Resonant Mass Gravitational Wave
Detectors
David Blair
University of Western Australia
•
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
•
Sao Paulo
•
Leiden
•
Frascati
•
Sphere
developments
Existing Resonant Bar Detectors
and sphere developments
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
Sphere
Torsional
Quadrupole
Oscillator
Weber’s suggestions
:
Earth: GW at 10

3
Hz.
Piezo crystals: 10
7
Hz
Al bars: 10
3
Hz
Detectable flux spec
density: 10

7
Jm

2
s

1
Hz

1
( h~ 10

22
for 10

3
s pulse)
Gravity Wave Burst Sources and Detection
2
2
3
16
h
h
G
c
S
Energy Flux of a
gravitational wave:
Short Bursts of duration
t
g
Assume
g
h
h
t
/
2
2
2
3
4
16
g
h
G
c
S
t
J m

2
s

1
g
G
h
G
c
E
t
2
3
4
16
Total pulse energy density
E
G
= S.
t
g
J m

2
s

1
Jm

2
Flux Spectral Density
Bandwidth of short pulse:
Dw
~ 1/
t
g
Reasonable
to assume flat spectrum: F(
w
) ~ E/
Dw ~E.t
g
ie:
G
h
c
F
w
4
)
(
2
3
J.m

2
.Hz

1
For short bursts:
F(
w
) ~ 20
x
10
34
h
2
Gravitational wave bursts with
t
g
~10

3
s 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 L
G
:
jk
jk
G
dt
D
d
c
G
L
2
3
3
5
5
where the quadrupole moment
D
jk
is defined as:
x
d
x
x
x
t
D
jk
k
j
jk
3
2
3
1
Notice: for a pair of point masses D=ML
2
,
for a spherical mass distribution D=0
for a binary star system in circular orbit D varies as sin2
w
t
Burst Sources Continued
Notice also that
represents
non

spherical
kinetic energy
ie the kinetic energy of non

spherically symmetric motions.
D
For binary stars (simplest non sperically symmetric source),
projected length (optimal orientation) varies sinusoidally,
D~ML
2
sin
2
2
w
t,
6
4
2
5
5
16
~
w
L
M
c
G
L
G
3
2
~
w
ML
D
The numerical factor
comes from the time
average of the third
time derivative of
sin2
w
t.
Now assume
isotropic radiation
2
4
r
L
S
G
2
3
16
h
G
c
S
but also use
Note that KE=
1
/
2
M
v
2
=
1
/
8
M
L
2
w
2
To order of magnitude
2
2
2
5
3
2
r
E
c
G
c
G
h
ns
w
and
r
E
c
G
h
ns
4
Maximal source: E
ns
=Mc
2
……merger of
two black holes
r
r
r
Mc
c
G
h
s
~
2
4
In general
for black
hole births
r
r
h
s
Here
is conversion
efficiency to gravitational
waves
•
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 10
3
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.
Weber’s Research
Energy deposited in a resonant mass
Energy deposited in a resonant mass E
G
w
w
w
d
F
E
G
is the frequency dependent cross section
F is the spectral flux density
Treat F as white over the instrument bandwidth
Then
w
w
w
d
F
E
a
G
2
8
c
v
c
Gm
d
s
w
w
Paik and Wagoner
showed for fundamental
quadrupole mode of bar:
x
y
z
Energy deposited in an
initially stationary bar U
s
U
s
=F(
w
a
).sin
4
sin
2
2
M
c
v
c
G
s
2
2
8
Incoming wave
Energy and Antenna Pattern for Bar
Sphere is like a
set of orthogonal
bars giving
omnidirectional
sensitivity and
higher cross
section
Detection Conditions
•
Detectable signal U
s
Noise energy U
n
•
Transducer: 2

port device:
Current
velocity
Z
Z
Z
Z
Voltage
Force
22
12
21
11
computer
•
Amplifier , gain G, has effective current noise spectral density
S
i
and
voltage noise spectral density S
e
Mechanical
input
impedance
Z
11
Forward
transductance
Z
21
(volts m

