gravitational wave interferometry

23/05/06 VESF School 1

Gravitational Wave Interferometry

Jean-Yves VinetARTEMIS

Observatoire de la Côte d’AzurNice (France)


Summary

• Shot noise limited Michelson

• Resonant cavities

• Recycling

• Optics in a perturbed space-time

• Thermal noise

• Sensitivity curve


0.0 Introduction

Resonant cavity

splitter

photodetectorr

Laser recycler 3km

20W

1kW

20kWRecyclingcavity

Virgo principle(LIGO as well)


0.1 Introduction

R. Weiss Electromagnetically coupled broadband gravitational antenna.

Quar. Prog. Rep. in Electr., MIT (1972), 105, 54-76

R.L. Forward Wideband laser-interferometer gravitational-radiation experiment. Phys. Rev. D (1978) 17 (2) 379-390

J.-Y. Vinet et al. Optimization of long-baseline optical interferometers for gravitational-wave detection. Phys. Rev. D (1988) 38 (2) 433-447

B.J. Meers Recycling in a laser-interferometric gravitational-wave detector. Phys. Rev. D (1988) 38 (8) 2317-2326

The firstidea

The firstExperiment

theory

MIT

Hughes


1.1 Shot noise

Detection of a light flux (power P) by a photodetector

Integration time : t

Number of detected photons : .P t

nh

In fact, n is a random variable, of statistical moments

00[ ]

P tE n n

h

0[ ]V n n

The photon statistics is Poissonian


We can alternatively consider the power P(t) as a randomProcess,

of moments

0[ ]E P P2 2

0 0[ ] [ ]P t P hh h

V P V nt t h t

( )( )

n t hP t

t

t May be viewed as the inverse of the bandwidth of the photodetector

0[ ]V P P h

0P h Is the spectral density of power noise

1.2 Shot noise


1.3 Shot noise

In fact, the « one sided spectral density » is

0( ) 2PS f P h

(white noise)

The « root spectral density » is 1/ 2

0( ) 2PS f P h

P=20W, = 1.064 m

1/ 2 9 -1/2( ) 2.7 10 W.HzPS f


1.4 Michelson

Light source

Mirror 2

Mirror 1

Split

ter

photodetector

a

b

1r

2r

, s sr tA

B


1.5 Michelson

Amplitude reaching the photodetector:

2 21 2

ika ikbs sB Ar t re r e

Detected power:

2 2 2 21 2 1 22 cos 2 ( - )out in s sP P r t r r r r k a b

Assume 0 ( )a a x t with | ( ) |x t

Linearization in x

2 /k

( )x t

( )out DCP P P t


1.6 Michelson

2 21 2 1 2

1( 2 cos )

4DC inP P r r r r

1 2( ) sin ( )inP t r r P kx t

02 ( )k a b

: tuning of the output fringe.

0 : bright fringe : dark fringe

2 1

2 1

contrastbright

dark

P r r

P r r


1.7 Michelson

The signal must be larger than the shot noise fluctuations of of spectral density:

( )P t

DCP

2 21 2 1 2

1( ) ( 2 cos )

2DCP inS f P h r r r r

Spectral density of signal:2 2 2 2 2

1 2( ) sin ( ) ( )P in xS f r r P k S f Signal to noise ratio:

2 2 21 2( ) 2 ( ) ( )

DC

inPx

P

PSf r r F k S f

S h

2

2 21 2 1 2

sin( )

2 cosF

r r r r


1.8 Michelson1 1 2

1 2

min( , )cos

max( , )opt

r r

r r

( )F 21 21/ min( , )r r


1.9 Michelson

The interferometer must be tuned near a dark fringe.

