Studies of materials for use in futureinterferometric gravitational wave detectors

Iain William Martin
Department of Physics and Astronomy,
University of Glasgow

Presented as a thesis for the degree of Ph.D.
in the University of Glasgow, University Avenue,
Glasgow G12 8QQ.

February 14, 2009

Contents
Contentsi

List of Figuresvi

List of Tables

xvii

Acknowledgements

xviii

Prefacexx

Summaryxxii

1 Gravitational wave detection

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

The nature of gravitational radiation . . . . . . . . . . . . . . .

3

1.3

Sources of Gravitational Waves . . . . . . . . . . . . . . . . . .

4

1.3.1

Periodic sources . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.2

Burst sources . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3.3

Stochastic gravitational wave background . . . . . . . . .

8

Gravitational wave detection . . . . . . . . . . . . . . . . . . . .

9

1.4.1

Resonant bar detectors . . . . . . . . . . . . . . . . . . .

9

1.4.2

Interferometric detectors . . . . . . . . . . . . . . . . . . 12

1.4

1.5

Limits to the sensitivity of long-baseline detectors . . . . . . . . 13
1.5.1

Gravitational gradient noise . . . . . . . . . . . . . . . . 14

1.5.2

Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.3

Seismic noise . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.4

Photoelectron shot noise . . . . . . . . . . . . . . . . . . 16

1.5.5

Radiation pressure noise . . . . . . . . . . . . . . . . . . 17

1.5.6

The Standard Quantum Limit . . . . . . . . . . . . . . . 18i

Contents
1.6

1.7

1.8

Interferometric techniques . . . . . . . . . . . . . . . . . . . . . 20
1.6.1

Delay line interferometers . . . . . . . . . . . . . . . . . 20

1.6.2

Faby-Perot arm cavities . . . . . . . . . . . . . . . . . . 21

1.6.3

Power recycling . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.4

Signal recycling . . . . . . . . . . . . . . . . . . . . . . . 22

Current interferometric detectors . . . . . . . . . . . . . . . . . 24
1.7.1

Second generation detectors . . . . . . . . . . . . . . . . 28

1.7.2

Future detectors. . . . . . . . . . . . . . . . . . . . . . 29

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Thermal noise in gravitational wave detectors
2.1

2.2

31

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1

Brownian motion . . . . . . . . . . . . . . . . . . . . . . 31

2.1.2

The Fluctuation-Dissipation Theorem . . . . . . . . . . . 32

Sources of mechanical dissipation . . . . . . . . . . . . . . . . . 35
2.2.1

Dissipation from internal friction in materials . . . . . . 35

2.2.2

Harmonic oscillator with internal friction . . . . . . . . . 36

2.2.3

The form of internal dissipation - Debye theory . . . . . 38

2.2.4

Thermal noise associated with a single resonant mode . . 39

2.3

Relationship between mechanical loss and the quality factor . . 42

2.4

Thermoelastic damping . . . . . . . . . . . . . . . . . . . . . . . 44

2.5

Thermal noise in a multi-resonance system . . . . . . . . . . . . 46
2.5.1

Inhomogeneous loss and coating thermal noise . . . . . . 48
2.5.1.1

2.6

Thermoelastic noise . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1

2.7

Calculation of coating thermal noise . . . . . . 49

Coating thermoelastic noise . . . . . . . . . . . . . . . . 52

Suspension thermal noise sources . . . . . . . . . . . . . . . . . 53
2.7.1

Pendulum mode . . . . . . . . . . . . . . . . . . . . . . . 53

2.7.2

Violin modes of the suspension ﬁbres . . . . . . . . . . . 55

2.8

Total thermal noise in a detector mirror . . . . . . . . . . . . . 55

2.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
ii

Contents
3 Inﬂuence of temperature on the settling time of hydroxidecatalysis bonds
58
3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2

The chemistry of hydroxide-catalysis bonding . . . . . . . . . . 61

3.3

Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4

Variation of settling time with temperature . . . . . . . . . . . . 64

3.5

3.4.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . 64

3.4.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.3

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Loss measurements using silica cantilevers at room temperature72
4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2

Mechanical loss of silica ribbons . . . . . . . . . . . . . . . . . . 73

4.3

4.2.1

Measuring mechanical loss . . . . . . . . . . . . . . . . . 74

4.2.2

Experimental method . . . . . . . . . . . . . . . . . . . . 74

4.2.3

Semi-empirical model for intrinsic loss in fused silica

4.2.4

Mechanical loss of ribbons with ‘necks’ . . . . . . . . . . 78

4.2.5

Gas damping . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.6

Welded silica ribbon . . . . . . . . . . . . . . . . . . . . 81

4.2.7

Coating loss measurements using silica cantilevers . . . . 84

. . 77

Conductive coatings for charge mitigation in gravitational wave
detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1

Charging issues in gravitational wave detectors . . . . . . 86

4.3.2

Charge mitigation . . . . . . . . . . . . . . . . . . . . . . 87

4.3.3

Experimental technique . . . . . . . . . . . . . . . . . . 87

4.3.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.4.1

Optical requirements and absorption measurements . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.4.2
4.3.5

Mechanical loss measurements of SnO coatings

89

Coating loss analysis . . . . . . . . . . . . . . . . . . . . 90
iii

Contents

4.4

4.3.6

Tin oxide coating loss results . . . . . . . . . . . . . . . 93

4.3.7

Thermal noise modelling . . . . . . . . . . . . . . . . . . 94

4.3.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Dissipation in optical coatings . . . . . . . . . . . . . . . . . . . 98
4.4.1

Reﬂective dielectric coatings . . . . . . . . . . . . . . . . 99
4.4.1.1

4.4.2

Mechanical dissipation in coating materials . . . . . . . . 101
4.4.2.1

4.4.3

Coating production . . . . . . . . . . . . . . . . 101

Previous measurements of coating dissipation . 102

Coating loss measurements using silica cantilever substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.4
4.5

Coating loss measurements . . . . . . . . . . . . . . . . . 107

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Temperature dependence of the mechanical dissipation in tantalum pentoxide coatings
112
5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.1

Measurements of the temperature dependence of coating
dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2

Initial measurements of the loss of doped tantalum pentoxide . . 116
5.2.1

Sample preparation . . . . . . . . . . . . . . . . . . . . . 116

5.2.2

Experimental method . . . . . . . . . . . . . . . . . . . . 119

5.2.3

5.3

5.2.2.1

Readout system . . . . . . . . . . . . . . . . . . 120

5.2.2.2

Thermoelastic loss as a function of temperature 123

Results for cantilever A . . . . . . . . . . . . . . . . . . . 125
5.2.3.1

Room temperature results . . . . . . . . . . . . 125

5.2.3.2

Measured loss as a function of temperature . . 126

5.2.4

Coating loss results for cantilever A . . . . . . . . . . . . 127

5.2.5

Coating loss results for cantilever B . . . . . . . . . . . . 131

5.2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Dissipation of a TiO2 -doped Ta2 O5 coating between 7 K and
300 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
iv

Contents

5.4

5.3.1

Sample preparation . . . . . . . . . . . . . . . . . . . . . 135

5.3.2

Experimental method . . . . . . . . . . . . . . . . . . . . 137

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.4.1

5.5

Analysis and Interpretation . . . . . . . . . . . . . . . . . . . . 141
5.5.1

5.6

Calculation of the coating loss . . . . . . . . . . . . . . . 141

Possible interpretation . . . . . . . . . . . . . . . . . . . 146

Undoped tantala loss measurements . . . . . . . . . . . . . . . . 149
5.6.1

Experimental setup for cooling with liquid helium . . . . 149

5.6.2

Loss measurements of undoped tantala . . . . . . . . . . 151

5.6.3

Arrhenius plot analysis . . . . . . . . . . . . . . . . . . . 156

5.6.4

Analysis of the distribution of potential battier heights . 159

5.6.5

Comparison of tantala coatings at room temperature . . 163

5.7

The eﬀect of heat treatment . . . . . . . . . . . . . . . . . . . . 164

5.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6 Temperature dependence of the mechanical dissipation in silica coatings
173
6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.2

Coating thermoelastic loss . . . . . . . . . . . . . . . . . . . . . 174

6.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4

Coating loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.4.1

Dissipation peak analysis . . . . . . . . . . . . . . . . . . 186

6.4.2

Silica coating loss above 200 K . . . . . . . . . . . . . . 187

6.5

Discussion - low temperature loss in silica . . . . . . . . . . . . 190

6.6

Multi-layer coating loss calculation . . . . . . . . . . . . . . . . 192

6.7

6.6.1

The mechanical loss of a multi-layer coating . . . . . . . 193

6.6.2

Coating thermoelastic loss . . . . . . . . . . . . . . . . . 196

6.6.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7 Conclusions

201

Bibliography

205v

List of Figures
1.1

The eﬀect of a gravitational wave with (a) the h+ and (b) h×polarisation, incident normal to the page, on a ring of test particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

3

A schematic diagram of a simple Michelson interferometer, consisting of a laser, beam splitter and two end mirrors. . . . . . . 13

1.3

Schematic diagram of two possible layouts for a delay line interferometer: (a) shows a folded arm scheme, while (b) illustrates
a scheme in which the laser beam is reﬂected along each arm
multiple times. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4

A schematic diagram of a Fabry-Perot interferometer, whichuses Fabry-Perot cavities in each arm. The inner mirrors are
partially transmissive to allow light to enter, and leave, the cavity. 21

1.5

A schematic diagram of a Michelson interferometer with a power
recycling mirror at the input port, forming an optical cavity with
the interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6

A schematic diagram of a Michelson interferometer with a signal
recycling mirror at the output port. This mirror forms an opticalcavity with the interferometer and allows the peak sensitivity to
be tuned to speciﬁc frequency ranges. . . . . . . . . . . . . . . . 24

1.7

The VIRGO interferometer at Cascina, Italy. Both 3 km longarms of the detector can be seen, meeting at the central station(right hand side) which houses the input and output optics. . . . 24vi

List of Figures
1.8

Left: A view along one 600 m long arm of the GEO 600 detector, with the end station housing the end test mass visible inthe background. Right: A GEO 600 suspension, consisting of a
triple pendulum, with a monolithic silica lowest stage. . . . . . . 25

1.9

The design sensitivity of GEO 600, showing the various noise
curves for operation with signal recycling at 250 Hz de-tuning. . 26

1.10 The typical amplitude strain sensitivity of GEO 600 in each of
the science runs between 2002 and 2007. . . . . . . . . . . . . . 27
2.1(a) A schematic diagram of a mechanical oscillator with a complex spring constant k(1 + iφ). (b) A schematic diagram representing a standard anelastic solid, with an ideal spring connectedin parallel with a spring-dashpot combination, where k1 and k2
are spring constants and α is the impedance of the dashpot. . . . 37

2.2

Thermal noise displacement spectra for two mechanical oscillators each of f0 =1 kHz, m =20 kg at T =290 K and having
diﬀering mechanical loss. The solid, red line shows the thermal
noise for a loss of φ = 1×10−9 while the black, dashed line is for
φ = 1 × 10−7 The lower loss material has lower oﬀ-resonancethermal noise, as more of the thermal motion is concentrated
close to the resonant frequency. . . . . . . . . . . . . . . . . . . 42

2.3

A schematic diagram of a bending mode of a rectangular crosssection bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4

Calculated levels of Brownian and thermoelastic thermal noise
in a single suspended fused silica mirror, in an interferometerwith a laser beam of radius r0 = 3.9 cm, assuming a test massloss of 5 × 10−9 and an ion-beam sputtered coating formed fromalternating multi-layers of SiO2 and Ta2 O5 . . . . . . . . . . . . 57

3.1

The planned orbit of the three satellites forming the LISA gravitational wave detector. . . . . . . . . . . . . . . . . . . . . . . . 59
vii

List of Figures
3.2

The engineering model of the Lisa Pathﬁnder optical bench,
which utilises fused silica optical components bonded to a Zerodur base plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3

A simpliﬁed diagram showing the surface of fused silica in a
fully-hydrated state after cleaning. Note that the surface consistsof OH groups attached to silicon atoms. . . . . . . . . . . . . . . 62

3.4

A simpliﬁed diagram of the etching stage of the bonding reaction.
Hydroxide ions can etch into the surface of the silica by forming
weak bonds with silicon atoms, weakening the Si-O bonds (a)resulting in silicate ions being released into solution (b). . . . . . 63

3.5

Once the pH of the solution between the silica components has
dropped below 11, the liberated Si(OH)−
5 molecules dissociate toform SiOH4 , which can hydrolyse to form polymer chains, eventually linking the two silica surfaces.

3.6. . . . . . . . . . . . . . . 64

The mechanical excitation system developed to gently shake theupper silica block during bonding. The holder for the silica block
was supported on one side by a rigid connection to a loudspeaker
and on the other side by an elastic connection to a rigid clamp.
A ‘ﬂag’ attached to the top of the holder was used to monitor
the motion of the block throughout the bonding process. . . . . . 65

3.7

Schematic diagram of the temperature controlled glove box inwhich the experiment was carried out. The glove box was 1 mlong, 40 cm deep and 50 cm high. . . . . . . . . . . . . . . . . . 66

3.8

Typical plot of the measured oscillation amplitude of the uppersilica block as a function of time during bonding. The point
where the bond set can be clearly seen at approximately 360 seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.9

Plot of the measured settling time as a function of temperaturefor hydroxide-catalysis bonds made using 0.1 mol l−1 KOH solution, with an exponential ﬁt. . . . . . . . . . . . . . . . . . . . 69
viii

List of Figures
3.10 Arrhenius plot of the settling time data, showing the logarithmof the settling time as a function of the reciprocal of the temperature, with linear ﬁt excluding the room temperature measurement. 70
4.1

A schematic diagram of a silica ribbon (not to scale). . . . . . . 73

4.2

A schematic diagram of the experimental apparatus used to measure the mechanical loss of cantilever samples - not to scale. . . 75

4.3

A schematic diagram of the split photodiode arrangement used
to measure the amplitude of oscillation of a cantilever . . . . . . 76

4.4

A ringdown measurement for the the third bending mode at 1051
Hz. A sampling rate of 15 Hz was used, and the mechanical loss
calculated from the ﬁtted exponential decay constant. . . . . . . . 76

4.5(a) The ﬂame-pulled ribbon section in the clamp. (b) The laserpulled ribbon in the clamp. . . . . . . . . . . . . . . . . . . . . . 79

4.6

Frequency dependence of the mechanical loss of a ribbon witha short neck fabricated with a laser pulling machine, comparedto that of a ribbon with a longer neck fabricated with a ﬂame
pulling machine. Also shown are the losses predicted by Penn’s
model for both samples. . . . . . . . . . . . . . . . . . . . . . . . 80

4.7

Left: Photograph of CO2 laser welding. Right: Schematic diagram of laser welding apparatus. . . . . . . . . . . . . . . . . . . 82

4.8(a) Photograph of the cantilever immediately after welding. (b)Photograph of the welded cantilever in the clamp. . . . . . . . . 83

4.9

Frequency dependence of the mechanical loss of a ribbon section
welded to a silica slide, compared to that of a similar ribbon
clamped at the neck region. Also shown are the losses predicted
by Penn’s model for both samples. . . . . . . . . . . . . . . . . . 84

4.10 Microscope images of the welded silica cantilever. . . . . . . . . 85
4.11 Plot of the measured loss of four modes of sample 1 before andafter coating with 25 nm of SnO. . . . . . . . . . . . . . . . . . 91
ix

List of Figures
4.12 A beam of thickness a and width b, with a thin coating of thickness t applied to one face, bent into an arc of radius R. . . . . . 92
4.13 Plot of the calculated coating loss for each of the three SnO coatings studied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.14 The mechanical loss measured as a function of coating thickness,with a linear ﬁt to the data. Each point indicates the mean loss
measured for several modes of a sample. . . . . . . . . . . . . . 95
4.15 (a) Microscope images of the surface of sample 3 after coatingwith tin oxide. (b) Higher magniﬁcation image of the tin oxide
coating showing evidence of the formation of large crystals. . . . 96
4.16 Calculated levels of the Brownian thermal noise associated witha 100 nm and a 4 nm thick SnO coating on a fused silica mirror
illuminated by a laser beam of radius r0 =3.9 cm, compared tothe thermal noise arising from a multilayer HR coating. . . . . . 97
4.17 A schematic diagram of a multi-layer reﬂective coating, composed of alternating layers of a high refractive index (nH ) and a
low refractive index (nL ).

. . . . . . . . . . . . . . . . . . . . . 100

4.18 A schematic diagram of a simple ion-beam sputtering chamber. . 100
4.19 Plot of the lowest losses measured in the 7 welded silica membranes, compared with the losses measured in a welded ribbon
section.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.20 Plot of the coating loss factors for LMA F5** and F5*** tantala, measured using silica membrane substrates welded to silica
clamping blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.21 Plot of the coating loss factors for undoped tantala, LMA F5*
doped tantala and silica coatings, measured using unwelded membranes in the clamp developed by LMA. . . . . . . . . . . . . . . 109
5.1

Schematic diagram of cantilever sample A, 34 mm long, 10 mm
wide and 92μm thick. The cantilever length is parallel to the110 crystal axis and the surface is on the (100) plane. . . . . . 116x

List of Figures
5.2(a) Cantilever A before coating. (b) Cantilever A after coating,clamped in preparation for mechanical loss measurements. . . . . 118

5.3

Schematic diagram of the cryostat, with an inset showing detailsof the clamp used to hold the cantilever samples. . . . . . . . . . 119

5.4

Diagram of the photodiode sensor used to measure the displacement of the cantilever. The mask was made from electrical insulating tape, and the maximum width of the exposed photodiode(labelled a) was 1 mm. The laser spot diameter D was measured
to be approximately 5 mm.

5.5. . . . . . . . . . . . . . . . . . . . 120

Circuit diagram for the photodiode circuit used to measure theamplitude of the cantilever displacement. . . . . . . . . . . . . . 121

5.6

Plot of the response of the photodiode readout system used tomeasure the amplitude of the cantilever displacement. . . . . . . 122

5.7

Noise plot for the photodiode sensor used to sense the motion of
the cantilever. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.8

The frequency dependence of the calculated thermoelastic loss
for cantilevers A and B (92 μm and 52 μm thick respectively).
The arrows mark the ﬁrst 6 bending frequencies of the samples. . 123

5.9

Calculated thermoelastic loss as a function of temperature forthree modes of silicon cantilever B. . . . . . . . . . . . . . . . . 124

5.10 Plot of room temperature loss measurements of cantilever A before and after coating with a 500 nm thick ﬁlm of Ta2 O5 dopedwith TiO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.11 Plot of the measured mechanical loss against temperature for thethird bending mode of cantilever A before and after coating. . . . 126
5.12 Plot of the measured mechanical loss of the fourth bending modeof cantilever A, before and after coating. . . . . . . . . . . . . . 127
5.13 Plot of the calculated coating loss as a function of temperaturefor 3rd mode of cantilever A. . . . . . . . . . . . . . . . . . . . . 128
5.14 Plot of the calculated coating loss as a function of temperaturefor 4th mode of cantilever A. . . . . . . . . . . . . . . . . . . . . 129
xi

List of Figures
5.15 Plot of the measured coating loss at approximately 1.9 kHz andthe amplitude of the piezo signal detected at the start of each
ringdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.16 Plot of the measured loss for the fourth mode of cantilever B
and its identical uncoated control sample. . . . . . . . . . . . . . 131
5.17 Plot of the measured loss for the ﬁfth mode of cantilever B and
its identical uncoated control sample. . . . . . . . . . . . . . . . 132
5.18 Plot of the coating loss for cantilever B at 777 Hz and 1280 Hz. 133
5.19 Plot of the coating loss for cantilever A at 1920 Hz and 3768 Hz,
calculated using the lowest loss factors measured for the coatedcantilever at each temperature. . . . . . . . . . . . . . . . . . . . 134
5.20 Schematic diagram of the cryostat used to measure the doped
tantala coating at Jena University, showing the laser interferometer readout system used. . . . . . . . . . . . . . . . . . . . . 136
5.21 Plot showing the temperature dependence of the measured mechanical loss of the ﬁrst mode (∼ 55 Hz) of the coated and uncoated cantilevers. . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.22 Temperature dependence of the measured mechanical loss of thesecond mode (∼ 350 Hz) of the coated and uncoated cantilevers. 138
5.23 Temperature dependence of the measured mechanical loss of thethird mode (∼ 989 Hz) of the coated and uncoated cantilevers. . 139
5.24 Temperature dependence of the measured mechanical loss of the
fourth mode (∼ 1939 Hz) of the coated and uncoated cantilevers. 139
5.25 Temperature dependence of the measured mechanical loss of theﬁfth mode (∼ 3184 Hz) of the coated and uncoated cantilevers. . 140
5.26 Temperature dependence of the dissipation of the doped Ta2 O5
coating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.27 The mechanical loss peak measured in the doped Ta2 O5 coating.
For clarity, error bars are only shown for the ﬁfth mode: these
are typical of the errors in all of the points. . . . . . . . . . . . . 143
xii

List of Figures
5.28 The ﬁts used to determine the position of the mechanical loss
peak in the doped Ta2 O5 coating. Note the temperature at which
the peak occurs increases with increasing mode frequency. . . . 144
5.29 Arrhenius plot, showing a linear ﬁt of ln(ω0 ) against 1/Tpeak forthe dissipation peak. . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.30 Comparison of the measured dissipation peak to the theoreticalpeak obtained from the Arrhenius ﬁt. . . . . . . . . . . . . . . . 146
5.31 (a) Bright ﬁeld TEM image of a tantala layer (dark) and a silica
layer (light) in a SiO2 / Ta2 O5 multi-layer coating, where theTa2 O5 layers are doped with 8% TiO2 (b) convergent beam electron diﬀraction from a Ta2 O5 layer showing just diﬀuse rings ofintensity, conﬁrming that this layer is amorphous. Both images
were taken by Ian MacLaren of the solid state physics group at
Glasgow University. . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.32 The addition of radiation shielding across the feed-through ports
in the roof of the experimental chamber (a) and thermally anchoring wires to the base-plate (b) allowed the clamp to cool to
78 K, 6 degrees lower than the previous lowest temperature obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.33 Comparison of the loss of an undoped Ta2 O5 and a Ta2 O5 coating doped with 14.5 % TiO2 at ∼ 1000 Hz. . . . . . . . . . . . . 152
5.34 Comparison of the loss peak for an undoped Ta2 O5 and a Ta2 O5
coating doped with 14.5 % TiO2 at ∼ 1000 Hz. . . . . . . . . . 153
5.35 Comparison of the loss of an undoped Ta2 O5 and a Ta2 O5 coating doped with 14.5 % TiO2 at ∼ 1900 Hz. . . . . . . . . . . . . 153
5.36 Comparison of the loss peak for an undoped Ta2 O5 with a Ta2 O5
coating doped with 14.5 % TiO2 at ∼ 1900 Hz. . . . . . . . . . . 154
5.37 Comparison of the loss peak for an undoped Ta2 O5 with a Ta2 O5
coating doped with 14.5 % TiO2 at ∼ 1900 Hz. . . . . . . . . . . 154
5.38 Comparison of the loss of an undoped Ta2 O5 and a Ta2 O5 coating doped with 14.5 % TiO2 at ∼ 3200 Hz. . . . . . . . . . . . . 155
xiii

List of Figures
5.39 Comparison of the loss peak for an undoped Ta2 O5 with a Ta2 O5
coating doped with 14.5 % TiO2 at ∼ 355 Hz. . . . . . . . . . . 155
5.40 The polynomial ﬁts used to determine the position of the dissipation peak for the second and third bending modes. . . . . . . . 156
5.41 The polynomial ﬁts used to determine the position of the dissipation peak for the fourth and ﬁfth bending modes. . . . . . . . . 157
5.42 Comparison of the Arrhenius plots of ln(ω0 ) against 1/Tpeak forthe dissipation peaks in TiO2 doped Ta2 O5 and undoped Ta2 O5 . . 158
5.43 A schematic diagram of an asymmetric double well potential,with a potential barrier V and an asymmetry Δ. . . . . . . . . . 159
5.44 Plot of the loss data used to calculate the barrier height distribution for pure tantala, titania doped tantala and silica. . . . . . 161
5.45 Plot of the calculated potential barrier height distribution function g(V )f0 as a function of barrier height for pure tantala,titantia-doped tantala and silica. . . . . . . . . . . . . . . . . . . 162
5.46 Comparison of the coating losses measured at room temperature
for Formula 5, Formula 5∗∗ and undoped Ta2 O5 coatings. . . . . 163
5.47 Measurements of the mechanical loss of the second bending mode
(at approximately 350 Hz) of three cantilevers coated with undoped Ta2 O5 and heated to 300, 600 and 800 ◦ C respectively. . . 166
5.48 Measurements of the mechanical loss of the third bending mode
(at approximately 1000 Hz) of three cantilevers coated with undoped Ta2 O5 and heated to 300, 600 and 800 ◦ C respectively. . . 166
5.49 Measurements of the mechanical loss of the fourth bending mode
(at approximately 1960 Hz) of three cantilevers coated with undoped Ta2 O5 and heated to 300, 600 and 800 ◦ C respectively. . . 167
5.50 Measurements of the mechanical loss of the ﬁfth bending mode
(at approximately 3200 Hz) of three cantilevers coated with undoped Ta2 O5 and heated to 300, 600 and 800 ◦ C respectively. . . 167
5.51 Arrhenius plot for the dissipation peak observed at ∼ 30 K for
an undoped tantala coating heat treated at 300 ◦ C . . . . . . . . 168
xiv

List of Figures5.52 Transmission electron microscope and EELS images of an undoped tantala coating annealed to 800◦ C. . . . . . . . . . . . . . 170
6.1

Comparison of the coating thermoelastic loss at 290 K calculated
for a 500 nm thick SiO2 coating and a 500 nm thick Ta2 O5
coating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.2

Temperature dependence of the coating thermoelastic loss calculated for three modes of the SiO2 coated silicon cantilever. . . . . 178

6.3

Plot of the measured loss for the ﬁrst mode of the cantilevercoated with silica and of the uncoated control sample. . . . . . . 178

6.4

Plot of the measured loss for the second mode of the cantilevercoated with silica and of the uncoated control sample. . . . . . . 179

6.5

Plot of the measured loss for the third mode of the cantilevercoated with silica and of the uncoated control sample. . . . . . . 179

6.6

Plot of the measured loss for the fourth mode of the cantilevercoated with silica and of the uncoated control sample. . . . . . . 180

6.7

Plot of the measured loss for the ﬁfth mode of the cantilevercoated with silica and of the uncoated control sample. . . . . . . 180

6.8

Calculated dissipation of the silica coating for the ﬁrst mode at
59 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.9

Calculated dissipation of the silica coating for the second mode
at 355 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.10 Calculated dissipation of the silica coating for the third mode at
995 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.11 Calculated dissipation of the silica coating for the fourth modeat 1920 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.12 Calculated dissipation of the silica coating for the ﬁfth mode at
3175 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.13 Arrhenius plot for the low temperature dissipation peak in thesilica coating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
xv

List of Figures
6.14 Plot comparing the initial measurements of the temperature dependence of the coating loss with the measurements repeated atroom temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.15 Plot of the repeated room temperature measurements of the coating loss. Shown for comparison are the loss factors presented in
Chapter 4 for a silica coating on a silica cantilever substrate andPenn’s coating loss data, measured at room temperature also using silica substrates. . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.16 A comparison of the dissipation measured for the ion-beam sputtered silica coating with dissipation measured in other types ofsilica.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.17 The measured coating loss for the silica and tantala coatings at∼ 1 kHz which was used in the multi-layer coating loss calculation.194
6.18 The expected loss in a 31 layer SiO2 /Ta2 O5 coating, calculated
from the single layer results, compared with the loss measured
in such a multi-layer coating by Yamamoto et al.

. . . . . . . . 195

6.19 Plot showing Yamamoto’s coating loss measurements at 1.1 kHz
and the calculated coating thermoelastic loss in (i) a 31 layer
SiO2 /Ta2 O5 coating on a sapphire substrate at room temperature and (ii) a SiO2 /Al2 O3 coating on a sapphire substrate as afunction of temperature. . . . . . . . . . . . . . . . . . . . . . . 197

xvi

List of Tables
4.1

Dimensions of the two ribbons with necks. . . . . . . . . . . . . 78

4.2

Summary of thickness and mean mechanical loss for the three
tin oxide coating samples. . . . . . . . . . . . . . . . . . . . . . 94

4.3

The average mechanical loss (calculated for all mode frequencies
studied) of several coatings applied to silica membranes. All
coatings were 0.5 μm thick: the membranes were 110 μm thick. 110

5.1

Properties of the silicon cantilevers A, B and C. . . . . . . . . . 118

5.2

The temperature at which the dissipation peak occurs and thepeak magnitude of the dissipation for each mode of the cantilevercoated with doped Ta2 O5 measured in Jena. . . . . . . . . . . . . 144

6.1

Material properties for silicon, silica and tantala at 290 K, usedin the calculation of the coating thermoelastic loss. . . . . . . . . 175

6.2

The coating loss factors for each mode at room temperature,
calculated using the data from the repeated loss measurements ofthe coated cantilever. Also shown is the calculated thermoelastic
coating loss for each mode frequency. . . . . . . . . . . . . . . . 189

6.3

Material properties of sapphire used to calculate coating thermoelastic loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xvii

Acknowledgements
Firstly, I would like to thank my supervisor Professor Sheila Rowan, for all ofher help and advice over the course of my PhD. My second supervisors Professor Jim Hough and Dr Caroline Cantley were also great sources of inspirationand encouragement. I thoroughly enjoyed my PhD, and am very grateful tohave had the opportunity to work in the Institute for Gravitational Research.
The work presented in this thesis was carried out with the support of a
PPARC (latterly STFC) CASE studentship in conjunction with the RutherfordAppleton Laboratory. Many thanks to Justin Greenhalgh, Tim Hayler and Ian
Wilmut at RAL. I would like to thank the cryogenics group of the Solid State
Physics Institute at Jena University for the opportunity to work with them fora month, and especially to Ronny Nawrodt and Christian Schwarz for theirwonderful hospitality.
I am very grateful to Stuart Reid and Alastair Heptonstall for all of their
good advice and for many useful discussions: both were a pleasure to workwith. Thanks also to Alan Cumming who shared an oﬃce with me from thestart and oﬀered much good-humoured practical assistance and support whenthings didn’t work. Colin Craig and Stephen Craig deserve special thanks foralways managing to get last-minute technical jobs done so well. I would also
like to thank Adam Woodcraft for his advice on all things cryogenic and forbeing willing to spend several late nights in the lab while showing me the ropes
with liquid helium transfers.
Others I would like to thank include Eleanor Chalkley, for sharing an oﬃce
and a lab; David Crooks, for his help at the start and for introducing me toxviii

Acknowledgements
espresso; Christian Killow and Ewan Fitzsimmons, fellow drinkers of lots ofcoﬀee; and all the occupants of room 427a. My friends Gordon, Craig, Robert,
Andy, Jen, Jenn and Jennifer and the regular Wednesday night pub trips did
a great job of keeping me sane. My sister Kirsty, though long-ago lost to
biochemistry, has often commiserated with me on the trials and tribulationsof doing an experimental PhD, which seem to be similar across all disciplines.
Thanks to my parents for all of their help, advice and support with everything
I’ve done over the years. I am particularly grateful to my Mum for proof
reading this thesis, and to my Dad for getting me interested in science in theﬁrst place. Finally, I would like to thank my partner Mairi for all of her love,support and understanding over the past few years.

xix

Preface
This thesis is an account of work carried out in the Institute for Gravitational
Research at the University of Glasgow between October 2004 and August 2008,
involving studies of materials for use in future gravitational wave detectors.
In Chapter 1 the nature and origin of gravitational waves is discussed,
various detection techniques are summarised and the status of current andplanned gravitational wave detectors reviewed.
In Chapter 2 the theory of thermal noise, its relationship to mechanicaldissipation and its inﬂuence on the displacement sensitivity of interferometricgravitational wave detectors is discussed.
In Chapter 3 an investigation into the extension of the settling time ofhydroxide catalysis bonds by cooling is presented. This work was carried out
jointly with S. Reid, who additionally constructed the low temperature environment and optical readout system which were used.
In Chapter 4 the development of a technique to measure the mechanicaldissipation of optical and conductive coating materials applied to silica cantilever substrates is discussed. A. Heptonstall fabricated the silica ribbon ﬁbres
used and assisted in welding them to silica slides using a CO2 laser. Some of
the measurements of the dissipation of optical coatings were carried out jointlywith P. Holt. The conductive SnO coatings were fabricated by S. Reid, who
also measured the coating thickness. L. Cunningham measured the resistivityof the coatings, and the optical absorption was measured by A. Markosyan
at Stanford University. The mechanical loss measurements and analysis werecarried out by the author.
xx

Preface
In Chapter 5 studies of the temperature dependence of mechanical dissipation in Ta2 O5 (tantala) coatings applied to thin silicon substrates are presented. This work was largely carried out in Glasgow using a cryostat initially
set up by S. Reid and modiﬁed by the author to allow cooling to liquid heliumtemperatures. In addition, measurements of the dissipation of a TiO2 -doped
Ta2 O5 coating were carried out at the University of Jena by the author, R.
Nawrodt and C. Schwarz. Analysis of the data was performed by the author,
with input from S. Rowan and J. Hough. One data point used in the analysis was derived from similar measurements at Glasgow by E. Chalkley. TEM
microscope and electron diﬀraction images of the coatings were taken by I.
MacLaren, R. Bassiri and J. Scott.
In Chapter 6 measurements of the temperature dependence of mechanicaldissipation in an SiO2 (silica) coating are presented. These measurements werealso performed at the University of Jena, by the author, R. Nawrodt and C.
Schwarz. Calculations of the expected level of coating thermoelastic loss in thecoated sample were carried out by the author, using published literature. The
analysis was carried out by the author following discussions with S. Rowan
and J. Hough.

xxiSummary
Gravitational waves, predicted by the theory of General Relativity, are ﬂuctuations in the curvature of space-time which arise from the asymmetric acceleration of mass. While gravitational waves have yet to be detected directly,
measurements of the inspiral rate of a binary pulsar system have provided
strong evidence for their existence and a world-wide eﬀort to develop moresensitive detectors is ongoing. In addition to testing predictions of General
Relativity, observation and analysis of gravitational waves from astrophysicalsources will provide new insights into a wide range of phenomena including
black holes, neutron stars and supernovae.
Gravitational waves are quadrupole in nature, and therefore produce ﬂuctuating tidal strains on space. Long baseline gravitational wave detectors aim
to measure this eﬀect using laser interferometry to measure ﬂuctuations in
the relative separation of free masses, coated to form highly reﬂective mirrors
and suspended as pendulums at the ends of perpendicular arms up to 4 km
in length. There are currently several long baseline gravitational wave detectors in operation around the world, including the three LIGO detectors in theUS, the UK/German GEO600 detector near Hannover and the French/ItalianVirgo detector near Pisa.
The strain expected from gravitational waves is very small, of order ≈10−22 . The magnitude of the resultant displacement is such that the thermal motion of the mirrors and their suspensions forms an important limit todetector sensitivity. The level of thermal noise is related to the mechanicaldissipation of the materials used in the test mass and the mirror coatings.
xxii

Summary
The research presented focuses on the characterisation and reduction of
the mechanical loss of materials for use in the construction of the interferometer mirrors, with the aim of reducing the level of thermal noise in future
gravitational wave detectors. An introduction to gravitational waves, possible
astrophysical sources and the techniques used in both interferometer and bar
detectors is given in Chapter 1. Chapter 2 discusses thermal noise, one of the
major limits to sensitivity in interferometric gravitational wave detectors, andits relationship to the mechanical loss of the materials used in the detector
mirrors.
The technique of hydroxide catalysis bonding was used to allow the construction of low thermal noise monolithic silica mirror suspensions in the GEO
600 detector. Hydroxide catalysis bonding is also of interest for the space based
gravitational wave detector LISA, which requires the rigid attachment of many
precisely aligned optical components to a Zerodur base plate. In Chapter 3,
an investigation of the dependence of the settling time of the bond on temperature is presented, and the results used to calculate the activation energy
associated with the bonding reactions.
Recent research has suggested that the thermal noise associated with theion-beam sputtered mirror coatings applied to the test masses will limit thesensitivity of planned ‘second generation’ detectors in their most sensitive frequency band. Current gravitational wave detectors use multi-layer dielectric
mirror coatings formed from alternating layers of ion-beam sputtered silica(SiO2 ) and tantala (Ta2 O5 ). Research carried out by the Institute for Gravitational Research in Glasgow in collaboration with colleagues in the US has
shown that the mechanical dissipation of these coatings arises predominantlyin the tantala layers, and that the dissipation can be reduced by ∼ 40 to 50% by doping the tantala with titania (TiO2 ). However, neither the process
responsible for mechanical dissipation in tantala or the mechanism by whichtitania doping reduces the dissipation is clearly understood.
xxiii

Summary
In Chapter 4, the fabrication of silica cantilevers for use in measurementsof the mechanical loss of various coatings is discussed. In addition to measurements of the dissipation of single layer coatings of silica and tantala atroom temperature, a study of the dissipation of conductive SnO coatings ispresented. SnO coatings may be of interest for the mitigation of electrostatic
charging, which can result in excess noise and control problems in gravitational
wave detectors. The results indicate that thin layers of SnO may meet the mechanical and optical speciﬁcations for use in the second generation Advanced
LIGO detectors.
Studies of the temperature dependence of the mechanical dissipation of
tantala coatings are presented in Chapter 5. A peak in the dissipation was
found at approximately 20 K. Analysis of this peak provides insight into possible dissipation mechanisms in ion-beam sputtered tantala, suggesting thatit may involve the reorientation of molecules in double-well potential systems.
The width of the dissipation peak suggests that there is a broad distributionof potential barrier heights, probably arising from the amorphous structureof the coating. Doping tantala with titania is shown to increase the average
activation energy of the dissipation process by ∼ 33%, in addition to reducing
the magnitude of the dissipation. Further experiments to study the eﬀect ofheat treatment on the low temperature mechanical dissipation of tantala coatings are described. An improved understanding of mechanical dissipation in
ion-beam sputtered tantala, based on the results presented here, may enable
the sensitivity of future gravitational wave detectors to be increased throughthe development of coatings with lower mechanical loss.
In Chapter 6 measurements of the temperature dependence of the dissipation of a silica coating are presented, and the possible eﬀect of coating thermoelastic loss on the measurements is assessed. Using the measurements fromthis chapter and Chapter 5, the expected temperature dependence of the dissipation in a silica/tantala multi-layer coating is calculated. The results suggestxxivSummarythat the multi-layer coating loss increases with decreasing temperature to apeak at approximately 20 K.

xxv

Chapter 1
Gravitational wave detection
1.1

Introduction

The existence of gravitational waves is a prediction of Einstein’s General Theory of Relativity [1], which describes the force of gravity as arising from curvature in space-time created by the presence of mass. A direct consequenceof this theory is that the acceleration of asymmetric masses should produce
ﬂuctuations in the curvature of space-time, or gravitational waves, propagating at the speed of light. Due to the nature of gravity, very large masses andaccelerations are required to produce detectable gravitational waves. Thus the
only likely sources of detectable gravitational radiation are astrophysical events
such as supernovae, orbiting and coalescing binary neutron stars and merging
black holes.
Gravitational waves produce ﬂuctuating tidal forces on masses in their path.
Due to the relative weakness of gravitational interaction with matter, thesetidal forces are predicted to be very small, making gravitational radiation exceptionally challenging to detect. However, as a consequence of the same weakinteraction with matter, gravitational waves are not as susceptible to scattering and absorption as electromagnetic radiation, making them a potentiallyinvaluable tool in astronomy.
Gravitational waves have not yet been directly detected, although extensive
experimental eﬀorts to do so are ongoing, originating from work by Joseph11.1 Introduction

2

Weber in the 1960s [2]. However, compelling evidence for the existence ofgravitational radiation has been provided by studies of the binary pulsar PSR
1913 + 16, discovered in 1974 by Hulse and Taylor [3, 4]. Analysis of the decayrate of the binary orbital period has shown that this system is losing energy
and angular momentum at almost precisely the rate predicted by models forthe emission of gravitational radiation [5, 6, 7]. In the decades following this
discovery, considerable progress has been made towards the direct detectionof gravitational waves. The ﬁrst generation of broad-band gravitational wave
detectors began to acquire observational data in 2002. While no detection
has been achieved after several years of observation, the data obtained haveput scientiﬁcally meaningful constraints on astrophysical models of various
sources. Signiﬁcant upgrades to improve the sensitivity of the ﬁrst generationof detectors are in the advanced planning stages, and it is widely expected thatthese upgrades will enable the ﬁrst detection of gravitational waves.
Detection of gravitational waves, and observation of their polarisation statesand velocity, will allow the predictions of General Relativity to be directly veriﬁed. Of greater signiﬁcance is the exciting prospect of gravitational wave
astronomy. Signals from a wide variety of astrophysical sources including neutron stars, binary compact objects, supernovae, black holes and gamma ray
bursts may be detected, greatly increasing our knowledge of these phenomena. A gravitational background radiation ﬁeld, analogous to the the cosmic
microwave background but originating earlier in the history of the universe,
has been predicted. Detecting these primordial gravitational waves would beof great cosmological signiﬁcance, perhaps placing constraints on inﬂation theories and providing information on the large scale structure of the universe.
Finally, it should be remembered that previous major advances in astronomy
have often resulted in new and unexpected observations, such as the discoveryof pulsars. Gravitational waves will provide a radically new way of observing
the universe, and it is possible that unexpected sources will be discovered.