1
s

1
)
Reverse
transductance
Z
12
(kg

amp

1
)
Electrical
output
impedance
Z
22
X
1
=Asin
Resonant
mass
transducer
Vsin
w
a
t
~
X
G
b
X
2
=Acos
Reference oscillator
multiply
0
o
90
o
Bar, Transducer and Phase Space Coordinates
b
determines time for transducer to
reach equilibrium
•
X
1
and X
2
are symmetrical phase
space coordinates
•
Antenna undergoes random walk in
phase space
•
Rapid change of state measured by
length of vector (P
1
,P
2
)
•
High Q resonator varies its state
slowly
Asin(
w
a
t+
Two Transducer Concepts
Parametric
Direct
•
Signal detected as modulation
of pump frequency
•
Critical requirements
:
low pump noise
low noise amplifier at
modulation frequency
•
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
Two normal modes split by
eff
a
M
m
D
w
w
Bending flap
secondary resonator
Microwave
cavity
j
j
j
a
Data
Acquisition
Mixers
Phase
shifters
Filter
Electronically
adjustable
phase shifter
& attenuator
S
D
SO
Filter
Phase
servo
Frequency
servo
m
W

amplifier
Primary
m
W

amplifier
Spare
m
W

amplifier
Microstrip
antennae
Microwave
interferometer
Cryogenic components
Bar
Bending
flap
Transducer
RF
9.049GHz
451MHz
9.501GHz
Composite
Oscillator
Microwave Readout System of NIOBÉ
(upgrade)
Secondary Resonator
(“mushroom”) and
Transducer
Pickup Coil
DC SQUID
(Amplifier.
Its output is
proportioanl
to the motion of
the mushroom)
Direct Mushroom Transducer
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
Series Noise
Back Action Noise
2
2
2
2
1
4
w
w
w
w
a
eff
a
th
M
kT
x
Low loss angle
compresses thermal
noise into narrow
bandwidth at
resonance.
Decreases for high
bandwidth.(small
t
i
)
Broadband
Amplifier
noise
, pump phase
noise or other
additive noise
contributions.
Series noise is
usually reduced if
transductance Z
21
is
high.
Always increases
with bandwidth
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:
i
e
eff
i
i
eff
a
i
a
n
S
Z
M
S
M
Z
kT
U
t
w
t
w
t
t
)
(
2
)
(
2
2
2
21
2
12
Reduces as
t
i
/
t
a
because of
predictability of
high Q oscillator
Reduces as
t
i
/M
because
fluctuations take
time to build up
and have less
effect on massive
bar
Increases as M/
t
i
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.
Noise equation may be rewritten
where A is Noise Number: equivalent number of quanta.
The sum A
B
+A
S
cannot reduce below~1: the Standard
Quantum Limit
S
B
T
a
n
A
A
A
U
A
w
s
a
s
eff
a
SQL
v
kms
M
tonne
kHz
f
v
M
h
1
5
.
0
5
.
0
21
5
.
0
2
2
10
1
1
10
1
.
1
~
2
w
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>10
10
(100Hz bandwidth)
5
.
0
2
2
Q
v
M
kT
h
s
eff
a
i
th
w
t
(Q=
t
a
/
w
5
.
0
9
2
10
21
100
1
.
0
10
10
1
10
B
Hz
K
T
Q
v
M
J
kHz
f
h
s
eff
th
Thermal noise makes it difficult to exceed h
SQL
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
frequency
For
example
w
a
=
2
x
700
Hz
w
pump
=
2
x
9
.
2
GHz
T
n
= 10K: Hence
and T
eff
= 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
n
p
a
eff
T
T
w
w
pump
n
kT
A
w
BPF
LOOP OSCILLATOR
Microwave
Interferometer
LO
RF
LNA
Circulator
Phase error
detector
mixer
Loop filter
Sapphire loaded
cavity resonator
Q
e
~3
10
7
j
varactor
DC Bias
m
W

amplifier
m
W

amplifier
Filtered
output
+
+
Non

filtered
output
Pump Oscillators for Parametric Transducer
A low noise oscillator is an
essential component of a
parametric transducer
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
w
a
w
p
w
+
=
w
p
+
w
a
w

