aspects, techniques and design of advanced interferometric ...€¦ · chapter five was undertaken...

Aspects, techniques and design of

advanced interferometric

gravitational wave detectors

by

Pablo J. Barriga Campino

Submitted in partial fulfilment of

the requirements for the degree of

Doctor of Philosophy

School of Physics

The University of Western Australia

2009

To:

Sam and our future...

Abstract

The research described in this thesis investigates some of the key technologies required

to improve the sensitivity of the next generation of interferometric gravitational wave

detectors. A complete optical design of a high optical power, suspended mode-cleaner

was undertaken in order to reduce any spatial or frequency instability of the input

laser beam. This includes a study of thermal effects due to high circulating power, and

a separate study of a vibration isolation system. In the last few years the Australian

Consortium for Interferometric Gravitational Astronomy (ACIGA) has developed an

advanced vibration isolation system, which is planned for use in the Australian Inter-

national Gravitational Observatory (AIGO). A local control system originally devel-

oped for the mode-cleaner vibration isolator has evolved for application to the main

vibration isolation system. Two vibration isolator systems have been assembled and

installed at the Gingin Test Facility (∼ 80 km north of Perth in Western Australia)

for performance testing, requiring installation of a Nd:YAG laser to measure the cav-

ity longitudinal residual motion. Results demonstrate residual motion at nanometre

level at 1 Hz.

Increasing the circulating power in the main arm cavities of the interferometer

can amplify the photon-phonon interaction between the test mass and the circulating

beam, enhancing the three-mode parametric interaction and creating an optical spring

effect. As part of a broader study of parametric instabilities, in collaboration with the

California Institute of Technology, a simulation of the circulating beam in the main

arms of Advanced LIGO was completed to determine the characteristics of the higher

order optical modes. The simulation encompassed diffraction losses, optical gain,

optical mode Q-factor and mode frequency separation. The results are presented for

varying mode orders as a function of mirror diameter. The effect of test mass tilt

on diffraction losses, and different coatings configurations are also presented. These

simulations led to a study of the effect of power recycling cavities in higher order mode

i

suppression. Current interferometric gravitational wave detectors in operation have

marginally stable recycling cavities, where higher order modes are enhanced by the

power recycling cavity, effectively increasing the parametric gain. A study of stable

recycling cavities was undertaken in collaboration with the University of Florida,

which has evolved into a proposed design for a 5 kilometre AIGO interferometer.

This thesis concludes with an analysis of the work here presented and an out-

line of future work that will help to improve the design of advanced interferometric

gravitational wave detectors.

ii

Acknowledgements

This thesis would have not been possible without the support, help and contribution

of many people. First and foremost I would like to thank my supervisors Prof. David

Blair, Dr. Li Ju (Juli) and Dr. Chunnong Zhao for their support and encouragement

throughout this research.

I am grateful to David for welcoming me into the gravitational waves group fresh

from Chile, for finding the time to discuss and review different technical aspects of

this research within his always busy schedule and for driving this project. I thank

him for his confidence in me representing the group at international conferences and

his understanding of ‘Spanglish’.

I thank Juli for always being an open and approachable person, ready to help in

different aspects of this research, including lending me her desktop computer to run

simulations for several weeks.

I also thank Zhao for his support and knowledge in the more technical aspects of

this research, his guidance in the Gingin experiments were invaluable. It is unfortu-

nate the busy work schedule worked against us having more football games.

There are also a large number of people that have helped me in one way or another

throughout this research. Thank you Riccardo DeSalvo and Phil Willems for being

great hosts during my visit to the California Institute of Technology and P. Willems

for his time in discussing the FFT code, which he helped to improve. Thanks also

to Guido Mueller for his support and patience during the development of the optical

design for recycling cavities and useful discussions during the Gingin workshop and

afterwards. Also a big thanks to the people at LIGO Hanford for making those days

and night shifts more enjoyable (especially during the ‘owl’ shift) and to to the people

at ANU and Adelaide University with whom I shared some good moments in various

conferences.

I would like to thank many people for their friendship and support; Jerome and

iii

Sascha; Andrew and Jean Charles (long days and nights in Gingin making things

work); the regular Gingin pilgrims Fan, Lucianne and Sunil; and David Coward and

Eric. Special thanks to Slawek for sharing enormous amounts of coffee and invaluable

discussions about gravitational waves and any other topics that came to mind.

Of course none of these would have been possible without the love and support of

my family (Francisco, Linda and Laura, Rodrigo, Daphne and Emilia) and my parents

Pablo and Victoria. We went through difficult times when the distance between Chile

and Australia did not help at all. My sister Alejandra is dearly missed. I would also

like to thank Nigel and Christine for their support and interest in what their favourite

son in law is doing.

And last, but definitely not least, thank you to my wife Sam for her support,

encouragement, love, faith, patience, for a few well-deserved kicks every now-and-

then, and for reading this thesis (more than once) and for her interest in understanding

what I have been doing all these years.

iv

Preface

This Preface presents an overview of the content of this thesis. The chapters predom-

inantly correspond to published papers, submitted papers and a report to the LIGO

Scientific Collaboration. The chapters in this thesis present advanced vibration iso-

lation systems, optical design and simulation of optical sub-systems for advanced

interferometric gravitational wave detectors. The order of the chapters is not related

to the time order of the work, nor to the order in which the papers were published.

The Introduction (Chapter One) presents a review of interferometric gravitational

wave detectors including the history behind the development of this field and its fu-

ture direction. It also describes a number of other projects around the world that are

aiming to directly detect gravitational waves and presents some of the noise sources

that need to be overcome.