3

1.2 The nature of gravitational radiation

1.2

The nature of gravitational radiation

Gravitational radiation is produced by the acceleration of mass, in a similar way to the production of electromagnetic radiation by the acceleration of
charge. However, the dominant component of a radiated electromagnetic ﬁeld
is dipole radiation. A gravitational dipole moment analogous to an electric
dipole can be deﬁned, but the conservation of momentum implies that it mustremain constant, precluding the existence of gravitational dipole radiation.
Fundamentally, this diﬀerence arises from the existence of both positive andnegative electric charge, whereas only positive mass can occur. There are noconservation laws to prevent higher moments of mass distributions from varying, and variations in gravitational quadrupole moment are predominantly
responsible for the emission of gravitational radiation.
The quadrupole nature of gravitational waves implies that they can only begenerated by the acceleration of an asymmetric mass and are typically emitted
at twice the oscillation frequency of the source. A gravitational wave exerts
diﬀerential strains along perpendicular paths in space, as shown in Figure 1.1.
This ﬁgure shows the eﬀect of a gravitational wave travelling perpendicular tothe page on a ring of masses of diameter L. The perpendicular axes of the ring

(a)
L

L+ΔL

L

L-ΔL

(b)

Figure 1.1: The eﬀect of a gravitational wave with (a) the h+ and (b) h× polarisation, incident normal to the page, on a ring of test particles.

4

1.3 Sources of Gravitational Waves

are alternatively extended and compressed by ΔL. The total strain is given
by:
h=

2ΔL
,
L

(1.1)

where h is the amplitude of the gravitational wave.
Long baseline gravitational wave detectors aim to measure this eﬀect using laser interferometry to measure ﬂuctuations in the relative separation offree masses located at the ends of perpendicular arms. Currently there areseveral such interferometric detectors in operation (some are currently being
up-graded) including the 600 m long GEO 600 detector in Germany [8], thetwo 4 km long and one 2 km long LIGO detectors in the United States [9],the French-Italian VIRGO detector in Italy [10] and the TAMA detector inJapan [11]. These detectors have all reached, or closely approached, their de1

sign sensitivities, with the lowest strain sensitivity of ∼ 2×10−23 Hz− 2 being
achieved by one of the LIGO detectors at its most sensitive frequencies. The
principles of laser interferometry and the design of these detectors is discussed
in Section 1.4.2. First, a more detailed overview of various possible sources ofgravitational waves is given.

1.3

Sources of Gravitational Waves

Many astrophysical and cosmological sources of gravitational radiation havebeen predicted. These can be conveniently grouped according to the type ofsignal they are expected to emit.

1.3.1

Periodic sources

Periodic sources emit gravitational waves continuously at an essentially constant frequency, with the most likely sources in the detection band of current detectors being rotating single neutron stars and low mass X-ray binaries(LMXBs) [12]. Neutron stars are formed from the supernova remnants of stars

5

1.3 Sources of Gravitational Wavesof between ∼ 1.4 and ∼ 24 solar masses. As a neutron star cools, a solid crust
forms, while the inner regions of the star are thought to remain in a liquid
phase. For a single neutron star to emit gravitational radiation, it must have aquadrupole moment resulting from an axial asymmetry. Three major mechanisms of gravitational wave emission have been postulated: non-axisymmetricdistortions of the crust, internal modes of the star and free precession of thestar.
Non-axisymmetric distortions of a single neutron star could arise from either elastic strains or from magnetic ﬁelds. Elastic strains may be caused by
the gradual spin-down of the star or may be ‘frozen’ into the crust during theformation of the star. The gravitational wave amplitude, h, of waves producedby these distortions is given by:h=

4π 2 G Izz f 2
,
c4
d

(1.2)

where Izz is the principal moment of inertia of the star,  is the ellipticity, dis the distance of the star from Earth, G is the gravitational constant, c is the
speed of light, and f is the gravitational wave frequency, which is twice therotational frequency of the neutron star [12]. The actual ellipticity of neutronstars is highly uncertain, but is likely to be limited by the breaking strain of thecrust. Certain proposed exotic stars, such as postulated ‘solid strange quarkstars’ [13], could potentially support larger ellipticities than a neutron star [14].
Indeed, it is possible that such stars may be identiﬁed for the ﬁrst time fromtheir gravitational wave signal. Another possible source of ellipticity is a strong
internal magnetic ﬁeld [15], which can be caused by diﬀerential rotation as theneutron star forms. The ellipticity can be increased if the toroidal componentof the magnetic ﬁeld is aligned towards the equator of the star.
Oscillations in the ﬂuid interior of a neutron star, most likely axial pulsation
modes (also known as ‘r-modes’), could also emit gravitational waves [16, 17].
This is thought to be particularly likely in newly formed or rapidly accreting
neutron stars.

1.3 Sources of Gravitational Waves

6

The third possible emission mechanism for a single neutron star is free
precession [18], a wobble of the star which occurs when the symmetry axis androtation axis do not coincide. The expected gravitational wave amplitude is:
 

θw
1kpc  ν 2
−27
hpreccession = 10
,
(1.3)
0.1
d
500Hz
where θw is the amplitude of this wobble in radians and ν is the orbital frequency [12, 19].
One neutron star, the Crab pulsar, is thought to be one of the most promising candidates as a local source of gravitational waves. Radio observations haveshown that the spin frequency of this pulsar is decreasing at a rate of 3.7 ×
10−10 Hz s−1 . If it is assumed that this spin-down is entirely due to the loss ofenergy by gravitational radiation, then an upper limit of h = 1.4 ×10−24 canbe placed on the gravitational wave amplitude at Earth [20].
Finally, LMXBs are thought to be good candidate sources of gravitationalradiation [21]. These are systems in which a neutron star accretes matter froma binary companion star (often a main sequence star or a red giant). The
stream of superheated matter emits intense X-rays as it falls into the neutronstar. Guided by magnetic ﬁelds, the accreted material could accumulate toform denser regions on the surface of the neutron star. The resulting distortionof the neutron star would increase the quadrupole moment leading to theemission of gravitational waves.
Another possible emission mechanism was predicted by Wagoner [22], whodemonstrated that an accreting neutron star could gain enough angular momentum to surpass the Chandrasekhar-Friedman-Schutz (CFS) limit [23, 24],at which point its rotation would become non-axisymmetric, leading to theemission of gravitational waves. These neutron stars, known as Wagoner stars,
are particularly interesting as the gravitational wave amplitude would be related to the observed X-ray ﬂux from the system.

7

1.3 Sources of Gravitational Waves

1.3.2

Burst sources

Certain rapid events, such as supernovae and the coalescence of compact binarystars [25], will produce short bursts of gravitational waves rather than continuous signals. Bursts of gravitational waves are signiﬁcantly more likely to be
detected if the expected waveform can be calculated and used as a template
for the data analysis procedures.
Over half of all stars are thought to occur in binary systems, in which twostars orbit their common centre of mass. Binaries consisting of a pair of compact objects (i.e. neutron stars or black holes) have been widely studied assources of gravitational waves. As mentioned in the introduction, the orbital
decay of the binary pulsar PSR 1913+16 has been measured to be within 0.2% of the rate predicted from the emission of gravitational waves [7]. As gravitational radiation carries away orbital energy and angular momentum from abinary system, the two compact objects move closer together and orbit more
rapidly until they ﬁnally coalesce. During the last few thousand orbits before coalescence (corresponding to a time of only a few seconds) the frequency
sweeps up from ∼ 10 to ∼ 300 Hz [25]. This rapid ‘chirp’ signal occurs in thefrequency band of current ground based detectors. The approximate strain
amplitude expected from inspiralling neutron stars can be shown to be [26]:−23h ≈ 10100Mpc
d



Mb
1.2M

 53 

f
200Hz

 23
,

(1.4)

where Mb = (M1 M2 )3/5 /(M1 + M2 )1/5 is known as the mass parameter of thebinary, and M1 and M2 are the masses of the two neutron stars.
The ﬁnal coalescence event is not yet well understood, making it diﬃcult tosearch for the emitted gravitational wave signal. However, the inspiral phasecan be modelled [27] and by tracking the changing frequency of the inspiral, it
may also be possible to detect the ﬁnal merger event.
Binary systems consisting of a black hole and a neutron star, or of two black
holes, have not been observed; however, they are expected to exist, although1.3 Sources of Gravitational Waves

8with a lower occurrence than neutron star binaries. Due to the greater density
and mass of a black hole, the gravitational wave amplitude from binary black
holes is expected to be about 4 times larger than from binary neutron stars,
making these inspirals promising candidates for detection.
An important property of inspiralling binary systems is that they are ‘standard candles’ for gravitational wave emission. If the time dependence of boththe frequency and amplitude of the signal are measured, then the distanceof the system can be calculated from the intensity of the detected gravitational waves. Thus, if enough events are measured to high enough red-shift,
the extracted distances may allow a new, independent measure of the Hubble
constant [26].
Bursts of gravitational waves may also be emitted by core collapse supernovae. These occur when a star of greater than ∼ 8 solar masses has used allof its available nuclear fuel, so that radiation pressure can no longer preventthe stars gravitational collapse. The core implodes rapidly, until the density of
nuclear material is reached. At this point the outer layers eﬀectively bounce
oﬀ the core in a violent explosion, leaving the remnant of the core exposed asa neutron star or perhaps a black hole. If there is any asymmetric distributionof mass during the supernova, perhaps related to the angular momentum of
the stellar core, then gravitational waves will be emitted. Detection of gravitational waves from supernovae may provide unique information, unavailable
through electromagnetic observations, about the processes involved in the collapse of stellar cores [28].

1.3.3Stochastic gravitational wave backgroundThe superposition of gravitational waves from many unresolved sources is predicted to form a stochastic, or random, background signal, signiﬁcantly below
the instrument noise level in ﬁrst generation interferometers. This backgroundwill include signals from astrophysical sources, such as faint and unresolved

1.4 Gravitational wave detection

9

neutron star signals. It is also expected to contain gravitational radiationfrom sources in the very early universe. The cosmic microwave backgroundradiation, produced approximately 105 years after the Big Bang, was famously
detected by Penzias and Wilson in 1964 [29, 30]. An analogous cosmological
gravitational background ﬁeld has been predicted, originating from an earlier
time, possibly as little as 10−35 seconds after the Big Bang. Detection of thisbackground radiation would allow unique insight into the very early universe.
Many possible cosmological sources of gravitational waves have been proposed, including the ampliﬁcation of quantum vacuum ﬂuctuations during inﬂation, pre-big bang models, the decay of cosmic strings and phase transitionsin the early universe. Reviews of these, and other possible sources and methods for their detection, can be found in Allen [31] and in Abbott et al [32].
Inﬂationary models have been studied extensively and many make detailed
predictions about the expected gravitational wave spectrum, while it has been
calculated that any gravitational waves produced at ≈10−25 seconds would be
red-shifted into the detection band of current ground based interferometers.
The amplitude of the stochastic background is expected to be signiﬁcantly
smaller than the sensitivity of current detectors. However, it may be possible
to detect it by cross-correlating the data from two detectors. Assuming thedetectors are widely separated, so that the instrument noise is uncorrelated,
this technique can signiﬁcantly increase the sensitivity of the search. Thismethod has already been used to place constraints on cosmic string models
[32].

1.4

Gravitational wave detection

1.4.1

Resonant bar detectors

The ﬁrst gravitational wave detectors were proposed by Joesph Weber in 1960[2]. Weber’s technique involved monitoring the oscillations of solid bodies

1.4 Gravitational wave detection

10

for evidence of interaction with a gravitational wave. Initially, Weber andhis colleagues looked for evidence of excitation of the normal modes of the
Earth by low frequency gravitational waves [33]. At the same time, Weberbegan attempts to detect tidal strains in large cylindrical aluminium bars,
which were isolated from external mechanical and acoustic noise sources [34].
Piezoelectric crystals bonded around the centre of the bar were used to monitor
the oscillations of its fundamental mode. A gravitational wave of suﬃcient
amplitude passing through the bar would be expected to excite this mode,with the changes to the amplitude and phase of the motion being detected by
the piezoelectric sensors.
In 1968 Weber observed multiple coincident events between two bars separated by 2 km [35], which he believed were possibly due to gravitational waves.
The following year, Weber took data from larger bars, approximately 66 cm
in diameter and up to 1.5 m long, located approximately 1000 km apart in
Maryland and Illinois. The bars were resonant at 1660 Hz, a frequency atwhich gravitational waves were believed to be emitted during a core collapse
supernova. Weber again reported coincident readings between these detectors,
at a rate of approximately one per day [36, 37], which he claimed were due togravitational waves.
With interest in the ﬁeld growing, several laboratories around the world
attempted to repeat Weber’s measurements. However, various research groups
including those at Yorktown Heights [38, 39], Bell Labs [40], Moscow [41]and Glasgow [42] failed to verify Weber’s results. Calculations suggested that
Weber’s bar detectors had a strain sensitivity of ∼ 10−16 for millisecond pulses,
whereas sensitivities in the order of 10−23 were thought to be required to enable
regular detection of gravitational waves from astrophysical sources. It was
clear that the sensitivity of the detectors had to be improved by many ordersof magnitude if gravitational waves were to be discovered.
Two strategies were adopted to develop more sensitive detectors: the ﬁrst

1.4 Gravitational wave detection

11

was to reduce the level of noise in bar detectors; the second involved moving
to a new type of detector using laser interferometry as discussed in the following section. The sensitivity of resonant bars was limited by thermally excited
vibrations of the bar and by electronic noise in the detector circuits, and research focused on the use of cryogenic cooling to reduce these eﬀects. Several
low temperature bar detectors were constructed including the Allegro detector
at Louisiana State University [43], the EXPLORER and NAUTILUS detectors
built by the University of Rome [44, 45], the NIOBE detector at the Universityof Western Australia [46] and the AURIGA detector built by the Universities
of Padua and Trento [47, 48]. In recent years, strain sensitivities approaching
10−20 for millisecond pulses have been reached [45, 49]. Observational runs
involving two or more bars have been undertaken, but, although coincident
events have been observed, there has been no compelling evidence of the detection of gravitational waves. Spherical bar detectors, which give superior
performance in determining the directionality of any signal, have been developed at the University of Leiden in the Netherlands [50] and at the Universityof Sao Paulo, Brazil [51].
Bar detectors have relatively narrow operational bandwidths centered on
the resonant mode of the bar; the resonant frequency is typically chosen to
optimise the sensitivity for a particular type of source. While Cerdonio et al[52] have suggested a method of increasing the bandwidth using a dual resonantmass detector, lack of broadband sensitivity remains a signiﬁcant disadvantage
of bar detectors. As broadband interferometric detectors (discussed in thefollowing section) have surpassed the sensitivity of bar detectors continued
funding of the bars has become limited, and only AURIGA, EXPLORER and
NAUTILUS are still in operation [53].

1.4 Gravitational wave detection

1.4.2

12

Interferometric detectors

At the same time as the sensitivity of bar detectors was being increased a new
type of detector using laser interferometry to measure the distance between two
widely separated masses was developed. Since gravitational waves produce
strains in space (i.e. a fractional change in length) the resulting absolute
displacement can be increased by increasing the separation of the masses. This
technique, ﬁrst proposed by Gertsenshtein and Pustovoit in 1962 [54], wasalso very attractive as it could provide high sensitivities over a large range of
frequencies [55].
The design of these detectors is based on a Michelson interferometer, shownin Figure 1.2. A beam splitter is used to divide the incident laser light into
two beams, directed along the two perpendicular arms of the interferometer.
At the end of the arms, the light is reﬂected back towards the beam splitterby the test mass mirrors, which are suspended as pendulums to provide seismic isolation. At the beam splitter the reﬂected light is recombined and theresulting interference pattern directed onto a photodiode.
Due to their quadrupole nature, polarised gravitational waves propagating
perpendicular to the plane of the interferometer will increase the length of
one arm by δL and reduce the length of the perpendicular arm by δL, as
illustrated in Figure 1.1. The change in arm lengths results in a relative phaseshift between the two reﬂected beams, causing a change in the intensity of
the interference fringe on the photodiode. In practice, feedback signals areused to control the positions of the mirrors to keep the output ‘locked’ onto
a certain point on the interference pattern; usually a dark fringe (see Section
1.5.4). Information about the diﬀerential motion of the mirrors, including any
gravitational wave signals, will therefore be encoded in the control signal.
Robert Forward and his colleagues constructed the ﬁrst prototype gravitational wave detector using laser interferometry in the early 1970s [56, 57],
although the sensitivity was limited by the low power of the He-Ne laser and13

1.5 Limits to the sensitivity of long-baseline detectors

end
mirrorbeam
splitterend
mirrorlaser

photodetector

Figure 1.2: A schematic diagram of a simple Michelson interferometer, consistingof a laser, beam splitter and two end mirrors.

the short arm length. In 1975 a 3 m prototype interferometer was developed
at the Max Planck Institute for Astrophysics in Munich [58], while a 1 m detector was built at the University of Glasgow [59]. These initial prototypes
were followed by a 10 m interferometer at Glasgow [60], a 40 m instrument
at the California Institute of Technology and a 30 m prototype at the Max
Planck Institute of Quantum Optics in Garching [61]. After the developmentof these interferometers, the technology was thought to be suﬃciently advanced
to allow the construction of longer base-line interferometers which would havea realistic chance of detecting gravitational waves. Thus, in the mid 1990s,
construction of several detectors with arms between 600 m and 4 km long
was begun at various sites around the world. A brief summary of this ﬁrst
generation of long baseline gravitational wave detectors is given in Section 1.7.

1.5

Limits to the sensitivity of long-baseline
detectors

In principle the sensitivity of a laser interferometer monitoring strains in spaceshould be limited by the Heisenberg Uncertainty Principle. In practice, however, a number of other factors can limit the sensitivity of an interferome-1.5 Limits to the sensitivity of long-baseline detectors

14ter, and reaching the quantum limit poses a signiﬁcant challenge. The major
sources of noise are described in the following sections.

1.5.1

Gravitational gradient noise

Fluctuations in the local gravitational ﬁeld can directly induce motion in theinterferometer test masses, resulting in a a form of displacement noise in theinterferometer known as gravity-gradient noise [62, 63]. This noise source ispotentially signiﬁcant at frequencies  20 Hz, and, since the test masses cannot
be shielded from gravitational ﬁelds, eﬀectively sets a lower limit on the lowfrequency sensitivity achievable on Earth. Indeed, due to the level of low
frequency gravity-gradient noise on Earth, experiments to detect gravitational
waves at frequencies below ∼ 1 Hz have to be carried out in space (see Chapter
3).
Locally ﬂuctuating gravitational ﬁelds can be produced by either density
ﬂuctuations in the ground or air surrounding the interferometer, or by the
motion of other masses (perhaps passing cars, people, or aeroplanes ﬂying
overhead) in the vicinity. Surface waves in the top-most layers of the Earth’s
crust, created by the impact of ocean waves, are a major source of gravitygradient noise [64]. The gravity-gradient noise spectrum is below the initial
sensitivity of the ﬁrst generation of long baseline interferometers [63], although
it is expected to limit the low frequency sensitivity of the planned second
generation of detectors.

1.5.2

Thermal noise

Thermal noise, arising from the thermally driven motion of the molecules in thetest masses and their suspensions, is one of the major limits to the sensitivity ofgravitational wave interferometers at frequencies up to several hundred Hertz.
Thermal excitation drives the various resonant modes of the suspended mirror,
with each mode having on average 12 kB T of thermal energy associated with it.

1.5 Limits to the sensitivity of long-baseline detectors

15

Since the internal modes are typically at high frequencies outwith the gravity
wave detection band, the thermal noise results from the tails of the resonancepeaks. It is possible to shape the thermal noise spectrum and concentrate themajority of the thermal motion close to the resonances, thus reducing the oﬀresonance motion in the operation band of the detector. This is achieved byusing materials of ultra-low mechanical loss, typically fused silica, for the test
mass mirrors. Fused silica suspension ﬁbres are also used in the GEO 600, andare planned for use in future gravitational wave detectors such as Advanced
LIGO and Advanced VIRGO. Thermal noise and mechanical dissipation are
discussed more extensively in Chapter 2.

1.5.3

Seismic noise

The eﬀects of the continual motion of the ground, with amplitudes in the
micrometer range, is a major low-frequency limit which must be overcome ina gravitational wave detector. Seismic noise between approximately 1 and 10
Hz largely arises from wind and human activity such as traﬃc and industry[64]. Below 1 Hz vibrations from ocean waves travelling as surface waves in the
Earth’s crust produce the micro-seismic background, with peaks in amplitude
at 0.08 Hz and 0.16 Hz [64]. Even at a relatively low-noise location on the1

Earth’s surface, the seismic noise spectrum is approximately 10−7 f −2 m Hz− 2in all dimensions [65]. If the disturbance to each test mass must be less than1

approximately 3 × 10−20 mHz− 2 [66], then at 20 Hz for example, the seismic
noise in the horizontal direction must be reduced by a factor of approximately
8×109 . This horizontal isolation can be achieved relatively easily by suspending
the test masses as pendulums. Above the resonant frequency of the pendulum,
the transfer function between horizontal motion of the suspension point and thependulum mass is proportional to 1/f 2 , providing a high degree of horizontal
isolation. In any pendulum system there is likely to be a degree of coupling
between vertical and horizontal motion, typically of order 10−3 [66, 67], making

1.5 Limits to the sensitivity of long-baseline detectors

16

it important to also provide a signiﬁcant level of vertical isolation. This can beachieved by suspending the test mass on a spring, most usually of the cantilevertype.
The requirements of horizontal and vertical seismic isolation lead to complex pendulum systems consisting of several stages. In GEO 600, for example,
a triple pendulum design is used, with the test mass suspended, via an intermediate mass, from cantilever springs on an upper mass [68]. This upper mass isitself suspended from cantilever springs at the top of the pendulum structure.
This entire structure is hung from a plate resting on both passive and active
anti-vibrational damping stages. The VIRGO detector, which was designed tooperate to frequencies below 10 Hz, uses a seven stage pendulum system forhorizontal isolation, with six of the upper stages hung from cantilever springsfor vertical isolation [69].
To reduce motion at the pendulum frequency, active damping of the pendulum mode is employed. Further ‘pre-isolators’ are required to reduce low
frequency motion at the micro-seismic peak and thus to reduce the necessary
dynamic range of the servo systems used to hold the optical cavities on resonance. Many forms of low frequency isolator are used. In VIRGO, tall inverted
pendulums are used with seismometers and actuators to provide feedback in
the horizontal direction and with cantilever springs of reduced stiﬀness in thevertical direction. GEO 600 also uses a system of seismometers and actuators
below the passive isolation stage, while a new active hydraulic pre-isolation
system, situated outside the vacuum, is being developed for Advanced LIGO
[70].

1.5.4

Photoelectron shot noise

Gravitational wave detectors operate with the interferometer locked at a particular point on an interference fringe. Any intensity changes at the interferometer output port are sensed with a photodiode, allowing a feedback signal

1.5 Limits to the sensitivity of long-baseline detectors

17

to be derived which is then applied, with appropriate phase, to a transducer
capable of moving one of the interferometer mirrors. Changes in the lengths of
the interferometer arms can be detected by monitoring the feedback signals. It
has been shown [71] that the best signal-to-noise ratio is obtained by locking
the interferometer close to a dark fringe.
Shot noise in the detected photo-electrons is one limit to the sensitivity ofthis optical readout scheme, where the photo-electrons are assumed to obey
Poisson statistics. If the laser has a mean power P and wavelength λ, then themean number of photons passing through the interferometer in an observing
time τ , corresponding to a bandwidth Δf = 1/2τ , is N = P τ /hν. Statisticalﬂuctuations in the number of photons detected by the photodiode can limit
the sensitivity of the detector. If the photoelectrons follow Poisson statistics,√
the average magnitude of these ﬂuctuations is N , leading to a limit to theaccuracy with which the diﬀerential displacement Δx of the mirrors can be
determined. In a simple Michelson interferometer this limit is given by [71]:
 12

cλΔf
Δx =
,
(1.5)
4πP cos2 (φ/2)
where φ is the phase diﬀerence between the light in the two arms of the interferometer and  = h/(2π) is the reduced Planck constant. It can be seenthat the eﬀects of shot noise can be reduced by increasing the light power andthat, as mentioned above, the best sensitivity is obtained by operating theinterferometer close to a null point in the interference pattern where φ = 0.

1.5.5

Radiation pressure noise

Fluctuations in radiation pressure on the interferometer mirrors can produce
a diﬀerential displacement between the two mirrors. For a simple Michelsoninterferometer the resulting diﬀerential displacement sensitivity limit, Δx, ata frequency ω can be shown to be:1
16πP Δf 2
Δx =,λm2 ω 4 c

(1.6)

1.5 Limits to the sensitivity of long-baseline detectors

18

where P is the laser power and m is the mass of the mirror. A formal quantum
mechanical treatment of an interferometer has shown that this ﬂuctuatingradiation pressure force arises from zero-point ﬂuctuations of the amplitudeof the vacuum electromagnetic ﬁeld, entering the interferometer at the beamsplitter via the output port [72, 73]. If this light has the correct phase it will
increase the light intensity in one of the interferometer arms while decreasing
the intensity in the other arm, producing anti-correlated ﬂuctuations of thelight intensity in each arm. These intensity ﬂuctuations are the diﬀerential
driving force which results in radiation pressure noise.
The same quantum mechanical treatment of an interferometer shows thatthe shot noise actually arises from ﬂuctuations in the phase of the vacuum ﬁeld
entering the interferometer at the beam-splitter [72, 73].

1.5.6

The Standard Quantum Limit

It has been shown, as discussed in the previous section, that the radiationpressure ﬂuctuations and shot noise in an interferometer arise from orthogonal
ﬂuctuations in the vacuum ﬁeld entering the dark port of the beam-splitter[72, 73]. They are therefore statistically independent of each other and thus
Equations 1.5 and 1.6 can be combined additively to give a total quantum
sensitivity limit of:(Δx)2 =cλΔf
16πP Δf+,4πP cos2 (φ/2)λm2 ω 4 c

(1.7)

Since radiation pressure noise is proportional to the laser power, while the shotnoise varies as the inverse of the laser power, it follows that at any frequencyof observation there will be an optimum power for which the total quantum
noise is a minimum [71]. By minimising the above expression with respect toP, this optimum power can be shown to be:
Poptimum =λmω 2 c
8π

(1.8)

19

1.5 Limits to the sensitivity of long-baseline detectors

At this power, the apparent total ‘quantum’ limit to the detectable diﬀerential
displacement can be found from Equation 1.7 [71]:

Δx =

4Δf
mω 2

 12
.

(1.9)

This is often referred to as the Standard Quantum Limit (SQL). In fact, it is
not necessarily a true limit to sensitivity.
The SQL arises only if the optical noise sources - photon shot noise andradiation pressure noise - are uncorrelated. These optical noise sources area result of ﬂuctuations in the vacuum ﬁeld entering the dark port of the interferometer, with ﬂuctuations in the phase quadrature responsible for shotnoise and ﬂuctuations in the amplitude quadrature responsible for radiationpressure noise. If the balance of the ﬂuctuations in the two quadratures of thevacuum ﬁeld can be changed, it is possible to reach the SQL at a lower power
than derived above. This can be achieved through the use of squeezed statesof light, which reduce the noise in one quadrature while increasing the noisein the orthogonal quadrature [73]. The injection of a squeezed vacuum ﬁeld
at the dark port of the beam-splitter can therefore reduce the dominant formof optical noise at a particular frequency, allowing the SQL to be reached at alower power than derived above.
In principle the SQL can actually be overcome by introducing correlations
between the shot noise and radiation pressure [74]. This is an example ofthe principle of quantum non-demolition, ﬁrst suggested by Braginsky [75], inwhich the quantum properties of a measurement apparatus are prevented from‘demolishing’ the quantum state of the system which they are measuring.
Correlations between the two optical noise sources in an interferometer canbe introduced using at least two methods. The optical cavities can be designed
so that the optical ﬁeld is strongly coupled to the mechanical system throughan optical spring eﬀect, described for example by Corbitt and Mavalvala [76].
These eﬀects can be enhanced through the use of optical bars and optical1.6 Interferometric techniques20

levers [77, 78]. Signal recycling (see Section 1.6.4), as used currently in the
GEO 600 detector, can also introduce correlations between the shot noise andradiation pressure noise, leading to the possibility of achieving sensitivities
below the SQL [79]. Another possible method of introducing such correlations
is to measure the output signal after suitable ﬁltering at optical frequencies
through the inclusion of a long ﬁltering Fabry-Perot cavity [80].

1.6Interferometric techniques

A number of techniques can be used to improve the sensitivity of the basic
Michelson interferometer layout shown in Figure 1.2. As discussed above,long interferometer arms are desirable to increase the absolute displacement
caused by a gravitational wave. However, the curvature of the Earth and othergeographical (and cost) factors limit the practical arm length to kilometerscales. Additionally, extremely high laser powers are required to overcome theshot noise sensitivity limit in a simple Michelson detector. The laser power
requirement can be signiﬁcantly reduced if the laser beam is allowed to pass up
and down each arm many times, as the apparent displacement of the mirror ismultiplied by the number of times the light is reﬂected from it. This can beachieved by a delay line conﬁguration or by the use of Fabry-Perot cavities in
the interferometer arms.

1.6.1

Delay line interferometers

The eﬀective arm-length of an interferometer can be increased by folding theoptical path so that the light is reﬂected from the test mass mirror more thanonce. This is known as a delay-line interferometer [55]. Two possible delayline conﬁgurations are shown in Figure 1.3. The total phase change due to a
gravitational wave builds up with each reﬂection, as long as the total storage
time of the light in the arm is less than half of the period of the gravitational

21

1.6 Interferometric techniques

folding
mirror(a)(b)
mirrorsend
mirrorend
mirrorfolding
mirrorlaserhole inmirrormirrors

laser

beam
splitterbeam
splitterphotodetector

photodetector

Figure 1.3: Schematic diagram of two possible layouts for a delay line interferometer:(a) shows a folded arm scheme, while (b) illustrates a scheme in which
the laser beam is reﬂected along each arm multiple times.

wave. Multiple beam delay lines were used in some prototype gravitational
wave detectors [58, 59, 61] but scattering proved to be a signiﬁcant noise sourcein these designs. The GEO 600 detector, however, makes use of a folding mirror
design (Figure 1.3 (a)) to double the eﬀective arm-length of the detector [8].

1.6.2

Faby-Perot arm cavities
end
mirrorarm cavities
partially transmissive
mirrorend
mirrorlaser

beam
splitterpartially transmissive
mirrorphotodetector

Figure 1.4: A schematic diagram of a Fabry-Perot interferometer, which uses FabryPerot cavities in each arm. The inner mirrors are partially transmissive
to allow light to enter, and leave, the cavity.

An alternative scheme, used in the LIGO, VIRGO and TAMA detectors,

1.6 Interferometric techniques22

is a Fabry-Perot Michelson interferometer. In this conﬁguration, Fabry-Perot
optical cavities are used to form the arms of the interferometer, as shown in
Figure 1.4. The laser beam is reﬂected back and forwards along itself in thecavity, resulting in light building up in the cavity. The input mirror of eachcavity is partially transmissive, to allow some light to enter and leave the arm.
The eﬀective number of bounces along the cavity is determined by the opticalproperties of the mirrors. If the arm length changes, perhaps due to interaction
with a gravitational wave, the resulting phase change is enhanced with eachbounce the light makes along the cavity.

1.6.3

Power recycling

An interferometer is typically operated close to a dark fringe to minimise thephoton shot noise, with most of the input light returned towards the laser.
However, this light can by reused by placing another mirror (the power recycling mirror) between the laser and the beam splitter, thus reﬂecting the output
light back into the interferometer, as shown in Figure 1.5. The interferometercan eﬀectively be considered as a mirror when operated on a dark fringe, andthe presence of the power recycling mirror forms a cavity between it and theinterferometer. As light builds up in this cavity, more power is incident on
the interferometer, resulting in higher light powers in the arms and increased
sensitivity.

1.6.4

Signal recycling

In addition to recycling the light at the input port of the interferometer, it
is possible to recycle the ‘signal’ light at the output port [81, 82]. A detector is normally operated locked close to a null in the interference pattern; agravitational wave, or indeed any other diﬀerential motion of the test masses,
produces a change in the phase of the light being recombined at the beam splitter, resulting in light leaking out of the output port. In the frequency domain,

23

1.6 Interferometric techniquesarm cavities
power
recyclingmirrorlaser

photodetector

Figure 1.5: A schematic diagram of a Michelson interferometer with a power recycling mirror at the input port, forming an optical cavity with the interferometer.

this phase change creates sidebands on the ‘carrier’ laser light at the frequencyof the gravitational wave. These sidebands do not interfere destructively, andso appear at the output of the interferometer.
A mirror placed between the output port of the beam splitter and thephotodiode detector will eﬀectively form a cavity with the interferometer, as
shown in Figure 1.6. A signal sideband will be reﬂected back into the interferometer, where it resonates, increasing the signal size in a certain bandwidth.
By changing the position of the signal recycling mirror to adjust the length of
the signal recycling cavity, the frequency at which the interferometer is mostsensitive can be tuned.
Three modes of signal recycling operation can be obtained by precise tuningof the position of the signal recycling mirror: broad-band signal recycling,resonant sideband extraction and de-tuned operation [83, 84, 85].
Narrow band signal recycling (from de-tuned operation) can be useful toprovide increased sensitivity at frequencies where sensitivity is normally limitedby shot noise and can be used in searches for continuous sources at speciﬁcfrequencies. Signal recycling with a larger bandwidth can be used to optimise
the detector frequency response to the signals expected from broadband and‘chirp’ sources. Currently only the GEO 600 detector uses signal recycling [86].

24

1.7 Current interferometric detectors

arm cavities
power
recyclingmirrorlaser

signalrecyclingmirrorphotodetector

Figure 1.6: A schematic diagram of a Michelson interferometer with a signal recycling mirror at the output port. This mirror forms an optical cavity with
the interferometer and allows the peak sensitivity to be tuned to speciﬁcfrequency ranges.

1.7

Current interferometric detectors

The construction of a number of long-baseline gravitational wave detectors
was begun in the mid-1990s at several sites around the world. The Laser
Interferometer Gravitational Wave Observatory (LIGO) in the US consists of
two widely separated detector facilities, one near Baton Rouge, Louisiana andthe other near Hanford, Washington. Both sites have an interferometer with

Figure 1.7: The VIRGO interferometer at Cascina, Italy. Both 3 km long arms ofthe detector can be seen, meeting at the central station (right hand side)which houses the input and output optics.

1.7 Current interferometric detectors

25

Figure 1.8: Left: A view along one 600 m long arm of the GEO 600 detector, with theend station housing the end test mass visible in the background. Right: A
GEO 600 suspension, consisting of a triple pendulum, with a monolithic
silica lowest stage.

4 km long arms, while the Hanford site also houses a 2 km long detector
within the same vacuum system. The French/Italian VIRGO detector near
Pisa, shown in Figure 1.7, has 3 km long arms and makes use of a sevenstage mirror suspension system to reduce seismic noise, allowing operation at
frequencies down to 10 Hz. Two prototype Japanese detectors called TAMA
and CLIO have been constructed [11, 87]. TAMA is located at the Tokyo
Astronomical Observatory and has an arm length of 300 m, CLIO is situatedin Kamioka mine where the seismic noise is two orders of magnitude lower thanat the TAMA site. CLIO is intended to demonstrate the use of cryogenicallycooled sapphire mirrors, and there are plans to adopt the design for a 3 km
detector known as LCGT, also to be constructed in Kamioka mine [88].
The detectors discussed so far all make use of Fabry-Perot arm cavities
and power recycling to increase the circulating laser power, and the mirrors
are suspended by metal wires. The German / British GEO 600 detector (seeFigure 1.8), situated near Hannover, makes use of several diﬀerent techniques.
GEO 600 uses a four pass delay line optical layout, with an eﬀective arm length

26

1.7 Current interferometric detectorsof 1200 m, and includes a signal recycling mirror, allowing the peak sensitivity
and detection bandwidth to be tuned to diﬀerent frequencies. Thermal noiseis reduced through the use of ultra low loss silica ﬁbres to suspend the mirrors. Figure 1.9 shows the design sensitivity of GEO 600, operated with signal
recycling de-tuned at 250 Hz.
Since the late summer of 2002, the LIGO detectors have completed ﬁve‘science runs’, in which coincident data was taken over extended periods oftime. The most recent science run involved observations by the LIGO detectors, GEO 600 and VIRGO, and ended in October 2007 after almost two
years of data taking. GEO 600, TAMA and the ALLEGRO and AURIGA
bar detectors took part in previous runs. Between the science runs, commissioning work was undertaken and the sensitivities of the various detectors was
improved. Figure 1.10 shows the typical sensitivity of the GEO 600 detector
during each science run.

The results of the science runs have been used to-18

Amplitude spectral density (h Hz -1/2)

10

Seismic
Suspension TN
Substrate TN
Coating TN
Thermorefractive
Shot 250Hz
Total 250Hz

-19

10

-20

10

-21

10

-22

10

-23

10

-24

10

1

10

2

3

10

10

Frequency (Hz)Figure 1.9: The design sensitivity of GEO 600, showing the various noise curves for
operation with signal recycling at 250 Hz de-tuning.

set new limits on various astrophysical sources. For example, data from the27

1.7 Current interferometric detectors-16

Amplitude spectral density (h Hz -1/2)

10

S1 Aug 26 ‘02
S3I Nov 5 ‘03
S3II Dec 31 ‘03
S4 Feb 22 ‘05
S5 N&W Mar 23 ‘06
S5 Jun 3 ‘06

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

2

3

10

10

Frequency (Hz)Figure 1.10: The typical amplitude strain sensitivity of GEO 600 in each of the science runs between 2002 and 2007.

ﬁfth science run have been used to place a new upper limit on gravitational
wave emission from the Crab pulsar [20], beating the spin-down limit, derived
by assuming that all of the energy loss is due to gravitational radiation, bya factor of ∼ 4. This result, which assumes that the gravitational wave andelectromagnetic emissions are phase-locked, implies that less than 6 % of theenergy loss responsible for the spin-down is emitted as gravitational waves [20].
Many other interesting analyses have been performed, including searches forgravitational waves associated with gamma-ray bursts [89, 90] and SGR 180620, a hyperﬂare originating from a galactic neutron star [91]. In addition,
cross-correlation searches for a stochastic background of gravitational waves
have begun to explore the parameter space of models of gravitational radiationfrom cosmic strings [32].