=
w
p

w
a
?
transducer
Pump
photons
Signal
phonons
Output
sidebands
Treat transducer as a photon scatterer
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
0
w
w
w
P
P
P
a
a
0
w
w
w
P
P
P
p
p
Formal
solution but
results are
intuitively
obvious
Upper mode
Lower mode
Cold damping of
bar modes by
parametric
transducer
Bar mode
frequency
tuning
by pump tuning
Parametric transducer damping and elastic stiffness
Electromechanical Coupling of Transducer to Antenna
signal energy in transducer
signal energy in bar
b
•
In direct transducer
b
= (
1
/
2
CV
2
)/M
w
2
x
2
•
In parametric transducer
b
=(
w
p
/
w
a
)(
1
/
2
CV
2
)/M
w
2
x
2
•
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
w
p
>>
w
a
, P
p
~

(P
+
+P

).
If
P
+
/
w
+
<
P

/
w

,then P
a
< 0…..
negative power flow…instability
If
P
+
/
w
+
>
P

/
w

,then P
a
> 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 Q
e
>
w
p
/
w
a
, then
b is maximised near
the cavity resonance
or at the sideband
frequencies. Strong
cold damping is
achieved for
w
p
=
w
cavity

w
a
.
Thermal noise contributions from bar and secondary
resonator
Thermal noise components for a
bar Q=2 x10
8
(antiresonance at
mid band) and secondary resonator
Q=5 x 10
7
Frequency Hz
bar
Secondary
resonator
Low
b,
high
series noise,
low back
action noise
Spectral
Strain
sensitivity
SNR/Hz/mK
Transducer Optimisation
This and the following
curves from M Tobar
Thesis UWA 1993
Reduced
Am noise
Spectral
Strain
sensitivity
SNR/Hz/mK
Higher
secondary
mass Q

factor
Spectral
Strain
sensitivity
SNR/Hz/mK
Reduced
back
action
noise
from
pump
AM noise
Spectral
Strain
sensitivity
SNR/Hz/mK
High Q
e
, high
coupling
Spectral
Strain
sensitivity
SNR/Hz/mK
Allegro Noise Theory and Experiment
Relations between Sensitivity and Bandwidth
eff
T
T
Q
f
Bandwidth
4
2
bw
M
kT
v
L
f
S
h
g
s
a
h
g
t
w
t
2
2
2
)
(
1
D
Minimum detectable energy is defined by the ratio of
wideband noise to narrow band noise
Express minimum detectable energy as an effective
temperature
f
noise
narrowband
ise
widebandno
T
E
D
D
2
min
Optimum spectral sensitivity depends on ratio
MQ
T
Independent of readout noise
Bandwidth and minimum
detectable burst depends
on transducer and
amplifier
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
High
b
, low
noise,3 mode
Two mode, low
b
,
high series noise
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
w
w
w
w
d
S
F
j
G
SNR
x
)
(
)
(
)
(
2
1
2
2
Fourier tfm of impulse
response of displacement
sensed by transducer for
force input to bar
Fourier tfm of
input signal force
Double sided spectral
density of noise refered to
the transducer displacement
Monochromatic and Stochastic Backgrounds
Both methods allow the limits to bursts to be easily exceeded.
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.
Allegro Pulsar Search
Niobe Noise Temperature
Excess Noise and Coincidence Analysis
Log number of
samples
Energy
•
All detectors show non

thermal 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)
Measure noise performance by noise temperature.
Typically h~(few x 10

17).T
n
1/2
Coincidence Statistics
r
R
P
t
1
Probability of event above threshhold:
(
Event rate R, resolving time
t
r
)
Prob of accidental coincidence in
coincidence window
t
c
If all antennas have same background
Hence in time t
tot
the number of accidental
coincidences is
N
i
i
N
c
N
R
P
,
1
t
N
c
N
N
R
P
t
1
N
c
N
ac
R
N
t
0
5
10
15
20
10
14
10
12
10
10
10
8
10
6
10
4
10
2
10
0
10
2
10
4
events/day
1 bar 1982
2 bars 1991
3 bars 1999
4 bars 1999
(not enough data)
h
burst
x
10
18
Improvements through coincidence analysis
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
isolation stages
Dead bug
cables
Nodal
point
Important tool: Finite element
modelling
Suspension choices
Niobe: 1.5 tonne Niobium Antenna with Parametric Transducer
Niobe Cryogenic System
Niobe Cryogenic Vibration Isolation
vibration isolation
•
Nodal suspension
•
Integrated secondary
and tertiary resonators
for reasonable
bandwidth
•
non

superconducting
for efficient cooldown
•
mass up to 100
tonnes
Sphere
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.
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