23/05/06VESF School1 Gravitational Wave Interferometry Jean-Yves Vinet ARTEMIS Observatoire de la...
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Transcript of 23/05/06VESF School1 Gravitational Wave Interferometry Jean-Yves Vinet ARTEMIS Observatoire de la...
23/05/06 VESF School 1
Gravitational Wave Interferometry
Jean-Yves VinetARTEMIS
Observatoire de la Côte d’AzurNice (France)
23/05/06 VESF School 2
Summary
• Shot noise limited Michelson
• Resonant cavities
• Recycling
• Optics in a perturbed space-time
• Thermal noise
• Sensitivity curve
23/05/06 VESF School 3
0.0 Introduction
Resonant cavity
splitter
photodetectorr
Laser recycler 3km
20W
1kW
20kWRecyclingcavity
Virgo principle(LIGO as well)
23/05/06 VESF School 4
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
23/05/06 VESF School 5
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
23/05/06 VESF School 6
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
23/05/06 VESF School 7
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
23/05/06 VESF School 8
1.4 Michelson
Light source
Mirror 2
Mirror 1
Split
ter
photodetector
a
b
1r
2r
, s sr tA
B
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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
23/05/06 VESF School 10
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
23/05/06 VESF School 11
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
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1.8 Michelson1 1 2
1 2
min( , )cos
max( , )opt
r r
r r
( )F 21 21/ min( , )r r
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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
23/05/06 VESF School 14
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
23/05/06 VESF School 15
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
23/05/06 VESF School 16
2.2 Resonant cavities
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
23/05/06 VESF School 17
2.3 Resonant cavities2
/E A
linewidth
FreeSpectralRange
= FSR/Full width at half maxFinesse :
1 2
1 21
r r
r r
F=
f
23/05/06 VESF School 18
2.4 Resonant cavities
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
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2.5 Resonant cavities
A
B
Fabry-Perot cavity
2/B A Arg( / )B A
f f
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2.6 Resonant cavities
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
23/05/06 VESF School 21
2.7 Resonant cavities
Gain factor at resonance for length
2L
50F=
100 kmeffectiveL
8d
d L
F
f
Arg( / )B A
effective actual(2 / )L F L
23/05/06 VESF School 22
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
23/05/06 VESF School 23
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
23/05/06 VESF School 24
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
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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.
23/05/06 VESF School 26
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
23/05/06 VESF School 27
If the round trip is a light ray’s
4.2 Optics in a perturbed Space Time
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
23/05/06 VESF School 28
4.3 Optics in a perturbed Space Time
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
23/05/06 VESF School 29
4.4 Optics in a perturbed Space Time
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
23/05/06 VESF School 30
4.5 Optics in a perturbed Space Time
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!
23/05/06 VESF School 31
4.6 Optics in a perturbed Space Time
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)
23/05/06 VESF School 32
4.7 Optics in a perturbed Space Time
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
23/05/06 VESF School 33
4.8 Optics in a perturbed Space Time
In particular, at resonance
2
4 1| |
1 4 g
LG
f
F
Showing the decreasing efficiency at high frequency
23/05/06 VESF School 34
4.9 Optics in a perturbed Space Time
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)
23/05/06 VESF School 35
4.10 Optics in a perturbed Space Time
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
23/05/06 VESF School 36
4.11 Optics in a perturbed Space Time
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
23/05/06 VESF School 37
4.12 Optics in a perturbed Space Time
Spectral sensitivity : (SNR=1)
1/ 2 2 2( ) 1 (2 / )
8h FWHMin
S f fL P
F
23/05/06 VESF School 38
1/ 2 ( )h gS
(Hz)g
4.13 Optics in a perturbed Space Time
Spectral sensitivity of a power recycled ITF
50F=
100F=
23/05/06 VESF School 39
4.14 Optics in a perturbed Space Time
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)
23/05/06 VESF School 40
4.15 Optics in a perturbed Space Time1/ 2 ( )h gS
(Hz)g
50F=
150F=
With optimal recycling rate
23/05/06 VESF School 41
Reasons for readingThe
VIRGO PHYSICS BOOK
(Downloadable from the VIRGO site)
23/05/06 VESF School 42
4.15 Optics in a perturbed Space Time
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)
23/05/06 VESF School 43
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
4.16 Optics in a perturbed Space Time
23/05/06 VESF School 44
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
23/05/06 VESF School 45
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
23/05/06 VESF School 46
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
23/05/06 VESF School 47
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!
23/05/06 VESF School 48
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
23/05/06 VESF School 49
5.5 Thermal noiseThermoelastic damping
1 2T T2T
By thermal conductanceHeat flux
M
TE oscillator
Gazspring
M
x
'x x
equibrium
23/05/06 VESF School 50
Thermoelastic dampingIn suspension wires
1T2 1T T
Thermoelastic dampingIn mirrors
5.6 Thermal noise
23/05/06 VESF School 51
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
23/05/06 VESF School 52
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:
23/05/06 VESF School 53
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
23/05/06 VESF School 54
5.10 Thermal noise
1/ 2 ( )xS f
(Hz)f
thermoelastic RSD of thermal noise
1.2f
2.5f
23/05/06 VESF School 55
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:
23/05/06 VESF School 56
5.12 Thermal noiseDirect approach:
Find the resonances of the solid by solving the elastodynamicalproblem. No exact solution.
23/05/06 VESF School 57
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
23/05/06 VESF School 58
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
23/05/06 VESF School 59
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
23/05/06 VESF School 60
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
23/05/06 VESF School 61
The sensitivity curve (Virgo)
pendulum
Shot noise
mirrors
23/05/06 VESF School 62