1.7 Current interferometric detectors

1.7.1

28

Second generation detectors

There are plans in place for signiﬁcant upgrades to both the LIGO and VIRGO
detectors, known as Advanced LIGO and Advanced VIRGO. The Advanced
LIGO project [92] aims to increase the peak sensitivity of the LIGO detectors by more than a factor of ten, increasing the volume of space from which
gravitational waves could be detected by a factor of order one thousand. The
second detector at the Hanford site will be increased in length from 2 km to 4
km. Each Advanced LIGO detector will incorporate a signal recycling mirror
at the output port, allowing the interferometer response to be tailored to thetype of source being searched for. The input laser power will be increased by afactor of 20, and the circulating power in the interferometer arms will be ∼ 830
kW, compared to the initial LIGO value of ∼ 10 kW. The 11 kg silica mirrors
used in the initial LIGO detectors will be replaced with 40 kg mirrors of larger
radius to reduce thermal noise and radiation pressure noise. The mirrors will
be suspended from fused silica ﬁbres based on those used in GEO 600, which
will be supplied by the Institute for Gravitational Research in Glasgow. A
complete suspension will consist of four pendulum stages, with an improved
seismic isolation system designed to reduce the seismic cut-oﬀ frequency from40 Hz to 10 Hz.
The installation of Advanced LIGO is planned to begin in 2011, with theinitial tuning of the detectors expected to occur in 2013. Prior to the installation of Advanced LIGO facilities, a series of upgrades are being implemented
to create Enhanced LIGO detectors which will be used for more science runs
between 2009 and 2011. Similar plans are in place for the upgrade of the
VIRGO detector to Advanced VIRGO.
GEO 600, with its shorter arms, will not be able to match the sensitivityof the second generation detectors. However, GEO 600 will operate (withsome low-level commissioning work) while the LIGO detectors are oﬀ-line for
the Enhanced LIGO installation to provide an ‘Astrowatch’ capability. In

1.7 Current interferometric detectors

29

the future, GEO 600 may be used to test new technologies, materials andinterferometric techniques for use in the proposed third generation of detectors.

1.7.2

Future detectors

A number of factors suggest that improvements to the sensitivity of the second
generation detectors will be challenging. At frequencies below ∼ 400 Hz, thesensitivity is expected to be limited by thermal noise, particularly that arisingfrom the optical coatings, while photon shot noise will be the major limitation
at higher frequencies. Thermal noise can, in principle, be reduced by coolingthe mirrors and by using materials of low mechanical loss at low temperature.
Fused silica is not a promising candidate material for cooling as it is known tohave a large mechanical loss peak at approximately 40 K [93].
The eﬀects of shot noise can be reduced by increasing the laser power. However, the thermo-mechanical properties of silica make it likely that any increaseof laser power beyond the Advanced LIGO design would result in signiﬁcantthermal deformation of the mirrors, making stable operation of the interferometer diﬃcult [94, 95]. Thus a material with a lower thermal expansion
co-eﬃcient and higher thermal conductivity would be desirable if substantiallyhigher laser powers are to be used.
The use of silicon optics has been proposed, as silicon is likely to meetboth the low temperature thermal noise requirements and to have the ability
to withstand high laser powers [94]. Silicon is not transparent to 1064 nm
light, which is used in current gravitational wave detectors. However, current
interferometer designs could incorporate silicon optics if a laser wavelength of
1.5 μm was used. Alternatively, interferometers using an all reﬂective topology
based on diﬀractive optics have been proposed [96, 97].
Improved low-frequency sensitivity could be obtained by building future
detectors underground, where levels of seismic noise are signiﬁcantly lower.
This approach, along with the use of cryogenic cooling, has been used for the1.8 Conclusion

30

Japanese prototype CLIO, which is located in Kamioka mine. The same mineis the proposed location for the planned Large-Scale Cryogenic Gravitational
Wave Telescope (LCGT) detector [88]. A major European Commission design
study for a third generation detector known as the Einstein Telescope (ET) is
currently underway with participation from groups across Europe [98].

1.8

Conclusion

Over the past eight years, the ﬁrst generation of long-baseline interferometricgravitational wave detectors have been commissioned and have closely approached or achieved their design sensitivities. Several science runs have takenplace in which various instruments in the world-wide network of detectors haveoperated in coincidence for extended periods. While gravitational waves have
not yet been detected, analysis of the data obtained has allowed astrophysicallyuseful upper limits to be placed on the gravitational wave emission of several
sources. The next generation of detectors, consisting of major upgrades to thecurrent LIGO and VIRGO detectors, are expected to begin operation before
2015, and it is widely expected that they will detect the ﬁrst gravitational wave
signals, heralding a new era in astronomy. Several challenges must be overcome to signiﬁcantly improve upon the sensitivities of the second generation
detectors. In particular, thermal noise is likely to limit detector sensitivity at
low frequencies, while improved sensitivity at high frequency will require thethermal loading eﬀects associated with high laser power to be overcome.

Chapter 2
Thermal noise in gravitational
wave detectors
2.1

Introduction

Thermal noise arising from the thermal motion of the molecules in the mirrors
and their suspensions sets an important limit to the sensitivity of interferometric gravitational wave detectors at frequencies up to approximately 500
Hz. The equipartition theorem states that any mechanical system will have amean value of 12 kB T of thermal energy associated with each degree of freedom.
This energy is stored in vibrations of the molecules making up the system, resulting in thermally induced motion of the macroscopic system. In the mirror
suspensions of a gravitational wave detector, the thermal energy will excite
the resonant modes of the test mass and the suspension ﬁbres, resulting in asource of displacement noise in the measured position of the mirror faces. This
noise can be of the same order as the displacement which would be caused bya gravitational wave, making reduction of thermal noise within the operational
bandwidth of gravitational wave detectors an important area of research.

2.1.1

Brownian motion

Thermal motion in a mechanical system was ﬁrst reported by the botanist
Robert Brown in 1828 after he observed the irregular motion of pollen grains31

32

2.1 Introduction

and dust suspended in water [99]. Despite studies by a number of scientists,
the source of this ‘Brownian motion’ remained unknown for several decades.
Experiments by Gouy and Exner established that the motion was more rapid
in less viscous liquids, and that the velocity of the dust particles rose as thetemperature of the liquid was increased [100, 101]. However, it was not until
1905 that a full mathematical treatment of Brownian movement was presented
in Einstein’s well-known paper [102]. Einstein realised that the motion arosefrom ﬂuctuations in the rate of collisions between water molecules and thepollen grains, and showed that as a result of these impacts the grains lost
kinetic energy. This was the ﬁrst time that a dissipation process, in this casethe loss of energy due to the viscosity of the ﬂuid, had been related to random
ﬂuctuations, here the random displacement of the pollen grain.
An electrical equivalent to Brownian motion was discovered in 1927, when
J. B. Johnson observed random voltage ﬂuctuations across a resistor [103].
These ﬂuctuations were shown to result from the thermal motion of the atoms
in the conductor [104, 105], and to be related to the temperature T , the resistance of the wire R and the bandwidth of the measurement Δf as follows:
V 2 (f ) = 4kB T RΔf.

2.1.2

(2.1)

The Fluctuation-Dissipation Theorem

Both Brownian motion and Johnson noise in a resistor show a link between
ﬂuctuations (in the position of a particle or of the voltage across a resistor)and dissipation (energy loss due to the viscosity of the ﬂuid or the electricalresistance). More generally, all of the parameters which characterise a linear
system in thermal equilibrium are found to undergo spontaneous thermally
driven ﬂuctuations, whose magnitude and frequency spectrum are related tothe dissipative (i.e. real) part of the impedance of the system. The FluctuationDissipation Theorem, derived by Callen et al in a series of papers in 1951-52[106, 107, 108], states that the power spectral density Sf (ω) of the ﬂuctuating

33

2.1 Introduction

thermal force on a system is given by:
Sf (ω) = 4kB T {Z(ω)},(2.2)

where Z(ω) is the impedance of the system, deﬁned as:Z(ω) ≡ F (ω)/v(ω),(2.3)

for a force F (ω) applied to the system resulting in motion with a velocity of
amplitude v(ω). Alternatively, the theorem can be written in terms of thepower spectral density of the ﬂuctuating motion of the system Sx (ω):
Sx (ω) =4kB T {Y (ω)}
,
ω2

(2.4)

where Y (ω), the mechanical admittance of the system, is equal to Z −1 (ω). It
is therefore possible to estimate the thermal noise spectrum of a mechanicalsystem if the macroscopic impedance of the system can be determined.
It has been shown that the level of thermal noise in a system is related
to the real (dissipative) part of the mechanical admittance of the system also known as the mechanical conductance. By deﬁnition, this is related tothe damping coeﬃcient of the system. Thus the level of thermal noise is also
related to the level of damping, or dissipation, present in the system. Thiscan be illustrated clearly by considering the example of a damped harmonic
oscillator [65]. Neglecting the ﬂuctuating force responsible for thermal noise,
the equation of motion of a damped harmonic oscillator of resonant frequency
ω0 and mass m is:
F = mẍ + bẋ + kx,(2.5)

where b is the damping coeﬃcient and k = mω02 is the spring constant of
the oscillator. The displacement x and acceleration ẍ of the oscillator can be
expressed in terms of the velocity ẋ as follows:
x=

ẋ
iω

and

ẍ = iω ẋ.

(2.6)

34

2.1 Introduction

The impedance can then be calculated from its basic deﬁnition:
Z ≡

=

F
ẋ

(2.7)

k ẋ
miωx bẋ
+
+
ẋ
ẋ
iω ẋ

(2.8)

= b + iωm +

k
,
iω

(2.9)

from which the real part can be seen to be simply b. Application of theFluctuation-Dissipation Theorem (Equation 2.2) allows the power spectral
density of the ﬂuctuating thermal force to be calculated:
Sf (ω) = 4kB T b.

(2.10)

To calculate the power spectral density of the thermal displacement of thesystem, the mechanical admittance Y = Z −1 of the system must be calculated:Y=1
b + iωm − ik/ω

(2.11)=

b − iωm + ik/ω.
b2 + (ωm − k/ω)2(2.12)The real part of the admittance is therefore given by:
{Y } =bb2 + (ωm − k/ω)2,(2.13)and can be substituted into Equation 2.12, to give the power spectral densityof the thermal noise:
Sx (ω) =kB T b.
22
ω (b + (ωm − k/ω)2 )(2.14)

This simple example illustrates that the real part of the impedance of a mechanical system is related to the level of damping in the system. The conceptof mechanical dissipation and its relationship to thermal noise in gravitational
wave detectors will be discussed in detail throughout the rest of this chapter.

35

2.2 Sources of mechanical dissipation

2.2

Sources of mechanical dissipation

It has been shown that the level of thermal noise in a mechanical system isrelated to the level of dissipation in the system. Broadly speaking there are two
types of mechanical dissipation: external dissipation, in which energy is lost
into the environment and internal dissipation, in which energy is ultimately
lost as heat within the system.
A suspended test mass in a gravitational wave detector has many potential
external sources of energy dissipation. Perhaps the most obvious analogy tothe original pollen grain observations is gas damping [93], in which friction
from gas molecules dissipates energy from the test mass pendulum. Other
sources of external dissipation may include friction at the pendulum suspension points and recoil loss, in which energy is lost from the pendulum into thesurrounding support structure [109]. External sources of damping can be minimised by careful design of the pendulum system. Gas damping, for instance,is reduced by operating the interferometer in ultra-high vacuum conditions. If
these external sources of damping are suitably reduced, the dominant sourceof thermal noise will be the internal dissipation which occurs in the mirror andsuspension ﬁbres.

2.2.1

Dissipation from internal friction in materials

Internal dissipation arises from a process called anelasticity [110]. In an ideal
elastic solid, an applied stress σ instantaneously creates a strain  in the material, with Hooke’s law deﬁning the relationship between stress and strainas σ = Y where Y is the elastic modulus. For an anelastic material (i.e.
most real materials), the stress-strain relationship is not instantaneous andthe strain response will lag behind the applied stress, developing over a ﬁnite
relaxation time. A periodic stress applied to the material can be written as:σ = σ0 eiωt ,(2.15)

36

2.2 Sources of mechanical dissipation

where σ0 is the stress amplitude and ω is the angular frequency of the oscillation. Since the stress-strain relationship is linear, the resulting strain will
also be periodic with the same angular frequency, but with a phase lag withrespect to the stress. This can be expressed as: = 0 eiωt−φ ,(2.16)

where φ is the angle by which the strain lags behind the stress, known as theloss angle. It can be shown that the loss angle, also described as the internal
friction of the material, is a measure of the fractional energy dissipated per
cycle of oscillation [110].
Anelasticity, and hence internal friction, can arise from many internal properties of a material which are a function of the internal stress [110, 111]. These
internal degrees of freedom include the temperature and the density of point
defects, dislocations and grain boundaries. If the internal stress in the material
is altered, these properties typically require a ﬁnite period of time over which
to respond to the new state of stress. The process by which internal degrees of
freedom change often involve energy barriers which must be crossed, resulting
in an exponential relaxation to the new equilibrium value, with a characteristic
time τ . In many cases, the relaxation time obeys the Arrhenius equation [112]:τ = τ0 eΔE/kB T ,(2.17)

where ΔE is the energy barrier which must be overcome and τ0 represents thecharacteristic time between attempts at crossing the barrier.

2.2.2

Harmonic oscillator with internal friction

Internal friction in a material has been shown to arise from the time delay
between an applied stress and the development of the associated strain. In
the frequency domain this appears as a phase lag φ between the applied force
F and the response of the system x. This can be represented by a modiﬁed37

2.2 Sources of mechanical dissipationαk(1+iφ)

(a)

k1

m

(b)

k2
m

Figure 2.1: (a) A schematic diagram of a mechanical oscillator with a complex spring
constant k(1 + iφ). (b) A schematic diagram representing a standard
anelastic solid, with an ideal spring connected in parallel with a springdashpot combination, where k1 and k2 are spring constants and α is the
impedance of the dashpot.

version of Hooke’s law, involving a complex spring constant, as shown in Figure
2.1(a):
F = −k(1 + iφ(ω))x.

(2.18)

If the resonant modes of the suspended mirror system in a gravitational wave
detector are modelled as harmonic oscillators with internal damping, the equation of motion can be written as:mẍ = −k(1 + iφ(ω))x + F.

(2.19)

Using the standard expressions relating the acceleration and displacement tothe velocity of a harmonic oscillator (see Equation 2.6), the force F on theoscillator can be expressed as:

F = iωmẋ +k(1 + iφ(ω))ẋ,iω(2.20)

from which the impedance Z = F/ẋ can be seen to be:Z=k − ω 2 m + ikφ(ω).
iω(2.21)

38

2.2 Sources of mechanical dissipation

This expression can be inverted to give the admittance, which, after rationalising the denominator is:Y =ωkφ(ω) + i(ωk − mω 3 ),(k − mω 2 )2 + k 2 φ2 (ω)(2.22)

from which the real part can be seen to be:
{Y } =

ωkφ(ω)(k − mω 2 )2 + k 2 φ2 (ω)(2.23)

Application of the Fluctuation-Dissipation Theorem shows that the thermal
noise power spectral density is given by:

Sx (ω) ==4kB T {Y (ω)}ω2(2.24)

4kB T kφ(ω).
ω[(k − mω 2 )2 + k 2 φ2 (ω)](2.25)

This expression can be simpliﬁed by noting that k = ω02 m, to give the following
expression for the thermal noise spectral density of an oscillator of resonant
frequency ω0 , in terms of its mass and mechanical loss:
4kB T
ω02 φ(ω).
Sx (ω) =ω m[φ2 (ω)ω04 + (ω02 − ω 2 )2 ]2.2.3

(2.26)

The form of internal dissipation - Debye theory

The complex spring constant given in Equation 2.18 is usually an excellent
approximation of anelastic eﬀects [111]. It is not, however, exact, and a moreaccurate model of an anelastic solid can be obtained by considering an ideal

39

2.2 Sources of mechanical dissipation

spring connected in parallel with a combination of a spring and a dashpot of
impedance α, as shown in Figure 2.1 (b). Zener [113] showed that, for ananelastic solid with no other similar relaxation times, this model results in aloss angle φ with a characteristic frequency dependence, given by:φ≈Δωτ
1 + ω2τ 2(2.27)

where Δ ≡ k2 /k1 is the relaxation strength, and τ = α/k2 is the relaxation
time. The dissipation peaks at a frequency ωpeak = τ −1 : this is known as a
Debye peak. Many authors have reported that the internal friction in a wide
variety of materials is largely independent of frequency [114, 115, 116, 117,
118, 119, 120]. Since an anelastic relaxation process results in the dissipation
having a Debye peak at a characteristic frequency, this result may at ﬁrst be
surprising. However, a material may have several Debye peaks at frequencies
separated by many orders of magnitude. At frequencies far from any of thepeaks, the combined eﬀect of the tails of these peaks is eﬀectively constant withfrequency. The operating frequency band of gravitational wave detectors (from∼ 10 Hz to several thousand Hz) usually lies in such a region. One anelastic
relaxation process, known as thermoelastic damping, gives rise to a dissipation
peak which can lie in or close to this frequency band. Thermoelastic damping
is discussed in Section 2.4.

2.2.4

Thermal noise associated with a single resonantmode

A mirror suspension system in a gravitational wave detector has many resonantmodes which can be thermally excited. These modes include the pendulum
mode of the suspension, the bending modes (or ‘violin’ modes) of the suspension ﬁbres and the internal modes of the mirror itself. Of these, usually only
the violin modes occur at frequencies which lie within the sensitive detection
band of the interferometer. The mirror suspension is typically designed such

40

2.2 Sources of mechanical dissipationthat its pendulum modes occur at frequencies below the detection band, while
the internal modes of the mirrors occur at several tens of kilohertz, well above
the detection band. For the pendulum modes and the internal mirror modes,
since the peak thermal noise occurs far from the interferometer detection band,
it is therefore the oﬀ-resonance thermal noise which sets a limit to detector
sensitivity.
Equation 2.26 can be simpliﬁed by approximation to examine the threecases of interest discussed above: that is, the thermal noise at frequencies far
below, equal to and far above the resonant frequency of a mode. First, consider
the thermal noise at frequencies far below the resonance, ie ω
ω02 φ(ω)
4kB T
ω m[φ2 (ω)ω04 + (ω02 − ω 2 )2 ](2.28)≈4kB T
ω02 φ(ω)ω m(φ2 (ω)ω04 + ω04 )(2.29)≈

4kB T φ(ω).
mωω02 (φ2 (ω) + 1)

(2.30)Sx (ω) =Now if φ2 (ω)ω0 :

1, which is a valid assumption for the low loss materials used in

the mirrors of gravitational wave detectors, the thermal noise spectral density
can be approximated as:
Sx (ω) ≈4kB T φ(ω),
mω02 ω

(2.31)

This equation can be used to calculate the thermal noise in the gravitational
wave detection band from an internal resonance of the mirror. The thermal
noise at frequencies well above a resonance can be calculated in a similar way.
In this case ωω0 and thus Equation 2.26 can be approximated as:
Sx (ω) ≈Again assuming that φ2 (ω)

4kB T φ(ω)ω02,mω[ω 4 + ω04 φ2 (ω)]

1, this can be simpliﬁed to:(2.32)

41

2.2 Sources of mechanical dissipation

4kB T ω02 φ(ω)
Sx (ω) ≈.
mω 5(2.33)

Equations 2.31 and 2.33 show that at frequencies far from a resonance, thethermal noise power spectral density due to that resonance is directly proportional to the mechanical loss φ(ω). Oﬀ-resonance thermal noise can therefore
be reduced by using materials with a low mechanical loss. However, since thetotal thermal energy associated with the resonance is 12 kB T , it follows that thenoise spectral density close to the resonance frequency must be very high. In
this case, ω = ω0 and Equation 2.26 becomes:
Sx (ω) =4kB T,mω03 φ(ω0 )(2.34)and it can be seen that the power spectral density of thermal motion at theresonant frequency of a mode is inversely proportional to the mechanical loss.
Reducing the mechanical loss eﬀectively conﬁnes more of the thermal motion into a narrow frequency band centred on the resonance, resulting in anarrower and higher thermal noise peak at the resonant frequency and lower
oﬀ-resonance thermal noise. This is demonstrated in Figure 2.2, which shows
the thermal displacement noise spectrum for a mode of two otherwise identicalmechanical oscillators with diﬀerent mechanical loss factors. It was mentioned
above that several violin modes of the suspension ﬁbres typically occur within
the gravitational wave detection band. The use of low loss suspension materials ensures that the thermal noise peaks from the violin modes are suitably
narrow, and thus they can be easily removed from the interferometer signal bynotch ﬁltering, with minimal reduction of the useful bandwidth of the detector.

42

10

-11

1/2

Thermal displacement noise spectral density (m/Hz )2.3 Relationship between mechanical loss and the quality factor

10

10

10

10

10

10

10

φ = 10 -9
φ = 10 -7

-13

-15

-17

-19

-21

-23

-25

10

100

1000

10000

Frequency (Hz)Figure 2.2: Thermal noise displacement spectra for two mechanical oscillators eachof f0 =1 kHz, m =20 kg at T =290 K and having diﬀering mechanicalloss. The solid, red line shows the thermal noise for a loss of φ = 1×10−9
while the black, dashed line is for φ = 1 × 10−7 The lower loss material
has lower oﬀ-resonance thermal noise, as more of the thermal motion isconcentrated close to the resonant frequency.

2.3

Relationship between mechanical loss andthe quality factor

The mechanical quality factor, Q is a dimensionless quantity related to thedissipation of an oscillator at a resonant frequency, f0 :Q(f0 ) ≡ 2π

EstoredElost per cycle

≡

f0
,
Δf

(2.35)

where Estored is the total energy stored in the oscillator, Elost per cycle is theenergy dissipated with each cycle of oscillation and Δf is the width of the resonance peak measured at half of its maximum power. Thus it can be seen that
oscillators with low levels of energy dissipation will have narrow resonances.

2.3 Relationship between mechanical loss and the quality factor

43

From this deﬁnition, the quality factor can be related to the loss angle (theangle by which the induced strain lags behind the stress applied to a system)by considering the energy stored and dissipated in an oscillating system. Fora mechanical oscillator of angular frequency ω0 with displacement x = Aeiω0 twhere the restoring force is given by FR = −kx, the stored potential energy is
given by:1
Estored = kA2 ,2(2.36)

where A is the amplitude of the oscillation. The energy lost per cycle canbe found by including the eﬀect of the dissipative part of the spring constant(using Equation 2.18).
FR = −k(1 + φ(ω0 ))x= −kAei(ω0 t+φ(ω0 ))= −kA{cos(ω0 t + φ(ω0 )) − ikA sin(ω0 t + φ(ω0 ))}.

(2.37)The real part of this expression can be integrated to ﬁnd the work done in one
period of oscillation, T :
W = FR dx
−kA cos(ω0 t + φ(ω0 ))dx=

 T
kA cos(φ(ω0 )t + φ(ω0 ))Aω0 sin ω0 tdt=
01= kω0 A2 T sin φ(ω0 ).
2(2.38)

44

2.4 Thermoelastic damping

Therefore the quality factor (Equation 2.35) is given by:
Q=

2πkA2
kω02 A2 T sin φ(ω0 )=1
sin φ(ω0 )≈1,
φ(ω0 )(2.39)

when φ(ω0 ) is small. Thus the mechanical loss can be determined by measuring
the quality factor of an oscillator at its resonant frequencies. If the damping isassumed to be essentially frequency independent (i.e. structural damping) thenthe mechanical loss at frequencies of interest for gravitational wave detectors
can be approximated from these measurements. Alternatively, the loss can bemeasured over a range of frequencies by measuring many modes of a varietyof samples of diﬀerent sizes.

2.4

Thermoelastic damping

Thermoelastic damping is an anelastic relaxation process related to the ﬂow ofheat in a material, and arises from the coupling of temperature and strain in abody via the thermal expansion coeﬃcient. Thermoelastic dissipation in thinbeams and ﬁbres was studied by Zener, who calculated the diﬀerential heating
across a ﬁbre as it vibrates [121, 122]. If a thin beam is deﬂected into an arc,
the part of the beam under compression will heat up, while the opposite sideof the beam will expand under tension and cool down, as shown in Figure 2.3.
A thermal gradient is therefore created, leading to the ﬂow of heat across the
thickness of the beam. If, instead of undergoing a static deﬂection, the beam
is vibrating there will be an oscillating heat ﬂow as each side of the beam iscompressed and stretched in turn. This heat ﬂow is a source of dissipation,
and will have a peak at a frequency related to the time taken for heat to ﬂow
across the beam.

45

2.4 Thermoelastic dampingt

Heating due tocompression

Cooling due toexpansion

Heating due tocompression

Cooling due toexpansionHeat flow

Heat flow

(c)

(b)

(a)

Figure 2.3: A schematic diagram of a bending mode of a rectangular cross-section
bar. Initially (a), the bar is at maximum deformation with one side
warming up under compression while the other side cools under tension.
As the bar oscillates through the un-deﬂected position (b) and bends inthe opposite direction (c), the heating and cooling, and thus the directionof heat ﬂow, is reversed.

When the beam, or indeed any body, is at rest, localised statistical temperature ﬂuctuations throughout the material result in deformation of the material,with the strain produced being governed by the thermal expansion coeﬃcient,dl
α = l−1 dT. The level of thermoelastic dissipation in a ﬂexure is calculated by

Zener [121, 122] to be:φ(ω) =Y α2 Tωτ,ρC 1 + ω 2 τ 2(2.40)

where Y is Young’s modulus, ρ is density, C is speciﬁc heat capacity and τ isthe relaxation time. The relaxation time is related to the time taken for heat
to ﬂow across the beam, and sets a characteristic frequency fchar = (2πτ )−1 atwhich the maximum thermoelastic loss occurs. For a rectangular cross sectionﬁbre, known as a ribbon, the relaxation time can be shown to be:
τ=

ρCt2
,
π2κ

(2.41)

where t is the thickness of the ribbon and κ is thermal conductivity [110]. Fora circular cross section ﬁbre of diameter d, the thermoelastic relaxation time46

2.5 Thermal noise in a multi-resonance system

is given by [110]:
ρCd2
τ=
.
13.55κ

(2.42)

This theory has direct application in the estimation of the thermoelastic thermal noise in the mirror suspension ﬁbres used in gravitational wave detectors.
Recent work has extended this theory to include the eﬀects of the temperature dependence of the Young’s modulus, with the interesting result that theeﬀective thermal expansion coeﬃcient of a suspension ﬁbre material could be
set to zero if an appropriate stress is applied to a ﬁbre [123]. It is therefore
possible that careful design of the suspension ﬁbres could lead to the nullingof suspension thermoelastic noise.

2.5

Thermal noise in a multi-resonance system

The thermal noise model described by Equation 2.26 is appropriate for a single
resonance of a mechanical system. In practice, the laser beam reﬂected fromthe front face of the mirror in a gravitational wave detector senses the total
thermal displacement arising from a large number of the resonant modes of thesuspended mirror. Early models assumed that the motion of each mode was
uncorrelated and that the total thermal noise power spectral density could
be calculated simply by summing the contribution of each individual mode[111] as calculated using Equation 2.31. Gillespie and Raab [124] derived thefollowing expression for the total power spectral density of the thermal motion
associated with the internal resonant modes of a mirror:
Sx (ω) = 4kB T φn (ω)αn mωn2 ω,(2.43)

where φn is the loss factor of the nth mode of resonant frequency ωn . αnrepresents the coupling between a particular mechanical mode of the mirror
and the laser beam [124].
However, the assumption that the modes are uncorrelated implies that eachmode has a separate Langevin thermal driving force, which is only the case if

2.5 Thermal noise in a multi-resonance system47

the mechanical dissipation is homogeneously distributed throughout the test
mass [125]. In the case of spatially inhomogeneous dissipation which occurs,for example, due to the presence of lossy mirror coatings or at the attachment
points of the suspension ﬁbres, it is possible that the thermal motion of diﬀerent
modes may have some correlation, thus making the simple mode summation
technique inadequate to predict the expected thermal noise levels correctly.
This was pointed out by Levin [125], who along with others [126, 127] hasdeveloped an alternative, direct method of calculating the internal thermal
noise which can account for the distribution of loss throughout the system andfor the precise shape of the laser beam.
Levin’s approach involves the direct application of the ﬂuctuation dissipation theorem to the interferometer readout of the position of the test massface, weighted by the Gaussian intensity proﬁle of the sensing laser beam. To
calculate the thermal noise a notional pressure, of the same spatial proﬁle as
the intensity of the laser beam spot on the test mass, is applied to the front
face of the test mass and the resulting power dissipated in the mass is calculated. Application of the ﬂuctuation dissipation theorem then allows the power
spectral density of thermal displacement, Sx (f ), to be calculated as:
Sx (f ) =2kB T Wdiss,
π 2 f 2 F02(2.44)

where F0 is the peak amplitude of the notional oscillatory force and Wdiss isthe power dissipated in the mirror. The dissipated power is given by:
Wdiss = 2πf(x, y, z)φ(x, y, z, f )dV,

(2.45)

V

where  is the energy density of the elastic deformation under the peak appliedpressure F0 . For spatially homogeneous loss, the formula for Wdiss becomes:
Wdiss = 2πf Umax φ(f ),(2.46)

where Umax is the total energy associated with the peak elastic deformation of
the test mass.

48

2.5 Thermal noise in a multi-resonance system

If the laser beam radius is considerably smaller than the radius of the test
mass, the test mass can be modelled as being half-inﬁnite and Umax can becalculated from elasticity theory [126, 127]. In this case, the power spectral
density of the Brownian thermal noise, SxIT M (f ), can be shown to be [126]:
SxIT M (f ) =2kB T 1 − σ 2φsubstrate (f ),π 3/2 f Y w0

(2.47)

where φsubstrate (f ) is the mechanical loss of the test-mass material, Y and σ
are the Young’s modulus and Poisson’s ratio of the material respectively and
w0 is the radius of the laser beam where the electric ﬁeld has fallen to 1/e ofthe maximum intensity. 1
An expression for the thermal noise power spectral density of a ﬁnite sized
test mass, SxFTM (f ), was also derived by Bondu et al [126], and corrected by
Liu and Thorne [127]:
SxFTM (f ) =4kB T
φsubstrate (f )(Uo + ΔU ),πf(2.48)

where the term (Uo + ΔU ) represents the elastic deformation energy, deﬁned
explicitly by Liu and Thorne [127].
The half-inﬁnite mass approximation is found to over-estimate the thermal
noise power spectral density, SxIT M (f ). Liu and Thorne calculate a correction factor, CFTM , to Equation 2.47 for the ﬁnite test-mass case, such that
2
SxFTM (f ) = CFTM
SxIT M (f ). For an Advanced LIGO type mirror, they foundthat this correction factor varied between 0.96 and 0.66 for beam radii ranging
from 1 to 6 cm respectively.

2.5.1

Inhomogeneous loss and coating thermal noise

Equations 2.44 and 2.45 show that the level of thermal noise is directly related
to the power dissipated in the test mass when a notional oscillating pressure
1

The radius at the point where the intensity has fallen to 1/e of the maximum value,√r0 = w0 / 2, is also widely used in thermal noise literature.

2.5 Thermal noise in a multi-resonance system49

is applied to the surface. The power dissipated at any point in the massis proportional to both the elastic energy associated with the deformations
caused by the pressure and the mechanical loss at that point [125, 127]. Since
most of the deformation occurs close to the point where the pressure is applied,
it follows that a source of dissipation located close to the front face of the mass
will contribute more to the thermal noise read out by the laser beam than an
identical source of dissipation located further from the front face. The spatial
distribution of mechanical loss throughout the volume of a test-mass must
therefore be considered when calculating the expected level of thermal noise.
Thus to reduce thermal noise, sources of dissipation should ideally be locatedas far as possible from the front surface of the test mass.
This realisation is of great signiﬁcance, as interferometric gravitational wave
detectors require the use of multi-layer dielectric coatings to form highly reﬂecting mirrors on the front faces of the test masses. The dissipation of thesecoatings has been shown to be in the order of 2 − 4 × 10−4 [128, 129, 130, 131].
This is several orders of magnitude higher than the dissipation of the silica
substrate material, which may be as low as ∼ 1 × 10−9 [132]. Thermal noise
associated with the reﬂective coatings will therefore potentially be the dominant source of thermal noise in a gravitational wave detector. This has led
to a programme of research, involving a number of research groups worldwide,
aimed at quantifying and studying ways of reducing the mechanical dissipationof the coating materials, and is the motivation for much of the work presented
in this thesis.
2.5.1.1

Calculation of coating thermal noise

Nakagawa et al [133] applied Levin’s method to approximate the thermal noise
in a coated mirror by modelling a multi-layer reﬂective coating as a thin surface layer of thickness d and mechanical loss φcoating , with the same material
properties as the test-mass substrate. Nakagawa showed that the total power2.5 Thermal noise in a multi-resonance system

50

spectral density of the thermal noise associated with such a mirror is given by:
2kB T 1 − σ
Sxtotal (f ) = 3/2
π f w0 Y2
2 (1 − 2σ) dφsubstrate + √φcoating ,π (1 − σ) w0(2.49)

where φsubstrate is the mechanical loss of the mirror substrate material.
In Harry et al [128] an expression incorporating the eﬀects of the layer
structure of the reﬂective coating and the diﬀering material properties of thecoating and the substrate is derived. Since the reﬂective coating is not homogeneous, but is made of of alternating layers of two materials (SiO2 and Ta2 O5 ),possible anisotropy of the coating mechanical loss factor must be taken into
account. Harry et al consider the coating mechanical loss factor to have twocomponents, φ⊥ and φ , associated with strains parallel and perpendicular tothe coating surface respectively. The Brownian thermal noise power spectral
density of a coated mirror can be shown to be [128]:
1
2kB T 1 − σ 2 
1 d
Sx (f ) =φsubstrate + √3/2
π f w0 Y
π w0 Y Y (1 − σ  2)(1 − σ 2 )×[Y 2 (1 + σ)2 (1 − 2σ)2 φ
+Y Y  σ  (1 + σ)(1 + σ  )(1 − 2σ)(φ − φ⊥ )+Y 2 (1 + σ  )2 (1 − 2σ  )2 φ⊥ ] ,(2.50)

where f is the frequency in Hz, T is the temperature in Kelvin, Y and σ are theYoung’s modulus Poisson’s ratio of the substrate, Y  and σ  are the Young’s
modulus Poisson’s ratio of coating, φ and φ⊥ are the mechanical loss values
for the coating for strains parallel and perpendicular to the coating surface, dis the coating thickness and w0 is the laser beam radius. In the case whereY  = Y , σ  = σ and φ⊥ = φ , this equation agrees with the result of Nakagawa
et al (Equation 2.49).
Harry et al note that for fused silica substrates coated with alternating
layers of SiO2 and Ta2 O5 , the Poisson’s ratio of the coating is likely to besmall enough that Equation 2.50 can be approximated to within ≈ 30% by2.6 Thermoelastic noise

51

setting σ  = σ = 0 [128]:
2kB T 1 
1 d YY
Sx (f ) = 3/2
φsub + √φ +  φ⊥ .
π f w0 YY
π w0 Y(2.51)

This expression is a signiﬁcant simpliﬁcation of Equation 2.50, and is usefulfor estimating the expected level of thermal noise. It should be noted that
Equations 2.49 to 2.51 all represent the sum of the power spectral densityof Brownian thermal noise from the coating and the substrate. The thermal
noise power spectral density associated with the coating alone can be obtained
explicitly by expanding the second term in Equation 2.51:

2kB T d Y YSx (f ) = 2
φ +  φ⊥ .
π f Y w02 Y
Y(2.52)

It can be seen that the coating thermal noise is a function of the Young’s
modulus of the substrate material and thus the same coating may have adiﬀerent level of thermal noise when applied to diﬀerent substrates.
It is interesting to note that the coating thermal noise varies as w0−2 while
the substrate thermal noise, arising from the homogeneous dissipation in thetest mass itself, varies as w0−1 . Thus one possible way to reduce thermal noise ingeneral, and coating thermal noise in particular, is to use laser beams of larger
radius. In addition, the standard Gaussian beam proﬁle is not optimum froma thermal noise perspective, as it averages the thermal displacement of the test
mass over a relatively small region. The use of diﬀerent beam geometries is anarea of active research [134, 135, 136, 137, 138, 139].

2.6

Thermoelastic noise

Thermoelastic dissipation has been discussed in the context of ﬂexural vibrations of a thin beam, in which energy is dissipated as the result of heat ﬂow fromthe side of the beam under compression to the side under tension. Thermoelastic dissipation is also of relevance when considering the expected thermal noise

52

2.6 Thermoelastic noiseof the test mass mirrors in gravitational wave detectors. Deformations associated with statistical ﬂuctuations in the temperature of a mirror will result in
displacements of the front face of the mirror, through the coeﬃcient of thermal linear expansion of the mirror substrate. In certain crystalline materials
which are of interest for use as mirror substrates such as sapphire [140] and silicon [95], this thermoelastic thermal noise is likely to be more signiﬁcant thanBrownian thermal noise. The power spectral density of thermoelastic thermal
noise, STE (f ) in a mirror (approximated as half-inﬁnite with respect to theradius of the laser beam incident on it) has been calculated by Braginsky et al[140] to be:√

SxTE (f ) =2 kB T 2 α2 (1 + σ)2 κ,
π 5/2
ρ2 C 2 r03 f 2(2.53)

where α is the coeﬃcient of linear thermal expansion, C is speciﬁc heat capacity, κ is thermal conductivity, ρ is density and r0 is the radius of the laser beam
at which the intensity has fallen to 1/e of the maximum value. This expression
is strictly only accurate in the case where the test mass is large with respect tothe size of the laser beam spot. A correction factor for the case of a ﬁnite sized
test mass was derived by Liu and Thorne [127]. This factor is dependent on
the test mass dimensions, and for typical test masses is of the order of unity.

2.6.1

Coating thermoelastic noise

In a coated test mass additional thermoelastic eﬀects resulting from the diﬀering thermo-mechanical properties of the coating and the substrate have beenshown to occur [141, 142]. The power spectral density of the thermoelastic
thermal noise arising from the presence of the coating has been shown by Fejer
and Braginsky [141, 142] to be given by:
Sx (f ) ≈C2
αs2
8kB T 2 d2
2 avg
√
√
(1
+
σ
)
Δ̃2 ,
s
Cs2 κs Cs
π πf w02(2.54)

53

2.7 Suspension thermal noise sources

where the subscript s denotes a property of the substrate and where Δ̃2 isgiven by:
Δ̃ ≡
2



α 1+σCsY+ (1 − 2σs )−1
2αs Cavg 1 − σ 1 + σs
Ys avg2.

(2.55)In these equations the averaging takes into account the relative thicknesses of
the alternating coating layers. The form of the averaging operation is given by[142]:(X)avg ≡dadb+,
da + db da + db

(2.56)

where the coating is considered to consist of alternating layers of two types ofmaterial, labelled a and b, which are da and db in thickness respectively.

2.7

Suspension thermal noise sources

Thermal noise resulting from the test mass and coating materials was discussed
in the previous section. Thermal noise associated with the resonant modes ofthe suspension system is considered brieﬂy here.