The two following chapters are related to seismic noise and advanced vibration

isolation systems. Chapter Two presents a novel design for a mode-cleaner vibra-

tion isolation system. The content of this chapter corresponds to early work from

the author in vibration isolation and contains predictions on the performance of the

proposed isolation system. Research undertaken on the control electronics and the

local control system for the mode-cleaner vibration isolator was then upgraded for

use with an advanced vibration isolation system. This work is presented in Chapter

Three, which describes the assembly of two advanced vibration isolators at the High

Optical Power Facility in Gingin, 80 km north of Perth in Western Australia. This

chapter contains two recently submitted papers, which present the performance of

the vibration isolation systems and the the local control system developed for their

operation. The complete design of the control electronics for the advanced vibration

isolator is presented in Appendix C.

Chapter Four contains three papers which present the research undertaken for the

optical design of a high power mode-cleaner. This work was started during develop-

v

ment of the vibration isolation system for the mode-cleaner. The first paper presents

the basic optical design of the triangular ring cavity used as an optical mode-cleaner,

including a preliminary study of thermal effects in this cavity. The thermal effects

and their consequences are then studied thoroughly in the following paper. In the

final paper of this chapter a design for an astigmatism free mode-cleaner is proposed.

Continuing with the study of optical modes suppression in advanced interferome-

ters, Chapter Five simulates the behaviour of higher order optical modes in the main

arms of an advanced interferometer. Although the simulations were made specifically

for an Advanced LIGO type of cavity, they can be easily extended to any advanced

interferometric gravitational wave detector. The calculations of the diffraction losses

are an important component of the parametric instabilities studies carried out by the

University of Western Australia. The studies by the author also included the effects

of mirror tilt and how they affect diffraction losses. A study of the potential use

of an apodising coating to reduce the possibility of parametric instabilities is also

presented. This work was presented at the California Institute of Technology and

later published as a technical report (T060273-00-Z). Research forming the core of

Chapter Five was undertaken in 2006 using the design parameters of Advanced LIGO

at that time. Since then the design parameters for Advanced LIGO have changed and

an actualisation of the main calculations is presented as a postscript in this chapter.

Appendix B presents an overview of the theory behind parametric instabilities and

strategies for the control of parametric instabilities in advanced gravitational wave

detectors.

The studies of optical modes are then extended in Chapter Six to include recy-

cling cavities and the design of a dual recycled interferometer for gravitational wave

detection, utilising stable recycling cavities instead of the marginally stable cavities

used in current interferometric detectors. The publication at the core of this chapter

was developed in collaboration with the University of Florida following a paramet-

ric instabilities workshop held in Gingin. This chapter presents a proposed design

for the Australian International Gravitational Observatory (AIGO). This design is

complemented in Appendix A with a description of the science benefits of a southern

hemisphere advanced gravitational wave detector, work that was presented at the

vi

http://www.ligo.caltech.edu/docs/T/T060273-00.pdf

Amaldi 7 conference on gravitational waves.

The Conclusions present a discussion of the work presented in this thesis and future

work. This includes the proposed new optical design for an astigmatism-free mode-

cleaner using a novel and compact vibration isolation system. This design could also

be used for auxiliary optics in advanced gravitational wave detectors. An advanced

vibration isolator was developed through a combination of different pre-isolation tech-

niques and tested at the AIGO test facility. This included the development of the

control electronics and a local control system, and demonstration of a residual motion

of 1 nm at 1 Hz. However this performance could be improved with the addition of

actuators on the Roberts linkage stage. This addition will improve performance at

low frequencies with the use of a super-spring configuration. The analysis of recycling

cavities for advanced gravitational wave detectors gives promising results, although

further analysis including their effects in parametric instabilities is still necessary.

vii

Publications used for this thesis

P. Barriga, A. Woolley, C. Zhao, D. G. Blair, “Application of new pre-isolation

techniques to mode-cleaner design,” Class. Quantum Grav. 21 (2004) S951–

S958 (Chapter 2, sections 2.2–2.5).

P. Barriga, J.-C. Dumas, C. Zhao, L. Ju, D. G. Blair, “Compact vibration

isolation and suspension system for AIGO: Performance in a 72 m Fabry-Perot

cavity,” Rev. Sci. Instrum., 2009, submitted (Chapter 3, sections 3.2–3.6).

J.-C. Dumas, P. Barriga, C. Zhao, L. Ju, and D. G. Blair, ”Compact suspension

systems for AIGO: Local control system,” Rev. Sci. Instrum. 2009, submitted

(Chapter 3, sections 3.8–3.12).

P. J. Barriga, C. Zhao, D. G. Blair, “Astigmatism compensation in mode-cleaner

cavities for the next generation of gravitational wave interferometric detectors,”

Phys. Lett. A 340 (2005) 1–6 (Chapter 4, sections 4.2–4.5).

P. Barriga, C. Zhao, D. G. Blair, “Optical design of a high power mode-cleaner

for AIGO,” Gen. Relat. Gravit. 37 (2005) 1609–1619 (Chapter 4, sections 4.6–

4.9).

P. Barriga, C. Zhao, L. Ju, D. G. Blair, “Self-Compensation of Astigmatism in

Mode-Cleaners for Advanced Interferometers,” J. Phys. Conf. Ser. 32 (2006)

457–463 (Chapter 4, sections 4.10–4.13).

P. Barriga, B. Bhawal, L. Ju, D. G. Blair, “Numerical calculations of diffraction

losses in advanced interferometric gravitational wave detectors,” J. Opt. Soc.

Am. A 24 (2007) 1731–1741 (Chapter 5, sections 5.2–5.6)

P. Barriga and R. DeSalvo, “Study of the possible reduction of parametric

instability gain using apodizing coatings in test masses,” Technical Report,

ix

T060273-00-Z, LIGO, (2006) (Chapter 5, section 5.7).

