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