2.7.1

Pendulum mode

As noted in Chapter 1, the mirrors in a gravitational wave detector are suspended as pendulums to attenuate seismic noise. The thermal noise associatedwith the pendulum mode of the suspended mass is a function of the mechanical loss of the material of the suspension ﬁbres. However, most of the energy
in a pendulum is stored as potential energy via the Earth’s gravitational ﬁeld
rather than in the elastic deformation of the suspension wire material. Only
the fraction of the total energy which is stored in the suspension ﬁbres canthus be dissipated. This results in the loss factor of a pendulum being lower
than that of the material used in the ﬁbres suspending it. It can be shown[111] that the loss of a pendulum is related to the loss of the suspension ﬁbre

54

2.7 Suspension thermal noise sourcesmaterial by:
φpend (ω0 )φmaterial (ω0 )

kﬁbre
,
kgrav

(2.57)

where kﬁbre is the elastic spring constant of the wire and kgrav is the eﬀective‘spring constant’ associated with the gravitational ﬁeld, that is kgrav = mg/l
where m is the mass of the mirror and l is the length of the wire. The elastic
spring constant for a pendulum suspended from n wires is:√
n TY I,kf ibres =2l2(2.58)

where T is the tension in each wire, Y is the Young’s modulus and I is themoment of inertia of the wire cross section. Thus the mechanical loss of thependulum is found to be:
φpend (ω0 )√n TY I.
φmat (ω0 )2mgl(2.59)

The factor by which the loss of the material is reduced is known as the dilution
factor D which, noting that mg/n = T , can be expressed as:
√
2l T
.
D= √
YI

(2.60)

This analysis assumes that each wire bends only at the top, close to the suspension point. It can be generalised by introducing a factor ξ which takes thevalue ξ = 1 if the wires are constrained to bend only at the top and ξ = 2 if
the wires can bend at both the top and bottom [143]:√
ξn T Y I.
φpend (ω0 ) φmat (ω0 )2mgl(2.61)The power spectral density of the pendulum thermal noise in the gravitational
wave band (at frequencies signiﬁcantly higher than the pendulum frequency,
ω0 ) can be found using Equation 2.33:
4kB T φpend (ω) ω02
Sx (ω) ≈m
ω5√
2kB T ξn T Y I ωo2≈
φmat (ω),
m2 glω5(2.62)(2.63)

2.8 Total thermal noise in a detector mirror

55

where m is the mass of the test mass and ω is the angular frequency in theoperating band of the detector.

2.7.2

Violin modes of the suspension ﬁbres

Finally, the eﬀect on the displacement of the front face of the mirror of thesuspension ﬁbres vibrating in their ‘violin’ modes must be taken into account.
The violin modes form a harmonic series, with many modes typically lying
within the interferometer’s operating frequency band. As with the pendulum
mode, the loss of a violin mode is reduced from the intrinsic loss of the material
by some dilution factor. Assuming that the loss in the suspension ﬁbres is spatially homogeneous, and that the rocking mode of the pendulum is constrained,
then the loss of the ﬁrst violin mode will be twice that of the pendulum mode
[120, 143]. The violin mode thermal displacement couples to the displacementof the test mass with a suppression factor m/mﬁbre , where m is the mass ofthe pendulum and mﬁbre is the mass of the suspension ﬁbre [144].
For a low loss silica suspension, the thermally induced motion of the test
mass due to the violin modes will be concentrated in a narrow bandwidtharound the violin frequencies, resulting in several spikes of noise in the detector
readout. However, these violin mode thermal noise peaks can easily be notchﬁltered out of the interferometer signal [145], with only a small reduction in
the useable bandwidth of the detector.

2.8

Total thermal noise in a detector mirror

To calculate the total thermal noise in a single interferometer mirror, the thermally driven motion of the various resonances of the system must be appropriately combined. The various contributions to the total thermal displacement
noise can be summarised as follows:• Substrate thermal noise2.9 Conclusion

56– Due to intrinsic mechanical loss of the test mass material (Equation
2.47).
– Due to thermoelastic eﬀects arising from temperature ﬂuctuationsin the test mass (Equation 2.54).
• Coating thermal noise– Due to the intrinsic mechanical loss of the coating materials (Equation 2.52).
– Due to thermoelastic eﬀects arising from temperature ﬂuctuations
and the diﬀering thermodynamic properties of the coating and substrate (Equation 2.54).
• Suspension thermal noise– Due to the pendulum mode (Equation 2.63).
– Due to violin modes of the suspension ﬁbres (Section 2.7.2).
The thermal noise expected in a typical mirror for use in the second generation
gravitational wave detectors has been calculated by Rowan et al [146], using alaser beam radius of r0 = 3.9 cm [92], and is shown in Figure 2.4.

2.9

Conclusion

Thermal noise from the mirrors and the lower stages of their suspensions is
one of the most important limits to the sensitivity of interferometric gravitational wave detectors. The level of thermal noise in the detector operating
band is related to the mechanical loss of the materials used in the test-mass
pendulums, with low loss materials resulting in lower thermal noise. However,the spatial distribution of the dissipation is also extremely important. Speciﬁcally, sources of dissipation located close to the reﬂecting surface of the mirror

57

Linear spectral density of thermal displacement [m/√Hz]

2.9 Conclusion

10

10

10

10

10

-18

-20

-22

substrate brownian
substrate thermoelastic
coating
total (substrate + coating)suspension .

-24

-26

10

1

10

2

Frequency [Hz]

a

10

3

us

10

4

Figure 2.4: Calculated levels of Brownian and thermoelastic thermal noise in a single
suspended fused silica mirror [146], in an interferometer with a laser
beam of radius r0 = 3.9 cm, assuming a test mass loss of 5 × 10−9 [132],
and an ion-beam sputtered coating formed from alternating multi-layersof SiO2 and Ta2 O5 [130].

have a larger contribution to the total thermal noise sensed by the laser. The
multi-layer dielectric coatings applied to the front face of the test-mass, which
are known to be signiﬁcantly more lossy than the silica test-mass substrate,
are therefore expected to be a particularly important source of thermal noisein future detectors, limiting the achievable sensitivity over a signiﬁcant portion of the detection band. Finding methods of reducing the dissipation of thereﬂective coatings is therefore one of the most important challenges for future
generations of detectors.

Chapter 3
Inﬂuence of temperature on thesettling time ofhydroxide-catalysis bonds
3.1

Introduction

Hydroxide-catalysis bonding can be used to join rigidly materials which canbe hydrated and dehydrated in the presence of a hydroxide catalyst. Initially developed and patented by Jason Gwo at Stanford University for assembling the fused silica star-tracking telescope of the Gravity Probe-B mission[147, 148, 149], the technique was further developed in Glasgow University foruse in the construction of the quasi-monolithic suspensions in the GEO 600
gravitational wave detector [150, 151]. In this detector, fused silica ﬁbres suspend the silica test mass. To avoid welding the ﬁbres directly to the test mass,potentially introducing thermal stress and increasing the mechanical loss, they
are welded to small silica ‘ears’ attached to the sides of the test masses byhydroxide-catalysis bonding. Variants of this technique are planned for use inboth Advanced LIGO and Advanced VIRGO [152].
In addition, hydroxide-catalysis bonding is also planned for use in the space
based gravitational wave detector LISA (Laser Interferometer Space Antenna)[153, 154]. LISA is designed to search for gravitational waves at frequencies
583.1 Introduction

59

Figure 3.1: The planned orbit of the three satellites forming the LISA gravitational
wave detector.

below ∼1 Hz, which are not detectable on Earth due to the sensitivity barrier
at low frequencies (below ∼ 10 Hz) formed by gravity gradient ﬂuctuations[63]. In addition to eliminating terrestrial gravity gradient noise, operating
the detector in space allows the interferometer arms to be very long, matching
the wavelength of the gravitational waves being searched for. Possible sources
which may be detected by LISA include short period galactic binaries (e.g.
binary white dwarfs) and the coalescence of massive black holes at the centresof merging galaxies [155].
LISA will consist of three spacecraft located at the corners of an equilateral triangle with sides 5 million kilometers long, as shown in Figure 3.1.
Each spacecraft will have two optical assemblies inside arms at 60◦ to eachother. These assemblies consist of a telescope, pointing at one of the othertwo spacecraft, and an optical bench containing a test-mass cage and otheroptical components. The arm-length is measured as the distance between thefreely ﬂoating test masses within each spacecraft. Two separate lasers are located on each spacecraft, and can be phase-locked in order to behave like thebeam-splitter in a Michelson interferometer. Drag-free control techniques will
be used to keep each spacecraft ‘centered’ on the test masses without touching60

3.1 Introduction

bonded silica
opticalcomponents
Zerodur slab

212 mmFigure 3.2: The engineering model of the Lisa Pathﬁnder optical bench, which
utilises fused silica optical components bonded to a Zerodur base plate.

them, ensuring that the test masses are isolated from external disturbancesand remain ‘freely ﬂoating’. The whole interferometer will follow an Earth-likeorbit around the sun, lagging about 20◦ behind the Earth. In total, the threespacecraft form three separate, but not independent, interferometers, sensitive
to gravitational waves between approximately 1 and 1 × 10−5 Hz [154].
To verify aspects of the technology developed for LISA, a demonstration
mission known as LISA Pathﬁnder is planned [156]. This will involve the launchof a single spacecraft to test some of the critical components, including laser
stabilisation, the phase readout of interferometric signals and drag-free control.
The LISA spacecraft will each use local interferometers to measure the phaseof the light received from the other two spacecraft. A similar interferometer
will be required for the LISA Pathﬁnder mission. The success of both missions
relies on the mechanical stability of the optical benches on which these interferometers are mounted [157], with the bench having to survive launch into
space.
The design of the optical bench incorporates a low thermal expansion Zerodur base-plate to which silica optical components are attached using hydroxide-3.2 The chemistry of hydroxide-catalysis bonding

61

catalysis bonding, as shown in Figure 3.2. At room temperature, the bonds
set rigidly in a few tens of seconds [147, 158], allowing only a short time window for the precision alignment of the components. This chapter describes
an experiment investigating the possibility of extending the settling time bycarrying out the bonding process at lower temperatures. The experiment wasundertaken jointly by the author and another PhD student [159].

3.2

The chemistry of hydroxide-catalysis
bonding

Hydroxide catalysis bonding was developed for jointing oxide materials suchas fused silica. Ideally, the surfaces of the material to be bonded should be
ﬂat to approximately λ/10 (where λ is 633 nm), and thoroughly cleaned tobe free of any contaminants. A small volume (typically 0.4μl/cm2 [158]) of
aqueous hydroxide solution, such as sodium or potassium hydroxide, is placed
onto one surface immediately before the components to be bonded are brought
together. The bonding solution is normally centrifuged and ﬁltered to remove
any undissolved particles.
If the surfaces are suﬃciently ﬂat and clean, the bonding solution spreadsout between the surfaces, where it reacts with the silica to form a silicate gel
which hardens over time to form a very strong, yet very thin, bond between thetwo surfaces. Recent measurements suggest that bonds between fused silica
components are typically between 30 and 100 nm in thickness [160].
At room temperature, the silica components can be moved (parallel to theplane of the bond) for ≈ 1 minute after the bonding solution is applied; after
this time, the bond ‘sets’ quite rigidly, although it takes several weeks until
the bond reaches maximum strength.
A series of chemical reactions are involved in the formation of a hydroxidecatalysis bond. The ﬁrst stage is hydration of the silica surfaces. The surface

62

3.2 The chemistry of hydroxide-catalysis bonding

O
Si

O

Si

OH

Si

OH

O
Si

O
O

Figure 3.3: A simpliﬁed diagram showing the surface of fused silica in a fullyhydrated state after cleaning. Note that the surface consists of OH groups
attached to silicon atoms.

of silica is hydrophillic, and on exposure to water, or hydroxide ions, hydration will occur, with the attraction of OH− ions to ﬁll any open bonds. Fully
hydrated silica, as shown in Figure 3.3, typically has 4 to 6 silanol (Si-OH)
groups per nm2 [161]. Any contaminants on the surface will inhibit this hydration, potentially obstructing the formation of the bond. The silica surfaces are
therefore cleaned thoroughly prior to bonding to remove any contaminants, as
described in Section 3.3.
When the bonding solution comes into contact with the silica, the OH− ionsetch the surface, as shown in Figure 3.4. The etching process is initiated by
OH− ions forming weak bonds with silicon atoms on the surface of the silica.
As a result, the original Si-O bonds in the silica are weakened, leading to
the possibility of a silicate (Si(OH)−5 ) molecule breaking from the surface. As
the etching process continues, a quantity of these liberated Si(OH)−
5 molecules
become available in solution. When the pH of the solution drops below 11, thesilicate molecule hydrolyses to form monosilicic acid (Si(OH)4 ) and a hydroxideion, as shown in Equation 3.1:
Si(OH)-5  Si(OH)4 + OH- .

(3.1)

Once the concentration of Si(OH)4 reaches approximately 1 - 2 %, silanolgroups on adjacent Si(OH)4 molecules can link together by dehydration to63

3.3 Sample preparation

O

Si
Si

OH

OH
O Si
OH
OH
O

Si

Si

Si

O

OH

OH Si(OH)5
OH
O

O

O

(a)

(b)

Si

OH

Figure 3.4: A simpliﬁed diagram of the etching stage of the bonding reaction. Hydroxide ions can etch into the surface of the silica by forming weak bonds
with silicon atoms, weakening the Si-O bonds (a) resulting in silicate ionsbeing released into solution (b).

form a siloxane bridge (Si-O-Si) [162]. Repeated dehydration results in theformation of a network of silicate-like polymer chains, as shown in Figure 3.5.
At the surfaces of the silica, exposed silanol groups can similarly dehydrate
to link the surface silicon atoms to the polymer chains, resulting in a rigid
polymer network between the two surfaces. It should be noted that Figure 3.5
is signiﬁcantly simpliﬁed with only one polymer chain shown. In reality, manycross-linked chains would be present.
If it is assumed that the conversion of Si(OH)−
5 to Si(OH)4 and the subsequent polymerisation is fast in comparison to the rate of etching of Si(OH)−5from the silica surface, then the settling time can reasonably be deﬁned as
the time taken for the silicate to start converting to silicic acid, which occurs
when the pH drops below 11. The assumption that the etching process is therate determining step for the formation of the bond has been experimentallyveriﬁed [163].

3.3

Sample preparation

Appropriate preparation of the silica surfaces is critical to the success of thebonding process and the formation of strong, reliable bonds. A global surface

64

3.4 Variation of settling time with temperature

OH

O
Si

Si

O

O H HO

O
Si

O

OH
Si

O

Si

OH

OH
Si

O H HO

Si

H 2O

O

Si

HO
H 2O

O

Si

O

OH

OH

OH

O

O

Si

O

Si

O

Figure 3.5: Once the pH of the solution between the silica components has dropped
below 11, the liberated Si(OH)−
5 molecules dissociate to form SiOH4 ,which can hydrolyse to form polymer chains, eventually linking the twosilica surfaces.

ﬂatness of approximately λ/10 (where λ is 633 nm) is required to ensure successful bonding [151]. The ﬂatness is normally measured with an interferometerprior to bonding.
The cleanliness of the surfaces is also important, to ensure that the silica is
free from chemical and particulate contaminants and to enable full hydration.
To this end, a speciﬁc cleaning procedure has been developed. The samples
are rinsed with deionised water before the bonding surfaces undergo a lightabrasive cleaning with a paste of cerium oxide rubbed over the surface forapproximately 20 to 30 seconds. After cleaning the surface with deionised
water, any remaining cerium particles are removed with a paste of sodium
bicarbonate. After a ﬁnal rinse with deionised water, the samples are dried
with a single wipe from a non-abrasive cloth dampened with methanol.

3.4

Variation of settling time withtemperature

3.4.1

Experimental setup

This experiment was designed to investigate whether cooling could extend
the length of time during which the position of bonded components can be65

3.4 Variation of settling time with temperatureadjusted. The settling time was deﬁned as the time between the start of thebonding reaction and the point where the silica pieces could no longer be easily
moved relative to one another. This was measured by applying an oscillating
force of constant amplitude to one of the pieces and monitoring the amplitudeof the motion as the bond formed.
Small silica blocks, of dimensions 10 × 10× 5 mm, were bonded to silica
discs of 24.5 mm diameter. A silica block was placed in a small holder, attached
on one side via a rigid connection to a loudspeaker and on the other side by anelastic connection to a ﬁxed clamp, as shown in Figure 3.6. This setup allowed
an oscillating horizontal force to be applied to the silica block. A silica disk was
rigidly clamped beneath the block, and the bonding solution placed between
the samples as they were brought into contact. A split-photodiode readout
system (see Figure 4.3) was used to measure the amplitude of the motion of a‘ﬂag’ attached to the holder for the silica block.
oscillation of upper block
rigid clamp
holder for silica block

flag

loudspeakerrigid attachment
elasticconnection

surfaces tobe bondedsilica discsilica block

Figure 3.6: The mechanical excitation system developed to gently shake the uppersilica block during bonding. The holder for the silica block was supported
on one side by a rigid connection to a loudspeaker and on the other side
by an elastic connection to a rigid clamp. A ‘ﬂag’ attached to the top of
the holder was used to monitor the motion of the block throughout thebonding process.

An oscillation frequency of 14 Hz was used as this frequency was foundto couple well to the sample holder, providing both a large enough amplitude3.4 Variation of settling time with temperature66

of motion and a clean sinusoidal readout signal. The initial amplitude of the
motion of the upper sample was typically between 100 and 500 μm. The
amplitude was observed to slowly decrease by approximately a factor of ﬁve as
the bond solidiﬁed, before the motion suddenly stopped entirely as the bondset. The time taken to reach this point where motion stopped was taken as
the settling time of the bond. The bonding was carried out in a temperature

Figure 3.7: Schematic diagram of the temperature controlled glove box in which theexperiment was carried out. The glove box was 1 m long, 40 cm deep
and 50 cm high.

controlled glove-box constructed by S. Reid [159]. The interior of the glove box
was cooled using two internal cross-blower fans to circulate the air over heatsinks cooled by Peltier devices, as shown in Figure 3.7. A capillary thermostat
was used to sense the internal temperature, which was controlled by regulating
the power to the Peltier devices. Temperatures between 0 and 20◦ C could
be obtained, with a typical stability of ± 0.1◦ C. The silica samples and thebonding solution were placed within the glove-box at least ten minutes prior
to bonding, to allow them to achieve thermal equilibrium.

3.4 Variation of settling time with temperature

3.4.2

67

Results

A series of measurements of the bonding time were taken at 0, 5, 10, 15 and20◦ C, using 0.1 mol l−1 potassium hydroxide (KOH) bonding solution. KOH
was chosen as the literature suggests that this solution produces the strongest
bonds [149]. A typical plot of the amplitude of motion of the upper block as afunction of time is shown in Figure 3.8. The point where the bond sets can beclearly seen at the right of this graph, where the amplitude of the upper silica
block suddenly drops to zero. On several occasions, the sample was observed
to stop moving as if the bond had set when in fact the bonding solution was
still ﬂuid. This was possibly due to small rough areas (or high areas) of thesilica surfaces coming into contact, or due to particulate contaminants between
the samples. In these cases the silica block was tapped gently and it resumed
shaking: the settling times were found to be similar to those measured forsamples which did not experience this problem.
Between 8 and 10 settling time measurements were made at each temperature point. The mean settling time was found to increase with decreasing
temperature, as shown in Figure 3.9. The error bars in this plot are the standard error calculated from the data taken at each temperature.

3.4.3

Analysis

For a general chemical reaction:nA + mB → F + G(3.2)

the rate equation [164] links the rate at which the reaction occurs to the concentrations of the reactants as follows:τ = k(T )[A]α [B]β ,(3.3)

where [A] and [B] are the concentrations of reactants A and B, k(T ) is therate constant for the reaction, and the exponents α and β describe the order68

3.4 Variation of settling time with temperature

0.8

settling time result
taken at 0 oC

Silica ear amplitude readout (V)

0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0

100

200

300

400

Time (s)Figure 3.8: Typical plot of the measured oscillation amplitude of the upper silica block
as a function of time during bonding. The point where the bond set can
be clearly seen at approximately 360 seconds.

of the reaction with respect to each reactant. The temperature dependence ofthe rate constant k(T) is generally described by the Arrhenius equation:k(T ) = Ce−Ea /kB T ,(3.4)

where Ea is the activation energy of the reaction, kB is the Boltzmann constant,
T is temperature in Kelvin and C is a constant known as the frequency factor[112, 165]. Thus for the general reaction above, the rate can be expressed as:τ = Ce−Ea /kB T [A]α [B]β .

(3.5)

Assuming that hydroxide catalysis bonding is a ﬁrst order reaction, depending
only on the concentration of the OH ions, then the reaction rate is:τ = Ce−Ea /kB T [OH].

(3.6)

69

3.4 Variation of settling time with temperature

400

Settling time results
Exponential fit:y = 612.1*exp(-x/34.2) - 276.6

350

Settling time, ts (s)

300
250
200
150
100
50
0
0

5

10

15

20

o

Temperature ( C)
Figure 3.9: Plot of the measured settling time as a function of temperature forhydroxide-catalysis bonds made using 0.1 mol l−1 KOH solution, withan exponential ﬁt.

The settling time, ts can reasonably be assumed to be inversely proportional
to the reaction rate:ts ∝ τ −1 ,(3.7)

which, when substituted into Equation 3.6, gives the following expression forthe bonding reaction rate:
ts = C 

eEa /kB T
,
[OH]

(3.8)

where C  is a constant. Taking the natural logarithm yields the following:ln ts = ln C  − ln [OH] +

Ea
.
kB T

(3.9)

Since all of the bonds were made with the same concentration of potassiumhydroxide solution (0.1 mol l−1 ), the OH concentration at the start of eachreaction was the same. Similarly, since the settling point was deﬁned as the70

3.4 Variation of settling time with temperature

6.0

Settling time results
Linear fit y = 5513.6x -14.3

ln ts

5.5

5.0

4.5

4.0
0.003400.003450.00350

0.00355

0.00360

0.00365

0.00370

1/T (K-1)
Figure 3.10: Arrhenius plot of the settling time data, showing the logarithm of thesettling time as a function of the reciprocal of the temperature, withlinear ﬁt excluding the room temperature measurement.

time at which the pH dropped below 11, the concentration of OH at the endof each bonding reaction was the same. Therefore the average OH ion concentration throughout the bonding process can be considered constant for all of
the bonds which were made. Hence, from Equation 3.9, it can be seen that agraph of ln(ts ) against 1/T should be a straight line with gradient Ea /kB . This
plot (see Figure 3.10) shows a clear linear relationship, with the exception of
the data taken at 20◦ C, which lies some distance below the line ﬁtted to theother data points.
There were two diﬀerences in the experimental procedure used for the 20◦ C
measurement which might explain this discrepancy. Firstly, the settling times
at 20◦ C were not measured within the temperature controlled glove-box. It is
possible that, on transferring the apparatus into the glove-box, the actuation

3.5 Conclusion

71

system or the readout was accidentally altered, giving a slightly diﬀerent result for the settling time. Secondly, the potassium hydroxide solution used forthese bonds was not ﬁltered, leading to the possibility that particulate contaminants (perhaps undissolved potassium hydroxide) were present in the solution,
possibly aﬀecting the settling time. Given these factors and the strong linear
relationship of the remainder of the data, the room temperature measurement
was disregarded and a straight line ﬁtted to the data measured at 15, 10, 5 and
0 ◦ C. The line has a gradient of 5514, corresponding to an activation energy of
0.48 ± 0.01 eV per molecule of OH− , or alternatively (4.58 ± 0.08)×104 J permole. It should be noted this is not the activation energy reaction for a single
reaction, but rather for the compound reaction of etching and polymerisation
resulting in the formation of the bond.

3.5

Conclusion

It has been demonstrated that the settling time for a hydroxide catalysis bondcan be increased by reducing the temperature at which the bonding is carriedout. For the 0.1 mol l−1 KOH bonding solution studied here, the settling timeincreased by a factor of ∼5 to 370 seconds on cooling from 20 ◦ C to 0 ◦ C.
This method may have applications in increasing the time available for the
precise alignment of optical components during the construction of rigid optical benches in projects such as LISA. The calculation of the activation energy
for the bonding reaction may also be useful in comparative investigations of
the properties and chemistry of hydroxide catalysis bonds between other materials such as silicon, which is of interest for use in future gravitational wave
detectors.

Chapter 4
Loss measurements using silica
cantilevers at room temperature
4.1

Introduction

In Chapter 2, the relationship between the thermal noise in a gravitational
wave detector and the mechanical loss of the materials used in the suspended
mirrors was discussed. The thermal noise arising from any particular sourceof loss, which is sensed by the laser, is dependent on the proximity of thesource of loss to the front face of the mirror. Thus the mechanical loss of thereﬂective mirror coatings applied to the front face of the test mass is particularly signiﬁcant. In this chapter a method of studying the mechanical lossof coating materials is described, in which thin silica cantilever samples areused as substrates to which coatings are applied. Preliminary measurementsof the mechanical loss of both optically reﬂective, and conductive tin oxide
coatings for charge mitigation, are presented, and the same technique is applied to estimate the loss associated with a polished silica surface. Firstly, thedevelopment of silica cantilever samples and the experimental method used tomeasure the mechanical loss will be discussed.

72

73

4.2 Mechanical loss of silica ribbons

4.2

Mechanical loss of silica ribbons

Thin rectangular cross-section silica ﬁbres (known as ‘ribbons’) were developedin the Institute for Gravitational Research in Glasgow for possible use in theﬁnal stage of the test mass suspensions for Advanced LIGO [166, 167], but also
form useful substrates for the study of the mechanical loss of coating materials.
Coating loss can be calculated from measurements of the loss of the substrate
before and after application of the coating. Fused silica is a good choice of
substrate as it is known to have a low mechanical loss at room temperature[93, 132] thus ensuring high sensitivity to the loss of the coating material.
Most of the ribbons studied were fabricated using a machine developedby Alastair Heptonstall [166]. A hydrogen-oxygen ﬂame is used to melt thecentral portion of a silica slide (1 ×11 × 75 mm), which is then rapidly pulled
into a long, thin ribbon using motorised clamps. The resulting ribbons areapproximately 60 cm long, 1 to 2 mm wide and 110 μm thick, as shown in
Figure 4.1. The tapered regions at each end of the ribbon are known as ‘necks’.
Initial studies were carried out to assess the level of mechanical loss of silica
ribbons and determine the optimum geometry of sample for use in coating loss
measurements.
silica slide 1 mm
thick

neck
600 mm
11 mm
1.5 mmribbon ~ 0.1mm thick

15 mm

Figure 4.1: A schematic diagram of a silica ribbon (not to scale).

74

4.2 Mechanical loss of silica ribbons

4.2.1

Measuring mechanical loss

The mechanical dissipation (also known as the mechanical loss), φ(ω), of aresonant mode of system at angular frequency ω0 is deﬁned as:φ(ω0 ) =Edissipated,
2πEstored(4.1)

where Estored is the total energy stored in the vibrating system and Edissipatedthe energy dissipated with each cycle of oscillation.
Dissipation can be measured using a ‘ring-down’ method, in which a resonant mode of a sample is excited and the amplitude of the resulting motion isallowed to decay freely. For a resonant mode of angular frequency ω0 , excited
to an initial amplitude A0 , the time dependence of the amplitude decay A(t)is given by:
A(t) = A0 e−φ(ω0 )ω0 t/2 .

(4.2)

The loss factor of a resonance may thus be found from measurements of theamplitude of the decaying resonant motion as a function of time.

4.2.2

Experimental method

Cantilever samples consisting of a length of silica ribbon, the neck region andthe remnant of the silica slide were produced by carefully cutting a ribbon at
the desired distance from the neck (see Figure 4.1). The silica slide was held
horizontally in a stainless steel clamp, from which the ribbon section projected.
The mechanical loss factors of the bending modes of the ribbon sample could
then be measured. The frequency of the nth bending mode of a cantilever ofthickness a, length L, Young’s modulus Y and density ρ is given by [168]:aωn = (kn L) √2 3L22

 1/2
Y
,
ρ

(4.3)

in which kn L = 1.875 (n=1), 4.694 (n=2), 7.853(n=3), 10.996, (n=4) and14.137 (n=5). For n > 5, kn l can be approximated as kn l = (2n − 1)π/2 [169].

75

4.2 Mechanical loss of silica ribbons

To vacuumpumps
Cantilever
sampleViewport

Clamp
base

Upper clampingblock

Signalgenerator
Lock-in
amplifier

Laser

Split photodiodesensorElectrostatic
drive plate

Vacuum tank
Signalgenerator

ADC
Computerdata recording

High voltage
amplifierFigure 4.2: A schematic diagram of the experimental apparatus used to measure themechanical loss of cantilever samples - not to scale.

Figure 4.2 shows a schematic diagram of the apparatus used to measure
the mechanical loss factors of the bending modes of silica cantilevers. After acantilever was clamped, the clamp was placed on a suspended table inside avacuum tank, which was evacuated to a pressure of approximately 1 × 10−6
mbar. The bending modes of the cantilever were excited in turn by applying
an oscillating high voltage signal at the frequency of a resonant mode, witha d.c. oﬀ-set of approximately 700 V, to an electrostatic drive plate locatedapproximately 5 mm below the cantilever.
The amplitude ring-down of an excited mode was recorded by illuminating
the cantilever with a diode laser beam and monitoring the resulting shadow
with a split photodiode sensor, illustrated in Figure 4.3. The signals fromthe two halves of the photodiode were subtracted and ampliﬁed. A lock-in
ampliﬁer was used to multiply the resulting signal with a generated sine wave
diﬀering in frequency by 2 Hz. This produced a signal, proportional to the76

4.2 Mechanical loss of silica ribbons

Cantilever shadow
Photodiodes

Laser spot

Figure 4.3: A schematic diagram of the split photodiode arrangement used to measure
the amplitude of oscillation of a cantilever .

displacement of the cantilever, at a beat frequency of 2 Hz, which was digitally
sampled and recorded with a computer. The dissipation was calculated froman exponential ﬁt to the amplitude ring-down using Equation 4.2; an example
of a typical ring-down is shown in Figure 4.4.

Ringdown data
Exponential fit y = 3.8 e-3.39E-4 x

Shadow sensor signal (V)

4

3

2

1

0
0

1000

2000

3000

4000

5000

6000

Sample number

Figure 4.4: A ringdown measurement for the the third bending mode at 1051 Hz. A
sampling rate of 15 Hz was used, and the mechanical loss calculated fromthe ﬁtted exponential decay constant.

The loss measured in this way is the sum of all of the energy dissipation

4.2 Mechanical loss of silica ribbons

77

mechanisms in the sample, including intrinsic dissipation within the material
and any excess dissipation, which can arise from a number of sources and mustbe suitably minimised. A model for the intrinsic dissipation of fused silica is
described in the next section, after which the mechanical loss results for theﬁrst two ribbon samples is presented.

4.2.3

Semi-empirical model for intrinsic loss in fusedsilicaPenn et al [170] have formulated a semi-empirical model for the dissipationin fused silica, based on a study of the experimental loss factors measured byseveral groups. The intrinsic loss, φ f, VS , is considered to have contributions
from three dissipation processes: dissipation in the bulk of the material, inthe surface of the material and thermoelastic dissipation, which is discussed indetail in Section 2.4. The total loss is given by:


S
S
φ f,
= C1 + C2 f C3 + C4 φthermoelastic ,
V
V

(4.4)

where φthermoelastic is the thermoelastic loss, S and V are the surface area in
mm2 and the volume in mm3 respectively. The coeﬃcients Cn are obtained
empirically and depend on the type and condition of the silica. The term C1 VS
describes the loss associated with the surface of a silica sample. The mechanicalloss of the surface is often signiﬁcantly diﬀerent to that of the bulk material[171]. This is postulated to arise from factors such as small surface cracks,
surface roughness and the increased likelihood of the presence of impurities.
The term C2 f C3 represents the bulk loss of fused silica. The bulk loss is thought
to be related to certain molecules (or larger structural units) which can have
two energetically stable states (e.g. one model involves two possible Si-O bondangles [172]), separated by a potential barrier [171, 173]. In response to an
applied strain these atoms can move between the stable states, resulting in the

78

4.2 Mechanical loss of silica ribbons

dissipation of energy 1 . The amorphous structure of fused silica results in a
distribution of the heights of the potential barriers. The frequency dependenceof the loss can be found from measurements of this distribution [173].
The results presented here are for ribbons fabricated from Heraeus Suprasil
312, for which the values of the coeﬃcients Cn are taken to be: C1 = 6.5×10−9m, C2 = 7.6 × 10−12 , C3 = 0.77 and C4 = 1 [170].

4.2.4

Mechanical loss of ribbons with ‘necks’

Initially the loss factors of two silica ribbon cantilevers were studied. The ﬁrst
ribbon was fabricated with the ﬂame pulling machine as described in Section
4.2, while the second ribbon was fabricated using a similar system in which a
CO2 laser rather than a ﬂame was used to heat the silica. The dimensions of
the ribbons are given in Table 4.1. The ribbon from the laser-based pullingmachine had a signiﬁcantly shorter neck than the ribbon produced with theﬂame pulling machine (see Figure 4.5).
Flame-pulled ribbon

Laser-pulled ribbon

Length60 ±0.5 mm

48 ±0.5 mm

Width

1.11 ±0.01 mm

5.98 ±0.01 mm

Thickness

110 ±5 μm

120 ±5 μm

Table 4.1: Dimensions of the two ribbons with necks.

It was possible to measure the lengths of the cantilevers to an accuracy of± 0.5 mm without contacting the surfaces. The width and thickness of thecantilevers were measured with a micrometer after the mechanical loss factors
had been measured.
The loss factors of the ﬁrst six bending modes of each ribbon were measured
using the ring-down technique described in Section 4.2.1. Each measurement
1

Asymmetric double-well potential models for the dissipation mechanism in silica are

discussed in more detail in Section 5.5.1.

79

4.2 Mechanical loss of silica ribbons

Long tapered neckregionRapid transition between silica slide andthin ribbon section

(a)

(b)

Figure 4.5: (a) The ﬂame-pulled ribbon section in the clamp. (b) The laser-pulled
ribbon in the clamp.

was repeated ﬁve times, and the standard error calculated from the spread ofthe data. The measured mechanical losses of these ribbons were compared to
the values obtained from Penn’s model, as shown in Figure 4.6. In both cases,
the measured loss factors were substantially higher than those predicted by
the model, although the loss of the laser-pulled ribbon was generally closer tothe predicted values at high frequency.
This excess loss may arise from some degree of movement of the cantileverin the clamp, resulting in frictional stick-slip loss [174]. Measurements by
Quinn et al have shown that stick-slip losses can be reduced by having a large
diﬀerence in thickness between the clamping block and the cantilever itself[174], which ensures that the bending point occurs outwith the jaws of theclamp. This reduces the shear stress at the edge of the clamp, which in turn
reduces the tendency of the ribbon to slip [175]. Both of the samples had
an extensive neck region over which the clamping block gradually thins andnarrows to form the ribbon. Thus there was very little diﬀerence in thickness
between the clamping point and the adjacent ribbon section, suggesting thatstick-slip eﬀects may indeed be a signiﬁcant loss mechanism. Furthermore,
due to the diﬀerent manufacturing processes, the laser-pulled ribbon had a

80

4.2 Mechanical loss of silica ribbons

Ribbon pulled with CO2 laser
Modelled loss for CO2 ribbon

Mechanical loss

10

10

10

Ribbon pulled with flame machine
Modelled loss for flame ribbon

-5

-6

-7

0

1000

2000

3000

4000

5000

Frequency (Hz)Figure 4.6: Frequency dependence of the mechanical loss of a ribbon with a short neckfabricated with a laser pulling machine, compared to that of a ribbon witha longer neck fabricated with a ﬂame pulling machine. Also shown are
the losses predicted by Penn’s model for both samples.

signiﬁcantly shorter neck, while the ﬂame-pulled ribbon had a long neck overwhich the thickness decreases more gradually (see Figure 4.5). This suggeststhat the lower loss of the laser-pulled ribbon at high frequency may be due
to reduced levels of friction in the clamp, arising from the larger diﬀerence inthickness on either side of the clamping point. The high loss measured at 1014
Hz for the laser-pulled ribbon is likely to be the result of energy coupling intoa resonant mode of the clamping structure. This eﬀect is discussed in moredetail by Reid [159] and in Section 5.2.4.

4.2.5

Gas damping

To ensure that the measured loss factors are not limited by frictional damping
from residual gas molecules, suitably low vacuum pressure must be reached.

4.2 Mechanical loss of silica ribbons

The loss due to gas damping, φgas , can be approximated to be [175]:
M
AP,
φgas ≈
mω RT

81

(4.5)

where A is surface area, P is pressure, m is the mass of the cantilever, ωis the angular frequency of the resonant mode, M is the mass of one moleof the gas, R is the gas constant and T temperature. The loss associatedwith gas damping is proportional to 1/ω: gas damping is therefore more likely
to inﬂuence loss measurements at low frequency. The ﬁrst bending mode of
the sample studied were at approximately 50 Hz. At this frequency the loss
predicted by the semi-empirical model is of the order of 1.4×10−6 . To ensurethat the gas damping is at least an order of magnitude lower than this, a gas
pressure of less than ≈ 2 × 10−3 mbar is required, assuming that the residual
gas molecules are nitrogen. The pressure in the vacuum chamber routinely
reached ≈ 1 × 10−6 mbar, so gas damping eﬀects are considered to have been
negligible for all of the modes studied.

4.2.6

Welded silica ribbon

One possible source of the diﬀerence between the loss factors measured in the
ribbon cantilevers and the values predicted by the semi-empirical model was
frictional loss in the clamp. In an eﬀort to improve the isolation of a vibrating
ribbon from the clamp, a new type of cantilever which did not have a ‘neck’
region was fabricated. First, a 1 mm wide ribbon was pulled using the ﬂame
machine and cut carefully in half with clean pliers. The free end of the ribbon
was welded onto a section of silica slide using a CO2 laser, as shown in Figure
4.7. The section of silica slide was thoroughly cleaned with chloroform andmethanol, and placed in a clamp. A second clamp was used to hold a lengthof silica ribbon in the correct position for welding. Both clamps were mounted
on horizontal and vertical translation stages, to allow the samples to be moved
relative to the laser beam. The best results were obtained by heating the slide

82

4.2 Mechanical loss of silica ribbons

SilicaribbonHeight adjustable
upper clamp

Weld point
on silica
slide
Lower clamp
CO2 laser
beam

Figure 4.7: Left: Photograph of CO2 laser welding. Right: Schematic diagram oflaser welding apparatus.

immediately below the weld region for approximately 20 seconds with a low
power (∼ 15 - 20W) before the samples were lowered so that the end of theribbon was directly heated by the laser beam. After a further 10 to 20 secondsof heating, the laser power was increased incrementally over 30 seconds toa maximum of 30 W. The silica ribbon was then cut to form a 7 cm longcantilever sample welded to a clamping block. The welded sample is shown in
Figure 4.8.
The section of silica slide was used as a clamping block, with the sample clamped directly behind the weld region. This sample therefore had asigniﬁcantly sharper transition between the ﬂexing cantilever and the thicker
clamping block with the aim of reducing the level of stick-slip loss in the clamp.
The ribbon was measured to be 110 ± 5 μm thick with a micrometer. The
thickness was also calculated from the measured resonant frequencies of eachof the ﬁrst 9 modes using Equation 4.3. The average thickness obtained was
113 ± 3 μm.
The loss factors measured for this ribbon were considerably lower than theprevious measurements for the necked ribbon samples, as shown in Figure 4.9.

83

4.2 Mechanical loss of silica ribbons

At low frequencies, the loss factors of the welded ribbon follow the modelledcurve closely, while at higher frequencies they are approximately a factor of
two above those predicted by the model.
There are several possible sources of this residual excess loss. During thewelding process, parts of the sample were contaminated with silica vapour,which may increase the loss associated with the cantilever surface (see Figure4.10). The ribbon section was cut with pliers and it is possible that the surface
close to the location of the cut was damaged slightly, again producing a sourceof excess loss. It is also likely that the weld itself is a source of some additionalloss, as measurements of welded cylindrical ﬁbres have shown a degree of loss
associated with the weld [176]. The modes at 1056 Hz and 2728 Hz show
evidence of higher levels of excess loss. The 1056 Hz mode is close to the frequency at which excess loss was observed in the laser pulled ribbon, providing
a further suggestion that there may have been energy loss into a mode of theclamp at a nearby frequency.
The results for this welded sample suggest that frictional losses in the clamp
can be reduced by the use of a thick clamping block with a sharp transition in

(a)

(b)

Weld

Silica vapour

Figure 4.8: (a) Photograph of the cantilever immediately after welding. (b) Photograph of the welded cantilever in the clamp.

84

4.2 Mechanical loss of silica ribbons

Ribbon welded to slide
Ribbon clamped at neck
Modelled loss

Mechancial loss

10

10

10

-5

-6

-7

0

1000

2000

3000

4000

5000

Frequency (Hz)Figure 4.9: Frequency dependence of the mechanical loss of a ribbon section welded
to a silica slide, compared to that of a similar ribbon clamped at the neckregion. Also shown are the losses predicted by Penn’s model for both
samples.

thickness to the cantilever.