P. Barriga, M. A. Arain, G. Mueller, C. Zhao, D. G. Blair, “Optical design of the

proposed Australian International Gravitational Observatory,” Opt. Express 17

(2009) 2149–2165 (Chapter 6, sections 6.2–6.7)

D. G. Blair, P. Barriga, A. F. Brooks, P. Charlton, D. Coward, J-C. Dumas,

Y. Fan, D. Galloway, S. Gras, D. J. Hosken, E. Howell, S. Hughes, L. Ju, D. E.

McClelland, A. Melatos, H. Miao, J. Munch, S. M. Scott, B. J. J. Slagmolen,

P. J. Veitch, L. Wen, J. K. Webb, A. Wolley, Z. Yan, C. Zhao, “The Science

benefits and Preliminary Design of the Southern hemisphere Gravitational Wave

Detector AIGO,” J. Phys. Conf. Ser. 122 (2008) 012001 (6pp) (Appendix A)

L. Ju, D. G. Blair, C. Zhao, S. Gras, Z. Zhang, P. Barriga, H. Miao, Y. Fan

and L. Merrill, “Strategies for the control of parametric instability in advanced

gravitational wave detectors,” Class. Quantum Grav. 26 (2009) 015002 (15pp)

(Appendix B)

x

Contents

Abstract i

Acknowledgements iii

Preface v

Publications used for this thesis ix

Contents xi

List of Figures xvii

List of Tables xxiii

1 Introduction 1

1.1 A bit of history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Gravitational wave detection . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Interferometric gravitational wave detectors . . . . . . . . . . . . . . 5

1.3.1 Seismic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Laser source noise . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Optical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.4 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.5 Readout scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.6 Output mode-cleaner . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Sensing and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 A network of gravitational wave detectors . . . . . . . . . . . . . . . 21

1.5.1 Resonant detectors . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.2 Interferometric detectors . . . . . . . . . . . . . . . . . . . . . 22

xi

1.5.3 Space interferometry . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.4 Pulsar timing . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Mode-Cleaner Vibration Isolator 37

2.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Isolation and suspension design . . . . . . . . . . . . . . . . . . . . . 40

2.4 Noise and locking predictions . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Advanced Vibration Isolator 55

3.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Vibration isolation design . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.1 Pre-isolation components . . . . . . . . . . . . . . . . . . . . . 61

3.3.2 Isolation stages . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.3 Control mass and test mass suspension . . . . . . . . . . . . . 64

3.3.4 Integrated system . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.1 Cavity parameters . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.2 Suspension system transfer function . . . . . . . . . . . . . . . 69

3.4.3 Laser control system . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Measurements and results . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . 77

3.7 Local control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.8 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.9 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.9.1 Isolator components . . . . . . . . . . . . . . . . . . . . . . . 82

xii

3.9.2 Control hardware . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.9.3 Control implementation and degrees of freedom . . . . . . . . 87

3.9.4 Optical lever . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.9.5 The digital controller . . . . . . . . . . . . . . . . . . . . . . . 90

3.10 Control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.10.1 Pre-isolation feedback . . . . . . . . . . . . . . . . . . . . . . 94

3.10.2 Control mass feedback . . . . . . . . . . . . . . . . . . . . . . 96

3.10.3 Optimised feedback for pre-isolation . . . . . . . . . . . . . . . 97

3.11 Initial cavity measurements . . . . . . . . . . . . . . . . . . . . . . . 100

3.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.13 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 Mode-Cleaner Optical Design 111

4.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.3 The mode-cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.4 Mode-cleaner thermal lensing . . . . . . . . . . . . . . . . . . . . . . 120

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.7 Mode-cleaner intrinsic astigmatism . . . . . . . . . . . . . . . . . . . 128

4.8 Astigmatism free mode-cleaner . . . . . . . . . . . . . . . . . . . . . . 131

4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.11 Substrate deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.12 Thermal lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5 Diffraction Losses and Parametric Instabilities 147

5.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xiii

5.3 Diffraction losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.4 FFT simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.5.1 Diffraction losses . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.5.2 Optical gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.5.3 Mode frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.5.4 Mode shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.7 Apodising coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.7.2 Apodising coating . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.8 Mirror tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.8.2 Mirror geometry . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.8.3 Mirror tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.9 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6 Stable Recycling Cavities 199

6.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.3 Dual recycling interferometers . . . . . . . . . . . . . . . . . . . . . . 204

6.4 Higher order modes suppression . . . . . . . . . . . . . . . . . . . . . 205

6.5 Possible solutions for stable recycling cavities . . . . . . . . . . . . . 209

6.5.1 Straight stable recycling cavity . . . . . . . . . . . . . . . . . 209

6.5.2 Folded stable recycling cavity . . . . . . . . . . . . . . . . . . 212

6.5.3 Comparison between designs . . . . . . . . . . . . . . . . . . . 215

6.6 Sidebands and the stable recycling cavities . . . . . . . . . . . . . . . 220

6.6.1 Modulation frequencies calculations . . . . . . . . . . . . . . . 220

xiv

6.6.2 Signal recycling cavity . . . . . . . . . . . . . . . . . . . . . . 222

6.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.8 Beam-splitter thermal effects . . . . . . . . . . . . . . . . . . . . . . . 228

6.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6.8.2 Astigmatism in folded design . . . . . . . . . . . . . . . . . . 229

6.8.3 Thermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.8.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 234

6.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7 Summary and Conclusions 239

Appendices 243

A Science Benefits of AIGO 245

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

A.2 Scientific benefits of the AIGO observatory . . . . . . . . . . . . . . . 247

A.3 Preliminary conceptual design for AIGO . . . . . . . . . . . . . . . . 251

A.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 253

A.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

B Control of Parametric Instabilities 257

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

B.2 Parametric instabilities theory and modelling . . . . . . . . . . . . . . 261

