Resonant Mass Gravitational Wave
Detectors
David Blair
University of Western Australia
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Historical Introduction
Intrinsic Noise in Resonant Mass Antennas
Transducers
Transducer-Antenna interaction effects
Suspension and Isolation
Data Analysis
Sources and Materials
• These notes are about principles and not projects.
• Details of the existing resonant bar network may be found
on the International Gravitational Events Collaboration
web page.
• References and some of the content can be found in
• Ju, Blair and Zhou Rep Prog Phys 63,1317,2000.
• Online at www.iop.org/Journals/rp
• Draft of these notes available www.gravity.uwa.edu.au
•Sphere
developments
Existing Resonant Bar Detectors
and sphere developments
•Leiden
•Frascati
•Sao Paulo
AURIGA
EXPLORER
Weber’s Pioneering Work
• Joseph Weber Phys Rev 117, 306,1960
• Mechanical Mass Quadrupole Harmonic
Oscillator: Bar, Sphere or Plate
• Designs to date:
Bar
Weber’s suggestions:
Earth: GW at 10-3 Hz.
Sphere
Piezo crystals: 107 Hz
Al bars: 103 Hz
Detectable flux spec
density: 10-7Jm-2s-1Hz-1
( h~ 10-22 for 10-3 s pulse)
Torsional
Quadrupole
Oscillator
Gravity Wave Burst Sources and Detection
Energy Flux of a
gravitational wave:
c3  2  2
S
h  h
16G
Short Bursts of duration tg
Assume
h  2h / t g
Total pulse energy density
EG = S.tg
J m-2 s-1
c 3 4h 2
S

16G t g 2
J m-2 s-1
c 3 4h 2
EG 

16G t g
Jm-2
Flux Spectral Density
Bandwidth of short pulse: Dw ~ 1/tg
Reasonable to assume flat spectrum: F(w) ~ E/ Dw ~E.tg
ie:
For short bursts:
c 3h 2
F (w ) 
4G
J.m-2.Hz-1
F(w) ~ 20 x 1034 h2
Gravitational wave bursts with tg~10-3s were the original
candidate signals for resonant mass detectors.
However stochastic backgrounds and monochromatic
signals are all detectable with resonant masses.
Black Hole Sources and Short Bursts
Start with Einstein’s quadrupole
formula for gravitational wave
luminosity LG:
LG 
G
5
2
3

5c jk
d D jk
dt
3
where the quadrupole moment
Djk is defined as:
 j k 1 2 jk  3
D jk    t  x x  x   d x

3

Notice: for a pair of point masses D=ML2 ,
for a spherical mass distribution D=0
for a binary star system in circular orbit D varies as sin2wt
Burst Sources Continued
Notice also that
 represents non-spherical kinetic energy
D
ie the kinetic energy of non-spherically symmetric motions.
For binary stars (simplest non sperically symmetric source),
projected length (optimal orientation) varies sinusoidally,
 ~ ML2w 3
D~ML2sin22wt, D
16 G
LG ~
  M 2 L4w 6
5 c5
Now assume
LG
isotropic radiation S 
4r 2
but also use
Note that KE=1/2Mv2= 1/8ML2w2
The numerical factor
comes from the time
average of the third
time derivative of
sin2wt.
c3  2
S
h
16G
To order of magnitude
G Ens
h

c4 r
and
Maximal source: Ens
two black holes
In general
for black
hole births
2 2
w
G G Ens
h   
c3 c5
r2
2
=Mc2……merger
rs
h 
r
of
G Mc 2 rs
h
~
4 r
r
c
Here  is conversion
efficiency to gravitational
waves
Weber’s Research
•Weber used arguments such as the above to show that
gravitational waves created by black hole events near the galactic
centre could create gravitational wave bursts of amplitude as high
as 10-16.
•He created large Al bar detectors able to detect such signals.
•He identified many physics issues in design of resonant mass
detectors.
• His results indicated that 103 solar masses per year were being
turned into gravitational waves.
•These results were in serious conflict with knowledge of star
formation and supernovae in our galaxy.
•His data analysis was flawed.
•Improved readout techniques gave lower noise and null results.
Energy deposited in a resonant mass
Energy deposited in a resonant mass EG
EG    w F w dw
 is the frequency dependent cross section
F is the spectral flux density
Treat F as white over the instrument bandwidth
Then
EG  F w a   w dw
Paik and Wagoner
showed for fundamental
quadrupole mode of bar:
8Gm  vs 
 
  w dw 
c  c 
2
Energy and Antenna Pattern for Bar
z
Energy deposited in an
initially stationary bar Us
2
v
8
G
Us=F(wa).sin4sin22   s2  M
 c c

y
Sphere is like a
set of orthogonal
bars giving
omnidirectional
sensitivity and
higher cross
section

