Reciprocity relationships for gravitational-wave interferom - PowerPoint Presentation

Slide1

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. DiscussionSlide2

Part 1: creep noise

Ageev et al. 97Cagnoli

et al.97De Salvo et al. 97, 98, 05, 08Slide3

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 2012Slide4

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

t

he coupling in both

directions is the sameSlide5

Reciprocity relations

Force

density

Readout

variable

displacement

f

orm-factor

f

orm-factorSlide6

Reciprocity relations

Force

density

Readout

variable

displacement

f

orm-factor

f

orm-factor

i

s invariant with

r

espect to interchange

o

f

andSlide7

stress

Part 3: the creep noise againSlide8

The response to a single event:

Location of

the creep event Pendulum

mode

Violin

modeSlide9
Slide10

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 independentSlide11
Slide12

Conclusions for creep:

Simple method to calculate elasto-dynamic response to creep eventsDirect coupling to transverse motion

Response the strongest for creep events near fibers’ ends=> Bonding!Slide13

Part 4: thermal deformations of mirrors

High-temperature region

Not an issue for

advanced KAGRA.

Major issue for LIGO

& Virgo

Zernike polynomials

New

coordinates

cf. Hello &

Vinet

1990

Treat this as a readout variableSlide14

How to calculate

Apply pressure to the mirror faceCalculate trace of the induced deformation tensor

Have to do it only once!Calculate the thermal deformation

Young

modulus

Thermal

expansion

Temperature

perturbation

King, Levin,

Ottaway

,

Veitch

in prep.Slide15

Check:

axisymmetric case (prelim)

Eleanor King,U. of Adelaide Slide16

Off-axis case (prelim)

Eleanor King,

U. of Adelaide Slide17

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 afternoonSlide18

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)Slide19

Part 6: opto

-mechanics with interfaces

Linear optical readout, e.g. phase measurements

Carrier light

+

Perturbation

Phase

Form-factorSlide20

Part 6: opto

-mechanics with interfaces

Linear optical readout, e.g. phase measurements

Photo-diode

Phase

Form-factorSlide21

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 phaseSlide22

Conclusions

Linear systems (elastic, optomechanical) feature reciprocity relationsThey give insight and ensure generality

They simplify calculations