B.2.1 Summary of theory . . . . . . . . . . . . . . . . . . . . . . . . 261

B.2.2 Modelling approach . . . . . . . . . . . . . . . . . . . . . . . . 264

B.2.3 Modelling results . . . . . . . . . . . . . . . . . . . . . . . . . 265

B.3 Possible approaches to PI control . . . . . . . . . . . . . . . . . . . . 266

B.3.1 Power reduction and thermal radius of curvature control . . . 266

B.3.2 Ring dampers or resonant acoustic dampers . . . . . . . . . . 269

B.3.3 Acoustic excitation sensing and feedback . . . . . . . . . . . . 272

B.3.4 Global optical sensing and electrostatic actuation . . . . . . . 273

B.3.5 Global optical sensing and direct radiation pressure . . . . . . 274

xv

B.3.6 Global optical sensing and optical feedback . . . . . . . . . . . 275

B.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

B.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

C Vibration Isolator Control Electronics 283

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

C.2 Control electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

C.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

D Publications List 313

D.1 Principal author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

D.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

D.3 ACIGA collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

D.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

D.5 LIGO Scientific Collaboration . . . . . . . . . . . . . . . . . . . . . . 317

D.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

xvi

List of Figures

1.1 An artist’s representation of a gravitational wave. . . . . . . . . . . . 2

1.2 The two polarisations of gravitational wave radiation. . . . . . . . . . 3

1.3 Simple Michelson interferometer. . . . . . . . . . . . . . . . . . . . . 5

1.4 Interferometer with Fabry - Perot cavities as arms. . . . . . . . . . . . 7

1.5 Dual recycled interferometer. . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Classical mode-cleaner layout. . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Dual recycled interferometer. . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 Distribution of the different sensing apparatus of a GW detector. . . 18

1.9 Dual-recycled advanced interferometric GW detector. . . . . . . . . . 19

1.10 Measured sensitivity of different GW detectors. . . . . . . . . . . . . 23

1.11 Angular uncertainty maps. . . . . . . . . . . . . . . . . . . . . . . . . 25

1.12 An artist’s representation of the LISA mission. . . . . . . . . . . . . . 27

2.1 Mode-cleaner suspension design. . . . . . . . . . . . . . . . . . . . . . 41

2.2 Frequency space representation of the isolator transfer function. . . . 41

2.3 The AIGO mode-cleaner layout. . . . . . . . . . . . . . . . . . . . . . 42

2.4 Mode-cleaner suppression factor. . . . . . . . . . . . . . . . . . . . . . 44

2.5 Horizontal and vertical seismic noise and predicted system response. . 45

2.6 Mode-cleaner residual motion. . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Mode-cleaner suspension (top view). . . . . . . . . . . . . . . . . . . 48

2.8 Comparison between theoretical and measured transfer function. . . . 49

2.9 Mode-cleaner vibration isolator. . . . . . . . . . . . . . . . . . . . . . 51

3.1 Full vibration isolator system for AIGO. . . . . . . . . . . . . . . . . 61

3.2 Inverse pendulum and LaCoste schematic. . . . . . . . . . . . . . . . 62

3.3 The Roberts linkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xvii

3.4 Detail of the multi-stage pendulum. . . . . . . . . . . . . . . . . . . . 64

3.5 Diagram of one self-damped pendulum stage. . . . . . . . . . . . . . . 65

3.6 Euler Spring vertical stage diagram. . . . . . . . . . . . . . . . . . . . 65

3.7 The combined 3D stage. . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.8 Control mass stage with test mass suspended. . . . . . . . . . . . . . 67

3.9 ETM during the assembly of the second suspension system. . . . . . . 68

3.10 Average of the cavity decay time. . . . . . . . . . . . . . . . . . . . . 70

3.11 ITM horizontal mechanical transfer function. . . . . . . . . . . . . . . 71

3.12 Laser control system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.13 Semi-theoretical transfer function. . . . . . . . . . . . . . . . . . . . . 73

3.14 Control loop frequency response. . . . . . . . . . . . . . . . . . . . . 74

3.15 Locked cavity frequency response. . . . . . . . . . . . . . . . . . . . . 75

3.16 Locked cavity residual motion. . . . . . . . . . . . . . . . . . . . . . . 75

3.17 Pitch and yaw angular residual motion for the ITM. . . . . . . . . . . 76

3.18 Isolation stages of the AIGO suspension chain. . . . . . . . . . . . . . 83

3.19 The shadow sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.20 The magnet-coil actuator. . . . . . . . . . . . . . . . . . . . . . . . . 86

3.21 Schematic of the inverse pendulum control. . . . . . . . . . . . . . . . 88

3.22 Schematic of the LaCoste control. . . . . . . . . . . . . . . . . . . . . 88

3.23 Schematic of the Roberts linkage control. . . . . . . . . . . . . . . . . 89

3.24 Schematic horizontal control of the control mass. . . . . . . . . . . . . 89

3.25 Schematic of the pitch control of the control mass. . . . . . . . . . . . 90

3.26 The optical lever setup. . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.27 Block diagram of the isolation local control system. . . . . . . . . . . 93