Incoming wave
x
Detection Conditions
• Detectable signal Us  Noise energy Un
 Force 
 
•Transducer: 2-port device: 
Voltage 
Z11 Z12  velocity 


Z 21 Z 22  Current 
computer
•Amplifier , gain G, has effective current noise spectral density
Si and voltage noise spectral density Se
Forward
Mechanical
Reverse
Electrical
transductance transductance
input
output
impedance Z11 Z21 (volts m-1s-1)
Z12 (kg-amp-1)
impedance Z22
Bar, Transducer and Phase Space Coordinates
Resonant
mass
Asin(wat+
b
multiply
transducer
G
b determines time for transducer to
reach equilibrium
•X1 and X2 are symmetrical phase
space coordinates
•Antenna undergoes random walk in
phase space
•Rapid change of state measured by
length of vector (P1,P2)
•High Q resonator varies its state
slowly
X
0o
Vsinwat
~
X1=Asin
90o X2=Acos
Reference oscillator
Two Transducer Concepts
Parametric
•Signal detected as modulation
of pump frequency
•Critical requirements:
low pump noise
low noise amplifier at
modulation frequency
Direct
•Signal at antenna frequency
•Critical requirements:
low noise SQUID
amplifier
low mechanical loss
circuitry
Mechanical Impedance Matching
•High bandwidth requires good impedance matching between
acoustic output impedance of mechanical system and transducer
input impedance
•Massive resonators offer high impedance
•All electromagnetic fields offer low impedance (limited by
energy density in electromagnetic fields)
•Hence mechanical impedance trasformation is essential
•Generally one can match to masses less than 1kg at ~1kHz
Mechanical model of transducer with
intermediate mass resonant transformer
Resonant transformer creates two mode system
m
Two normal modes split by Dw  w a 
M eff
Microwave
cavity
Bending flap
secondary resonator
Microwave Readout System of NIOBÉ
(upgrade)
9.049GHz 451MHz
Composite
Oscillator
Filter
RF
SO
Frequency
servo
9.501GHz
Cryogenic components
Phase
Transducer
Bar
Bending
flap
Microstrip
antennae
Microwave
interferometer
S
Spare
mW-amplifier
servo
Filter
a
j
Mixers
j
Electronically
adjustable
phase shifter
D & attenuator
Primary
mW-amplifier
Phase
shifters
mW-amplifier
j
Data
Acquisition
Direct Mushroom Transducer
Secondary Resonator
(“mushroom”) and
Transducer
Pickup Coil
DC SQUID
(Amplifier.
Its output is
proportioanl
to the motion of
the mushroom)
A superconducting persistent current is modulated by the
motion of the mushroom resonator and amplified by a
DC SQUID.
Niobium Diaphragm Direct Transducer
(Stanford)
Three Mode Niobium Transducer (LSU)
•Two secondary
resonators
•Three normal modes
•Easier broadband
matching
•Mechanically more
complex
Three general classes of noise
Brownian Motion
Noise
kT noise energy
2
xth

4kTw a
 w 2 

2


wM eff  1  2   
 w a 

Low loss angle 
compresses thermal
noise into narrow
bandwidth at
resonance.
Decreases for high
bandwidth.(small ti)
Series Noise
Broadband Amplifier
noise, pump phase
noise or other
additive noise
contributions.
Series noise is
usually reduced if
transductance Z21 is
high.
Always increases
with bandwidth
Back Action Noise
Amplifier noise acting
back on antenna.
Unavoidable since
reverse transductance
can never be zero.
A fluctuating force
indistingushable from
Brownian motion.
Noise Contributions
Total noise referred to input:
Z12
2
2M eff Se (w )
ti
U n  2kTa