4.2.7

Coating loss measurements using silica cantilevers

It has been shown that a level of mechanical loss close to that predicted by
Penn’s semi-empirical model can be measured in silica cantilever samples produced by laser-welding a silica ribbon to a 1 mm thick section of silica slide.
The low loss of these samples makes them of interest for use in experiments tomeasure losses associated with both surface eﬀects in silica and with coatingsof various types. In the following section, loss measurements of conductive tinoxide coatings, which may be of interest for the mitigation of charging eﬀects
in the test masses and suspensions of future gravitational wave detectors, are4.3 Conductive coatings for charge mitigation in gravitational wave detectors

85

Region polishedby laser beam

1 mm

RibbonDeposited silica
vapourWeld region

1 mmFigure 4.10: Microscope images of the welded silica cantilever.

described. Silica cantilevers were also used to measure the loss of single layers of the optical coating materials used in gravitational wave detectors; theresults are presented in Section 4.4.

4.3

Conductive coatings for charge mitigationin gravitational wave detectors

Electrostatic charging of the test masses can be a signiﬁcant problem in a gravitational wave detector, potentially resulting in increased levels of displacement
noise [177] and in diﬃculties with the interferometer control systems [178]. A
fused silica test mass suspended in ultra-high vacuum is a very well electricallyisolated system, particularly if suspended on silica ﬁbres, as is the case with
GEO 600 and the planned Advanced LIGO and Advanced VIRGO detectors.
It is therefore possible that high levels of electrostatic charge with long decay times will arise. Indeed, in laboratory experiments the charge of fused4.3 Conductive coatings for charge mitigation in gravitational wave detectors

86

silica masses suspended in a vacuum has been observed to increase continuously over several months, with occasional large jumps of charge [179, 180].
There are many mechanisms by which the mirrors could become charged including friction from evacuating and venting of the vacuum tanks, by touchingan earthquake stop 2 or from interactions with cosmic rays [181]. One possible solution is the application of a conductive coating to the test mass andsuspension ﬁbres, which would allow the dispersal of any charges as they are
accumulated.

4.3.1

Charging issues in gravitational wave detectors

Charging of the test masses and suspensions can lead to several potential problems in a gravitational wave detector. Electrostatic damping may occur if
charges on the mirror interact with surrounding structures, reducing the quality factor of the modes of the suspended mirror [182, 183, 184] and therefore
resulting in higher levels of thermal noise in the detector. Fluctuating electricﬁelds associated with charge migration across the mirror or suspension surfaces
can lead to an additional source of displacement noise [177, 185].
Charging can also lead to control issues, with electrostatically induced displacement of the mirrors potentially resulting in calibration errors, diﬃculties
in controlling the mirror position and locking the interferometer. Chargingrelated control problems have recently been experienced at the GEO 600 detector [178]. In GEO 600, electrostatic actuators located on the reaction masssuspended immediately behind each test mass, are used to control the longitudinal position of the test mass. After a power cut at the GEO 600 site in 2006,
a test mass was found to have touched the electrostatic actuator and becomecharged. The charged mirror was found to signiﬁcantly aﬀect the performanceof the electrostatic actuators, and the detector could not be locked using thestandard locking parameters.
2

Earthquake stops are designed to limit the swing of the test-mass in the event of excessive

seismic excitation.

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

87

Similar charging events are thought to have been observed at the LIGO
Livingston detector [186], with the charging occurring when the test mass
made contact with an earthquake stop. The suspension resonant frequencies
were observed to increase after this event, and the noise spectrum was alteredsigniﬁcantly. After venting the vacuum chambers, both the suspension frequencies and the noise spectrum returned to normal. It is clear therefore thatthe development of charge mitigation techniques is of considerable importancefor future gravitational wave detectors.

4.3.2

Charge mitigation

There are several possible methods of reducing the eﬀects of charging of themirror suspension in a gravitational wave detector. Experiments carried out
in Glasgow demonstrated that illumination with UV radiation could control
the level of charge on a Macor pendulum suspended on silica ﬁbres [182]. The
same technique was used to discharge the positively charged GEO 600 test
mass referred to above.
A method of charge mitigation commonly used for dielectric materials under vacuum is the application of a thin, slightly conductive coating of a material
such as tin oxide [187]. This technique would have the advantage of preventing
charge accumulating on the test mass. For such a system to be implemented ina gravitational wave detector, the conductive coating would have to meet stringent thermal noise requirements, and, for application to the front face of themirror, optical loss requirements. In the following sections measurements ofthe mechanical loss of tin oxide coatings will be presented, and their potential
thermal noise contribution in a gravitational wave detector calculated.

4.3.3

Experimental techniqueFused silica cantilevers, similar to those described in Section 4.2.6, were used
to investigate the mechanical loss of the SnO coatings. Several 50 mm sections

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

88of ribbon, 1.5 mm wide and ≈ 0.11 mm thick, were welded to fused silica
clamping blocks of approximate dimensions (10 × 10 × 1) mm, using a CO2
laser as described in Section 4.2.6. The mechanical loss of several modes ofeach ribbon was measured before and after coating with tin oxide.
Tin oxide coatings were produced using pulse spray pyrolysis, a techniquein which a thin ﬁlm is deposited by spraying a solution onto a heated surface
where the constituents react to form the desired coating compound [188]. Spraypyrolysis is often used for depositing high quality, low cost transparent oxideﬁlms [187, 189, 190], and is an established method for applying conductive tinoxide coatings to glass [191].
Various precursor solutions can be used to deposit tin oxide ﬁlms including
SnCl2 , SnCl4 and SnF2 [192, 193]. The coatings measured here were depositedusing a pyrolysis solution of 0.1 mol l−1 tin (II) chloride dissolved in methanol,
which reacted on, or immediately above, the silica surface to form tin oxide asshown below:
SnCl2 + H2 O → SnO + 2HCl.

(4.6)

The deposition was carried out by Stuart Reid with assistance from the author.
The silica samples were heated to 600◦ C in a ceramic tube oven. Small quantities of the tin chloride solution were sprayed into the oven using an airbrushnozzle with dry nitrogen gas as the carrier gas, in pulses of 1 second duration.
The distance between the nozzle and the samples was approximately 20 to 30
cm, and the pulses were separated by 10 second intervals to minimise evaporative cooling of the samples. It was found that approximately 10 to 15 ml oftin chloride solution was required to deposit ∼100 nm of coating.

4.3.4

Results

Measurements with a four point probe carried out by Liam Cunningham
showed that the resistivity of the coatings were between 3.4×10−3 and 9.5×10−3
Ω cm. The variations in resistivity are possibly explained by variations in the4.3 Conductive coatings for charge mitigation in gravitational wave detectors

89

temperature of gas carrying the SnCl2 vapour resulting in variations in the
thickness or quality of the coating, as observed by Ishida et al when depositing coatings under similar conditions [194]. The lowest resistivity measured isconsistent with the values reported by Shalama et al in similar coatings [195].
4.3.4.1

Optical requirements and absorption measurements

To achieve desired levels of shot-noise limited performance in gravitational
wave detectors substantial light powers must be supported in the arms of
the interferometers. The optical mirror coatings used place a limit on the
maximum achievable power, as heating due to absorption in the coatings canlead to thermal deformation of the optics [94], resulting in instabilities within
the laser cavities. For example, the planned Advanced LIGO detectors havea design requirement that the mirror coatings within the Fabry-Perot arm
cavities have an absorption of less than 1 ppm to allow the cavity to maintain830 kW of circulating laser power [92].
Several silica disks coated with SnO were sent to collaborators at Stanford
University for measurements of the SnO absorption at 1064 nm. Values as
low as 60 ppm absorption were measured for the ∼100 nm thick SnO coatings[196]. This level of absorption clearly rules out the use of these SnO coatings
on top of the HR mirror coatings. However a coating placed beneath the
HR mirror coating would only be exposed to the laser power external to thecavity, approximately 1 kW in the case of Advanced LIGO [92], resulting in a
reduction in the absorbed power by a factor of ≈ 800. It is thus possible, in
principle, to place a conductive tin oxide coating below the multi-layer opticalcoating without exceeding the absorption requirements for Advanced LIGO.
4.3.4.2

Mechanical loss measurements of SnO coatings

In this section, measurements of the mechanical loss of SnO coatings are presented and used to estimate the thermal noise contribution of a tin oxide

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

90

coating on the front face of a test mass. Three silica cantilevers were coatedwith SnO in separate pyrolysis runs. Several partially masked ‘witness samples’ (25 mm diameter silica disks) were included in each coating run. The
coating thickness was obtained from measurements of the step height between
the masked and the coated regions of the witness samples carried out by Stuart
Reid using a Talysurf stylus proﬁler. Thicknesses ranging from 25 nm to 1.6
μm were measured for coatings produced at diﬀerent times, with the typicalvariation of coating thickness between samples in a single coating run being ∼10%.
The loss of several bending modes of each cantilever (up to the 6th mode
where possible) was measured before and after coating. The loss of the tinoxide layer was calculated using the diﬀerence in the measured loss of eachmode, as explained in the following section. The measured loss factors for oneof the cantilevers, which was coated with 25 nm of SnO, are shown in Figure
4.11. The loss of this cantilever after coating generally tended to decrease withincreasing frequency, with the exception of loss measured for the 3.6 kHz mode,
which is signiﬁcantly higher. No large increase in loss was observed at similarfrequencies in the other coated cantilevers, so it seems possible that the highvalue of loss measured was due to energy loss into the clamping structure at
this frequency.

4.3.5

Coating loss analysis

The measured loss of a coated cantilever arises from the loss of both the cantilever substrate and the loss of the coating material. As the coating is thinin comparison to the cantilever, only a small fraction of the energy associatedwith a resonant mode of the cantilever is stored in, and thus can be dissipatedin, the coating layer. It can be shown that the measured loss of the coatedcantilever φ(ω0 )coated is related to the loss of the substrate φ(ω0 )substrate and91

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

3E-6

Coated loss
Uncoated loss

Mechanical loss

2E-6

1E-6

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency (Hz)Figure 4.11: Plot of the measured loss of four modes of sample 1 before and after
coating with 25 nm of SnO.

the intrinsic loss of the coating material φ(ω0 )coating , as follows [197]:
φ(ω0 )coated = φ(ω0 )substrate +Ec
φ(ω0 )coating ,
Es

(4.7)

where Ec /Es is the ratio of the energy stored in the coating layer to the energy
stored in the cantilever substrate. This energy ratio can be calculated byconsidering a bar of length L, thickness a and width b, with a thin coating ofthickness t on one surface, bent into an arc of a circle of radius R, as shown in
Figure 4.12.
The energy stored in the coating layer is given by:
Ec =

Yc A
ΔL2
2L

(4.8)

=

Yc tb
ΔL2 ,
2L

(4.9)

where the cross sectional area of the coating, A, is given by A = tb, Yc is theYoung’s modulus of the coating and the change in length, ΔL, is given by:4.3 Conductive coatings for charge mitigation in gravitational wave detectors

92

θ
R

a

L - ΔLdr

L

Coating layerof thickness t

L + ΔL

Figure 4.12: A beam of thickness a and width b, with a thin coating of thickness t
applied to one face, bent into an arc of radius R.

a
ΔL = θ.
2(4.10)

Similarly, the energy stored in the half of the beam under compression may be
written as:dE =Ys bΔL2
dr,
2L(4.11)

where Ys is the Young’s modulus of the substrate and ΔL = (r-R)θ. Integratingover half of the beam gives: R+ a2 Y bθ 2Es
1 Ys bθ2 a3s2=(r − R) dr =
.
2
2L
48 L
R

(4.12)

If it is assumed that the energy stored in the half of the beam under tension
is equal to the energy stored under compression, the total stored energy in the
beam is twice that calculated in Equation 4.12. From Equations 4.12 and 4.9,
the ratio of the energy stored in the coating to that stored in the substrate canbe shown to be:
Ec
3Yc t
.
=
Es
Ys a

(4.13)

Since the shape of each bending mode of a cantilever is made up of a numberof such arcs, the energy ratio could be expected to remain constant for each4.3 Conductive coatings for charge mitigation in gravitational wave detectors

93

mode. This has been conﬁrmed by ﬁnite element analysis [198]. Rearranging
Equation 4.7 and substituting for the energy ratio gives the following expression
for the coating loss:
φ(ω0 )coating =

4.3.6

Ys a
(φ(ω0 )coated − φ(ω0 )substrate ).
3Yc t(4.14)

Tin oxide coating loss results

Equation 4.14 was used to calculate the coating loss of the three SnO coatings from the loss factors of the cantilevers before and after deposition of thecoating. The Young’s modulus of the tin oxide coating was taken to be 100± 10 GPa [199]. The calculated coating loss factors are shown in Figure 4.13.
The loss of the 25 nm thick SnO ﬁlm was signiﬁcantly lower than that of the
thicker ﬁlms below 2.5 kHz. The high coating loss measured at 3.6 kHz for the
25 nm coating arises from the anomalously high loss measured for this mode
after the cantilever was coated. As discussed in Section 4.2.4, this high loss ispossibly related to frequency dependent energy loss into the clamping structure. The large error in the coating loss in Figure 4.11 arises primarily fromthe combination of the uncertainties in the Young’s modulus and thickness ofthe tin oxide ﬁlms.
Table 4.2 summarizes the thickness, energy ratio and average loss acrossall of the measured modes for the three coatings measured. It can be seen in
Figure 4.14 that the average loss appears to have a linear dependence on thecoating thickness. It is possible that this apparent dependence of the loss on
coating thickness is due to structural diﬀerences in the thicker coatings. On
examination under an optical microscope, the thicker coatings were found tocontain many large grain-like structures, shown in Figure 4.15. Particularly
large crystal structures were observed near the free end of the cantilever (i.e.
the opposite end from the weld). These structures were not observed in the 25
nm thick coating.

94

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

Coating loss

5x10

-3

4x10

-3

3x10

-3

2x10

-3

1x10

-3

25 nm SnO film
135 nm SnO film
101 nm SnO film

0

1000

2000

3000

4000

5000

Frequency (Hz)Figure 4.13: Plot of the calculated coating loss for each of the three SnO coatingsstudied.

The mean mechanical loss calculated for all of the modes of the three samples studied was (2.4 ± 0.2) × 10−3 . This is of the same order as measurementsof the dissipation of SnO2 ﬁlms at higher frequencies (between 5 and 20 kHz)
reported by Mitrokhin et al [200].

SnO coating thickness

Sample 1

Sample 2

Sample 3

25 ± 2.5 nm

101 ± 10 nm

135 ± 14 nm−3−33.07 × 10−3

Ecoating /Esubstrate

0.69 × 10

Mean mechanical loss of SnO(1.5 ± 0.3) × 10−3

2.72 × 10(2.6 ± 0.2) × 10−3

(3.0 ± 0.2) × 10−3

Table 4.2: Summary of thickness and mean mechanical loss for the three tin oxide
coating samples.

4.3.7

Thermal noise modelling

The level of thermal noise associated with an SnO coating applied to the front
surface of a test mass in a gravitational wave interferometer can be calculated

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

3.5x103.0x10Mechanical loss

2.5x102.0x101.5x101.0x105.0x1095-3Mean mechanical loss
Linear Fit y = 1.36E-5x + 1.18E-3

-3

-3

-3

-3

-3

-4

0.0
0

20

40

60

80

100

120

140

Coating thickness (nm)Figure 4.14: The mechanical loss measured as a function of coating thickness, witha linear ﬁt to the data. Each point indicates the mean loss measured
for several modes of a sample.

from the measured mechanical loss using Equation 2.52. Assuming that themechanical losses associated with strains parallel and perpendicular to thecoating surface are identical, φ = φ⊥ , this equation becomes:
 

Y
Y
2kB T dSx (f ) = 2+  φ,
π f Y w02 Y
Y(4.15)

where f is frequency, T is temperature in Kelvin, Y and Y  are the Young’s
modulus values for the substrate and coatings respectively, φ is the mechanicalloss of the SnO coating, d is the coating thickness and w0 is the radius of theinterferometer laser spot in terms of electric ﬁeld amplitude. The loss of thecoating was taken to be the mean mechanical loss measured for all of the modesof the three samples: φ = 2.4 × 10−3 .
Since thermal noise associated with the optical coatings is expected to set
the primary limit to the sensitivity of future detectors in their most sensitivefrequency band, it is desirable for the magnitude of any additional thermal

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

96

(a)

2 mm

(b)

0.08 mmFigure 4.15: (a) Microscope images of the surface of sample 3 after coating with tinoxide. (b) Higher magniﬁcation image of the tin oxide coating showing
evidence of the formation of large crystals.

noise (e.g. from SnO coatings) to be at least an order of magnitude lower than
this. This is a general design requirement of the Advanced LIGO detectors.
Figure 4.16 shows the calculated linear spectral density of the Brownian thermal noise associated with two diﬀerent thicknesses of SnO coating applied to
the front face of a detector mirror. Also shown is the Brownian thermal noise
arising from a standard silica-tantala multi-layer reﬂective coating. It can beseen that for a 100 nm thick SnO coating the associated thermal noise is onlya factor of ∼ 2 lower than the noise from the reﬂective coating. To ensure
the thermal noise associated with an SnO coating would be a factor of ten
lower than the reﬂective coating thermal noise, as required for the Advanced
LIGO design, a thickness of ≈ 4 nm of SnO would be necessary. This is approximately ﬁve times thinner than the thinnest coating produced by spraypyrolysis to date, and further work would be needed to establish if such a thincoating could be produced and to study its optical, mechanical and electrical97)

4.3 Conductive coatings for charge mitigation in gravitational wave detectors

Linear spectral density of thermal dsiplacement (mHz

-1/2

Multilayer reflective coating
100 nm SnO
4 nm SnO

10

10

10

10

-19

-20

-21

-22

1

10

100

1000

Frequency (Hz)Figure 4.16: Calculated levels of the Brownian thermal noise associated with a 100
nm and a 4 nm thick SnO coating on a fused silica mirror illuminatedby a laser beam of radius r0 =3.9 cm, compared to the thermal noise
arising from a multilayer HR coating.

properties. Studies of mechanical loss in SnO coatings deposited by alternative methods may also be of interest, although it should be noted that similar
losses to those presented here have been measured in SnO ﬁlms deposited bymagnetron sputtering [200].
As noted previously, the mechanical dissipation of the SnO coatings wasobserved to have some dependence on the ﬁlm thickness. Further investigation
to conﬁrm this relationship is required. However, if conﬁrmed it would potentially allow the target thermal noise to be met by a slightly thicker coating.
This may be of some importance if the optical and electrical properties arefound to also depend on the thickness of the coating. The linear ﬁt in Figure 4.14 gives the following relationship between the mechanical loss, φ, and98

4.4 Dissipation in optical coatings

thickness in meters, t, of an SnO coating:φ = 1.36 × 104 t + 1.18 × 10−3 ,(4.16)

from which the loss of a 4 nm thick coating can be found to be 1.24 × 10−3 ,a factor of 1.9 lower than the mean loss used above. This would result in a
reduction in the linear spectral density of thermal displacement associated withthe coating by a factor of 1.38 (since the linear thermal spectral density, x, isdeﬁned as x(f ) = Sx (f )). The loss-thickness relationship can be substituted
into Equation 4.15 to calculate the coating thickness required for a particularlevel of thermal noise. The maximum coating thickness compatible with theAdvanced LIGO noise target is found to be 7.4 nm.

4.3.8

Conclusion

Initial investigations have shown that conductive tin oxide coatings applied tothe test-mass mirrors beneath the reﬂective mirror coatings could potentially
fulﬁll both the optical absorption and thermal noise requirements for future
gravitational wave detectors. The use of these coatings could mitigate bothcontrol problems and increased levels of thermal noise associated with electrostatic charging of the mirrors. Coatings of less than 10 nm in thickness
would be required to meet the thermal noise speciﬁcations of future detectors,
necessitating further studies of the properties of very thin SnO coatings.

4.4

Dissipation in optical coatings

As discussed in Chapter 2, the thermal noise arising from the mechanical loss
in the mirror coatings is expected to limit the sensitivity of the next generationof gravitational wave detectors between several tens and several hundreds ofHertz. In this section, experiments using silica cantilevers to measure the loss
factors of single layers of the individual coating materials will be described.

99

4.4 Dissipation in optical coatings

4.4.1

Reﬂective dielectric coatings

Interferometric gravitational wave detectors require mirrors with a very highreﬂectivity, as discussed in Chapter 1. Metallic coatings are not suitable due tohigh levels of absorption [201], potentially resulting in excessive thermal deformation in the mirror and damage to the coating. Dielectric coatings, however,tend to have lower absorption levels and are more durable than metallic coatings [201]. Highly reﬂecting dielectric coatings consist of alternating layers of
two dielectric materials with diﬀerent refractive indices. The optical thicknessof each layer is generally λ/4, where λ is the wavelength to be reﬂected: 1064
nm in the case of the Nd:YAG lasers used in gravitational wave detectors.
Optical thickness, δ, is related to the actual thickness of the layer, t, by:δ = nt,(4.17)

where n is the refractive index of the layer. An illustration of a quarterwavelength reﬂective coating is shown in Figure 4.17.
The total reﬂectivity of the coating is related to the diﬀerence in the refractive indices of the alternating layers. For a laser beam in a medium withrefractive index n0 incident on a multi-layer coating with an even number, 2N ,of layers of refractive indices nH and nL with nH > nL , the reﬂectivity of thecoating is given by [202] :


R2N =

ns f − n0
ns f + n0

2,(4.18)

where ns is the refractive index of the substrate and f = (nH /nL )2N . For an
odd number, 2N + 1, of layers this expression becomes:
2
 2
nH f − n0 ns.
R2N +1 =n2H f + n0 ns(4.19)The reﬂectivity of the coating can therefore be increased either by increasing
the number of pairs of high and low index layers or by increasing the diﬀerence
between the refractive indices of the layers. For a given reﬂectivity the number of layers, and thus the thickness of the coating, can be reduced by usingmaterials with a large diﬀerence in refractive index.

100

4.4 Dissipation in optical coatings

Figure 4.17: A schematic diagram of a multi-layer reﬂective coating, composed ofalternating layers of a high refractive index (nH ) and a low refractive
index (nL ).

Ion beam source

To vacuum
pumps
Ion beam

Substrate

Vacuumchamber

Coating target

Sputteredatoms

Figure 4.18: A schematic diagram of a simple ion-beam sputtering chamber.

4.4 Dissipation in optical coatings

4.4.1.1

101

Coating production

All current gravitational wave detectors use multi-layer dielectric coatings
formed from alternating layers of silica (SiO2 ) and tantalum pentoxide (Ta2 O5 ).
These layers are deposited by ion-beam sputtering. In this technique, a beamof ions is accelerated towards a target of the coating material. On impact, theions eject atoms from the surface of the target, which travel to the substrate
where they condense, gradually forming a uniform coating layer. A simple
schematic of an ion beam sputtering system is shown in Figure 4.18. Tantalum pentoxide coatings are usually produced using a tantalum metal target
with a small amount of oxygen present in the deposition chamber [203]. The
sputtered tantalum atoms react with the oxygen atoms and are subsequently
deposited as tantalum pentoxide. The coatings studied in this these were deposited using argon ions as the sputtering species.
Ion-beam sputtering has several advantages over other thin ﬁlm deposition
methods. Since the sputtered ions have a relatively high energy (several eV),
the coating atoms adhere more strongly to the substrate and the coatings
tend to have a denser structure [204]. In addition, the optical absorption ofion-beam sputtered coatings is signiﬁcantly lower than for coatings deposited
with other techniques such as electron beam deposition [205]. Post-deposition
thermal annealing is used in most optical coatings to reduce optical absorption
losses [203].

4.4.2

Mechanical dissipation in coating materials

As discussed in Chapter 2, recent research has shown that thermal noise associated with the Brownian motion of the ion-beam sputtered cavity mirror coatings of amorphous oxide materials sets a fundamental limit to the sensitivity ofinterferometric gravitational wave detectors [128, 129, 206, 207]. In addition to
gravitational wave detectors, similar amorphous oxide coatings are utilised in
the laser cavities used in the development of highly frequency-stabilised lasers4.4 Dissipation in optical coatings

102

for high-resolution optical spectroscopy [208], fundamental quantum measurements, optical frequency standards [209, 210] and quantum information science[211]. These systems are currently limited, or expected to soon be limited, by
thermal noise in the cavity mirror coatings.
It is clearly of major importance to a range of precision measurement systems to determine the exact level of dissipation, and thus thermal noise, expected from speciﬁc coatings, to understand the mechanism responsible forthis dissipation and to study possible methods of reducing it.
4.4.2.1

Previous measurements of coating dissipation

Previous studies have shown that dissipation in the commonly-used SiO2 /Ta2 O5 coatings is dominated by the dissipation in the coating materials, andis not signiﬁcantly aﬀected by friction at the layer interfaces or by the coating/substrate interface [130, 131]. It has also been shown that the dissipationof the Ta2 O5 layers is signiﬁcantly larger than that of the SiO2 layers, and someevidence of a small frequency dependence of the coating loss has been observed[131, 212]. Currently a number of research groups worldwide are involved in a
programme of research aimed at understanding the mechanical dissipation of
the coating materials and studying ways of reducing it. One approach has involved the reduction of the total Ta2 O5 content of a multi-layer coating, using
an optimisation technique which varies the number and thickness of coating
layers to minimise the mechanical loss for a given reﬂectivity [213]. Modelling
has suggested that a reduction of approximately 14 % in the coating dissipation
may be achievable using this technique.
Another approach has been to attempt to reduce the mechanical loss of
the Ta2 O5 layers by doping. It has been shown that the dissipation of a SiO2
/ Ta2 O5 coating can be reduced by up to ∼40% by doping the Ta2 O5 withtitanium dioxide (TiO2 ) [214]. The lowest coating loss factor measured byHarry et al [214] was 1.7×10−4 , for a coating which was doped with TiO2
to a level of (24 ± 2)% Ti cation concentration. However, there is no clear

4.4 Dissipation in optical coatings103

correlation between the concentration of TiO2 dopant and the magnitude of
the reduction in dissipation and, as yet, the exact mechanism by which thisdoping reduces the dissipation is not clearly understood.

4.4.3

Coating loss measurements using silica cantileversubstrates

Most coating loss experiments to date have focussed on multi-layer coatings
applied to samples with a comparatively small surface to volume ratio. The
experiments presented in this thesis were designed to study the dissipationof single layers of either Ta2 O5 or SiO2 , thus allowing these materials to bestudied independently of each other. The coatings studied were applied tothin substrates, between approximately 50 and 100 nm in thickness. Thin
substrates are desirable to ensure a relatively high fraction of the total elastic
energy associated with a resonant mode is stored in the coating, thus increasing
the eﬀect of the intrinsic loss of the coating on the loss factor of a coatedcantilever.
In this section, measurements of the mechanical loss of single layers of
optical coating materials applied to silica cantilevers will be described. This
work was carried out in collaboration with LMA, a coating vendor in Lyon
[215]. Colleagues at LMA had developed an experiment to measure the loss of
coatings applied to silica membranes (45 mm long, 10 mm wide and ≈ 0.11 mm
thick) fabricated by mechanically polishing silica wafers. After manufacture,
the membranes were annealed at 1000◦ C for approximately ten hours.
The measurement technique used by LMA involved directly clamping themembranes in a highly polished steel clamp. In an eﬀort to obtain better isolation from the clamp, six of these membranes were sent to the lab in Glasgow,
where they were attached to clamping blocks of approximate dimensions 10
mm × 10 mm × 1 mm using the CO2 laser welding technique described in
Section 4.2.6.

104

4.4 Dissipation in optical coatings

The loss of the ﬁrst six bending modes of the uncoated welded samples
was measured at room temperature, initially by the author and latterly withthe assistance of another student, P. Holt [216]. To test the level of clamping
loss in these samples, one membrane was welded to a 5 mm thick silica block.
This resulted in no signiﬁcant diﬀerence to the measured loss, suggesting that
adequate isolation from the clamping structure was provided by the 1 mm
thick clamping blocks.
The lowest loss factors measured in the set of seven membranes are compared to data for an identical un-welded membrane clamped directly in the
LMA clamp in Figure 4.19. The loss of the welded ribbon discussed in Section
4.2.6 is also included for comparison. Welding the membranes to clamping
blocks can be seen to signiﬁcantly reduce the measured mechanical loss, suggesting that the clamping block may reduce recoil and frictional losses into
the clamp. However, the loss of the welded LMA membranes was signiﬁcantly
higher than the loss of the welded silica ribbon. It is possible that this difference is related to the diﬀerent surfaces of the two types of sample. The
polishing process used to fabricate the LMA membranes may have left a damaged surface which could be more lossy than the smooth surface of a ﬂame
pulled ribbon.
The loss of the damaged surface layers can be estimated by comparing
the results to the Penn model (Equation 4.4) and the experimental data for
a ﬂame-pulled silica ribbon (Section 4.2.6). The modelled loss obtained fromthe bulk and thermoelastic terms of Penn’s model were subtracted from themeasured loss of each mode to ﬁnd the level of ‘excess loss’ present. Excess
loss is deﬁned here as the sum of the loss associated with the surface, φS , andany loss due to the weld region or the clamp, φother :φexcess = φS + φother .

(4.20)

Thus the total measured loss can be represented by:φmeasured = φthermoelastic + φbulk + φexcess ,(4.21)

105

4.4 Dissipation in optical coatings

Mechanical loss

10

10

10

Welded membrane
Welded ribbon
Un-welded membrane
Penn model

-5

-6

-7

0

1000

2000

3000

4000

5000

Frequency (Hz)Figure 4.19: Plot of the lowest losses measured in the 7 welded silica membranes,
compared with the losses measured in a welded ribbon section.

where φmeasured is the measured loss for a mode and φthermoelastic and φbulk are
the thermoelastic and bulk loss terms from Penn’s model.
A simple model for the surface loss assumes that there is a surface layer ofthickness h, which is lossier than the bulk of the material, with a dissipation
factor of φsurface [217]. It is important to distinguish between φsurface , which isthe actual loss of the surface layer (which would be obtained if the surface could
be removed from the sample and measured independently) and φS , the part of
the total measured loss which is due to surface loss. Following the argument
in Section 4.3.5 for a coated beam, the contribution φS of the surface loss tothe total loss of the cantilever is related to the loss factor of the surface layer,
φsurface , by the ratio of the energy stored in the surface layer to the energy
stored in the bulk of the cantilever:φS =Esurfaceφsurface .
Ebulk(4.22)

Assuming that the Young’s modulus of the surface layer is identical to that of

106

4.4 Dissipation in optical coatings

the bulk of the ribbon, the energy ratio given by Equation 4.13 can be correctedby a factor of two to account for the presence of the lossy surface layer on bothsides of the ribbon, giving the following expression for the surface loss:h
φS  6 φsurface ,a(4.23)

where a is the thickness of the ribbon. Here, the portions of the surface associated with the thin edges of the ribbons and membranes are ignored.
Measurements by Heptonstall et al suggest the product hφsurface in ﬂamepulled silica ribbons is ≈ 3 ×10−12 m [217]. Using this value, the weld or
clamp loss associated with the ribbon cantilever, φother can be estimated using
Equation 4.21, using the average experimental loss factor at frequencies above
the thermoelastic peak. Further assuming that this weld loss is identical forthe ribbon and the membrane sample, the surface loss, φS in the membrane
can be calculated using Equation 4.21, to give a value of 1.23×10−6 . The valueof hφsurface can then be calculated using Equation 4.23, giving hφsurface = (2.3
± 0.2) ×10−11 m.
Studies of fused silica suggest that an appropriate thickness for the lossy
surface layer in ﬂame-pulled silica ﬁbres is ∼ 1 μm [218]. This lossy layer
could arise from chemical contaminants or from absorption of water from theﬂame used to heat the silica prior to pulling the ﬁbre. It is also possible thatstress arising as the molten ribbon cools could alter the loss of a surface layer[217, 218].
Studies of the thickness of a damaged layer in silica caused by polishinghave been carried out as part of the Gravity Probe B mission, with the layer
thickness found to be approximately twice the diameter of the polishing particles used [219]. Thus for the polished surface of the membrane, which waspolished with particles of diameter ∼1 μm [216], h = ∼ 2μm. These values forthe thickness of the lossy surface layers give the following surface losses:
φsurface(ribbon) = (3 ± 0.2) × 10−6 ,(4.24)

4.4 Dissipation in optical coatings

φsurface(membrane) = 1.1 ± 0.1 × 10−5 .

107(4.25)

Given these assumptions, the loss of the surface layer damaged by polishing,after annealing, is found to be approximately a factor of 4 higher than the lossof the surface of a ﬂame-pulled ribbon. Heat treatment is known to reduce theloss of silica [171], therefore it is likely that a mechanically polished surface
which was not heat treated would have a higher loss. This calculation is only
an estimate, and relies on several assumptions. More accurate measurementsof the surface loss of a polished surface could be obtained by using identically
sized samples to ensure that the weld loss and clamp loss between the two
samples would be closer to being identical. An alternative method used byanother postgraduate student in Glasgow is to attempt to remove the damaged
surface layer by chemical etching, and measure the resulting change in loss ofthe sample [216]. The loss of a polished silica surface is of interest to allow
accurate calculation of the thermal noise associated with the polished front
face of the test mass mirrors in gravitational wave detectors, as discussed by
Reid [159].

4.4.4

Coating loss measurements

After the loss factors of the uncoated welded membranes discussed above had
been measured, they were sent to LMA for coating with single layers of tantala and silica. Most of the coated membranes were studied by colleagues at
LMA; however, the loss factors of two of the coated samples were measured
in Glasgow. These membranes, whose loss factors had been measured prior
to coating, were coated with variants of the LMA Formula 5 coating (Ta2 O5
doped with 14.5 % TiO2 ) known as Formula 5** (F5**) and Formula 5***(F5***). These variants were produced using a method in which a commercially sensitive deposition parameter was varied in an attempt to reduce themechanical loss of the coating [220], and are otherwise identical to the doped
Formula 5 coating measured elsewhere [214]. The loss factors measured for the4.4 Dissipation in optical coatings

108

F5** and F5*** coatings are shown in Figure 4.20. It can be seen that lower
loss factors were measured for the F5** coating for three of the four modes
measured. At 350 Hz, the F5*** coating has a marginally lower lower loss,
although it is within experimental error of the loss factor for the F5** coating.
It is likely that the high loss of ∼ 1 × 10−3 measured for the F5*** coating at
2 kHz was inﬂuenced by energy coupling to a mode of the clamp during themeasurements of the coated sample.
In addition to these welded membranes, several coated silica membranes
which were not welded to clamping blocks were measured by the author andPaul Holt during a visit by collaborators from LMA to the lab in Glasgow. A
silica coating, an undoped tantala coating and a further variant of the LMA
Formula 5 titania-doped coating known as Formula 5* were measured; eachcoating was 0.5 μm thick. All of the coated cantilevers which were studied arelisted in Table 4.3. The coating loss was calculated using the loss measurements
for the uncoated cantilever in the LMA clamp shown in Figure 4.19. The
calculated coating losses are shown in Figure 4.21. The silica coating loss wasfound to be signiﬁcantly lower than the loss of the tantala as expected, with amean value (for the two modes studied) of ∼ 9.7 × 10−5 . The loss factors for
the F5* coating are approximately a factor of 1.4 lower lower than those forthe undoped tantala.
If the loss measurements using the welded and unwelded substrates arecompared, it can be seen that all of the TiO2 -doped Ta2 O5 coatings have alower loss than the undoped coating, with Formula 5∗∗ having the lowest loss.
While this comparison is not ideal, as the welded cantilevers have been shown
to be better isolated from the clamp, it may be valid if it is assumed thatthe frictional clamping losses are approximately equal for both the coated anduncoated cantilevers. If this is the case, the eﬀects of the frictional loss could
be expected to be removed when the loss factors of the coated and uncoatedcantilevers are subtracted.

109

4.4 Dissipation in optical coatings

Coating mechanical loss

10

10

tantala F5** (welded substrate)
tantala F5*** (welded substrate)

-3

-4

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency (Hz)Figure 4.20: Plot of the coating loss factors for LMA F5** and F5*** tantala, measured using silica membrane substrates welded to silica clamping blocks.

Coating mechanical loss

10

10

un-doped tantala (unwelded substrate)
tantala F5* (unwelded substrate)
silica (unwelded substrate)

-3

-4

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency (Hz)Figure 4.21: Plot of the coating loss factors for undoped tantala, LMA F5* doped
tantala and silica coatings, measured using unwelded membranes in theclamp developed by LMA.

110

4.5 Conclusion

Coating

Substrate

Mean mechanical loss

Undoped Ta2 O5

Un-welded SiO2 membrane(4.2 ± 0.3)×10−4

Doped Ta2 O5 F5∗Un-welded SiO2 membrane(3.0 ± 0.01)×10−4

Doped Ta2 O5 F5∗∗

Welded SiO2 membrane(1.5 ± 0.1)×10−4

Doped Ta2 O5 F5∗∗∗

Welded SiO2 membrane(2.0 ± 0.4)×10−4

SiO2

Un-welded SiO2 membrane(0.97 ± 0.02)×10−4

Table 4.3: The average mechanical loss (calculated for all mode frequencies studied)of several coatings applied to silica membranes. All coatings were 0.5 μm
thick: the membranes were 110 μm thick.

The results indicate that the deposition parameter which is altered to produce the diﬀerent variants of Formula 5 Ta2 O5 tantala may signiﬁcantly aﬀect
the mechanical loss of the coating. Without knowing what this parameter is,
the results are hard to analyse further: however, it is clear that the loss of
Ta2 O5 coatings with the same level of doping can be signiﬁcantly reduced by
optimising the conditions under which they are deposited.
The following chapter describes a series of experiments designed to study
the mechanism by which energy is dissipated in Ta2 O5 coatings, and the eﬀectof TiO2 doping on this mechanism. In addition, this technique may be of value
for a systematic study of the eﬀects of various deposition parameters on thedissipation in these coatings.

4.5

Conclusion

Thermal noise arising from dissipation in the mirror coatings is expected tolimit the sensitivity of the next generation of gravitational wave detectors attheir most sensitive frequencies. In recent years, studies have shown that thedissipation in SiO2 /Ta2 O5 coatings is dominated by the Ta2 O5 layers, and thatthe dissipation can be signiﬁcantly reduced by doping the Ta2 O5 with TiO2 .
However, further studies of the level of dissipation in the various coating mate-4.5 Conclusion

111

rials are of considerable interest. Charging of the test masses in gravitational
wave detectors can cause excess noise and aﬀect the operation of control systems in the interferometers. The mechanical properties of conductive coatings
for use in mitigating charging eﬀects is thus of interest.
Methods of fabricating and clamping silica cantilevers for mechanical loss
measurements of coating materials have been investigated. It has been shownthat welding a section of silica ribbon to a thicker silica clamping block canprovide isolation from the clamping structure, with evidence of the reductionof slip-stick losses in the clamp. These cantilevers have been used to study theloss of conductive tin oxide coatings, the loss of single layers of the materials
used in the mirror coatings and the loss associated with the damaged silica
surface layer caused by polishing.
Results have shown that tin oxide coatings applied by spray pyrolysis witha maximum thickness of 4 - 8 nm would meet the thermal noise requirements
for Advanced LIGO. Further study of a possible relationship between the dissipation and the coating thickness is required.
Preliminary measurements of the dissipation in single layers of SiO2 andTa2 O5 at room temperature have been carried out. The results suggest thatvarying a particular (undisclosed) deposition parameter can reduce the dissipation in Ta2 O5 coating layers doped with 14.5 % TiO2 by up to ≈ 50%. The
variant of this coating found to have the lowest loss at room temperature is
Formula 5∗∗ . In the remainder of this thesis, investigations of the temperaturedependence of the dissipation in single layers of ion-beam sputtered silica andtantala coatings are presented.