3.28 Loop gain and closed loop transfer function of the inverse pendulum. 95

3.29 Test mass frequency response. . . . . . . . . . . . . . . . . . . . . . . 98

3.30 Test mass yaw angular motion. . . . . . . . . . . . . . . . . . . . . . 99

3.31 Illustration of the proposed pre-isolator feedback control. . . . . . . . 99

3.32 Measured integrated residual motion of the cavity. . . . . . . . . . . . 101

3.33 Time distribution of the power inside the cavity. . . . . . . . . . . . . 102

3.34 Test mass angular motion. . . . . . . . . . . . . . . . . . . . . . . . . 103

xviii

3.35 Test mass angular oscillation histograms. . . . . . . . . . . . . . . . . 104

4.1 Simplified schematic showing the layout of the AIGO mode-cleaner. . 113

4.2 Transmission of the higher order modes. . . . . . . . . . . . . . . . . 119

4.3 Detail of transmission of the higher order modes. . . . . . . . . . . . 120

4.4 Triangular ring cavity layout used as a mode-cleaner. . . . . . . . . . 127

4.5 Spots eccentricity M1/M2, M3 and waist for AIGO mode-cleaner. . . 129

4.6 Eccentricity variation with input power. . . . . . . . . . . . . . . . . 131

4.7 Mode-cleaner optical setup. . . . . . . . . . . . . . . . . . . . . . . . 136

4.8 Mode-cleaner spot size simulation. . . . . . . . . . . . . . . . . . . . . 136

4.9 Waist and M3 spot eccentricity variation with input power. . . . . . . 138

4.10 Steady state solution for the bulk absorption case. . . . . . . . . . . . 139

4.11 Steady state temperature distributions for coating absorption. . . . . 140

5.1 Advanced LIGO substrate dimensions. . . . . . . . . . . . . . . . . . 158

5.2 Diffraction losses for different higher order modes. . . . . . . . . . . . 159

5.3 Intensity profile at the ITM, mirror of diameter 34 cm. . . . . . . . . 160

5.4 Comparison of diffraction losses. . . . . . . . . . . . . . . . . . . . . . 161

5.5 Cavity optical gain for some HG modes of different orders. . . . . . . 162

5.6 Optical gain variation for higher order modes. . . . . . . . . . . . . . 163

5.7 Intensity profile variation of mode HG40. . . . . . . . . . . . . . . . . 163

5.8 Modes of order 7 as they would appear in an infinite sized mirror. . . 165

5.9 Frequency variations from the theoretical value. . . . . . . . . . . . . 167

5.10 Optical Q-factor for the higher order modes in the proposed cavity. . 168

5.11 Intensity profile of mode HG33 in a infinite sized mirrors. . . . . . . . 169

5.12 Diffraction losses for an Advanced LIGO type cavity. . . . . . . . . . 172

5.13 Proposed apodising coatings design for ITM and ETM. . . . . . . . . 173

5.14 Diffraction losses of higher order modes. . . . . . . . . . . . . . . . . 174

5.15 Ratio between the different coatings. . . . . . . . . . . . . . . . . . . 175

5.16 Diffraction losses comparison between homogeneous apodising coatings. 176

5.17 Different dielectric absorption for higher order modes. . . . . . . . . . 177

5.18 Optical mode LG21 in a perfectly aligned cavity with circular mirrors. 181

xix

5.19 Optical mode LG21 in a perfectly aligned cavity with flat sides. . . . 181

5.20 Optical mode LG03 showing the effect of the flat sides. . . . . . . . . 182

5.21 Influence of mirror geometry in diffraction losses. . . . . . . . . . . . 184

5.22 Frequency change with mirror size. . . . . . . . . . . . . . . . . . . . 185

5.23 Frequency change with mirror size for modes of order 7. . . . . . . . . 186

5.24 Change in the optical path for the laser beam due to mirror tilt. . . . 186

5.25 Change in the path length of the laser beam. . . . . . . . . . . . . . . 187

5.26 Spot size tuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.27 Frequency variation with mirror tilt for modes of order 4. . . . . . . . 191

5.28 Transverse mode frequency variation with mirror tilt. . . . . . . . . . 192

5.29 Comparison of diffraction losses between Advanced LIGO designs. . . 194

5.30 Comparison of higher order modes optical gain. . . . . . . . . . . . . 194

5.31 Q-factor comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.32 Frequency shift comparison. . . . . . . . . . . . . . . . . . . . . . . . 195

6.1 Configuration of the proposed AIGO interferometer. . . . . . . . . . . 203

6.2 Transmission of HOM as function of the Gouy phase shift. . . . . . . 207

6.3 Comparison of the intensity suppression of HOM. . . . . . . . . . . . 208

6.4 Proposed stable PRC design for AIGO advanced interferometer. . . . 210

6.5 Mode-matching drop as a function of PR1. . . . . . . . . . . . . . . . 211

6.6 Proposed stable PRC design for AIGO advanced interferometer. . . . 213

6.7 Mode-matching as a function of proposed radius of curvature. . . . . 215

6.8 Variation of the accumulated Gouy phase. . . . . . . . . . . . . . . . 217

6.9 Two possible solutions for the AIGO stable recycling cavity. . . . . . 228

6.10 Astigmatism comparison between AIGO and Adv. LIGO designs. . . 230

6.11 Temperature profile in the BS due to the PRC circulating power. . . 232

6.12 Comparison of the in-line arm waist position and size. . . . . . . . . . 233

6.13 Comparison of the perpendicular arm waist position and size. . . . . 233

A.1 Angular area maps for world array. . . . . . . . . . . . . . . . . . . . 250

A.2 Expected average number of galaxies. . . . . . . . . . . . . . . . . . . 251

B.1 Parametric scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . 261

xx

B.2 Three-mode interactions . . . . . . . . . . . . . . . . . . . . . . . . . 262

B.3 Number of unstable modes as function of ETM radius of curvature. . 266

B.4 Maximum parametric gain with different ETM radius of curvature. . 267

B.5 The parametric gain distribution of unstable modes. . . . . . . . . . . 267

B.6 Example of unstable modes suppression using a ring damper. . . . . . 270

B.7 Internal modes Q-factor of a test mass with small resonant damper. . 271

B.8 Fields of the fundamental mode, high order mode and cavity feedback. 276

B.9 Schematic diagram of the PI optical feedback control. . . . . . . . . . 277

C.1 Block diagram of the local control. . . . . . . . . . . . . . . . . . . . 284

C.2 Control electronics diagram . . . . . . . . . . . . . . . . . . . . . . . 285

C.3 Control electronics board distribution. . . . . . . . . . . . . . . . . . 286

C.4 Vibration isolator sensors and actuators distribution . . . . . . . . . . 287