The optimal SNR is now

2( ) 2 ( )inx

Pf k S f

h

The shot noise limited spectral sensitivity in x correspondsto SNR=1:

1/ 2 2( )

4xin

S fP

For 20W incoming light power and a Nd:YAG laser :1/ 2 17 -1/2( ) 1.2 10 m.HzxS f


1.10 Michelson

For detecting GW:

( ) ( )x t L h t (L : arm length, 3 km)

1/ 2 2( )

4hin

S fL P

1/ 2 21 -1/2( ) 3.8 10 HzhS f

Increase L ?Increase ? inP


2.1 Resonant cavities

Relative phase for reflection and transmission

1

RT

2 21R T p ,R T Z

A B

RA TB TA RB

2 2 2 2(1 )RA TB TA RB p A B

Arg( ) Arg( ) / 2R T

, , ,R ir T t r t



The Fabry-Perot interferometer

1 1 1, ,r t p 2 2,r pL

A

B

E1 1 2 exp(2 )E t A r r ikL E

2 /k

Intracavity amplitude: 21 2

1

1 ikLE A

r r e

Reflected amplitude: 21 1 2

ikLB ir A it r e E iFA

21 1 2

21 2

(1 )

1

ikL

ikL

r p r eF

r r e

resonances


2.3 Resonant cavities2

/E A

linewidth

FreeSpectralRange

= FSR/Full width at half maxFinesse :

1 2

1 21

r r

r r

F=

f



Assume 0

Free spectral range (FSR):2FSR

c

L

2FWHM

c

L

FLinewidth

with FSR 0 Being a resonance,

Reduced detuning / FWHMf

Total losses : 1 2p p p /p FCoupling coeff.

2

2

(2 )1

1 4F

f

1 12

Arg( ) tan tan 21

fF f



A

B

Fabry-Perot cavity

2/B A Arg( / )B A

f f



if 1 and 1f Arg( ) 4F f

0 02 2

FWHM

L Lf

c c

FSR

F FF

8Arg( )

dF

d L

F

Instead of 4 /d

d L

For a single round trip

2 /n F //Effective number of bounces



Gain factor at resonance for length

2L

50F=

100 kmeffectiveL

8d

d L

F

f

Arg( / )B A

effective actual(2 / )L F L


Perfectly symmetricalinterferometer

MicT

MicR b

( )(1 ) cos ( ) .ik a bMic sR i p e k a b F

F

F

,s s sr t psplitter

At the black fringe and cavity resonance:

2 2 2(1 ) (1 ) 1 2( )Mic s sR p p

3.1 Recycling

a


3.2 Recycling

Mic

, ,r r rr t p

Recycling mirror

2 2

2 2

1

1 (1 )1

r r rrec in

r sr Mic

t p rP P

r pr R

inP recP

Recycling cavity

Recycling cavityat resonance:

Optimal value of :rr 1rOPT r sr p p

Optimal power recycling gain: ,

1

2( )R OPTr s

gp p

Recycling gain Rg


3.3 Recycling

The cavity losses are likely much larger than other losses

22 ( )inputmirror endmirrorp p

F

Recycling gain limited by1

2Rg

New sensitivity to GW1/ 2 1/ 2

, ,

1 1h NEW h OLD

R

S Sn g

50501/ 2 23 -1/2( ) 10 HzhS f


3.4 Recycling

The spectral sensitivity is not flat (white). Efficiency is expectedto decrease when the GW frequency is larger than the cavitylinewidth.

Rough argument : for high GW frequencies, the round tripduration inside the cavity may become comparable to the GWperiod, so that internal compensation could occur.

Necessity of a thorough study of the coupling betweena GW and a light beam.

The heuristic (naive) preceding theory is not sufficient.


4.1 Optics in a perturbed Space Time

rt

t

2 /

2 1( )

2

t

r t L c

Lt t h u du

c

L

One Fourier component of frequency : / 2

2sinc / cos ( / )r

L hLt t L c t L c

c c

( ) cos( )h t h t


If the round trip is a light ray’s


t

rt( )A t

( )B t( ) ( )rB t A t

( ) i tA t Ae

( ) ( )0 1 2

1 1( )

2 2i t i t i tB t B e hB e hB e

if

then

Creation of 2 sidebands



20 0

2 ( ) (2 )1 1 0

2 ( ) (2 )1 2 0

sinc /

sinc /

ikL

i k K L i k K L

i k K L i k K L

B e A

LB e A i L c e A

cL

B e A i L c e Ac

Assume

0 1 2( , , )A A AA= 0 1 2( , , )B B BB

Round trip

GW

A B

Linear transformation

1



0 00 0

1 10 11 1

2 20 22 2

. .

.

.