Chapter 5
Temperature dependence of themechanical dissipation intantalum pentoxide coatings
5.1

Introduction

As discussed in Chapter 4, thermal noise arising from the mechanical loss in theion-beam sputtered mirror coatings is expected to limit the sensitivity of thenext generation of gravitational wave detectors at frequencies between several
tens and several hundreds of Hertz. In the case of the proposed Advanced LIGO
detectors, coating thermal noise is predicted to be the dominant noise source
between approximately 40 and 400 Hz. This frequency band will coincidewith the peak sensitivity of the detectors. Reduction of the level of coating
thermal noise has therefore become one of the most signiﬁcant challenges tobe overcome in the design and development of future generations of detectors.
In general, as can be seen from Equation 2.51, thermal noise can be reduced
by reducing the mechanical dissipation of the materials used or by reducing thetemperature of the mirror. These approaches aim to reduce the thermal motionof the mirror in the frequency band of interest. A third method currently
under investigation is the reduction of the eﬀect of the thermal motion on
the sensitivity of the interferometer through the use of diﬀerent laser beam112

5.1 Introduction

113

proﬁles, such as a ﬂat-topped ‘mesa beam’ instead of a standard Gaussian
beam [134, 135, 136, 137, 138, 139].
For the second generation detectors, such as Advanced LIGO, research has
focussed on reducing the dissipation of the current SiO2 /Ta2 O5 coatings. Studies have indicated that at room temperature the dissipation of these coatings
is dominated by the dissipation of the Ta2 O5 layers, and can be reduced by
up to 40% by doping the Ta2 O5 with TiO2 . However, neither the mechanism
responsible for the dissipation nor the process by which it is reduced by doping with TiO2 is clearly understood. A better understanding of the process
responsible for the dissipation in Ta2 O5 may allow it to be further reduced.
One method of probing the dissipation mechanisms in a material is tomeasure the dissipation as a function of temperature. Many materials exhibit
temperature dependent dissipation peaks, occurring at particular temperaturesassociated with the activation energy of the process responsible for the dissipation. A well-known example of such a material is fused silica, which has alarge dissipation peak at around 40 to 60 K. Studies by several authors haveassociated this peak with energy dissipation arising from thermally activatedtransitions of oxygen atoms between two energy states in the amorphous SiO2
network [93, 221, 222]. The broad nature of this peak has been associated withthe distribution of bond angles in the disordered network of SiO2 molecules[172, 173]. The form of the temperature dependence of the dissipation in fusedsilica is thus strongly associated with the dissipation mechanism. Low temperature dissipation peaks have also been observed in several other amorphousoxides and disordered crystalline materials, including GeO2 and B2 O3 [223].
The ion-beam-sputtered Ta2 O5 used in reﬂective coatings is also an amorphousoxide (see Figre 5.31), and it is therefore of considerable interest to study its
dissipation as a function of temperature.
Cooling the mirror is perhaps the most intuitive method of reducing thermal motion, and generally a direct reduction in thermal noise is expected on

5.1 Introduction

114

lowering the operating temperature of a material. However, as discussed above,mechanical dissipation is often strongly dependent on temperature and fusedsilica, the most commonly used test mass material in current detectors, has alarge, broad dissipation peak centered on approximately 40 to 60 K [93]. Silica
is therefore not a promising material for use in a low temperature gravitational
wave interferometer.
In contrast, other materials such as silicon are known to have low dissipation
at cryogenic temperature [224, 225], making a further reduction in substrate
thermal noise possible by operation at low temperatures. The thermal expansion coeﬃcient and thermal conductivity of silicon also make it attractive foruse as a mirror substrate in a low temperature detector using a high circulating laser power [94]. Currently, a European Commission design study for athird generation gravitational wave detector is underway [98], and this includes
signiﬁcant research into the use of cryogenically cooled silicon mirrors. It istherefore important to study the temperature dependence of the dissipation of
the appropriate mirror coatings to allow the coating thermal noise contribution
to the sensitivity of a low temperature interferometer to be calculated.
The study of coating loss as a function of temperature therefore has two
purposes: ﬁrstly, the performance of the coating materials at low temperatures
must be evaluated for use in possible future cryogenic detectors; secondly, thetemperature dependence of the dissipation of Ta2 O5 may help to understandthe underlying dissipation mechanism.

5.1.1

Measurements of the temperature dependence of
coating dissipation

As in Chapter 4, to ease interpretation, single layers of Ta2 O5 were applied tothin cantilever substrates. Ideally the loss of the substrate used for coating loss
measurements should be as low as possible, to ensure high sensitivity to theloss of the coating material. This precluded the use of silica cantilevers, as the5.1 Introduction

115

low temperature dissipation peak in silica may have made interpretation of theresults more diﬃcult. Thin silicon substrates of order 50 to 100 μm in thickness
were chosen, as the loss of silicon is known to decrease at low temperature[224], thus minimising the contribution of the substrate to the total measured
loss. Silicon is also a particularly interesting substrate material as it is alikely candidate material for the test masses future cryogenic gravitational
wave detectors [94, 95].
Yamamoto et al [226] have recently measured the temperature dependenceof the mechanical dissipation of multi-layer silica-tantala coatings applied tosapphire substrates, with data taken at room temperature (300 K) and at asmall number of temperatures between 80 K and 4 K . The tantala layers ofthese coatings were not doped. These results show a small variation in coating
loss across the entire temperature range, between approximately 2 × 10−4 and4 × 10−4 . Yamamoto suggests that the temperature dependence of the coating
loss is therefore essentially ﬂat: however, with only ﬁve temperature points
measured, this experiment is not conclusive.
In the remainder of this chapter studies of the mechanical loss of various
tantala coatings as a function of temperature are presented. The experimental apparatus is described, and initial measurements of two coating samples
between 290 and 80 K are presented. Measurements below 80 K were initially
carried out by the author at Freidrich-Schiller University in Jena, Germany.
The cryostat in Glasgow was then modiﬁed by the author to allow cooling toliquid helium temperatures and the loss factors of several additional samples
were measured at temperatures between 10 K and 300 K.

116

5.2 Initial measurements of the loss of doped tantalum pentoxide

5.2

Initial measurements of the loss of doped
tantalum pentoxide

5.2.1

Sample preparation

The silicon cantilever substrates were fabricated, by colleagues at Stanford
University, from approximately 0.5 mm thick single-crystal silicon wafers by ahydroxide etching process. Suitable masking during this process allowed one
end of the cantilever to remain un-etched, forming a thicker block by which thecantilever could be clamped, as shown in Figure 5.1. This clamping block was
intended to reduce energy loss into the clamping structure through frictional
processes such as stick-slip loss [174, 225].

34 mm
92 μm

10 mm
500 μmFigure 5.1: Schematic diagram of cantilever sample A, 34 mm long, 10 mm wideand 92μm thick. The cantilever length is parallel to the 110 crystal
axis and the surface is on the (100) plane.

Initial measurements were carried out on a doped tantala coating layer
provided by LMA [215], a vendor involved in the development of coatings
for the Advanced LIGO mirrors. The dissipation in the coating layer can becalculated from measurements of the mechanical loss of the cantilever before
and after the application of the coating, as described in Section 4.3.5. It should
be noted that the coating process involved two heat treatment steps which
could potentially change the loss of the silicon cantilever itself. Firstly, toenable the coating to adhere properly to the silicon surface, it was necessary
to grow a thermal oxide layer on the cantilever surfaces. A dry-oxidation
process was used, in which the cantilevers were heated to 900◦ C in air for ten
hours, resulting in an SiO2 layer (∼ 20 to 30 nm) thick [220]. Secondly, after5.2 Initial measurements of the loss of doped tantalum pentoxide

117

deposition of the coating, the cantilevers were put through a heat treatment
procedure at 600◦ C for 24 hours under atmospheric conditions. Post-deposition
heat treatment is standard practice to reduce absorption losses and to relieve
the internal stress in ion-beam sputtered multi-layer coatings [203].
Initially two cantilevers, known as cantilevers ‘A’ and ‘B’, were coated; theproperties of each cantilever are given in Table 5.1. It was possible to measure
the length of the cantilevers to an accuracy of ± 0.5 mm without contacting
the cantilever surface. The thickness of each cantilever was calculated frommeasurements of its resonant mode frequencies at room temperature using
Equation 4.3. Cantilevers A and B were found to be (92 ± 1) μm and (52 ± 2)
μm thick respectively. This method of calculating the thickness gave excellent
agreement with measurements using a Wyko NT1100 Optical Proﬁler, whichshowed cantilever A to be (92 ± 2) μm thick [159].
The loss of cantilever A had been studied as a function of temperature
by a colleague, prior to the application of the coating [227]. These loss values have been used as the uncoated measurements for this sample. However,both of the heat treatment steps described above may possibly change theloss of a cantilever substrate, so that the previously measured uncoated lossof cantilever A may not be appropriate to use as the substrate loss in thesestudies. While the eﬀect on the calculated coating loss was likely to be small,a second cantilever (cantilever B) with a nominally identical uncoated partner(cantilever C), etched from the same silicon wafer, was also coated. Cantilever
C remained uncoated but was subject to identical thermal oxidation and postdeposition heat treatment. This ensured that in principle the only diﬀerence
between cantilevers B and C was the presence of the coating layer. Previous
measurements have shown the mechanical loss of uncoated cantilevers to beconsistent to within 10 % for samples of the same dimensions etched from thesame wafer [227].
Cantilevers A and B were coated with a 0.5μm thick layer of Ta2 O5 doped

118

5.2 Initial measurements of the loss of doped tantalum pentoxidewith TiO2 to a level of (14.5 ± 1) % , applied by ion-beam sputtering using
argon as the sputtering gas. This coating is of interest as it is currently one of
the lowest loss tantala coatings at room temperature [214] and it was depositedin a coating chamber which is suitable to produce the Advanced LIGO mirror
coatings. During the coating procedure, the cantilevers were masked to preventthe coating extending onto the clamping block. After coating, both cantilevers
were heated at 600◦ C for 24 hours. As noted above, the uncoated partner tocantilever B underwent the same thermal oxidation process and heat treatment
as the coated cantilevers.
Cantilever A

Cantilever B

Cantilever C

Length

34 ±0.54 mm

57 ±0.5 mm

57 ±0.5 mm

Thickness

92 ±1 μm

52 ±1 μm

52 ±1 μm

Dopingp-typen-typen-typeDopantBoron−1

Phosphorous

Phosphorous−1

5-10 Ωcm−1

Resistivity

10-20 Ωcm

5-10 Ωcm

Clamping block thickness

500 μm

350μm

350μm

Coating thickness

500 nm

500 nmuncoated

Table 5.1: Properties of the silicon cantilevers A, B and C.

Figure 5.2: (a) Cantilever A before coating. (b) Cantilever A after coating, clamped
in preparation for mechanical loss measurements.

5.2 Initial measurements of the loss of doped tantalum pentoxide

5.2.2

119

Experimental method

Each silicon cantilever was held horizontally in a stainless steel clamp identical to the clamp used for the room temperature measurements described in
Chapter 4. Figure 5.2 shows cantilever A in the clamp, both before and after coating. The clamp was rigidly attached to the base-plate of the vacuum
chamber of a cryostat, which could be cooled by submersion in liquid nitrogen, as shown in Figure 5.3. As described in Section 4.2.4, the experimental
chamber was evacuated to a pressure of less than 1×10−6 mb to ensure thatgas damping eﬀects were negligible. A resistive heater mounted on the clamp

Figure 5.3: Schematic diagram of the cryostat, with an inset showing details of theclamp used to hold the cantilever samples.

was used to control the temperature of the clamp and cantilever. The temperature was measured using a silicon diode temperature sensor mounted within
the clamp directly below the cantilever. Typically the temperature variation
during measurements was less than ± 0.2 K. After the temperature of the5.2 Initial measurements of the loss of doped tantalum pentoxide120clamp had stabilised at each measurement point, the system was left for approximately ﬁve minutes before measurements commenced, to ensure that thecantilever was in thermal equilibrium with the clamp.
5.2.2.1

Readout system

For the room temperature measurements presented in Chapter 4, it was possible to position the clamped cantilever in the path of a laser beam and tomeasure the displacement of the resulting shadow. This arrangement was notstraightforward to implement in the cryostat, as the optical feed-through pipe
was located directly above the cantilever. Therefore the cantilever motion wassensed using a laser beam reﬂected from the surface of the cantilever and directed onto a photodiode sensor outside the cryostat. The movement of thecantilever displaced the reﬂected laser spot vertically. Due to the long path
length between the cantilever and the photodiode, the motion of the reﬂected
laser spot was typically too large to be sensed with a split-photodiode as usedin Chapter 4. Thus a single rectangular photodiode, masked so that only a thin
triangular strip of the surface was exposed, was used (see Figure 5.4). As the

Figure 5.4: Diagram of the photodiode sensor used to measure the displacement of
the cantilever. The mask was made from electrical insulating tape, andthe maximum width of the exposed photodiode (labelled a) was 1 mm.
The laser spot diameter D was measured to be approximately 5 mm.

121

5.2 Initial measurements of the loss of doped tantalum pentoxide

10 pF

1 kΩ

To photodiode

10 μF
+Vc

-

OP227

+

-Vc
10 μF

Figure 5.5: Circuit diagram for the photodiode circuit used to measure the amplitudeof the cantilever displacement.

cantilever vibrated, the reﬂected laser spot moved up and down, illuminatinga variable area of the photodiode and producing a photocurrent proportional
to the amplitude of motion of the spot. To ensure linearity, the base of thetriangle (labelled ‘a’ in Figure 5.4) was designed to be less than a ﬁfth ofthe diameter of the laser spot.

This readout system has the advantage of asigniﬁcantly larger dynamic range than a split photodiode, allowing the large
movement of the laser spot to be measured. The circuit used to amplify thesignal from the photodiode is shown in Figure 5.5. The photodiode readout
was constructed by S. Reid. A calibration plot and sensitivity curves for thesensor are shown in Figures 5.6 and 5.7 respectively.
The mechanical loss factors of several resonant modes of each cantileverwere measured over a range of temperatures between 85 K and 292 K, with anelectrostatic actuator being used to excite the modes as described in Chapter
4. Due to the long distance travelled by the reﬂected laser light between the
cantilever and the sensor, it was often not possible to excite the fundamental mode of a cantilever to a suitable amplitude above the background noise
without saturating the photodiode readout. The inclusion of a lens to reduce
the range of motion of the reﬂected laser spot was found to reduce this prob-

122

5.2 Initial measurements of the loss of doped tantalum pentoxide

0.08

Photodiode calibration data
Linear fit y = 237.23845 x + 0.00305

Photodiode readout signal (Vrms)

0.06

0.04

0.02

0.00

-0.02

-0.04

-0.06
-3x10

-4

-2x10

-4

-1x10

-4

0

1x10

-4

2x10

-4

3x10

-4

Cantilever displacement angle (radians)

Figure 5.6: Plot of the response of the photodiode readout system used to measure the amplitude of the cantilever displacement.
Readout noise with laser and lights off
Readout noise with laser on and lights off
Readout noise with laser on and lights on

Noise (rad Hz

-1/2

)

10-4

10-5

10-6

10-7

10-8
100

1000

10000

Frequency (Hz)Figure 5.7: Noise plots for the photodiode sensor. The noise level was measured three times:
with no incident laser spot and with the room lights turned oﬀ; with an incident
laser spot and the room lights turned oﬀ and with an incident laser spot withthe room lights switched on.

123

5.2 Initial measurements of the loss of doped tantalum pentoxidelem, but the ringdowns obtained were often noisy, possibly as a consequenceof low frequency background noise in the system. Also, the diameter of the
optical feed-through limited the maximum excitation amplitude which could
be achieved before the reﬂected laser spot was truncated by the walls of thebeam tube.
5.2.2.2

Thermoelastic loss as a function of temperature

Thermoelastic loss (see Section 2.4) is one of the dominant dissipation mechanisms in silicon cantilevers of these dimensions at room temperature. Figure
5.8 shows the thermoelastic loss at 290 K, calculated using Equation 2.40, forcantilevers A (92 μm thick) and B (52 μm thick). The peak thermoelastic loss
occurs at ∼ 16.4 kHz for the (thicker) cantilever A and at ∼ 51.2 kHz for the

Calculated thermoelastic loss

(thinner) cantilever B. Over the frequency range at which measurements were

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

10

Cantilever A (92 μm thick)Cantilever B (52 μm thick)

0

10

1

10

2

10

3

10

4

10

5

10

6

Frequency (Hz)
a1

b1

b2

a3 a4 a5 a6
a2
b3 b4 b5 b6

Figure 5.8: The frequency dependence of the calculated thermoelastic loss for cantilevers A and B (92 μm and 52 μm thick respectively). The arrows
mark the ﬁrst 6 bending frequencies of the samples.

124

5.2 Initial measurements of the loss of doped tantalum pentoxidetaken (< 6 kHz), far from the peak, the level of thermoelastic loss in cantilever
B is approximately a factor of three lower than that in cantilever A.
Thermoelastic loss in silicon is a strong function of the temperature, largely
due to the temperature dependence of the thermal expansion co-eﬃcient which
has two zeros at approximately 125 K and 18 K, leading to nulls in the thermoelastic loss at these temperatures. This is illustrated in Figure 5.9, whichshows the calculated temperature dependence of the thermoelastic loss forthree modes of cantilever B, again using material properties from Touloukian[228]. Thermoelastic loss eﬀects can also arise from the diﬀering thermomechanical properties of the coating and substrate [141, 142]. The magnitudeof this ‘coating thermoelastic loss’ is not signiﬁcant for the Ta2 O5 coatings

Thermoelastic loss

studied here, but is discussed in detail for silica coatings in Chapter 6.
10

-5

10

-6

10

-7

10

-8

10

-9

10

-10

10

-11

10

-12

10

-13

10

-14

10

-15

10

-16

Thermoelastic loss 22 Hz
Thermoelastic loss 383 Hz
Thermoelastic loss 1854 Hz

10

100

300

Temperature (K)Figure 5.9: Calculated thermoelastic loss as a function of temperature for threemodes of silicon cantilever B.

125

5.2 Initial measurements of the loss of doped tantalum pentoxide

5.2.3

Results for cantilever A

5.2.3.1

Room temperature results

The mechanical loss factors of three modes (the second, third and fourth bending modes at 670 Hz, 1.9 kHz and 3.7 kHz respectively) of cantilever A had
been studied as a function of temperature by a colleague [159, 227] before
application of the coating. These measurements were used as the uncoatedcontrol data for cantilever A. Initially, the losses of the coated and uncoatedMechanical loss

cantilevers were compared at room temperature, as shown in Figure 5.10. It
8.0x10-5

7.0x10-5

6.0x10-5

5.0x10-5

4.0x10-5

3.0x10-5

2.0x10-5

1.0x10-5

Coated loss
Min thermoelastic
Max thermoelastic
Uncoated loss

0

1000

2000

3000

4000

5000

Frequency (Hz)Figure 5.10: Plot of room temperature loss measurements of cantilever A before andafter coating with a 500 nm thick ﬁlm of Ta2 O5 doped with TiO2 .

can be seen that the presence of the coating increases the loss of the cantilever,
although at low frequencies the increase in loss of 8 % is relatively small andonly slightly larger than the uncertainty in the measured losses. The previous
data taken at low temperature for the second bending move (670 Hz) showedsigniﬁcant evidence of clamping loss [159], therefore only the third and fourth
bending modes of the coated cantilever were studied here.

126

Mechanical loss

5.2 Initial measurements of the loss of doped tantalum pentoxide

10

-5

10

-6

10

Coated loss 19.5.06
CoatedCoatedloss 15.6.06
loss 19.5.06
CoatedCoatedloss 16.5.06
loss 15.6.06
Coated loss 4.5.06
Coated loss 16.5.06
Coated loss 27.4.06
Coated loss 4.5.06
Uncoated loss
Errorthermoelastic loss
Calculated

-7

50

100

150

200

250

300

Temperature (K)Figure 5.11: Plot of the measured mechanical loss against temperature for the third
bending mode of cantilever A before and after coating.

5.2.3.2

Measured loss as a function of temperature

The mechanical loss factors of the third and fourth bending modes of the coatedcantilever were measured as a function of temperature between 84 K and 290
K. As with all of the measurements in this chapter, the cantilever was allowed
to cool to the lowest attainable temperature before measurements commenced,
with the heater attached to the clamp then being used to heat the sample toa desired temperature. At each temperature at which data were taken the lossof each mode was measured at least three times and an average loss calculated.
The uncertainties presented here in the average measured loss is the standard
error.
Figures 5.11 and 5.12 show the temperature dependence of the measured
loss factors of the third and fourth bending modes, at 1.9 kHz and 3.7 kHz
respectively, both before and after coating. At temperatures above ∼200 K the

127

Mechanical loss

5.2 Initial measurements of the loss of doped tantalum pentoxide

10

-5

10

-6

10

Coated loss 19.5.06
Coated loss 7.5.06
Coated loss 16.5.06
Coated loss 15.6.06
Uncoated loss
Calculated thermoelastic loss

-7

50

100

150

200

250

300

Temperature (K)Figure 5.12: Plot of the measured mechanical loss of the fourth bending mode of
cantilever A, before and after coating.

mechanical loss of the uncoated cantilever is dominated by thermoelastic loss,
while at lower temperatures some other loss mechanism, perhaps associatedwith the surface of the silicon [163, 225] is dominant. The application of
the coating to the sample signiﬁcantly increased the mechanical loss factor by
between a factor of 2 at room temperature and a factor of ∼10 at l20 K. The
losses measured for these modes of the coated sample over several temperature
cycles are shown in Figure 5.12 where it can be seen that the losses in eachrun were reasonably consistent, with small variations probably attributable toslight variations in the levels of clamping loss.

5.2.4

Coating loss results for cantilever A

Using the data shown in Figures 5.11 and 5.12, the mechanical loss of the coating was calculated for both the third and fourth modes using Equation 4.14.

128

5.2 Initial measurements of the loss of doped tantalum pentoxide

The Young’s modulus of tantala was taken to be (140 ± 15) GPa [229]. The
ratio of the energy stored in the coating to the energy stored in the cantileversubstrate was calculated to be Ecoating /Esubstrate = 1.39 × 10−2 using Equation
4.7.
The calculated coating loss for the third and fourth modes is shown in
Figures 5.13 and 5.14 respectively. The coating loss appears to decrease as thetemperature decreases from room temperature to 200 K. At lower temperatures
however, the coating loss shows a steady increase with decreasing temperature.
The large losses at room temperature, in the order of 9×10−4 , are larger by afactor of ∼ 3 than the losses measured for a tantala coating (with the samelevel of doping) applied to a silica cantilever presented in Chapter 4. The
measurements close to room temperature were less repeatable than those atlower temperatures, and showed a larger degree of scatter with no obvious

Mechanical loss

trend. These facts suggest that some form of excess loss associated with the
8.0x10-4

7.0x10-4

6.0x10-4

5.0x10-4

4.0x10-4

3.0x10-4

2.0x10-4

1.0x10-4

15.6.06
27.4.06
4.5.0616.5.06
19.5.06

50

100

150

200

250

300

Temperature (K)Figure 5.13: Plot of the calculated coating loss as a function of temperature for 3rd
mode of cantilever A.

129

Mechanical loss

5.2 Initial measurements of the loss of doped tantalum pentoxide

1.0x10-3

8.0x10-4

6.0x10-4

4.0x10-4

2.0x10-4

19.5.06
15.6.06
16.5.06
7.5.060.0-2.0x10

-4

50

100

150

200

250

300

Temperature (K)Figure 5.14: Plot of the calculated coating loss as a function of temperature for 4th
mode of cantilever A.

clamp might have been present at these temperatures, possibly due to energy
from the cantilever coupling into a nearby resonance of the clamp.
To test this hypothesis, a piezo-electric sensor was used to measure themagnitude of any motion of the clamp. The piezo was attached to the top
clamping block and the signal at the cantilever mode frequency was recordedat the start of each ring-down measurement. To account for variations in
excitation amplitude, the piezo signal was normalised by dividing it by themeasured amplitude of the excited motion of the cantilever immediately after
excitation.
At temperatures above 200 K, where the large and more variable losses were
measured, signiﬁcantly higher levels of clamp motion were detected. Figure5.15 shows both the measured loss of the 1.9 kHz mode over the course of
one temperature run and the amplitude of the clamp motion at the same
frequency. It can be seen that the clamp motion remained relatively constant

130

1x10

-3

1.9kHz bending mode - 19 May 2006
piezo response

Measured coating loss φcoating

8x10

6x10

4x10

2x10

1.4x10-3

1.2x10-3

1.0x10-3

8.0x10-4

6.0x10-4

4.0x10-4

2.0x10

-4

-4

-4

-4

-4

Normalised piezo response (arb. units)

5.2 Initial measurements of the loss of doped tantalum pentoxide

0.0
50

100

150

200

250

300

Temperature (K)Figure 5.15: Plot of the measured coating loss at approximately 1.9 kHz and theamplitude of the piezo signal detected at the start of each ringdown.

below ≈ 225 K, and that the mechanical loss follows quite a smooth trend in
this temperature range. At higher temperatures, both the motion in the clamp
and the measured loss factor increase substantially.
These results suggest that a resonance of the clamping structure may beclose in frequency to this mode of the cantilever, and at temperatures above
225 K, the two resonant frequencies become close enough to allow the motionof the cantilever to excite the mode of the clamp, resulting in excess loss ofenergy from the cantilever. Similar data were obtained for the fourth bending
mode, indicating that it was also aﬀected by energy loss into the clamping
structure at temperatures above ∼ 225 K. Thus it is likely that the calculated
coating loss above 225 K for both modes is higher than the intrinsic loss of thecoating material.

131

5.2 Initial measurements of the loss of doped tantalum pentoxide

5.2.5

Coating loss results for cantilever B

Cantilever B was coated in the same coating run as cantilever A. However
this cantilever had a separate, identical control sample (cantilever C) which
underwent the same oxidation and annealing processes as the coated cantilever.
The properties of these cantilevers are given in Table 5.1. The factor of ≈ 6.7
between the thickness of the cantilever and the clamping block (compared
to a factor of ≈ 5.4 for cantilever A) should enable greater isolation of thecantilever from the clamp. The energy ratio for cantilever B was calculated
to be Ecoating /Esubstrate = 2.46 × 10−2 . The mechanical loss factors of the thefourth and ﬁfth modes, at ∼777 Hz and ∼1280 Hz respectively, of cantilevers
B and C were measured as a function of temperature; the results are shown in
Figures 5.16 and 5.17. For both modes the loss of the uncoated sample follows

Mechanical loss

the trend of the expected thermoelastic loss above 200 K. The loss of the 777

10

-5

10

-6

10

-7

10

-8

Coated loss
Uncoated loss
Calculated thermoelastic loss
50

100

150

200

250

300

Temperature (K)Figure 5.16: Plot of the measured loss for the fourth mode of cantilever B and its
identical uncoated control sample.

132

Mechanical loss

5.2 Initial measurements of the loss of doped tantalum pentoxide

10

-5

10

-6

10

-7

10

-8

Coated loss
Uncoated loss
Calculated thermoelastic loss
50

100

150

200

250

300

Temperature (K)Figure 5.17: Plot of the measured loss for the ﬁfth mode of cantilever B and its
identical uncoated control sample.

Hz mode continues to decrease to a minimum of 3.1 × 10−7 at 120 K, where theloss begins to increase quite steeply. The loss of the mode at 1280 Hz appears
to have a small, broad peak at approximately 120 to 140 K. The piezo sensor
on the upper clamping block did not show any indication of excess loss into
the clamping structure. It is possible, therefore, that this small peak may be
intrinsic to the silicon. Other authors have observed peaks in the loss of siliconin this temperature range [224], and this would be of interest for further study.
The loss of the coated cantilever for both the 777 Hz and 1280 Hz modes wasfound to decrease smoothly with temperature to a minimum at approximately
160 K where the loss begins to increase with decreasing temperature.
Figure 5.18 shows the calculated coating loss for both of the modes. The
coating loss was found to increase as the temperature decreases, rising froma level of approximately 1.5× 10−4 at 290 K to 3.8× 10−4 at 85 K. Thereis evidence of a plateau in the loss between 175 and 225 K, although more

133

5.2 Initial measurements of the loss of doped tantalum pentoxide

5.0x10-4

Mechanical loss

Coating loss ~777 Hz mode
Coating loss ~1280 Hz mode
4.0x10-4

3.0x10-4

2.0x10-4

1.0x10

-4

0.0
50

100

150

200

250

300

Temperature (K)Figure 5.18: Plot of the coating loss for cantilever B at 777 Hz and 1280 Hz.

measurements would be required to examine this feature more thoroughly.
Below 175 K, the slope of the loss curve increases. The coating loss results forboth modes studied agree to within experimental error at every point except
at room temperature. However, below 150 K, the loss of the 1280 Hz mode
appears to increase more rapidly than that of the 777 Hz mode, suggestingthat the coating loss may have some frequency dependence.
The loss measurements for cantilever B were repeatable to within 5 % over
several measurement runs and did not show evidence of energy coupling to theclamp, unlike the results for cantilever A. This is possibly due to cantilever
B being better isolated from the clamp due to the larger clamping block tocantilever thickness ratio. It is also possible that the mode frequencies of cantilever B were further away from the resonances of the clamp, making coupling
between the sample and the support structure less likely. For ease of comparison, the lowest coating losses found at each temperature for cantilever A are
shown in Figure 5.19. Generally, the losses measured for cantilever A were134

5.2 Initial measurements of the loss of doped tantalum pentoxide

slightly higher than those measured in cantilever B. Several anomalously low
losses measured in cantilever A (e.g. at 87 and 160 K) can be attributed toexcess clamp loss in the data measured prior to application of the coating (seeMechanical loss

Figure 5.12).

8.0x10-4

6.0x10-4

4.0x10-4

2.0x10-4

Coating loss cantilever A ~1921 Hz mode
Coating loss cantilever A ~3768 Hz mode

0.0
50

100

150

200

250

300

Temperature (K)Figure 5.19: Plot of the coating loss for cantilever A at 1920 Hz and 3768 Hz, calculated using the lowest loss factors measured for the coated cantileverat each temperature.

5.2.6

Conclusion

These initial loss measurements demonstrated that it was possible to use thetechnique described here to measure the temperature dependence of the mechanical loss of a coating. The loss of the ion-beam sputtered Ta2 O5 ﬁlm dopedwith 14.5 % TiO2 was observed to increase signiﬁcantly as its temperature was
reduced from room temperature to 84 K. Since the loss of the Ta2 O5 layers
is known to be the main source of loss in SiO2 /Ta2 O5 multi-layer coatings atroom temperature, this result is potentially of signiﬁcance for the operation of5.3 Dissipation of a TiO2 -doped Ta2 O5 coating between 7 K and 300 K

135

gravitational wave interferometers at cryogenic temperatures.
If the coating loss continues to increase below 84 K, then it is possible thatthe coating thermal noise may actually be increased by operating a gravitational wave detector at these temperatures. It was therefore clearly of interest
to continue these measurements at lower temperatures. In addition, the results of similar studies of a silica coating are presented in Chapter 6, enablinga calculation of the expected loss in a multi-layer SiO2 /Ta2 O5 coating fromthe measured loss of the individual coating materials.
Initial measurements of the loss of single-layer coatings at temperaturesapproaching liquid helium temperature were carried out during a visit to collaborators at Jena University, and are detailed in Section 5.3. Concurrent
modiﬁcations (described in Section 5.6.1) were made to the cryostat experiment
in Glasgow to allow cooling with liquid helium, after which several undoped
Ta2 O5 coatings were studied as described in Sections 5.6 and 5.7.

5.3

Dissipation of a TiO2-doped Ta2O5 coating
between 7 K and 300 K

5.3.1

Sample preparation

The mechanical loss of a tantala coating of identical doping to that described in
the previous section was measured between 7 and 300 K during a visit to Jena
University. The previous pair of samples could not be used because cantileverC (the uncoated control sample) was found to have cracked in the clamp during
the ﬁnal measurement run. While only the clamping block was damaged, it
was no longer possible to clamp the cantilever properly or repeatably, and noother control sample for cantilever B was available.
Therefore a new pair of cantilevers, of approximately the same thickness
and with the same clamping block dimensions as cantilever B, were used. These
cantilevers were fabricated from n-type, antimony-doped single-crystal silicon

136

5.3 Dissipation of a TiO2 -doped Ta2 O5 coating between 7 K and 300 K

Cryostat
45° mirror

Clampedcantilever

Commercial
interferometer

Electrostatic
drive

Figure 5.20: Schematic diagram of the cryostat used to measure the doped tantala
coating at Jena University, showing the laser interferometer readout
system used.

of resistivity 0.005-0.25 Ωcm and were (34 ± 0.5) mm long. The thickness,again calculated from the measured mode frequencies of the uncoated cantilever using Equation 4.3, was found to be (47.7 ± 0.5) μm.
Subsequent to the previous coating run, LMA had developed the Ta2 O5
coating process by changing a commercially sensitive (and therefore undisclosed) deposition parameter. The new variant of coating, which LMA havefound to have a lower mechanical loss at room temperature [220], is known as
Formula 5∗∗ . The TiO2 doping level remained unchanged at a cation concentration of 14.5 %. A loss factor of (1.5 ± 0.1)×10−4 was measured for Formula
5∗∗ tantala applied to a silica cantilever, as presented in Chapter 4.
As before, two identical cantilevers were sent to LMA. A thermal oxide
layer approximately 20-30 nm thick was grown on both samples, before one
was coated with 500 nm of Formula 5∗∗ Ta2 O5 by ion-beam sputtering. Boththe coated cantilever and the control sample were then heat treated at 600◦ C
for twenty-four hours.

137

5.4 Results

5.3.2

Experimental method

The cryostat used in Jena University is shown in Figure 5.20. In place of thelaser and photodiode sensor described in Section 5.2.2.1, a commercial laser interferometer (SIOS model SPS-120/500) was used to measure the displacementof the cantilever, with the laser beam reﬂected from one surface of the sample. Due to the design of the cryostat, with a short optical path between the
cantilever and the interferometer, it was possible to measure the fundamental
mode of the cantilevers.

5.4

Results

The measured mechanical loss factors for the ﬁrst ﬁve bending modes of boththe sample coated with doped tantala and the uncoated control sample areshown in Figures 5.21 to 5.25. Each point represents the average loss calculated
from at least three ring-down measurements and typically has an experimental
error of less than 3 %. Error bars have been omitted for clarity.
The most striking feature of this plot is a clearly deﬁned peak in the dissipation of the coated sample centred at approximately 20 K. This peak ispresent for all of the modes studied, suggesting that it is unlikely to be related
to any frequency dependent energy loss into the clamp. This was conﬁrmed
through the use of a piezo-electric sensor attached to the clamp, as described in
5.2.4, which showed no evidence of energy coupling into the clamping structureassociated with the dissipation peak.
The ﬁrst two modes both show some evidence of a second, smaller peak inthe dissipation at approximately 90 K (see Figures 5.21 and 5.22). However,there is no evidence of this feature in the data for the third and fourth modes,
which show the loss increasing smoothly with decreasing temperature.
The measured dissipation of the ﬁfth mode appears to be anomalously highand to have apparent structure above 100 K. Unfortunately few measurements

138

5.4 Results-5

3.0 x 10

1st mode coated
1st mode uncoated-5

2.5 x 10-5

Mechanical loss

2.0 x 10-5

1.5 x 10-5

1.0 x 10-6

5.0 x 10

0

50

100

150

200

250

300

Temperature (K)Figure 5.21: Plot showing the temperature dependence of the measured mechanicalloss of the ﬁrst mode (∼ 55 Hz) of the coated and uncoated cantilevers.
-5

3.0 x 10

2nd mode coated
2nd mode uncoated-5

2.5 x 10

Mechanical loss

-5

2.0 x 10-5

1.5 x 10-5

1.0 x 10-6

5.0 x 10

0

50

100

150

200

250

300

Temperature (K)Figure 5.22: Temperature dependence of the measured mechanical loss of the second
mode (∼ 350 Hz) of the coated and uncoated cantilevers.

139

Mechanical loss

5.4 Results

3.5 x10-5

3.0 x10-5

2.5 x10-52.0 x10-5

1.5 x10-5

1.0 x10-55.0 x10-6

3rd mode coated
3rd mode uncoated

0

50

100

150

200

250

300

Temperature (K)Figure 5.23: Temperature dependence of the measured mechanical loss of the third

Mechanical loss

mode (∼ 989 Hz) of the coated and uncoated cantilevers.
3.5 x10-5

3.0 x10-5

2.5 x10-52.0 x10-5

1.5 x10-5

1.0 x10-55.0 x10-6

4th mode coated
4th mode uncoated

0

50

100

150

200

250

300

Temperature (K)Figure 5.24: Temperature dependence of the measured mechanical loss of the fourthmode (∼ 1939 Hz) of the coated and uncoated cantilevers.

140

Mechanical loss

5.4 Results

4.0 x10-5

3.5 x10-5

3.0 x10-5

2.5 x10-52.0 x10-5

1.5 x10-5

1.0 x10-55.0 x10-6

5th mode coated
5th mode uncoated

0

50

100

150

200

250

300

Temperature (K)Figure 5.25: Temperature dependence of the measured mechanical loss of the ﬁfth
mode (∼ 3184 Hz) of the coated and uncoated cantilevers.

of the clamp motion using the piezo sensor were obtained for this mode below
150 K, although excess motion of the clamp at this frequency was detected at
higher temperatures. It is therefore likely that the anomalously high dissipation measured close to room temperature was due to energy coupling into theclamp. However, the possibility that the high loss measured at 90 K - 100 K
may be intrinsic to the sample cannot be ruled out.
Each graph also shows the measured loss of the uncoated cantilever. For
each mode, the loss of the uncoated sample decreases to a plateau between
approximately 175 K and 110 K, below which the loss rises to a relatively
small broad peak at ∼ 25 K. Above 200 K, the loss of the second, third andﬁfth modes of the uncoated sample follow the trend of the thermoelastic loss.
The loss of the ﬁrst and fourth modes, however, do not follow the expected
thermoelastic trend, with the loss of the fourth mode in particular being signiﬁcantly higher and more scattered than would be expected. Indeed, the5.5 Analysis and Interpretation

141

measured losses were actually larger than those measured for the same modeof the coated sample. The piezo sensor clearly showed evidence of excess clamp
motion during ring-down measurements of this mode of the uncoated cantileverat temperatures above 200 K, suggesting that the high losses measured wereexplained by energy loss into the clamp.
It should be noted again that the peak in the dissipation of the coatedsample at 20 K is well deﬁned for all of the modes measured, and that noevidence of energy coupling to the clamping structure was observed aroundthis peak.

5.4.1

Calculation of the coating loss

The mechanical loss of the doped tantala layer was calculated for each of
the ﬁrst ﬁve modes of the cantilever using Equation 4.14. Due to its low
mechanical loss the ring-downs of the uncoated cantilever took a considerable
time, resulting in data being taken at fewer temperatures than for the coatedcantilever. At temperatures where the uncoated loss had not been measured,interpolated values were used.
As shown in Figure 5.26, the coating loss was generally found to steadily
increase with decreasing temperature, from approximately 2 × 10−4 at 292 K
to a peak of approximately 1×10−3 , at ∼ 20 K. The two exceptions to thegeneral trend occur for the fourth and ﬁfth modes at temperatures above 75
K. At these temperatures the calculated coating loss has been aﬀected by theclamping loss eﬀects observed for the fourth and ﬁfth modes, as discussed in
Section 5.4. The level of loss measured above 77 K is broadly consistent withthat measured in the other, similar coatings presented in Section 5.2.

