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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
Part 1: creep noiseAgeev et al. 97Cagnoli et al.97De Salvo et al. 97, 98, 05, 08
Quakes in suspension fibers
defects
• 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.
Levin 2012
Part 2: Reciprocity relations
If you flick the cow’s nose it will wag its tail.
If someone then wags the cow’s tail it will ram youwith its nose. Provided that the cow is non-dissipativeand follows laws of elastodynamics
the coupling in both directions is the same
Reciprocity relations
Forcedensity
Readoutvariable
displacement
form-factor
form-factor
Reciprocity relations
Forcedensity
Readoutvariable
displacement
form-factor
form-factor
is invariant with respect to interchangeof
and
stress
Part 3: the creep noise again
The response to a single event:
Location ofthe creep event
Pendulummode
Violin mode
Random superposition of creep events
parameters, e.g.location, volume,strength of the defect.
Fouriertransform
Probabilitydistributionfunction
Caveat: in many “crackle noise” system the events are not independent
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!
Part 4: thermal deformations of mirrors
( )x r
High-temperature region
Not an issue for advanced KAGRA.Major issue for LIGO& Virgo
Zernike polynomialsNew coordinates
cf. Hello & Vinet 1990
Treat this as a readout variable
How to calculate
• Apply pressure to the mirror face
• Calculate trace of the induced deformation tensor Have to do it only once!