C.5 6-way cross for wire feed-through. . . . . . . . . . . . . . . . . . . . . 288

C.6 The intermediate board at the ITM vibration isolator. . . . . . . . . 289

C.7 Control board block diagram. . . . . . . . . . . . . . . . . . . . . . . 290

C.8 LED circuit diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

C.9 Photo-detectors circuit diagram. . . . . . . . . . . . . . . . . . . . . . 292

C.10 Control signal circuit diagram. . . . . . . . . . . . . . . . . . . . . . . 293

C.11 Power supply circuit diagram. . . . . . . . . . . . . . . . . . . . . . . 294

C.12 P connector signal distribution diagram. . . . . . . . . . . . . . . . . 295

C.13 Filters board and power supply circuit diagram. . . . . . . . . . . . . 296

C.14 High current signal filters circuit diagram. . . . . . . . . . . . . . . . 297

C.15 Intermediate board block diagram. . . . . . . . . . . . . . . . . . . . 298

C.16 Horizontal axis connections circuit diagram. . . . . . . . . . . . . . . 299

C.17 Vertical axis connections circuit diagram. . . . . . . . . . . . . . . . . 300

C.18 Control mass connections circuit diagram. . . . . . . . . . . . . . . . 301

C.19 Electrostatic board connections circuit diagram. . . . . . . . . . . . . 302

C.20 Backplane board block diagram. . . . . . . . . . . . . . . . . . . . . . 303

C.21 Backplane horizontal axis signal distribution circuit diagram. . . . . . 304

C.22 Backplane vertical axis signal distribution circuit diagram. . . . . . . 305

C.23 Backplane Roberts linkage signal distribution circuit diagram. . . . . 306

xxi

C.24 Backplane filter signal distribution circuit diagram. . . . . . . . . . . 307

C.25 Backplane control mass horizontal signal distribution circuit diagram. 308

C.26 Backplane control mass tilt signal distribution circuit diagram. . . . . 309

C.27 Backplane DB–25 signal distribution diagram. . . . . . . . . . . . . . 310

C.28 Backplane Sub–D 100 pin signal distribution diagram. . . . . . . . . . 311

xxii

List of Tables

2.1 Design parameters of different mode-cleaners. . . . . . . . . . . . . . 39

2.2 Parameters affected by different radii of curvature. . . . . . . . . . . . 43

2.3 Theoretical parameters of the AIGO 12 m mode-cleaner. . . . . . . . 45

3.1 Parameters for the 72 m cavity. . . . . . . . . . . . . . . . . . . . . . 69

3.2 Cavity parameters derived from our measurements. . . . . . . . . . . 70

3.3 I/O channel allocation usage of DSP. . . . . . . . . . . . . . . . . . . 91

3.4 Stages and degrees of freedom in the control scheme. . . . . . . . . . 94

4.1 Mode-cleaner configurations used by other GW interferometers. . . . 114

4.2 Difference between hot and cold parameters for AIGO mode-cleaner. 122

4.3 Comparison of waist and M3 spot sizes. . . . . . . . . . . . . . . . . . 128

4.4 Output beam thermal lensing focal length. . . . . . . . . . . . . . . . 142

5.1 Diffraction losses and cavity optical gain of mode HG40. . . . . . . . 164

5.2 Frequency shift of modes of order 7 for different sized mirrors. . . . . 166

5.3 Diffraction losses for the HG00 mode using different coatings. . . . . . 174

5.4 Comparison between LG21 modes in mirrors with different geometry. 183

5.5 Comparison of Advanced LIGO design parameters. . . . . . . . . . . 193

6.1 Parameters of the arm cavities for the AIGO interferometer. . . . . . 205

6.2 Distance between the different optical components. . . . . . . . . . . 219

6.3 Focal length of the different optical components. . . . . . . . . . . . . 219

6.4 Development of the spot size radius on different optical components. . 220

6.5 Optical length of the different cavities proposed for the AIGO. . . . . 225

6.6 Distance between the different mirrors for the proposed AIGO. . . . . 225

6.7 Astigmatism induced in the stable folded cavity designs. . . . . . . . 229

xxiii

6.8 Comparison between thermal effects in both arms of the interferometer. 234

Chapter 1

Introduction

1.1 A bit of history

In November 1915 Albert Einstein finishes his work on the General Theory of Relativ-

ity and presents it in a 4-part speech at the Prussian Academy of Sciences. Einstein

has found a way to present the laws of physics independent of the frame in which they

were expressed. Consistent with Special Relativity he explained gravity as distortions

in the fabric of space, with the associated wave phenomenon being a gravitational

wave (GW). In this way he described the structure of space-time and gravity as

the simple manifestation of space-time curvature. The more massive the object, the

greater the curvature it causes, and hence the stronger the gravity. As massive objects

move around in space-time, this curvature will change. As a consequence, a moving

object or system of objects will cause fluctuations in space-time that spread outward

like ripples in the surface of a pond. These ripples are gravitational waves and like

any wave they carry energy (and therefore information) from a source.

The curvature of space-time is governed by the Einstein field equation:

Gµν =8πGN

c4Tµν (1.1)

Where Gµν is the Einstein tensor, GN is Newton’s gravitational constant, c is

the speed of light and Tµν is the stress-energy tensor. Put simply by John Wheeler

the equation can be interpreted as “matter tells space how to curve, in turn space

tells matter how to move”. The scalar constant linking the two tensors has a value

of ∼2×10−43, an extremely small value that shows that the interaction between the

distribution of matter and energy and the distortion of space-time is very weak.

1

2 CHAPTER 1. INTRODUCTION

Figure 1.1: An artist’s representation of a GW caused by two massive objects that are

orbiting each other. Gravitational waves are propagating gravitational fields, ‘ripples’ in

the curvature of space-time, generated by the motion of massive particles, such as two stars

or two black holes orbiting each other (figure courtesy of J. C. Dumas).