5.5

Analysis and Interpretation

A dissipation peak at a particular temperature usually arises from a speciﬁcloss mechanism within the material. If it is assumed that the dissipation has

142

5.5 Analysis and Interpretation

1.2x10-3
st

4 modend

5 mode1 mode2 modeMechanical loss of Ta 2 O5 layer

1.0x108.0x106.0x104.0x102.0x10

th
th

rd

-3

3 mode

-4

-4

-4

-4

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 5.26: Temperature dependence of the dissipation of the doped Ta2 O5 coating.

the form of a Debye peak, as is the case for most dissipation mechanisms [110],then the loss can be expressed as:φ(ω) = Δωτ
1 + (ωτ )2(5.1)

where Δ is a constant related to the magnitude of the dissipation[110]. For
clarity, the loss of each of the modes at temperatures around the peak is shownin more detail in Figure 5.27. The temperature at the point of maximum dissipation can be seen to increase with the mode frequency. This is characteristicof dissipation arising from a thermally activated relaxation process, in which
the relaxation time τ is related to the activation energy Ea of the process by
the Arrhenius equation:
τ −1 = τ0−1 e−Ea /kB T ,(5.2)

where τ0 is the relaxation constant of the dissipation mechanism and kB is
Boltzmann’s constant [93]. From Equation 5.1 it is clear that at the dissipation

143

Mechanical loss

5.5 Analysis and Interpretation

Temperature (K)Figure 5.27: The mechanical loss peak measured in the doped Ta2 O5 coating. For
clarity, error bars are only shown for the ﬁfth mode: these are typicalof the errors in all of the points.

peak, ωτ = 1 and thus Equation 5.2 can be written as:ω = τ0−1 e−Ea /kB Tpeak ,(5.3)

and thus :ln ω = ln τ0−1 −Ea.
kB Tpeak(5.4)

Therefore a plot of the natural logarithm of the angular frequency of eachmode against the reciprocal of the peak temperature (known as an Arrhenius
plot) should be a straight line of slope Ea /kB .
To determine the temperature of each peak accurately, a fourth order polynomial function was ﬁtted to each data set in the immediate vicinity of thepeak (approximately between 12 and 25 K), and the position of the peak wasfound by calculating the roots of the derivative of the ﬁt function. The ﬁts used
are shown in Figure 5.28. The temperature at which the dissipation peaks canbe seen to increase with increasing frequency. With the exception of the third144

5.5 Analysis and Interpretation

1.0x10Mechanical loss of coating

9.0x108.0x107.0x106.0x105.0x10-3

55 Hz mode
350 Hz mode
990 Hz mode
1920 Hz mode
3300 Hz mode

-4

-4

-4

-4

-4

12

14

16

18

20

22

24

26

Temperature (K)Figure 5.28: The ﬁts used to determine the position of the mechanical loss peak inthe doped Ta2 O5 coating. Note the temperature at which the peak occurs
increases with increasing mode frequency.
Mode

Frequency

Tpeak

φpeak

1

55 Hz

17.81K

7.9×10−4

2355 Hz18.91 K

8.3×10−4

3989 Hz19.78 K

8.6×10−441920 Hz20.72 K

8.5×10−453175 Hz21.33 K

9.4×10−4

Table 5.2: The temperature at which the dissipation peak occurs and the peak magnitude of the dissipation for each mode of the cantilever coated with doped
Ta2 O5 measured in Jena.

mode at ∼ 990 Hz, the magnitude of the loss peak can also be seen to increaseat higher frequencies. The third mode displayed the highest peak loss of all(although in the ﬁtted curve this is not apparent), possibly due to the eﬀectof clamping loss at this frequency. Table 5.2 shows both the temperature and145

5.5 Analysis and Interpretation

magnitude of the dissipation peak obtained from the ﬁt for each mode. The
Arrhenius plot of this data, which is shown in Figure 5.29, yields a straight linewith a ﬁtted gradient of 430.6 and intercept of ln(ω) = 30.3. Comparison with
Equation 5.4 allows an activation energy of (39.5 ± 2.7) meV and a relaxation
constant of (1.7 ± 0.1) ×10−14 s to be calculated.
Following the analysis of Cannelli et al [230], the values of Ea and τ0−1
can be substituted into Equations 5.2 and 5.1, which can be used to calculate
the theoretical shape of the mechanical loss peak, as shown in Figure 5.30.
The theoretical peak is substantially narrower than the observed experimental
peak. This is characteristic of strongly interacting or disordered systems inwhich there is a range of values of Ea [230]. As the Ta2 O5 coating is amorphous(see Figure 5.31), it is possible that a distribution of values of Ea could arise
from the disordered structure of the material. The calculated activation energyof 39.5 ± 2.7 meV would therefore correspond to the average activation energy
11Linear Fit y = -430.6 x + 30.3
10ln(ω0)

9

8

7

6

5
0.046

0.048

0.050

0.052

0.054

0.056

0.058

1/Tpeak

Figure 5.29: Arrhenius plot, showing a linear ﬁt of ln(ω0 ) against 1/Tpeak for thedissipation peak.

146

5.5 Analysis and Interpretationin this distribution.

1.0

Normalised coating loss

0.8

0.6

0.4
rd

3 mode measuredth
5 mode measuredrd
3 mode modelth
5 mode model

0.2

0.0
5

10

15

20

25

30

35

Temperature (K)Figure 5.30: Comparison of the measured dissipation peak to the theoretical peakobtained from the Arrhenius ﬁt.

5.5.1

Possible interpretation

It is interesting to note that the activation energy of the well-known low temperature dissipation peak in fused silica, also an amorphous oxide material,has been measured to be 44 meV using a method similar to the one describedabove [172]. However, the dissipation peak in silica is signiﬁcantly wider than
the simple Debye peak predicted to arise from a single activation energy of
44 meV (see Equation 5.1), implying that there is a distribution of activationenergies. The activation energy of 44 meV extracted from the Arrhenius-type
analysis therefore actually corresponds to the most probable activation energy
in this distribution.
Fine et al ﬁrst suggested that the spectrum of activation energies, and thus
the width of the dissipation peak, might arise from the disorder in the structure

147

5.5 Analysis and Interpretation

(a)

(b)

Figure 5.31: (a) Bright ﬁeld TEM image of a tantala layer (dark) and a silica layer(light) in a SiO2 / Ta2 O5 multi-layer coating, where the Ta2 O5 layers
are doped with 8% TiO2 (b) convergent beam electron diﬀraction froma Ta2 O5 layer showing just diﬀuse rings of intensity, conﬁrming that
this layer is amorphous. Both images were taken by Ian MacLaren of
the solid state physics group at Glasgow University.

of fused silica [231]. Over the following two decades, various models for the
speciﬁc dissipation mechanism were proposed. Anderson and Bommel [172]suggested that the dissipation arose from certain oxygen atoms (the ‘bridge’
oxygen atoms, linking together ring-like formations in the silica) moving between two possible bond angles. The width of the dissipation peak is attributed
to the distribution of Si-O bond angles in the disordered silica network.
Strakna proposed a similar model in which certain Si-O-Si bonds are naturally elongated [222]. If the two end atoms in a linear chain of three atoms
are suﬃciently far apart, the middle atom will have two possible equilibrium
positions separated by an energy barrier [169]. The loss arises from vibrationsof the oxygen atom between these two potential minima positions in the elongated bond. The width of peak is again related to the wide distribution ofbond angles and bond lengths in the disordered silica structure.
Vukcevich [232] proposed a third model, in which the low temperature dissipation occurs when neighbouring SiO4 tetrahedral structures rotate between

5.5 Analysis and Interpretation

148

two stable states. The activation energy of this process depends on the microscopic environment of individual tetrahedra, again resulting in a spectrum ofactivation energies and a correspondingly wide dissipation peak.
In all of the above models, changes in Si-O-Si bond angles are postulated
to be responsible for the low temperature dissipation peak in silica. The threemodels diﬀer only in describing which type of motion occurs and the nature of
the associated potential barrier which must be overcome. In each model, theatom, or group of atoms, involved in the dissipation process sits in a doublewell potential: that is, there are two possible positions which can be occupied
with almost equal probability. Energy dissipation can occur if an atom gains
enough energy (the activation energy) to surmount the potential barrier which
separates the two equilibrium positions.
The theory of dissipation in silica was further developed when Gilroy andPhillips [233] pointed out that for a dissipation peak to occur, the double-wellpotential had to be asymmetric. Although most previous analysis had assumed the potential well was symmetric, the asymmetric double-well potential(ADWP) was thought to be a more realistic physical model, and had alreadybeen used to explain various types of anomalous low temperature behaviourin amorphous solids, including silica [234, 235]. The ADWP model has since
been corroborated by low frequency light scattering measurements [173].
The average activation energy of 39.5 ± 2.7 meV obtained here for thedissipation mechanism in doped tantala is of a similar order to the activation energy of the dissipation mechanism in fused silica. Furthermore, several
other amorphous materials are known to exhibit dissipation peaks similar tothe peak in silica [223, 236]. Therefore, one possible dissipation mechanism
responsible for the peak observed in tantala may be thermally activated transitions of oxygen atoms between two states in a double well potential. As insilica, it is possible that these transitions may involve bond angles ﬂipping
between energetically stable states, or perhaps rotational movement of micro-

5.6 Undoped tantala loss measurements

149

scopic structures consisting of a small number of atoms. The wide distributionof activation energies, implied by the width of the dissipation peak (see Figure5.30), suggests that there is a distribution of potential barrier heights, possibly related to the distribution of Ta-O bond angles in the amorphous network.
However further studies are required to gain a better understanding of thedissipation mechanism.

5.6

Undoped tantala loss measurements

Doping ion-beam sputtered Ta2 O5 with TiO2 has been shown to signiﬁcantly
reduce the mechanical loss at room temperature [214]. Following the discovery of the low temperature dissipation peak in TiO2 -doped Ta2 O5 , subsequentstudies of an undoped Ta2 O5 coating were carried out in an attempt to determine what eﬀect, if any, the presence of doping has on the characteristics ofthe dissipation peak.

5.6.1

Experimental setup for cooling with liquid helium

Before cooling the cryostat with liquid helium, several modiﬁcations were madeto minimise unwanted heating and allow temperatures close to 4 K to be attained. Initially, when cooling with liquid nitrogen, the lowest clamp temperature obtained was ∼ 84 K. One possible source of heating is thermal radiationfrom warmer areas of the cryostat, especially the top parts of the three feedthrough pipes, which were always close to room temperature. To reduce thiseﬀect, a narrow band optical ﬁlter (transparent at 633 nm to allow the laser
beam to be transmitted) was attached to the top-plate of the experimental
chamber beneath the optical read-out port (see Figure 5.32), thus preventing
the cantilever from ‘seeing’ room temperature. Similarly, the lower ends ofthe electrical and vacuum feed-through pipes were screened with thin copper
plates thermally linked to the cold top-plate of the vacuum chamber. These

150

5.6 Undoped tantala loss measurements

Optical filter transparent at 633 nm

Clamped silicon cantilever

(b)

(a)

Copper radiation shielding

Copper heat sinks for wires

Figure 5.32: The addition of radiation shielding across the feed-through ports in theroof of the experimental chamber (a) and thermally anchoring wires tothe base-plate (b) allowed the clamp to cool to 78 K, 6 degrees lower
than the previous lowest temperature obtained.

radiation shields allowed the clamp to cool to 80 K, 4 K lower than the previous
minimum temperature.
The wiring in the cryostat was initially anchored onto the back of the clamp,
providing a direct conduction path between the sample and the ends of thewires at room temperature. Furthermore, to allow better temperature control
the clamp was thermally insulated from the base-plate by Macor (a machinableceramic) spacers. To prevent heating through the wires, the anchor point wasremoved from the clamp and the lower stages of the wiring replaced with verythin enamelled copper wire, which was wound around copper bobbins bolted
to the base plate and covered in thermally conductive cryogenic varnish. This
heat sinking method was used for the wires connecting the temperature sensors
and the resistive heater, but was not thought necessary for the high voltage
wires, which did not touch the clamp. On cooling with liquid nitrogen after
these wiring changes, the base plate of the experimental chamber reached 77.5
K while the clamp cooled to 78 K.

5.6 Undoped tantala loss measurements

5.6.2

151

Loss measurements of undoped tantala

In this section, measurements of the temperature dependence of the dissipation of undoped tantala are presented, and compared to the results for doped
tantala in Section 5.3. An identical silicon cantilever to the pair used in theprevious section was coated with a 0.5 μm thick layer of undoped Ta2 O5 , andheated to 600◦ C for 24 hours. The losses of the 2nd, 3rd, 4th and 5th bending
modes were measured between 11 and 300 K and the coating loss calculatedusing the uncoated control data from Section 5.4. The calculated coating loss
factors of each mode are shown in Figures 5.33 to 5.39.
A peak in the dissipation was found to occur at almost identical temperatures to the peak in the titania doped coating. However, for three of thefour modes studied, the peak dissipation was larger for the undoped coating.
For the third mode, the peak dissipation was approximately 1.1×10−3 for bothcoatings. It is interesting to note that for this mode the peak dissipation in thetitania doped coating was abnormally high in comparison to the other modes,
perhaps suggesting that the true dissipation was lower than that measured.
The second bending mode of the cantilever was measured only between
11 and 23 K. At higher temperatures this mode proved diﬃcult to measure
because the frequency of the ﬁrst torsional mode became very close to thefrequency of the second bending mode. While no evidence of direct energy
coupling between the two modes was observed, the torsional mode was often
excited in parallel with the bending mode, perhaps as a result of some asymmetry in the electrostatic drive plate. The resulting torsional vibration of thecantilever moved the laser spot from side to side on the photodiode readout,
making it very diﬃcult to obtain a clean ringdown of the amplitude of motionof the bending mode. This eﬀect is likely to be responsible for the large amountof scatter observed in the measurements of the loss of the second bending mode.
The data for the second mode of the cantilever coated with TiO2 -doped Ta2 O5
in Section 5.3 showed no evidence of this problem. It is possible that, due to152

5.6 Undoped tantala loss measurements

small diﬀerences in the geometry of the cantilever, the thickness of the coating or the rigidity of the clamp, the frequency of the torsional mode was more
widely separated from the frequency of the second bending mode. However, themost likely explanation is the longer optical path between the cantilever andthe sensor in the Glasgow cryostat eﬀectively ampliﬁed the apparent torsionalmotion of the cantilever as sensed at the photodiode.

1.4 x10

Mechanical loss

1.2 x10-3

Undoped tantala ~1000 Hz mode
Doped tantala ~ 1000 Hz mode-3

1.0 x10-3

8.0 x10-4

6.0 x10-4

4.0 x10-4

2.0 x10

-4

0

50

100

150

200

250

300

Temperature (K)Figure 5.33: Comparison of the loss of an undoped Ta2 O5 and a Ta2 O5 coating dopedwith 14.5 % TiO2 at ∼ 1000 Hz.

153

5.6 Undoped tantala loss measurements

1.2 x10-3

Mechanical loss

Undoped tantala ~1000 Hz mode
Doped tantala ~ 1000 Hz mode
1.0 x10-3

8.0 x10-4

6.0 x10-4

4.0 x10

-4

5

10

15

20

25

30

35

40

45

Temperature (K)Figure 5.34: Comparison of the loss peak for an undoped Ta2 O5 and a Ta2 O5 coating
doped with 14.5 % TiO2 at ∼ 1000 Hz.

Mechanical loss

1.4 x10-3

1.2 x10-3

1.0 x10-3

8.0 x10-4

6.0 x10-4

4.0 x10-4

2.0 x10-4

Undoped tantala ~1900 Hz mode
Doped tantala ~1900 Hz mode

0

50

100

150

200

250

300

Temperature (K)Figure 5.35: Comparison of the loss of an undoped Ta2 O5 and a Ta2 O5 coating dopedwith 14.5 % TiO2 at ∼ 1900 Hz.

154

5.6 Undoped tantala loss measurements

1.2 x10-3

Undoped tantala ~1900 Hz mode

Mechanical loss

Doped tantala ~ 1900 Hz mode
1.0 x10-3

8.0 x10-4

6.0 x10-4

4.0 x10

-4

5

10

15

20

25

30

35

40

45

Temperature (K)Figure 5.36: Comparison of the loss peak for an undoped Ta2 O5 with a Ta2 O5 coating
doped with 14.5 % TiO2 at ∼ 1900 Hz.
1.2 x10-3

Mechanical loss

Undoped tantala ~ 3200 Hz mode
Doped tantala ~ 3200 Hz mode
1.0 x10-3

8.0 x10-4

6.0 x10-4

4.0 x10

-4

5

10

15

20

25

30

35

40

45

Temperature (K)Figure 5.37: Comparison of the loss peak for an undoped Ta2 O5 with a Ta2 O5 coating
doped with 14.5 % TiO2 at ∼ 1900 Hz.

155

Mechanical loss

5.6 Undoped tantala loss measurements

1.2x10-3

1.0x10-3

8.0x10-4

6.0x10-4

4.0x10-4

2.0x10-4

Undoped tantala ~3200 Hz mode
Doped tantala ~3200 Hz mode

0

50

100

150

200

250

300

Temperature (K)Figure 5.38: Comparison of the loss of an undoped Ta2 O5 and a Ta2 O5 coating dopedwith 14.5 % TiO2 at ∼ 3200 Hz.
1.2 x10-3

Undoped tantala ~355 Hz mode

Mechanical loss

Doped tantala ~ 355 Hz mode
1.0 x10-3

8.0 x10-4

6.0 x10-4

4.0 x10

-4

5

10

15

20

25

30

35

40

45

Temperature (K)Figure 5.39: Comparison of the loss peak for an undoped Ta2 O5 with a Ta2 O5 coating
doped with 14.5 % TiO2 at ∼ 355 Hz.

156

5.6 Undoped tantala loss measurements

5.6.3

Arrhenius plot analysis

Polynomial ﬁtting was again used to enable the temperature at which the peakoccurs to be estimated. The polynomial ﬁts used are shown in Figures 5.40
and 5.41. The data show considerably more scatter than the previous data forthe doped coating, and the ﬁts are therefore correspondingly poorer. As withthe doped coating (see Section 5.5) the third bending mode proved diﬃcult
to ﬁt, seeming to have a diﬀerent shape with a sharper peak than the othermodes.
1.2x10Mechanical loss of coating

1.1x101.0x109.0x108.0x107.0x106.0x105.0x104.0x10-3355 Hz mode
994 Hz mode

-3

-3

-4

-4

-4

-4

-4

-4

14

16

18

20

22

24

26

Temperature (K)Figure 5.40: The polynomial ﬁts used to determine the position of the dissipation
peak for the second and third bending modes.

An Arrhenius plot for this peak is shown in Figure 5.42, from which anactivation energy Ea = (28.6 ± 1.2) meV and a rate factor of τ0 = (5.9 ± 0.2)
×10−12 s were calculated. Also shown is the data for the doped coating, forwhich the activation energy and relaxation constant were (39.5 ± 2.7) meVand (1.7 ± 0.1) ×10−14 s respectively. The results suggest that doping Ta2 O5
with TiO2 increases the average value of the activation energy of the process

157

5.6 Undoped tantala loss measurements

1.1x10Mechanical loss of coating

1.0x109.0x108.0x107.0x106.0x105.0x10-31948 Hz
3191 Hz

-3

-4

-4

-4

-4

-4

16

18

20

22

24

26

28

30

Temperature (K)Figure 5.41: The polynomial ﬁts used to determine the position of the dissipation
peak for the fourth and ﬁfth bending modes.

responsible for the dissipation by approximately 33 % and decreases the rate
factor by three orders of magnitude.
The peal temperature for the fundamental mode (1/Tpeak = 0.0625 K−1 )of the undoped coating was obtained from measurements taken by Eleanor
Chalkley [237] using a second cryostat with a shadow sensor readout similar
to that described in Section 4.2.2. This arrangement allowed the fundamental mode of the cantilever to be measured without the saturation problems
observed previously.
While the losses of this coating show signiﬁcantly more scatter than thosefor the doped coating, it is possible to be conﬁdent of the position of thepeak to within approximately 0.2 K for three of the ﬁve modes studied. It is
unlikely that the data for the second mode could be improved using the current
cryostat, as the proximity of the torsional mode at ∼ 365 Hz makes obtaining
a clean ringdown challenging.

158

5.6 Undoped tantala loss measurements

12
TiO2-doped tantala
Un-doped tantala
Linear fit for doped data
Linear fit for un-doped dataln(ω0 )

10

8

6

4

0.045

0.050

0.055
1/Tpeak

0.060

0.065

Figure 5.42: Comparison of the Arrhenius plots of ln(ω0 ) against 1/Tpeak for thedissipation peaks in TiO2 doped Ta2 O5 and undoped Ta2 O5 .

159

5.6 Undoped tantala loss measurements

5.6.4

Analysis of the distribution of potential battier
heights

As noted previously, mechanical dissipation peaks occurring at temperatures
above 10 K in many amorphous solids [238], such as fused silica, are thought
to arise from thermally activated transitions of atoms or molecules between
stable orientations. These stable orientations can be represented by a double
well potential, with two minimum energy states separated by a potential energy
barrier of height V [172]. The amorphous nature of these materials results ina wide distribution of potential barrier heights, explaining the broad nature of
the dissipation peaks observed in these materials. Initial models were basedon the assumption that the double-well potentials were symmetric, with thepotential barrier separating states of equal potential energy [172, 222, 232, 236].
Gilroy and Phillips [233] pointed out several drawbacks of this model. Firstly,
for a peak in dissipation to occur, a low energy cut-oﬀ in the distribution of
barrier heights must be included in the model. Secondly, the symmetric model

Potential energy

fails to explain the observed linear scaling of the peak loss in fused silica withV
Δ

Figure 5.43: A schematic diagram of an asymmetric double well potential, with apotential barrier V and an asymmetry Δ.

160

5.6 Undoped tantala loss measurements

the measurement frequency. Thirdly, Gilroy and Phillips suggest that the samestructural variations which give rise to the distribution of potential barrierheights would also be expected to produce asymmetric potential wells.
Figure 5.43 shows a model of an asymmetric double-well potential (ADWP),
with a barrier height V and an asymmetry Δ in the energies of the wells. In an
amorphous solid there will be a distribution of potential barrier heights, g(V ),and also a distribution of the asymmetries f (Δ) in the energies of adjacentpotential wells. Gilroy and Phillips showed that the dissipation predicted bythis model is given by the following double integral over Δ and V [233]:
γ2
φ=kB T Cii

 ∞ ∞
0

0

ωτ
sech2
1 + (ωτ )2



Δ
2kB T


f (Δ)g(V )dΔdV,(5.5)

where ω is the angular frequency of measurement, Cii is the appropriate elastic constant, γ is the elastic coupling constant which represents the coupling
between the defect (e.g. the atom re-orienting within the ADWP) and theapplied strain. The relaxation time τ associated with a barrier height V isgiven by the Arrhenius equation (Equation 5.2).
Starting with this integral, and assuming that g(V ) is independent of temperature and that f (Δ) = f0 = constant [233], Topp and Cahill [223] derive
the following expression relating the mechanical loss to the function g(V ):φ=πγ 2 f0
kB T g(V ),
Cii

(5.6)

1
).
ωτ0

(5.7)

where
V = kB T ln(

Thus the distribution of barrier heights, g(V ), can be calculated from themeasured temperature dependence of the mechanical loss. This calculation
was carried out for the doped and undoped tantala loss data measured at∼1900 Hz, using the values of τ0 obtained from the Arrhenius ﬁts: τ0 = 5.9
×10−12 s for undoped tantala and τ0 = 1.7 ×10−14 s for doped tantala. For

161

5.6 Undoped tantala loss measurements

1.2x101.0x10Mechanical loss

8.0x106.0x104.0x102.0x10-3

Silica (Cahill & Van Cleve)
Undoped tantala
Doped tantala

-3

-4

-4

-4

-4

0.0
0

20

40

60

80

100

Temperature (K)Figure 5.44: Plot of the loss data used to calculate the barrier height distribution.
The silica data is taken from Cahill and Van Cleve [239] and was measured at 100 kHz.

comparison, the calculation of g(V )f0 for fused silica in Topp and Cahill was
repeated, using experimental loss data from Cahill and Van Cleve [239]. The
data used for the calculation are shown in Figure 5.44.
The value of the coupling coeﬃcient γ for tantala could not be found in
the literature. However, experimental data in Topp and Cahill [223] suggeststhat for most amorphous solids 0.6 eV < γ < 1.1 eV. It was decided to use thevalue of γ measured in fused silica, γ = 0.9 eV. Data for the elastic constantsCii of various amorphous materials are given in Topp and Cahill, all falling in
the range 1.4 × 104 < Cii < 7.8 × 104 MPa. Cii for tantala was approximatedby scaling the value for silica, Cii = 3.3 × 104 MPa [240], by the ratio of theYoung’s modulii of tantala and silica, giving Cii = 6.42 × 104 . It should be
noted that, while every eﬀort was made to make reasonable approximations of
Cii and γ, these constants are likely to be similar for both doped and undoped
tantala, and the precise values are thus of little importance for comparing the162

5.6 Undoped tantala loss measurements

Silica (Cahill & Van Cleve)
Undoped tantala
Doped tantalag(V)f0 (10 20 cm -3 eV -2)

1000

100

10
10

100

Barrier height (meV)

Figure 5.45: Plot of the calculated potential barrier height distribution functiong(V )f0 as a function of barrier height for pure tantala, titantia-doped
tantala and silica. The silica data used in the calculation was takenfrom Cahill and Van Cleve [239].

two materials. Moreover, the constants do not aﬀect the shape of the barrierheight distribution function, which is of primary interest.
The calculated function g(V )f0 for doped tantala, undoped tantala andfused silica [223, 239] is shown in Figure 5.45. This plot allows the distributionof barrier heights (i.e. activation energies for the transition between potential
wells in the ADWP) to be compared.
Both forms of tantala have a peak in the barrier height distribution: in bothcases this peak corresponds closely to the average activation energy calculated
from the Arrhenius equation. The presence of titania doping appears to shift
peak in the distribution to a higher potential barrier energy. In addition, thetitania doping appears to increase the width of the peak in the distributionfunction.

163

5.6 Undoped tantala loss measurements

5.6.5

Comparison of tantala coatings at room temperature

The mechanical loss of three types of tantala coating supplied by LMA havebeen studied: TiO2 -doped Formula 5, TiO2-doped Formula 5∗∗ and undoped
tantala. Formula 5 tantala was used in one of the multi-layer coatings studied
by Harry et al [214] to determine the eﬀect of TiO2 doping on the dissipation at room temperature. LMA claimed to have reduced the dissipation of
the Formula 5 material by changing an unspeciﬁed coating deposition parameter; resulting in the Formula 5∗∗ coating. The results presented in Chapter 4,
measured using silica substrates, suggested that the Formula 5∗∗ coating did
indeed have a lower than than Formula 5 tantala. Figure 5.46 compares thedissipation measured in each of these coatings as a function of frequency atroom temperature. It can be seen that the Formula 5∗∗ coating measured on-3

Coating loss

10

Formula 5 (A)Formula 5 (B)Formula 5**
Un-doped

10

-4

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency (Hz)Figure 5.46: Comparison of the coating losses measured at room temperature for
Formula 5, Formula 5∗∗ and undoped Ta2 O5 coatings.

5.7 The eﬀect of heat treatment

164

a silicon substrate has slightly higher dissipation at room temperature thanthe Formula 5 coating. TiO2 doping can be seen to reduce the loss by approximately 11 % at 1 kHz and 28 % at 1.9 kHz. This is a signiﬁcantly smaller
decrease in loss than the factor of almost two reported by Harry et al [214] in
multi-layer coatings. At room temperature, the silicon cantilever experiment
has large errors because the loss diﬀerence between the coated and uncoatedcantilevers is typically very small. It is possible that these errors are responsible for the discrepancy between the single layer and multi-layer coating loss
results. Further investigation of the coating loss at room temperature would
be of interest.

5.7

The eﬀect of heat treatment

Heat treatment of fused silica can signiﬁcantly reduce the observed mechanicalloss at both room temperature and cryogenic temperature [171]. Heating isbelieved to reduce the internal stresses in silica, resulting in a narrowing of
the distribution of Si-O bond angles, which corresponds to a narrowing of the
distribution of potential barrier heights in the double-well systems. The height
and width of the dissipation peak are also related to the ﬁctive temperature[241] i.e. the temperature at which the glass formed by freezing from a molten
liquid structure. The dissipation peak is found to become higher and narrower
as the ﬁctive temperature decreases. It is thought that silica formed at lower
temperatures has a lower concentration of stressed ring structures, and thatthe distribution of bond angles and therefore potential barrier heights is also
reduced, resulting in a higher, narrower dissipation peak.
If the dissipation mechanism in tantala is indeed analogous to that in silica,
heat treatment may also be eﬀective in reducing the dissipation in this material.
To investigate this, it was decided to measure the mechanical loss of tantalaﬁlms which had been heat treated at diﬀerent temperatures. A set of nominallyidentical cantilevers were sent to CSIRO, a coating vendor in Australia [242].

5.7 The eﬀect of heat treatment

165

As before, a thermal oxide layer approximately 15 to 20 nm thick was grownon all of the samples. Eight cantilevers were coated with undoped tantala.
These coated cantilevers were heat treated in pairs at each of the following
temperatures: 300, 400, 600 and 800◦ C. An oxidised but uncoated control
cantilever was included in each of the heat treatment runs. The heat treatment
was carried out in standard atmosphere.
Ion-beam sputtered tantala ﬁlms are know to crystallise when heated to ≈650◦ C, to the detriment of their optical properties. However, it was thought
to be interesting to study the mechanical loss on either side of this transition
temperature in order to gain a better understanding of the physics involved.
As yet, uncoated control samples have not been measured and therefore
the loss of the coatings has not been calculated. However, the cantilevers arenominally identical and thus the only diﬀerence between them is the temperature at which the post-coating heat treatment was carried out. Each cantileverhad been heated to ∼ 900◦ C prior to coating for the growth of the thermaloxide layer, so it is perhaps reasonable to assume that the further post-coatingheat treatment will not substantially alter the mechanical loss of the silicon
substrate. Even if the substrate loss was changed substantially by the heat
treatment, it would only be expected to signiﬁcantly aﬀect the measured lossof the coated samples at temperatures above approximately 200 K. It is therefore of signiﬁcant interest to compare the losses of the coated samples at low
temperature.
Figures 5.47 to 5.50 show the measured loss of the second, third, fourthand ﬁfth modes of the coated cantilevers which had been heat treated at 300
and 800◦ C. Also shown are the results from Section 5.6 for the LMA undoped
tantala coating, which was heat treated at 600◦ C. The most striking features of
the results are the absence of a dissipation peak at ∼ 20 K for the coating heat
treated at 300◦ C and the large, broad peak in the dissipation of the coatingheated to 800◦ C.
The dissipation of the coated cantilever heat treated at 300◦ C peaks at

166

5.7 The eﬀect of heat treatment-4

1.0x10-5

8.0x10Loss of coated cantilever

Heat treated at 300 C
Heat treated at 600 C
Heat treated at 800 C-56.0x10-5

4.0x10-5

2.0x10

0
0

50

100

150

200

250

300

Temperature (K)Figure 5.47: Measurements of the mechanical loss of the second bending mode (atapproximately 350 Hz) of three cantilevers coated with undoped Ta2 O5
and heated to 300, 600 and 800 ◦ C respectively.
1.2x10Loss of coated cantilever

1.0x108.0x106.0x104.0x102.0x10-4

Heat treated at 300 C
Heat treated at 600 C
Heat treated at 800 C

-4

-5

-5

-5

-5

0
0

50

100

150

200

250

300

Temperature (K)Figure 5.48: Measurements of the mechanical loss of the third bending mode (at approximately 1000 Hz) of three cantilevers coated with undoped Ta2 O5
and heated to 300, 600 and 800 ◦ C respectively.

167

5.7 The eﬀect of heat treatment

1.0x10Loss of coated cantilever8.0x106.0x104.0x102.0x10-4

Heat treated at 300 C
Heat treated at 600 C
Heat treated at 800 C

-5

-5

-5

-5

0
0

50

100

150

200

250

300

Temperature (K)Figure 5.49: Measurements of the mechanical loss of the fourth bending mode (atapproximately 1960 Hz) of three cantilevers coated with undoped Ta2 O5
and heated to 300, 600 and 800 ◦ C respectively.
1.0x10Loss of coated cantilever8.0x106.0x104.0x102.0x10-4

Heat treated at 300 C
Heat treated at 600 C
Heat treated at 800 C

-5

-5

-5

-5

0
0

50

100

150

200

250

300

Temperature (K)Figure 5.50: Measurements of the mechanical loss of the ﬁfth bending mode (at approximately 3200 Hz) of three cantilevers coated with undoped Ta2 O5
and heated to 300, 600 and 800 ◦ C respectively.

168

5.7 The eﬀect of heat treatment

approximately 30 K for each mode studied. The peak is signiﬁcantly smaller
and broader than the peak at 20 K previously observed in tantala coatings
which had been heat treated at 600 ◦ C. Analysis of the Arrhenius plot forthis peak, which is shown in Figure 5.51, yielded an activation energy andrelaxation constant (138 ± 4) meV and (3.5 ± 0.1) ×10−27 s respectively. It
is possible that the two peaks arise from the same dissipation mechanism andthat heating the Ta2 O5 to 600◦ C altered the distribution of potential barrierheights, resulting in a lower average activation energy. However, the dissipation
at (and above) 30 K is almost identical for both samples, suggesting that the 30
K peak may also be present in the coatings treated at 600◦ C, although partiallyobscured by the larger peak at 20 K. At temperatures above the dissipation
11

Linear Fit y = -1597.1x + 60.9ln(ω0)

1098

70.03160.03200.03240.0328
1/T peak0.03320.0336

Figure 5.51: Arrhenius plot for the dissipation peak observed at ∼ 30 K for an undoped tantala coating heat treated at 300 ◦ C.
peak, between ∼40 and 200 K, the dissipation of the coatings treated at 300
and 600 ◦ C follow a similar trend and, for the third and fourth modes of thesamples, have the same magnitude to within experimental error. Between

5.7 The eﬀect of heat treatment

169∼200 K and room temperature, the loss of the cantilever treated at 600◦ C
appears to be signiﬁcantly lower. In this region, however, the thermoelastic
loss of the cantilever substrate becomes increasingly dominant and thus theeﬀect of the coating on the measured loss becomes smaller. It is therefore
diﬃcult to draw conclusions in this temperature range until the dissipation of
the uncoated control samples has been measured.
The cantilever treated at 800◦ C exhibits a large and very broad dissipation
peak at ∼ 80 to 90 K and a smaller, narrower peak at at ∼ 20 K. As noted
above, ion-beam sputtered tantala is known to crystallise at approximately
650◦ C. To conﬁrm that the coating was poly-crystalline, a silica disk which had
been coated and heat treated at the same time as the cantilever was studied
under a transmission electron microscope by R. Bassiri and J. Scott. Both themicroscope images and the results of electron diﬀraction measurements, shownin Figure 5.52, indicate that the tantala coating has a crystalline structure. It
is likely, therefore, that the large dissipation peak centered on 80 to 90 K is
associated with the polycrystalline structure of the tantala. Dissipation peaksin poly-crystalline materials can arise from phonon scattering at the grainboundaries [110], and this dissipation mechanism may be responsible for the
peak observed at 90 K.
The eﬀects of the dissipation peak at 20 K can be seen as a small plateau
in the low temperature edge of the 90 K peak in the loss data for each modeof the sample heat treated at 800◦ C. At temperatures between ≈ 200 and
250 K, this cantilever has a lower loss than those heat treated at 300 and
600 ◦ C, while closer to room temperature, it has the largest loss of the three
cantilevers. The lower loss between 200 and 250 K may be caused by theheat treatment at 800◦ C reducing the loss of the silicon substrate, perhaps
by removing contaminants from the surface or by driving water out of thethermal oxide layer. Measurements of the control samples which underwent
the same heat treatment will be required to fully understand the behaviour of170

5.8 Conclusions

(b)

(a)

Figure 5.52: (a) Transmission electron microscope image of the tantala coating annealed
to 800◦ C, showing evidence of crystal structure with features up to 15 nm
across. (b) Electron energy loss diﬀraction pattern from the same coating,showing clear diﬀraction spots conﬁrming the crystalline structure of the coating. Images taken by Riccardo Bassiri and Jamie Scott [243].

the dissipation above 200 K.

5.8

Conclusions

A peak in the dissipation of 500 nm thick ﬁlms of tantalum pentoxide heattreated at 600◦ C has been discovered at approximately 20 K. The peak ispresent in both undoped tantala and in tantala doped with (14.5 ± 1) % titania.
The frequency dependence of the temperature at which the peak occurs implies
that the dissipation arises from a Debye-type thermally activated relaxation
process which can be described by the Arrhenius equation.
The presence of titania doping in the tantala, which is known to reduce
the loss of multi-layer SiO2 /Ta2 O5 coatings at room temperature, was found
to reduce the dissipation throughout the entire temperature range studied,with the exception of the wings of the peak, where the loss dissipation of
the undoped coating was slightly higher. Analysis of the dissipation peaks
observed in the doped and undoped coatings suggests that the TiO2 doping
increases the activation energy of the dissipation mechanism by approximately

5.8 Conclusions

171

33 % from 28.6 ± 1.2 meV to 39.5 ± 2.7 meV.
Given the magnitude of the activation energy and the amorphous structure of ion-beam sputtered tantala, it has been postulated that the dissipation
mechanism may be similar to that observed in fused silica, where the dissipation is thought to arise from thermally activated transitions of oxygen atoms
between stable states in a double-well potential system. The dissipation peakobserved in tantala is signiﬁcantly wider than a Debye peak, suggesting that
there may be a wide distribution of activation energies for the energy loss
mechanism. This distribution may be related to the amorphous structure of
the coating. Following the asymmetric double-well potential model of Gilroy
and Phillips and its extension by Topp and Cahill, the distribution of activation energy barriers has been calculated for doped and undoped tantala. The
results suggest that the presence of TiO2 doping broadens this distribution andshifts it towards a higher average activation energy.
Heat treatment is believed to increase the height and reduce the widthof the dissipation peak in fused silica by reducing the distribution of Si-O
bond angles. Initial studies of ion-beam sputtered Ta2 O5 have shown that theposition, height and width of the dissipation peak are very strongly related tothe heat treatment temperature. A second, very broad dissipation peak has
been observed in a Ta2 O5 ﬁlm which had crystallised after heat treatment at
800◦ C.
The identiﬁcation of this low temperature dissipation peak provides theﬁrst evidence of the mechanism responsible for mechanical dissipation in ionbeam sputtered Ta2 O5 coatings. Further study of the eﬀects of doping andheat treatment on the dissipation, and particularly on the characteristics ofthe dissipation peak, may facilitate the development of lower loss coatings foruse at both room and cryogenic temperatures, thus enabling the sensitivity of
future gravitational wave detectors to be signiﬁcantly enhanced. In addition,these results are of direct interest to a variety of optical measurements and may,

5.8 Conclusions

172

for example, be of use in the choice of coatings and operating temperatures for
future ultra-stable laser cavities.

Chapter 6
Temperature dependence of themechanical dissipation in silica
coatings
6.1

Introduction

In the preceding chapter, the temperature dependence of the mechanical lossof single layers of ion-beam sputtered tantalum pentoxide was studied, resulting in a possible identiﬁcation of one of the loss mechanisms in this material.
To evaluate the coating thermal noise in a cryogenic gravitational wave detector, it is also necessary to study the low temperature loss in a silica coating
layer. For multi-layer silica/tantala coatings the tantala layers are known tobe the dominant source of the coating loss at room temperature [130, 131], as
discussed in the previous chapter. However, it is known that the loss of bulk
silica increases signiﬁcantly at low temperatures [93]. It is therefore important to investigate the temperature dependence of the dissipation in ion-beam
sputtered silica coatings.
Low temperature dissipation in thin silica ﬁlms grown by thermal oxidation and electron beam evaporation has been studied [244], but to the author’s
knowledge, the temperature dependence of the dissipation in single layers ofion-beam sputtered silica coatings has not previously been investigated. Mea173

174

6.2 Coating thermoelastic loss

surements of the loss of a single layer of silica deposited on a silicon cantileverby ion-beam sputtering are presented in this chapter. The losses measured atroom temperature are compared with those obtained by Penn for ion-beam
sputtered silica deposited on a silica disk and found to be higher, and any possible role of coating thermoelastic loss in this observed diﬀerence evaluated.
These results are combined with the tantala dissipation results from Chapter
5 to calculate the expected loss in a multi-layer coating as a function of temperature. Finally, these results are compared with direct measurements of theloss in a multi-layer coating by Yamamoto et al [226].