B O A

B O O A

B O O A

Operator « round trip »

Any optical element can be given an associated operator ofthis type.

. .

. .

. .

r

r

r

M =Example: reflectance of a mirror



0 00 0

1 10 11 1

2 20 22 2

. .

.

.

B O A

B O O A

B O O A

Transmissionof already existingsidebands

New contributionto sidebands : GW!



Evaluation of the signal-to-noise ratio for any optical setupamounts to a linear algebra calculation leading to the overalloperator of the setup.

The set of all operators having the structure

00

10 11

20 22

O

O O

O O

form a non-commutative field (all properties of R except commutativity)



Example of a Fabry-Perot cavity

F

G F

G F

F=1 2 ( )

1 2 ( )g

g

i f fF

i f f

1 2

1 2

ifF

if

2 2

(1 2 ) 1 2 ( )g

LG i

if i f f

F

/ FWHMf : reduced frequency detuning (wrt resonance)

/g GW FWHMf : reduced gravitational frequency



In particular, at resonance

2

4 1| |

1 4 g

LG

f

F

Showing the decreasing efficiency at high frequency



Computing the SNRGW

carrier Carrier + 2 sidebandsS

Shot noise : proportional to

A B

B=SA00

10 11

20 22

S

S S

S S

S

00S

Signal : proportional to 00 0iS S (root spectral density)



Computing the SNR

10 00 20 00

00

( ) ( )2

inP S S S SSNR f h f

S

Shot noise limited spectral sensitivity:

00

10 00 20 00

2( )

in

Sh f

P S S S S

General recipe: compute 00 10,S S



SNR for a Michelson with 2 cavities:

2

8 1 / 2( ) (1 ) ( )

21 4( / )in

Mic s

FWHM

PLSNR f p h f

f

F

SNR for a recycled Michelson with 2 cavities

1 (1 ) 1r

Micr s

tSNR SNR

r p

Optimal recycling rate:

,OPT (1 )(1 )(1 ) 1 ( )r r s r sr p p p p



Spectral sensitivity : (SNR=1)

1/ 2 2 2( ) 1 (2 / )

8h FWHMin

S f fL P

F


1/ 2 ( )h gS

(Hz)g


Spectral sensitivity of a power recycled ITF

50F=

100F=



Increasing the finesse leads to

A gain in the factor 8 /L F

A narrowing of the linewidth of the cavitya lower cut-off

An increase of the cavity losses /p F /

There is an optimal value depending on the GW frequency

Optimizing the finesse

(0)2 2gFSR

g L

F

But increasing the finesse is not only an algebrical game!(Thermal lensing problems)


4.15 Optics in a perturbed Space Time1/ 2 ( )h gS

(Hz)g

50F=

150F=

With optimal recycling rate


Reasons for readingThe

VIRGO PHYSICS BOOK

(Downloadable from the VIRGO site)



Other types of recycling: signal recycling (Meers)

Powerrecycler

Signalrecycler

FP

FP

Signal extraction (Mizuno)

FP

FP

Ringcavity

SynchronousRecycling(Drever)

Narrowing the bandwidth

broadband Resonant (narrowband)


All these estimations of SNR were done in the « continuousdetection scheme » .

In practice, one uses a modulation-demodulation scheme

ITF

USO

modulator

sidebands

GW

detector

demodulator

Low pass filtersignal

« video »+« audio »sidebands



5.0 Thermal noise

Mirrors are hanging at the end of wires. The suspension systemIs a series of harmonic oscillators.

At room temperature, each degree of freedom is excited withEnergy (k : Boltzmann constant)1

2 kT

Pendulum motionViolin modesof wires

ElastodynamicModes of mirrors


5.1 Thermal noise

m

x(t)

20

( )kT

xm

A few 1510 m

Harmonic oscillator

Dissipation due to viscous damping:

2202

( ) /d x dx

x F t mdt dt

Damping factor Resonance freq. Langevin force


5.2 Thermal noise

Fourier transform:

2 20

1 ( )( )

Fx

m i

Spectral density:

22 2 2 20

1 1( ) ( )x FS f S f

m

( )FS f A constant (white noise) is determined by the condition

200

( )x

kTS f df

m

2 f


5.3 Thermal noise

0

2 222 2 0

0 2

4( )x

kTS f

mQQ

0 0 /Q

Quality factor

1/ : damping time

30

40 : ( )x

kTS f

mQ

0 30

4 : ( )x

kTQS f

m

04

4 : ( )x

kTS f

mQ

Spectral density concentrated on the resonance.Increase the Q!