The only Lorenz-invariant speed is the speed of light. As a consequence and simi-

lar to electromagnetic waves it is expected that gravitational waves propagate at the

speed of light, they can not propagate with infinite speed. They also have different

polarisations, but unlike electromagnetic waves that are generated by a dipole, grav-

itational waves are generated by a quadrupole and therefore they can have two types

of polarisations, plus (+) or cross (×). However they can also be linearly or circularly

polarised, with any combination expressed in terms of the basic polarisations + or ×.

Consider a ring of particles floating in space as shown in figure 1.2. As a GW

passes through these particles it will perturb the distance between them. Due to

its transverse nature, a GW propagating into the page will perturb the distances

in the plane of the page. For the first quarter of the GW period the particles will

be stretched apart in one axis (chosen here for convenience as the vertical), and be

brought closer together in the perpendicular (horizontal) axis (h+ at π/2 radians).

During the second and third quarters of the GW period the distance between the test

particles will contract in the vertical axis and expand in the horizontal direction, as

illustrated in figure 1.2 (h+ at 3π/2 radians), passing through the initial distribution

at π radians. During the last quarter the particles are stretched in the vertical axis

1.1. A BIT OF HISTORY 3

0 π/2 π 3π/2 2π

h+

hx

Figure 1.2: The two polarisations of GW radiation. The major axis of the elliptical

displacement is determined by the polarisation of the GW. The axes of h+ and h× differ by

π/4 radians.

and contract in the horizontal returning to the initial distribution and completing

a full period (h+ at 2π radians). This is an example of one polarisation, normally

referred to as (+) or h+ polarisation. An orthogonally polarised wave, denoted by

(×) or h×, would have the axes of the distortions rotated by 45o.

The strength of a GW is measured by its strain, h, which gives an indication of

the fractional length change induced by the passing wave. The amount of distortion

shown in figure 1.2 is grossly exaggerated. The action of a GW will produce a relative

deformation ∆ l/l ∼ h, this dimensionless parameter is a measure of the deviation

of the metric from the Euclidean metric in the field of a GW, which is of the order

of 10−21 to 10−22. If such a wave was to pass between the earth and the sun, their

separation would change by less than the radius of a hydrogen atom. To detect the

presence of a GW passing between two objects, we must be able to measure the

changes in their separation with unprecedented accuracy.

In 1974 Russell Hulse and Joseph Taylor found indirect evidence for gravitational

waves in observations of a binary pulsar known as PSR1913+16, the Hulse-Taylor

Binary Pulsar [1]. Their efforts were recognised with a Nobel Prize in 1993, but

experiments designed to detect the waves directly have so far drawn a blank. However

upper limits have been set after each science run of the existing GW detectors.


1.2 Gravitational wave detection

The quest for the direct detection of gravitational waves started in the late 1960’s

when Joseph Weber developed the first GW detector, the resonant bar detector [2, 3].

He built two bar detectors, massive cylinders of metal whose resonant modes can be

excited by the passage of a GW. Weber claimed to have detected gravitational waves,

which were later dismissed after other detectors built around the world failed to detect

them. However he succeeded in starting a new field of research 50 years after Einstein

presented his theory.

Weber suspended a ∼1 tonne aluminium bar in a vacuum tank and bonded a ring

of piezoelectric transducers around its centre. Weber built more than one of these

bars, which were resonant at around 1600 Hz, a frequency where the energy spectrum

of signals from collapsing stars was predicted to peak. These detectors exploit the

narrow mechanical resonance of the bar to achieve high sensitivities over a bandwidth

of a few hertz around the mechanical resonance (typically several hundred hertz).

Modern bar detectors are currently about 1000 times more sensitive than Weber’s

original design [4, 5].

At about the same time, Soviet physicists Mikhail E. Gertsenshtein and Vladislav

I. Pustovoit argued a case for the use of the Michelson-Morley interferometer [6] as

an instrument of GW detection [7], but detailed studies of this technique were not

completed until a decade later. Early researchers in this field included G. Moss [8] and

R. Weiss [9]. Robert Forward and other scientists built the first interferometer for GW

detection in the early 1970’s in California at the Hughes Research Laboratories [10].

For a number of years resonant bar detectors were the main observational in-

strument for GW detection. These have since been surpassed in sensitivity by laser

interferometry based detectors.The concept behind the interferometers is to replace

the bar with free-falling test masses, that will interact with an incoming GW. By

placing a free-falling mass at the end of two very long, perpendicular arms the prin-

ciple of a Michelson interferometer like the one in figure 1.3 could be used to measure

the differential displacement of the end mirrors used as test masses. Such small dis-

placements would be measured by interference of the laser light returning from each

mirror along the arms.

1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS 5

Laser

ETMY

ETMXBS

To Detector Bench

Figure 1.3: Schematic diagram of a Michelson interferometer including a laser source, a

beam-splitter (BS) and two end test masses, ETMX and ETMY.

