1st VIRGO-SIGRAV School on Gravitational Waves - Australian ...

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.