5.4 Thermal noise

1/ 2 ( )xS f

(Hz)f

Q=10

Q=10000

Viscoelastic RSD of thermal noise

200

( )x

kTS f df

m

Const at low f

2f


5.5 Thermal noiseThermoelastic damping

1 2T T2T

By thermal conductanceHeat flux

M

TE oscillator

Gazspring

M

x

'x x

equibrium


Thermoelastic dampingIn suspension wires

1T2 1T T

Thermoelastic dampingIn mirrors

5.6 Thermal noise


5.7 Thermal noise

The Fluctuation-dissipation (Callen-Welton) theorem

Let Z be the mechanical impedance of a system described by the

degree of freedom x, i.e.

v /Z FWhere F is the driving force. Then

2

4( ) ( )x

kTS f e Z

In the viscous damping case, we had 2 20

/i mZ

i


5.8 Thermal noise

The FD theorem allows to treat the case of the thermo-elasticalDamping (the most likely).

In a not too low frequency domain we have:

2 20( ) (1 ) ( ) ( ) /x i x F m

: loss angle, analogous to 1/Q

We dont know much about ( )FS f

20

22 2 4 20 0

1( )e Z

m

But:


5.9 Thermal noise

Following the FD theorem:

20

22 2 4 20 0

4( )x

kTS f

m

20

40 : ( )x

kTS f

m

0 30

4: ( )x

kTS f

m

20

5

4: ( )x

kTS f

m


5.10 Thermal noise

1/ 2 ( )xS f

(Hz)f

thermoelastic RSD of thermal noise

1.2f

2.5f


5.11 Thermal noise

Main limitation to GW interferometers: Elastodynamic modesOf mirrors.

Coupling betweenSurface and light beam

Surface equation: ( , , )zz u t x y

Reflected beam:

A

2( , ) ( , )zikuB x y e A x y2, 1 2 ( , ) ( , , )ziku

zB A AAe ds ik I x y u t x y dx dy ( ) ( , ) ( , , )zz t I x y u t x y dx dyEquivalent displ:


5.12 Thermal noiseDirect approach:

Find the resonances of the solid by solving the elastodynamicalproblem. No exact solution.


5.13 Thermal noise

Interaction energy:( ) ( )F t z tE

or( , , ) ( ) ( , )zu t x y F t I x y dxdy E

( , , ) ( ) ( , )p t x y F t I x yIs analogous to a pressure of gaussian profile

2 2

22

2

2( , ) e

x y

wI x yw

For finding the mechanical impedance, we can regardzu As resulting from the pressure p

Heuristic point of view


5.14 Thermal noise

( ) ( , ) ( , , )zz t I x y u t x y dx dyAssume an oscillating force

-i t( ) eF t F If is much lower than the resonances of the solid, and if

( , )zu x y is the surface distortion caused by the pressure . ( , )F I x y

We have ( , , ) e ( , )i t iz zu t x y u x y

(One neglects the inertial forces). is the thermo elastical loss angle.

v( ) e ( , ) ( , )izi u x y I x y dxdy

2

( , ) ( , )e ( )

zu x y FI x y dxdyZ

F


5.15 Thermal noise

1( , ) ( , )

2 zu x y FI x y dxdy WW : elastical energy stored in the solid under pressure.

2/U W F : energy for a pressure normalized to 1 N

2

4 4( ) 2z

kT kTS f U U

f

The problem amounts to compute U

Result due to Levin


5.16 Thermal noise

For an infinite half space:

21

2U

Yw

Poisson ratio

Young modulus

Beam width

Fused silica: 10 -27.3 10 NmY 0.17

Input mirror: 0.02 mw1/ 2

1/ 2 18 -1/21 Hz( ) 10 m.HzzS f

f

BHV solution


The sensitivity curve (Virgo)

pendulum

Shot noise

mirrors

gravitational wave interferometry

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