Si (w )t i 
2
t a 2M eff
ti
Z
21
Reduces as ti/ta
because of
predictability of
high Q oscillator
Reduces as ti/M
because
fluctuations take
time to build up
and have less
effect on massive
bar
Increases as M/ti
reduces due to
increased bandwidth
of noise contribution,
and represents
increased noise
energy as referred to
input
Quantum Limits
Noise equation shows any system has minimum noise
level and optimum integration time set by the competing
action of series noise and back action noise.
Since a linear amplifier has a minimum noise level called
the standard quantum limit this translates to a standard
quantum limit for a resonant mass.
Un
 AT  AB  AS
Noise equation may be rewritten A 
w a
where A is Noise Number: equivalent number of quanta.
The sum AB+AS cannot reduce below~1: the Standard
Quantum Limit
 2w 
a

hSQL   2
  M eff vs2 


0.5
0.5
 21 f a   1tonne 
~ 1.110 
 

 1kHz   M 
0.5
 10kms1 


 v

s


Burst strain limit~10-22 (100t sphere) corres to h(w)~3.10-24
Thermal Noise Limit
Thermal noise only becomes negligible for Q/T>1010
(100Hz bandwidth)
 kTt w

i a 
hth  
 M  2v 2Q 
s 
 eff
0.5
(Q=ta/w


10
9



f
10
J
10
T
100
Hz








hth  10  21 




2



 1kHz  M eff vs  Q  0.1K  B 




Thermal noise makes it difficult to exceed hSQL
0.5
Ideal Parametric Transducer
Noise temperature characterises noise energy of any system.
Since photon energy is frequency dependent, noise number is
more useful.
Amplifier effective noise temperature must be referred to antenna
wa
frequency
Teff 
Tn
wp
For example wa = 2 x 700Hz
Tn = 10K: Hence A 
kTn
w
pum p
wpump= 2 x 9.2 GHz
and Teff = 8 10-7 K
Cryogenic microwave amplifiers greatly exceed the performance
of any existing SQUID and have robust performance
•Oscillator noise and thermal noise degrade system noise
Pump Oscillators for Parametric Transducer
A low noise oscillator is an
essential component of a
parametric transducer
j
DC Bias
+
+
Non-filtered
output
mWamplifier
BPF
LOOP OSCILLATOR
 varactor
Filtered
output
Circulator
Sapphire loaded
cavity resonator
Qe~3107
Microwave
Interferometer
mW-amplifier
RF
mixer
LNA
Loop filter
LO
Phase error
detector
A stabilised
NdYAG laser
provides a
similar low noise
optical oscillator
for optical
parametric
transducers and
for laser
interferometers
which are similar
parametric
devices.
Two Mode Transducer Model
Coupling and Transducer Scattering Picture
Treat transducer as a photon scatterer
Signal wa
phonons
w+=wp+wa
?
transducer
w-=wp-wa
Pump wp
photons
P
w
P
w
P P
 a   0
wa w

Pp
P
  0
w p w
Formal
solution but
results are
intuitively
obvious
Output
sidebands
Because transducer has negligible loss
use energy conservation to understand
signal power flow- Manley-Rowe
relations.
Note that power flow may be
altered by varying b as per
previous slide
Parametric transducer damping and elastic stiffness
Cold damping of
bar modes by
parametric
transducer
Bar mode
frequency tuning
by pump tuning
Upper mode
Lower mode
Electromechanical Coupling of Transducer to Antenna
b
signal energy in transducer
signal energy in bar
•In direct transducer b = (1/2CV2)/Mw2x2
•In parametric transducer
b=(wp/wa)(1/2CV2)/Mw2x2
•Total sideband energy is sum of AM and PM
sideband energy, depends on pump frequency
offset
Offset Tuning Varies Coupling to Upper and Lower
Sidebands
Manley-Rowe Solutions
If wp>>wa, Pp ~ -(P++P-).
If P+/w+ < P-/w- ,then Pa< 0…..negative power flow…instability
If P+/w+ > P-/w- ,then Pa> 0…..positive power flow…cold damping
By manipulating b using offset tuning can cold-damp
the resonator…very convenient and no noise cost.
Enhance upper sideband by operating with pump
frequency below resonance.
Offset tuning to vary Q and b in high Q limit
If transducer cavity
has a Qe>wp/wa , then
b is maximised near
the cavity resonance
or at the sideband
frequencies. Strong
cold damping is
achieved for
wp=wcavity-wa .
Thermal noise contributions from bar and secondary
resonator
Secondary
resonator
bar
Frequency Hz
Thermal noise components for a
bar Q=2 x108 (antiresonance at
mid band) and secondary resonator
Q=5 x 107
Transducer Optimisation
Spectral
Strain
sensitivity
Low b, high
series noise,
low back
action noise
SNR/Hz/mK
This and the following
curves from M Tobar
Thesis UWA 1993
Spectral
Strain
sensitivity
SNR/Hz/mK
Reduced
Am noise
Spectral
Strain
sensitivity
SNR/Hz/mK
Higher
secondary
mass Qfactor
Spectral
Strain
sensitivity
SNR/Hz/mK
Reduced
back
action
noise
from
pump
AM noise
Spectral
Strain
sensitivity
SNR/Hz/mK
High Qe, high
coupling
Allegro Noise Theory and Experiment
Relations between Sensitivity and Bandwidth
Minimum detectable energy is defined by the ratio of
wideband noise to narrow band noise
Express minimum detectable energy as an effective
temperature
DEmin  2T
widebandnoise
Df
narrowbandnoise
2
4f T
Bandwidth 