6.2

Coating thermoelastic loss

Thermoelastic loss in a material arises when heat ﬂows between regions ofdiﬀerent temperature. In a vibrating cantilever an oscillating diﬀerential temperature distribution occurs leading to energy dissipation from heat ﬂow acrossthe thickness of the cantilever, as described in Section 2.4.
A further source of thermoelastic loss occurs in a coated sample from thediﬀering thermo-mechanical properties of the coating and substrate materials.
This eﬀect is described in detail by Fejer et al [142], where the following equations for the ‘coating thermoelastic loss’ are derived. It should be noted that
this is a misnomer as this loss is not solely dependent on the properties of thecoating and arises from the diﬀerences in the properties of the coating and substrate materials. For a uniform ﬁlm of thickness l on a much thicker substrate,
the coating thermoelastic loss is described by the following four equations:
2Yc αc2 T
φte =Cc (1 − νc )

αs Ys (1 − νc )Cc
1−
αc Yc (1 − νs )Cs

2
g(ω),

(6.1)

where Y is Young’s modulus, α is the coeﬃcient of thermal linear expansion,
C is speciﬁc heat capacity and ν is Poisson’s ratio and the subscripts ‘s’ and‘c’ refer to properties of the substrate and the coating ﬁlm respectively. The

175

6.2 Coating thermoelastic loss

function g(ω) describes the frequency dependence:√1sinh( iωτc )√√g(ω) = Im − √iωτc cosh( iωτc ) + Rsinh( iωτc )

,

(6.2)

in which τc and R are deﬁned as:


kc Cc
,
ks Cs

(6.3)

τc = l2 /κc ,

(6.4)

R=
and

and κc = kc /C is the thermal diﬀusivity of the coating. Upon examinationof Equation 6.1 it can be seen that if the coating and substrate materials
are identical then the combination of material properties in brackets becomeszero and there is no thermoelastic loss from this eﬀect. Since the magnitudeof the dissipation has a quadratic dependence on the diﬀerence between theproperties of the coating and the substrate, the loss can be very sensitive tosmall changes in the properties of the coating.
SiliconYoung’s modulus
Linear thermal expansion coeﬃcient

Silica166 GPa−6

2.6 × 10

Tantala72 GPa−1

K−7

5.1 × 10140 GPa−1

3.6 × 10−6 K−1

KSpeciﬁc heat capacity

705 Jkg−1 K−1746 Jkg−1 K−1

306 Jkg−1 K−1Poisson’s ratio

0.27

0.17

0.23

Thermal conductivity
Density−1

140 Wm−1

K2330 kgm−3−1

K

33 Wm−1 K−12202 kgm−36850 kgm−3

1.38 Wm−1

Table 6.1: Material properties for silicon [228], silica [142] and tantala [142] at 290
K, used in the calculation of the coating thermoelastic loss.

The calculated magnitudes and frequency dependence of the coating thermoelastic loss for 500 nm thick single layer silica and tantala coatings on silicon substrates is compared in Figure 6.1. The material properties used in thiscalculation are shown in Table 6.1. The coating thermoelastic loss peaks at

176

6.3 Results

approximately 1 MHz for the cantilever coated with silica and 20 MHz for the
samples coated with tantala. The diﬀerent shape of the peaks is explained by
the diﬀering values of R, as discussed by Fejer [142]. The thermoelastic loss is
signiﬁcantly higher in the case of the cantilever coated with silica. At frequencies above 100 Hz it is of the same order as the measured loss of the uncoatedsilicon cantilever, and thus might be expected to have a small eﬀect on themeasured coating loss at room temperature. To allow correct calculation of
the intrinsic loss of the coating material, the coating thermoelastic loss mustbe subtracted from the measured coating loss. The formula for calculating the
intrinsic coating loss (Equation 4.14) now becomes:

φ(ω0 )coating =Ys a(φ(ω0 )coated − φ(ω0 )substrate ) − φ(ω0 )coating thermoelastic , (6.5)
3Yc twhere φ(ω0 )coating thermoelastic is the coating thermoelastic loss calculated from
Equation 6.1.
The temperature dependence of the coating thermoelastic loss was calculated for the cantilever coated with silica, and is shown in Figure 6.2. The
temperature dependent values of all of the material properties were taken from
Touloukian [228] and White [245]. Over the temperature range studied, thecoating thermoelastic loss is highest at room temperature.
In the frequency range at which loss measurements were carried out (below
5 kHz) the tantala coating thermoelastic loss is approximately two orders ofmagnitude smaller than the loss of the uncoated silicon cantilever, and thus
has no signiﬁcant eﬀect on the loss of the coated sample. It was therefore notnecessary to correct the measurements in Chapter 5 for coating thermoelastic
loss.

6.3

Results

A 492 nm thick silica coating was applied by ion-beam sputtering to a silicon cantilever by LMA [215]. The cantilever was identical to those used to177

6.3 Results

1E-3

Silica coating
Tantala coating

Thermoelastic loss

1E-4

1E-5

1E-6

1E-7

1E-8

1E-9
10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

Frequency (Hz)Figure 6.1: Comparison of the coating thermoelastic loss at 290 K calculated for a
500 nm thick SiO2 coating and a 500 nm thick Ta2 O5 coating.

measure the loss of a doped tantala coating to liquid helium temperature (seeSection 5.3) and underwent the same pre-coating thermal oxidation process
and post-coating heat treatment at 600 ◦ C. The temperature dependence ofthe mechanical loss of the coated cantilever was measured using the method
and apparatus described in Section 5.3.2. The measured dissipation is shown
as a function of temperature in Figures 6.3 to 6.7. Each point represents theaverage loss calculated from at least three ring-down measurements and has
a standard error of typically less than 3%. The uncoated control data set is
identical to that used in Section 5.3: it has, however, been interpolated onto
the same temperature points as the measurements of the silica coating sample.

Loss of the coated cantilever
Each mode of the coated cantilever shows some evidence of a broad dissipation
peak at approximately 21 K, which will be discussed further in the next section.

178

6.3 Results

1E-5

1st mode 59 Hz
2nd mode 1028 Hz
3rd mode 3332 Hz

1E-6

Thermoelastic loss

1E-7
1E-8
1E-9
1E-10
1E-11
1E-12
1E-13
1E-14
10

Temperature (K)

100

300

Figure 6.2: Temperature dependence of the coating thermoelastic loss calculated forthree modes of the SiO2 coated silicon cantilever.

Mechanical loss

1.5x101.0x105.0x10

1st mode coated
1st mode uncoated

-5

-5

-6

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 6.3: Plot of the measured loss for the ﬁrst mode of the cantilever coated withsilica and of the uncoated control sample.

179

6.3 Results

2nd mode coated
2nd mode uncoated-5

Mechanical loss

1.5x10-5

1.0x10-6

5.0x10

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 6.4: Plot of the measured loss for the second mode of the cantilever coatedwith silica and of the uncoated control sample.
3.0x10-5

3rd mode coated
3rd mode uncoatedMechanical loss

2.0x101.0x10

-5

-5

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 6.5: Plot of the measured loss for the third mode of the cantilever coated withsilica and of the uncoated control sample.

180

6.3 Results

Mechanical loss

3.0x102.0x101.0x10

4th mode uncoated
4th mode coated

-5

-5

-5

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 6.6: Plot of the measured loss for the fourth mode of the cantilever coatedwith silica and of the uncoated control sample.
2.5x10-5

5th mode coated
5th mode uncoatedMechanical loss

2.0x101.5x101.0x105.0x10

-5

-5

-5

-6

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 6.7: Plot of the measured loss for the ﬁfth mode of the cantilever coated withsilica and of the uncoated control sample.

6.3 Results

181

Limited data were obtained for the ﬁrst mode above 175 K, as clean ringdowns
were diﬃcult to obtain, possibly due to low frequency background noise. At
temperatures above 200 K the dissipation of each mode rises rapidly, withbroad peak or plateau structures apparent for all of the modes studied. The
data for the second and third modes show a broad peak centred on ∼ 235
K and a narrower, higher peak at ∼ 270 K. Immediately above 200 K, thedissipation of the fourth mode follows the trend of the uncoated sample moreclosely with no peak at 235 K, but does show evidence of the higher peak at∼ 270 K. The data for the ﬁfth mode show a steep increase in loss between
200 and 240 K. However, the loss does not peak, but continues to increase, lesssteeply, towards room temperature.
The very high losses measured for three of the modes at temperatures
around ∼ 270 K may be indicative of energy coupling into modes of the clamp.
However, it seems unlikely that this type of clamping loss would have such asimilar eﬀect on three modes separated widely in frequency. It is possiblethat the cantilever was clamped such that a degree of frictional stick-slip loss
was present close to room temperature. Thermal contraction of the clamp
could eliminate this eﬀect at lower temperatures: certainly the measured loss
is generally smooth below 200 K. It is possible that the broad peaks observed
between 200 and 240 K are intrinsic to the coating, although the absence of
this structure in the data for the fourth mode would weigh against this possibility.

Loss of the uncoated cantileverA small peak, centred on 20 - 25 K, was observed in the dissipation of the uncoated cantilever. It is possible that this peak arises from the 20-30 nm thickthermal oxide layer grown on the cantilevers by the coating vendor. Measurements of a 500 nm thick thermal oxide layer on a silicon substrate at 5 kHz
by White and Pohl [244] have shown a dissipation peak at 35 K, a somewhat

6.4 Coating loss

182

higher temperature than the peak observed in the dissipation of the uncoatedcantilever (see Section 6.5). However, in addition to being more than a factorof 10 thicker, the thermal oxide layer measured by White and Pohl was grownusing a wet oxidation process, whereas the cantilevers studied by the author
underwent a dry oxidation process. Given these diﬀerences, it may be possiblethat the peak at 25 K is related to the dissipation peak of the oxide layer. Further investigation of the loss of thermal oxide layers on silicon cantilevers at
cryogenic and ultra-cryogenic temperatures is planned, in collaboration with
the INFN laboratory at Legnaro, Italy. However, the peak should not aﬀect
the calculated coating loss, as the eﬀects of the peak would be expected to
be present in the dissipation of both the coated and uncoated cantilevers andthus will be removed when the losses are subtracted.

6.4

Coating loss

The dissipation of the silica coating was calculated using Equation 6.5. For thelevel of loss measured in the silica coating the thermoelastic correction term
had an almost negligible eﬀect. The coating thermoelastic loss was calculated
to increase with temperature and frequency. For the highest frequency modestudied (the ﬁfth mode at ∼ 3300 Hz) at room temperature, the thermoelastic correction was found to reduce the coating loss by ∼ 1.3%. As expected,
the eﬀect decreased at low temperatures, falling to a reduction of ∼ 0.015% at 100 K and ∼ 0.003 % at 20 K. While the thermoelastic correction had
an insigniﬁcant eﬀect on the results presented here, it is likely to be important for the correct interpretation of loss measurements of thicker coatings, ormeasurements carried out at higher frequencies.
The coating loss results for each of the modes of the cantilever are shownin Figures 6.8 to 6.12. The loss of the silica ﬁlm has quite a complex shape andfollows a similar trend below 225 K for each of the cantilever modes studied.
Generally, between 225 K and 190 K the dissipation of the ﬁlm drops sharply

183

6.4 Coating loss

to a minimum, before rising slightly to a low and broad peak centred on 150 K.
A second minima or an inﬂection point is reached at approximately 100 to 110
K. Below this temperature the loss rises steadily to a peak at approximately
17 K.

-3

Coating loss

10

10

-4

0

50

100

150

200

250

300

Temperature (K)Figure 6.8: Calculated dissipation of the silica coating for the ﬁrst mode at 59 Hz.

184

6.4 Coating loss

-3

Coating loss

10

10

-4

0

50

100

150

200

250

300

Temperature (K)Figure 6.9: Calculated dissipation of the silica coating for the second mode at 355
Hz.

-3

10

-4

Coating loss

10

0

50

100

150

200

250

300

Temperature (K)Figure 6.10: Calculated dissipation of the silica coating for the third mode at 995 Hz.

185

6.4 Coating loss

-3

Coating loss

10

10

-4

0

50

100

150

200

250

300

Temperature (K)Figure 6.11: Calculated dissipation of the silica coating for the fourth mode at 1920
Hz.

-3

Coating loss

10

10

-4

0

50

100

150

200

250

300

Temperature (K)Figure 6.12: Calculated dissipation of the silica coating for the ﬁfth mode at 3175
Hz.

186

6.4 Coating loss

6.4.1

Dissipation peak analysis

The method described in Section 5.5 was used to construct an Arrhenius plot
for the low temperature dissipation peak at ∼17 K, shown in Figure 6.13. As
shown on this graph, there is a reasonable linear relationship (R2 = 0.93) between ln(ω) and 1/Tpeak for the ﬁrst, second, third and fourth bending modes.
However the data for the ﬁfth mode at ∼3175 Hz did not ﬁt this trend. If theﬁfth mode is neglected, the data for the remaining four modes can be used
to calculate an activation energy associated with the dissipation peak of 54.4
± 5.5 meV. This is somewhat higher than the activation energy of 44 meV
measured in bulk silica by Anderson and Bommel [172].
12

10

ln(ω)

8

6

4
st

1 to 4th modes
2

0
0.040

th

5 modeLinear fit y = -630x + 48.5

0.045

0.050

0.055

0.060

0.065

0.070

1/Tpeak

Figure 6.13: Arrhenius plot for the low temperature dissipation peak in the silica
coating.

The coating loss for the ﬁfth mode was observed to peak at 21.9 K, approximately 5.5 K higher than predicted by the linear ﬁt for the other fourmodes. It is possible that this discrepancy is related to the eﬀect of the loss of
the uncoated cantilever on the calculated coating loss. The small peak in theloss of the uncoated cantilever centred on 20 to 25 K, which is present in the6.4 Coating loss

187

data for the ﬁrst four modes, is not clearly observed in the data for the ﬁfth
mode, possibly due to the eﬀect of clamping loss. For each of the other modes,
on substraction of the loss of the uncoated sample from the loss of the coatedsample, the temperature at which the peak occurred was found to decrease by
up to 2 K. This is due to the eﬀect of the small peak in the loss of the uncoatedcantilever. In contrast, on calculation of the coating loss for the ﬁfth mode thetemperature at which the peak occurred increased by approximately 2 K.
Further measurements of the ﬁfth mode of both the coated and uncoatedcantilevers are required to fully investigate this anomalous point. Study ofthe dissipation of higher order modes would also be of interest to investigate
if the linear trend for the ﬁrst four modes in the Arrhenius plot continues athigher frequency. However, measurements of the sixth (and higher frequency)
bending modes of the coated cantilever proved diﬃcult due to very short ringdown times.

6.4.2

Silica coating loss above 200 K

Recent measurements of the loss of a 1 μm thick silica coating (deposited by
CSIRO [242]) on a silica substrate carried out by Penn [246] suggest that atroom temperature the loss may be as low as 1 to 3 ×10−5 at ≈ 3-4 kHz. This isapproximately an order of magnitude lower than the room temperature losses
presented in this chapter at similar frequencies, and approximately a factor of
3 lower than the losses measured at 355 Hz and 1 kHz. This diﬀerence may berelated to the possible clamping loss peaks in the measurements of the coatedcantilever which were noted above. To investigate further, the cantilever was
re-clamped and the measurements repeated at room temperature. The losses
were found to be substantially lower than the previous results. For example, the
new measurement of the loss factor of the third mode at 290 K was 6.4×10−6 ,which is a factor of 1.9 lower than the previous measurement of 1.2×10−5 . The
calculated coating losses were also correspondingly lower. Figure 6.14 shows

188

6.4 Coating loss

Mode 2 coating loss
Mode 3 coating loss
Mode 2 room temperature repeat
Mode 3 room temperature repeat-3Coating loss

10

-4

10

0

50

100

150

200

250

300

Temperature (K)Figure 6.14: Plot comparing the initial measurements of the temperature dependenceof the coating loss with the measurements repeated at room temperature.

both the original temperature dependent coating loss and the repeated room
temperature measurements for the second and third modes. The new coating
loss factor at 290 K was found to be (1.0 ±0.1) ×10−4 for the second mode, (1.1
±0.2) ×10−4 for the third mode and (2.3 ±0.2) ×10−4 for the third mode (seeTable 6.2). These values are lower than the previous values measured in Jenaby factors of ∼2, ∼4.8 and ∼3.9 respectively. The repeated measurements of
the ﬁfth mode yielded the same loss as originally measured, and the coating
loss for this mode remained unchanged at (5.4 ±0.3) ×10−4 .
It seems likely therefore that the high losses initially measured above 200
K were indeed related to the clamp.
The repeated measurements of the coating loss factors are compared to
Penn’s data and to the silica coating loss factors presented in Chapter 4 (fora 500 nm thick SiO2 coating on a silica cantilever substrate) in Figure 6.15.
Between 2 kHz and 4 kHz, the coating loss measured using the silicon can-

189

6.4 Coating loss

Mechanical loss

Coating loss (290 K, repeated)
Penn coating loss
Coating loss on silica cantilever

10

10

-4

-5

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency (Hz)Figure 6.15: Plot of the repeated room temperature measurements of the coating loss.
Also shown is Penn’s coating loss data, measured at room temperatureusing silica substrates [246].
Mode

Coating loss

Thermoelastic correction

355 Hz(1.0±0.1) ×10−4

2.1×10−6

989 Hz(1.1±0.2) ×10−4

3.8×10−6

1920 Hz(2.3±0.2) ×10−4

4.8×10−63175 Hz(5.4±0.3) ×10−4

6.9×10−6

Table 6.2: The coating loss factors for each mode at room temperature, calculatedusing the data from the repeated loss measurements of the coated cantilever. Also shown is the calculated thermoelastic coating loss for eachmode frequency.

tilever substrate is signiﬁcantly higher than the loss measured using the silica
cantilever substrate, which in turn is 3 to 4 times higher than the coating loss
measured by Penn.
There are several possible explanations of these diﬀerences. The coatings

6.5 Discussion - low temperature loss in silica

190

were supplied by diﬀerent vendors (Penn’s coating was produced by CSIRO[242], while the coatings studied by the author were supplied by LMA [215])
so it is possible that the diﬀerences in the mechanical loss are related to variations in the coating material or in the deposition process. Penn’s coating wasapproximately twice as thick as the coatings applied to the cantilevers, so adependence of the loss on the thickness of the coating could also explain thediﬀerences in the results.
The coated silicon cantilever is likely to have a higher level of internal stressthan the other two samples, due to the diﬀerences in the material properties of
the coating and substrate. It is possible that this could increase the mechanicalloss of the coated cantilever, resulting in an artiﬁcially high coating loss.
Another possibility is that the coating thermoelastic loss for the siliconcantilever coated with silica is higher than was calculated. As noted previously,
the magnitude of the calculated thermoelastic loss is very sensitive to thematerial properties used. Thus if the properties of the ion-beam sputtered silica
ﬁlm diﬀer from those of bulk silica, it is possible that the coating thermoelastic
loss correction could be more signiﬁcant than shown in Table 6.2, reducing the
calculated intrinsic loss of the coating.
Further study of the dependence of the dissipation of silica coatings on
coating thickness and substrate material is clearly desirable to enable theseresults to be fully explained.

6.5

Discussion - low temperature loss in silica

The dissipation in bulk pieces of silica has been extensively studied as a function of temperature and a dissipation peak centred on approximately 40 to 60
K has been well characterised by many authors [93, 172, 222, 239]. White andPohl have studied the dissipation of silica coatings deposited by electron beam
evaporation and by thermal oxidation of silicon substrates [244]. This experiment used a silicon double paddle oscillator vibrating at 5.5 kHz to measure the191

6.5 Discussion - low temperature loss in silica

loss of several thicknesses of e-beam ﬁlms, and a 500 nm thick thermal oxide
layer, between 10−2 K and 100 K. The loss of the thermal silica was found to
closely resemble that observed in bulk silica samples, with a dissipation peakat 35 K. However, no evidence of a similar dissipation peak was observed in
the e-beam coatings. For coatings under 10 nm thick the loss was essentiallyconstant below 20 K. At approximately 5 K, the loss of bulk, thermal and thethin (under 10 nm) e-beam coatings was found to be almost identical, lying
between 3×10−4 and 4×10−4 . Thicker e-beam coatings were found to have asigniﬁcantly higher loss throughout the temperature range studied, with theloss increasing rapidly above 10 K, perhaps towards a peak at a temperature
above 100 K.
These results suggest that thicker e-beam coatings might have a diﬀerent-3

2.0x10-3

1.5x10-3

1.0x10-3

5.0x10-4

Ion beam sputtered SiO2 3.2 kHz
109 nm e-beam SiO2 (White & Pohl)
8.7 nm e-beam SiO2 (White & Pohl)
4.6 nm e-beam SiO2 (White & Pohl)
500 nm thermal SiO2 (White & Pohl)
Bulk SiO2 56 kHz (Cahill & Van Cleve)Loss

2.5x10

0.0
0

20

40

60

80

Temperature (K)Figure 6.16: A comparison of the dissipation measured for the ion-beam sputteredsilica coating at 3.3 kHz with the loss at 5.5 kHz of silica coatings grownby wet thermal oxidation and electron beam evaporation (from Whiteand Pohl [244]) and with the loss of bulk silica at 56 kHz (from Cahill
and Van Cleve [239].)

6.6 Multi-layer coating loss calculation

192

structure, and thus a diﬀerent dissipation mechanism, than the e-beam ﬁlms
less than 10 nm in thickness. White and Pohl suggest that macroscopic disorder, possibly related to porosity, in the thicker ﬁlms may be related to thehigher losses and diﬀerent temperature behaviour observed [247].
The mechanical loss of the ion-beam sputtered silica coating studied here,
also shown in Figure 6.16, can be seen to have signiﬁcantly lower loss than
the same thickness (500 nm) of thermal oxide measured by White and Pohl
at a similar frequency. Furthermore, the peak loss in the ion-beam sputteredcoating occurs at a lower temperature - about 21 K - than the peak observed inbulk silica and thermal oxide. Below 25 K, the loss of the ion-beam sputteredcoating is very close to the loss of the thinnest (less than 10 nm thick) e-beamcoatings, while the loss of the 100 nm thick e-beam coating is almost an orderof magnitude higher.
While more work is required, particularly to measure the dependence ofthe mechanical dissipation on coating thickness for ion-beam sputtered silica,
this comparison suggests that both thickness and coating deposition method
can have a very signiﬁcant eﬀect on the low temperature loss in silica coatings. The results also indicate that ion-beam sputtered coatings appear tohave signiﬁcantly lower loss than similar thicknesses of e-beam coatings. It
is known that ion-beam sputtered coatings tend to be more dense and havebetter stoichiometry than e-beam deposited ﬁlms [204], and it is possible thatthe diﬀerent losses are related to structural diﬀerences associated with thedeposition method.

6.6

Multi-layer coating loss calculation

If it is assumed that the loss of ion-beam sputtered silica and tantala are
independent of the coating thickness, the experimental results from this chapter
and Chapter 5 can be used to calculate the expected loss in a multi-layer
SiO2 /Ta2 O5 coating. This calculated loss is compared with measurements of193

6.6 Multi-layer coating loss calculation

a multi-layer coating on a sapphire disk by Yamamoto et al [226].

6.6.1

The mechanical loss of a multi-layer coating

For a multi-layer coating the loss φ, Young’s modulus Y and thickness t are
related to the properties of the individual component materials as follows [197]:
Ycoating tcoating φcoating = YSiO2 tSiO2 φSiO2 + YTa2 O5 tTa2 O5 φTa2 O5 ,(6.6)

where tSiO2 and tTa2 O5 are the total thickness of silica and tantala present in
the coating. Hence the loss of a multi-layer coating is related to the individual
losses of its constituent materials as follows:φcoating =YSiO2 tSiO2 φSiO2 + YTa2 O5 tTa2 O5 φTa2 O5.
Ycoating tcoating(6.7)

It can be shown from composite materials theory that the eﬀective Young’s
modulus in an isotropic multi-layer coating consisting of two materials is given
by [248]:
Ycoating =Y1 t1 + Y2 t2,
t1 + t2(6.8)

where the subscripts ‘1’ and ‘2’ refer to properties of the diﬀerent materials.
The optical thickness, δ = nt (see Section 4.4.1), of each layer of silica andtantala is 266 nm (i.e. λ/4 for a wavelength of 1064 nm), where the refractive
indices are nsilica = 1.45 and ntantala = 2.03 [131]. Thus the physical layer
thickness can be calculated to be ∼180 nm and ∼130 nm for silica and tantala
respectively.
If it is assumed that the loss of the coating materials is independent of thecoating thickness, Equation 6.7 can be used to calculate the loss in a multi-layer
coating from the measured losses of the individual coating materials.
The measured dissipation for the undoped tantala coating (see Section 5.6)
and the silica coating at 1 kHz was used to calculate the expected loss ina 31 layer silica / tantala coating (with the same reﬂectivity as the coating
measured by Yamamoto et al). The single layer coating loss data used are194

6.6 Multi-layer coating loss calculation

Coating loss

1.2x10-3

1.0x10-3

8.0x10-4

6.0x10-4

4.0x10-4

2.0x10-4

Loss of tantala coating
Loss of silica coating

0.0
0

50

100

150

200

250

300

Temperature (K)Figure 6.17: The measured coating loss for the silica and tantala coatings at ∼ 1 kHz
which was used in the multi-layer coating loss calculation.

shown in Figure 6.17. Figure 6.18 compares the predicted loss in a multi-layer
coating calculated using Equation 6.7 with the loss measured in three multilayer coatings by Yamamoto et al at approximately the same frequency. Twoof the multi-layer coatings were not heat treated and these display higher losses
at room temperature than the third coating, which was heat treated at several
hundred ◦ C. The exact temperature of the heat treatment is not known [249].
Close to room temperature, the multi-layer loss calculated from the single layer
results presented here is very similar to the loss measured by Yamamoto forthe heat treated coating.
At low temperature, the calculated multi-layer loss is dominated by thedissipation peak observed in Ta2 O5 . This peak does not appear to be present inYamamoto’s experimental results. While there are not enough data points from
Yamamoto’s work to allow a full comparison, it can be seen that the calculated
peak loss at approximately 20 K is almost a factor of two higher than the195

6.6 Multi-layer coating loss calculation

1.0x10Mechanical loss

8.0x106.0x104.0x102.0x10-3

Calculated multilayer loss 1 kHz
Yamamoto 2006 (no. 2) multilayer (no heat treatment) 1.1 kHz
Yamamoto 2006 (no. 4) multilayer (heat treated) 1.1 kHz
Yamamoto 2006 (no. 3) multilayer (no heat treatment) 1.1kHz

-4

-4

-4

-4

0

50

100

150

200

250

300

Temperature (K)Figure 6.18: The expected loss in a 31 layer SiO2 Ta2 O5 coating, calculated from thesingle layer results, compared with the loss measured in such a multilayer coating by Yamamoto et al [226]loss measured by Yamamoto in the heat treated coating at this temperature.
However, the experimental multi-layer data do show some evidence of a peakin the dissipation at higher temperatures, with the peak occurring at a lower
temperature in the heat treated coating.
The calculated multi-layer coating loss around 250 K is signiﬁcantly higher
than expected from the general trend of the data. These high loss values are a
result of the high losses measured in the silica coating at these temperatures,
which are likely to be due to clamping loss eﬀects and therefore not intrinsic
to the coating material.

6.6 Multi-layer coating loss calculation

6.6.2

196

Coating thermoelastic loss

It has been shown that, at the frequencies of interest, the coating thermoelastic
loss at room temperature in a 500 nm silica or tantala coating on a silicon
substrate does not have a signiﬁcant contribution to the measured coatingloss. Further, the coating thermoelastic loss of the silica sample has been shown
to be negligible at all temperatures between 10 and 300 K. This calculation
could not be carried out for tantala as the temperature dependence of therelevant material properties could not be found in the literature. However, atroom temperature the coating thermoelastic loss of the cantilever coated withtantala is approximately two orders of magnitude lower than that of the silica
coating sample, and it is therefore unlikely that it will become signiﬁcant at
low temperature.
However, Fejer et al [142] have shown that, at room temperature, the coating thermoelastic loss arising from a silica/tantala multilayer coating applied toa sapphire substrate may be of the same order as the intrinsic loss of the coating materials. As the coatings studied by Yamamoto had sapphire substrates,
it is of interest to calculate and correct for the expected level of thermoelastic
loss in these samples.
The coating thermoelastic loss of a multilayer coating can be calculatedusing a form of Equation 6.1, adapted to be applicable to a multilayer coating consisting of alternating layers of two materials, where a suitable volume
averaging operator (Equation 2.56) is applied to obtain the average material
properties of the multilayer [141, 142].
The thermoelastic contribution to the coating loss measured by Yamamotoat room temperature was calculated using the material properties of silica andtantala given in Table 6.1 and the properties of sapphire given in Table 6.3.
To allow an estimate of the thermoelastic loss at low temperatures tobe made, the properties of tantala were approximated by those of sapphire,
for which the required temperature dependent properties were obtained from197

6.6 Multi-layer coating loss calculation

Coeﬃcient of thermal linear expansion

5.4 ×10−6 K−1

Young’s modulus

4 ×1011 Nm−2

Speciﬁc heat capacity

777 Jkg−1 K−1

Thermal conductivity

33 Wm−1 K−1Poisson’s ratio

0.23

Density

3980 kgm−3

Table 6.3: Material properties of sapphire used to calculate coating thermoelastic loss
[142].

10

Mechanical loss

10

10

10

10

10

-2

coating TE loss for SiO2/Al2O3 coating
coating TE loss SiO2/Ta2O5
SiO2 / Ta2O5 coating loss (Yamamoto)

-3

-4

-5

-6

-7

0

50

100

150

200

250

300

Temperature (K)Figure 6.19: Plot showing Yamamoto’s coating loss measurements at 1.1 kHz[226] and the calculated coating thermoelastic loss in (i) a 31 layer
SiO2 /Ta2 O5 coating on a sapphire substrate at room temperature and(ii) a SiO2 /Al2 O3 coating on a sapphire substrate as a function of temperature.

Touloukian [228]. The density, Poisson’s ratio and Young’s modulus were assumed not to vary signiﬁcantly with temperature.
The calculated coating thermoelastic loss is shown in Figure 6.19, alongside Yamamoto’s measured coating loss factors. At room temperature, the6.6 Multi-layer coating loss calculation

198

thermoelastic loss calculated for the silica/tantala coating on sapphire is 3.8
×10−4 , more than twice as high as the coating loss measured by Yamamoto.
This implies that the residual intrinsic loss of the coating, φcoating intrinsic =φcoating −φcoating thermoelastic , is negative, which is not a physically possible result.
This implies that the material properties of the actual coating and substrate
may diﬀer from the properties used in the calculation.
When the properties of tantala are approximated by those of sapphire, thecalculated coating thermoelastic loss at room temperature is approximately 50% lower than the measured coating loss, resulting in a residual coating loss of
8.8 ×10−5 . While this is a substantial correction at room temperature, it canbe seen that below 100 K the calculated coating thermoelastic loss is more than
two orders of magnitude lower than the measured coating loss factors. For thisapproximation, the thermoelastic correction at low temperature is therefore
negligible.
It is clearly of signiﬁcant interest to measure the temperature dependenceof α, k and C for ion-beam sputtered tantala and silica ﬁlms to allow moreaccurate calculation of the coating thermoelastic loss.

6.6.3

Discussion

There are several possible explanations for the diﬀerences between the results.
It is possible that systematic experimental factors such as variations in thecoating materials from diﬀerent vendors or thermometry errors between diﬀerent experiments could result in the measured dissipation peaking at diﬀerent
temperatures. The multi-layer data of Yamamoto are not inconsistent withthere being a narrow dissipation peak located between 20 and 30 K. The individual layers of silica and tantala studied in this thesis were 500 nm thick,
whereas in a standard multi-layer coating the layers of silica and tantala areapproximately 180 nm and 130 nm thick respectively. The diﬀerence in coating thickness, and therefore in surface to volume ratio, may also result in a

6.7 Conclusion

199

diﬀerence in the mechanical loss. It is also possible that diﬀerences in thetemperature at which the coatings were heat treated could result in the loss
peaking at a diﬀerent temperature.
An alternative explanation is that a single layer of coating material does
not display the same dissipation properties as a layer of the same material aspart of a multi-layer stack. It has been shown that the coating layer interfaces
are not a signiﬁcant source of dissipation [130]. However, the stress exerted oneach layer of a multi-layer coating may be signiﬁcantly diﬀerent to the internal
stress in a single layer of silica or tantala, possibly resulting in diﬀering levelsof dissipation.
Further work is required to understand if the loss in a multi-layer coating
can be accurately modelled from measurements of the losses of single layers of
the individual materials. In addition it would be very interesting to repeat themulti-layer coating loss measurements in signiﬁcantly more detail.

6.7

Conclusion

The temperature dependence of the dissipation in a silica coating applied byion-beam sputtering to a silicon cantilever has been measured. The results
show that the dissipation decreases to two broad minima centred on approximately 180 K and 115 K, separated by a small and broad peak at approximately
140 K. Below 100 K the dissipation increases to a peak at between 16 K and
20 K. This activation energy of the dissipation mechanism responsible for thispeak was calculated to be 54.4 ± 5.5 meV using the data from the ﬁrst four
bending modes. However, the data from the ﬁfth bending mode did not lie on
the same line on the Arrhenius plot as the other modes. The most likely explanation is that the calculated coating loss was ‘contaminated’ by clamping loss
eﬀects in the data for the uncoated control sample, altering the temperature atwhich the coating loss appeared to peak. This requires further investigation.
Broad peaks in the loss of the coated cantilever close to room temperature6.7 Conclusion

200

and around 225 K were observed and the coating loss calculated at room
temperature was signiﬁcantly higher than expected. Repeated measurements
at room temperature yielded a lower loss, and it seems likely that the original
losses measured above 200 K were artiﬁcially high due to clamping eﬀects.
However, the loss factors measured at room temperature were still up to anorder of magnitude higher than Penn’s results for an ion-beam sputtered silica
coating on a silica substrate.
The single layer coating loss results for silica and tantala have been used
to calculate the expected loss in a multi-layer coating of the type measured by
Yamamoto et al. The calculated multi-layer coating loss peaks at ∼ 20 K, witha maximum loss of ∼ 9.2×10−4 . This is almost a factor of 2 higher than themulti-layer coating loss of ∼5×10−4 measured at 20 K by Yamamoto. Further
comparison is diﬃcult due to the limited experimental data for the multi-layer
coating. However, Yamamoto’s data are not inconsistent with the presenceof a similar dissipation peak between 20 and 30 K. Further investigations of
the temperature dependence of the dissipation in multi-layer coatings, and its
relationship to the dissipation of the individual coating materials, is clearly ofgreat interest.

Chapter 7
Conclusions
The ﬁrst generation of long baseline interferometric gravitational wave detectors have reached their design sensitivities and acquired data for extended periods of time. Data analysis from the various science runs has allowed constraints
to be placed on astrophysical models of sources of gravitational radiation. The
adoption of the silica suspension technology and signal recycling techniques
pioneered in the GEO 600 detector by the longer baseline LIGO and VIRGO
detectors, alongside an increase in the circulating light power, will allow thesensitivity of these instruments to be increased by approximately a factor often. It is widely expected that these second generation detectors will make theﬁrst direct detection of gravitational waves.
Investigations into some aspects of the materials and processes which may
be used in future gravitational wave detectors have been presented in thisthesis. In particular, much of the research was aimed at understanding themechanical dissipation associated with the dielectric mirror coatings, allowing
reduction of the coating thermal noise in future detectors.
Studies of hydroxide catalysis bonding have shown that the bond settling
time can be increased by carrying out the bonding procedure at 0◦ C. This
result may be of use in future space missions, such as LISA, to allow moretime for the alignment of bonded components on an optical bench. The settling
time data allowed calculation of the activation energy of the chain of chemical
201

202
reactions involved in the bond formation. Similar studies may be useful to allow
comparison of the chemistry of hydroxide catalysis bonds between oxidisedsilicon surfaces, which may be required in third generation gravitational wave
detectors.
Brownian thermal noise arising from the reﬂective mirror coatings is likely
to limit the sensitivity of the next generation gravitational wave detectors intheir most sensitive frequency band. The magnitude of the coating thermal
noise is related to the mechanical dissipation factor of the coating materials.
It is thus important to determine the exact level of dissipation expected from
speciﬁc coatings, to understand the mechanism responsible for the dissipationand to ﬁnd methods of reducing it.
A method of measuring the dissipation in various coating materials at room
temperature using silica cantilever substrates has been developed. It has beenshown that a silica ribbon welded to a thicker silica clamping block can provide isolation from the clamping structure, reducing excess energy loss into theclamp. Silica cantilevers have been used to measure the mechanical loss of tinoxide coatings deposited by spray pyrolysis. It has been shown that tin oxide
coatings applied by spray pyrolysis with a maximum thickness of 4 - 8 nm
would meet the thermal noise requirements for second generation detectors.
Preliminary measurements of the dissipation in single layers of ion-beam sputtered silica and tantala, the materials used in the reﬂective coatings, have been
carried out at room temperature. The results suggest that altering a particular (undisclosed) deposition parameter can reduce the dissipation in tantala
coating layers doped with 14.5 % TiO2 by 40 to 50%.
Silicon cantilevers, which have a low mechanical loss at cryogenic temperatures in comparison to silica, have been used to measure the temperaturedependence of the dissipation in 500 nm thick ﬁlms of silica and tantala. Studies of single layers of tantala, heat treated at the same temperature as thestandard multi-layer coatings, have shown the presence of a dissipation peak203
at approximately 20 K. Analysis of this peak has provided the ﬁrst evidenceof the nature of the mechanical dissipation mechanism in this material, suggesting that it may be analogous to the double-well potential loss mechanism
observed in fused silica, with the loss arising when molecules cross energy barriers whilst undergoing changes in conﬁguration. The width of the dissipation
peak is indicative of a broad distribution of energy barrier heights. Doping
tantala with titania to a level of 14.5 % cation concentration has been shown
to reduce the magnitude of dissipation at the peak, and to increase the average
activation energy by ∼ 33%.
Initial studies suggest that the form of the low temperature dissipation in
ion-beam sputtered tantala is strongly related to the temperature at which thepost-deposition heat treatment is carried out. Measurements of a coating heat
treated at 300 ◦ C showed no dissipation peak at 20 K and a smaller, broader
dissipation peak at ∼ 35 K. Both a peak at ∼ 20 K and a second, very broaddissipation peak at ∼ 90 K were observed in a tantala coating which had crystallised after heat treatment at 800 ◦ C. It is likely that the broad peak centredon 90 K is related to the onset of polycrystalline structure in this coating. The
temperature dependence of the loss of an ion-beam sputtered silica coating has
also been measured, and a dissipation peak at ∼ 17 K was observed. The peakis at a lower temperature, and has a somewhat higher activation energy, than
the well-known dissipation peak in bulk silica. Comparison with the literaturesuggests that the method of fabrication can signiﬁcantly aﬀect the mechanicaldissipation in silica ﬁlms.
The loss factors measured for silica and tantala coatings have been used to
calculate the expected loss in a multi-layer coating. The calculated multi-layer
coating loss peaks at ∼20 K, with a maximum loss of ∼ 9.2×10−4 . This isalmost a factor of 2 higher than Yamamoto’s multi-layer coating loss factor of∼ 5×10−4 measured at 20 K. However, Yamamoto’s data are not inconsistent
with the presence of a similarly high dissipation peak between 20 and 30 K.

204Further investigations of the temperature dependence of the dissipation in
multi-layer coatings, and its relationship to the dissipation of the individual
coating materials, is an important topic of further research.
In summary, the source of mechanical dissipation in ion-beam sputteredtantala coatings has been probed for the ﬁrst time and the results suggest thatthe dissipation mechanism may involve structural reorientations in doublewell potentials. It has been shown that the dissipation of the tantala layers is
likely to dominate the total dissipation of a multi-layer silica / tantala coating
between 10 K and 300 K. Further study of the eﬀects of doping and heat treatment on the low temperature dissipation peaks in tantala may facilitate thedevelopment of lower loss coatings for use at both room and cryogenic temperatures, thus enabling the sensitivity of future gravitational wave detectors
to be signiﬁcantly enhanced.

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