reciprocity relationships for gravitational-wave interferometers
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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!
• Calculate the thermal deformation
Youngmodulus Thermal
expansionTemperatureperturbation
King, Levin, Ottaway, Veitch in prep.
Check: axisymmetric case (prelim)
Eleanor King,U. of Adelaide
Off-axis case (prelim)
Eleanor King,U. of Adelaide
Part 5: thermal noise from local dissipation
( )x r Readout variable
Conjugate pressure
Uniform temperature
Local dissipation
Non-uniform temperature.Cf. KAGRA suspension fibers
See talk by Kazunori Shibata this afternoon
Part 6: opto-mechanics with interfaces
Question: how does the mode frequency change when dielectric interface moves?
Theorem:
Modeenergy
Interfacedisplacement
Optical pressureon the interface
Useful for thermal noise calculations from e.g. gratings(cf. Heinert et al. 2013)
Part 6: opto-mechanics with interfaces
Linear optical readout, e.g. phase measurements
Carrier light
+
Perturbation
Phase Form-factor
Part 6: opto-mechanics with interfaces
Linear optical readout, e.g. phase measurements
Photo-diode
Phase Form-factor
Part 6: opto-mechanics with interfaces
Photo-diode
1. Generate imaginary beam with oscillating dipoles
2. Calculate induced optical pressure on the interface
3. The phase
Conclusions
• Linear systems (elastic, optomechanical) feature reciprocity relations
• They give insight and ensure generality
• They simplify calculations