calculations from e.g. gratings (cf. Heinert et al. 2013)

  17. Part 6: opto-mechanics with interfaces Linear optical readout, e.g. phase measurements Carrier light + Perturbation Phase Form-factor

  18. Part 6: opto-mechanics with interfaces Linear optical readout, e.g. phase measurements Photo-diode Phase Form-factor

  19. Part 6: opto-mechanics with interfaces 1. Generate imaginary beam with oscillating dipoles Photo-diode 2. Calculate induced optical pressure on the interface 3. The phase

  20. Conclusions • Linear systems (elastic, optomechanical) feature reciprocity relations • They give insight and ensure generality • They simplify calculations

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