bw Q Teff
h
1
tg
S h (w a )
L
 2
2Df
2vs t g
Bandwidth and minimum
detectable burst depends
on transducer and
amplifier
kT
M
T
Optimum spectral sensitivity depends on ratio
MQ
Independent of readout noise
Burst detection:
maximum total
bandwidth important
Search for pulsar signals (CW) in
spectral minima.
More bandwidth=more sources
at same sensitivity
Stochastic background: use two
detectors with coinciding spectral
minima
Improving Bar Sensitivity with Improved Transducers
Two mode, low b,
high series noise
High b, low
noise,3 mode
Optimal filter
Signal to noise ratio is optimised by a filter which has
a transfer function proportional to the complex
conjugate of the signal Fourier transform divided by
the total noise spectral density
Fourier tfm of impulse
response of displacement
sensed by transducer for
force input to bar
Fourier tfm of
input signal force

1
SNR 

2  
2
G ( jw ) F (w )
S x (w )
2
dw
Double sided spectral
density of noise refered to
the transducer displacement
Monochromatic and Stochastic Backgrounds
Monochromatic (or
slowly varying) : (eg
Pulsar signals):Long term
coherent integration or
FFT
Very narrow bandwidth
detection outside the
thermal noise bandwidth.
Stochastic Background:
Cross correlate between
independent detectors.
Thermal noise is
independent and
uncorrelated between
detectors.
Both methods allow the limits to bursts to be easily exceeded.
Allegro Pulsar Search
Niobe Noise Temperature
Excess Noise and Coincidence Analysis
Measure noise performance by noise temperature.
•All detectors show nonthermal noise.
•Source of excess noise is
not understood
•Similar behaviour (not
identical) in all detectors.
•All excess noise can be
elliminated by coincidence
analysis between sufficient
detectors. (>4)
Log number of
samples
Typically h~(few x 10-17).Tn1/2
Energy
Coincidence Statistics
Probability of event above threshhold:
P1  Rt r
(Event rate R, resolving time tr)
Prob of accidental coincidence in
coincidence window tc
PN  t cN
If all antennas have same background
Nac 
i 1, N
PN  R Nt cN
Hence in time ttot the number of accidental
coincidences is
 Ri
N N 1
R tc
Improvements through coincidence analysis
10
4
10
2
10
0
1 bar 1982
2 bars 1991
-2
events/day
10
-4
10
3 bars 1999
-6
10
-8
10
10
-10
10
-12
10
-14
4 bars 1999
(not enough data)
0
5
h
burst
10
18
x 10
15
20
Suspension Systems
•General rule:Mode
control. Acoustic
resonance=short circuit.
• Low acoustic loss
suspension: many
systems.
•Low vibration coupling
to cryogenics:
•Cable couplings: Taber
isolators or non-contact
readout
•Multistage isolation in
cryogenic environment
•Room Temperature
Suspension choices
cables
Nodal
point
Dead bug
Important tool: Finite element
modelling
Niobe: 1.5 tonne Niobium Antenna with Parametric Transducer
Niobe Cryogenic System
Niobe Cryogenic Vibration Isolation
Sphere
•Nodal suspension
• Integrated secondary
and tertiary resonators
for reasonable
bandwidth
•non-superconducting
for efficient cooldown
•mass up to 100
tonnes
vibration isolation
Current limits set by bars
Bursts: 7 x 10-2 solar masses converted to gravity
waves at galactic centre (IGEC)
Spectral strain sensitivity: h(f)= 6 x 10-23/Rt Hz
(Nautilus)
Pulsar signals in narrow band (95 days): h~ 3 x 10-24
(Explorer)
Stochastic background: h~10-22
(Nautilus-Explorer)
Summary
Bars are well understood
Major sensitivity improvements underway
SQUIDs for direct transducers now making progress (see Frossati’s
talk)
All significant astrophysical limits have been set by bars.
At high frequency bars achieve spectral sensitivity in narrow bands
that is likely to exceed interferometer sensitivity for the forseeable
future.