1.3 Interferometric gravitational wave detectors

A GW interacting with the test masses at the end of the arms of an interferometer

will induce a phase shift in the carrier light. The phase change (φ(t)) incurred is given

by:

φ (t) = ωotr '2ωol

c± ωo

2

t∫t−2l/c

h+ (t) dt. (1.2)

Where tr is the round trip time of the light in the arm cavity, which is modulated

by the incoming GW signal, assuming that |h+| 1 tr can be defined as

tr '2l

c± 1

2

t∫t−l/c

h+ (t) dt, (1.3)

ωo is the angular frequency of the carrier light, 2l/c is the round trip light transit

down the arm cavity of length l, and h+(t) corresponds to a sinusoidal GW with

angular frequency ωg and peak amplitude h0,


h+ (t) = ho cos (ωgt) . (1.4)

This implies that:

δφ ' ωo2

t∫t−2l/c

ho cos (ωgt) dt (for ho 1)

' ho2

ωoωgsin (ωgt)− sin [ωg (t− 2l/c)]

' hoωo sin (ωgl/c)

ωgcos [ωg (t− 2l/c)] . (1.5)

As mentioned previously the amplitude of an incoming GW is of the order of

10−21, inducing a modulation of the order of 10−9 radians. Even though this value

depends on the arm length the modulation index can be approximated to:

δφ ' hoωo sin(ωgl/c)

ωg' hoωol

c. (1.6)

This is almost proportional to the initial distance l between the test masses, as

long as it is short compared to half a wavelength of the GW,

ωgl

c.π

2. (1.7)

In other words, to obtain an optimum sensitivity to gravitational waves it is nec-

essary that the light takes one half period of the GW to do a round trip in one arm of

the interferometer. Therefore the required time for a GW with a frequency of 1 kHz

is 0.5 msec, which implies an arm length of approx 75 km, and even longer for sources

of lower frequencies. These distances are impractical for a ground-based experiment.

However the arms of the interferometer can be effectively ‘folded’ by making the light

resonate or by folding the light path. The effective number of bounces can then be

traded off against the overall length to achieve a desired total path length or stor-

age time. One possible solution is to replace the arms of the interferometer with

delay-lines, effectively folding the path into multiple reflections, as in a Herriott delay

line [11, 12]. A different solution is to increase the light travel time by storing the laser

light in a resonant cavity. This requires keeping the round trip length of the cavity


an integer number of times the wavelength of the input monochromatic laser, which

can be achieved by using Fabry - Perot cavities as the arms of the interferometer as in

figure 1.4. The main advantage of a resonant optical cavity over delay lines is the size

of the mirrors. A delay line needs several reflections in different spots of a mirror (or

several mirrors to accommodate each reflection), while in a Fabry - Perot cavity by

superimposing the beam reflections the mirror size can be reduced, only to be limited

by diffraction losses. A disadvantage is that the light enters a Fabry - Perot cavity

through the substrate of an input test mass (ITM), which has to be of outstanding

optical quality. Effectively this design corresponds to a Michelson interferometer with

coupled Fabry - Perot cavities on the arms.

To Detector Bench

Fabry-Perot Cavity

ETMY

ETMXBS ITMX

ITMY

Laser

LX

LY

Fabry-Perot Cavity

Figure 1.4: Interferometer with Fabry - Perot cavities as arms. The addition of two input

test masses (ITMX and ITMY) form two Fabry - Perot cavities of lengths LX and LY. The

input laser light has to be resonant in the cavities in order to be able to detect gravitational

waves.

As in the Michelson interferometer case, a passing GW will induce changes in

the length of the arm cavities by relative displacement of the end mirrors. These

changes in length will induce changes in the phase of the reflected light. A passing


GW modulates the phase of the light at its frequency, generating sidebands at angular

frequencies from the laser frequency. The end test mass (ETM) has a much higher

reflectivity than the ITM. Assuming that the losses in the cavity are much lower than

the ITM transmission most of the light is transmitted back to the beam-splitter (BS)

where it recombines with the light from the other, perpendicular arm. The differential

changes on each arm are then transmitted to the output port (also known as dark

port or asymmetric port) where they will be detected.

By placing a mirror at the input port (also refer to as bright port or symmetric

port) of the Michelson interferometer, just before the BS, the bright fringe will be

reflected back into the interferometer rather than propagated towards the laser. The

bright fringe corresponds to the common mode strains in the two interferometer arms.

Therefore the presence of this additional mirror does not affect the frequency response

of the interferometer, but does increase the sensitivity by increasing the stored optical

power in the interferometer arms. This technique, known as power recycling and the

resulting cavity, known as the power recycling cavity (PRC), were independently

proposed in the early 80’s by R. Schilling and by R. Drever [13].

Another technique to further enhance the sensitivity of an interferometric GW

detector is to add a mirror of specifically selected reflectivity at the output. The

addition of this mirror adds an extra cavity to the interferometer known as the Signal

Recycling Cavity (SRC). As described, a passing GW will generate sidebands on the

carrier light of the arm cavities. Since these sidebands will not interfere destructively

at the BS, they will appear at the output. This Signal Recycling Mirror (SRM) can be

used to reflect the sidebands back into the interferometer. If the reflected sidebands

are in phase with the sidebands in the arm cavities they will add coherently, enhancing

the signal over a given bandwidth set by the mirror reflectivity. This technique is

known as Signal Recycling (SR) [14, 15, 16].

The transmittance and reflectivity of the compound mirror formed by the SRM

and the ITM is dependant on frequency. In signal recycling, this cavity is tuned so that

the GW signal will see a lower transmittance (higher reflectivity) than that of the ITM

alone. This controls the effective number of round trips over which the GW sidebands

are summed, increasing their storage time, and determining the bandwidth of the


interferometer. The position of the SRM controls which GW sideband frequencies

will add constructively and which will add destructively, thus determining the tuning

of the interferometer and its frequency response.

The bandwidth of an interferometric GW detector with Fabry-Perot arm cavities,

without a SRM, is set by the bandwidth of the cavities. High finesse arm cavities

are desirable in order to reduce photon shot-noise, simultaneously the storage time

for the signal sidebands must be kept short enough to obtain the desire detection

bandwidth. This determines what is referred to as storage time limit.

Resonant Sideband Extraction (RSE) was proposed by Mizuno et al [17] as a

new optical configuration to overcome this limitation. In this mode of operation

the extra cavity at the output of the interferometer is usually referred to as the

Signal Extraction Cavity (SEC). The purpose is to reduce the storage time for the

GW signal, allowing for long storage times in the arm cavities without sacrificing

the detector bandwidth [18, 19]. The tuning of this SEC in this case results in a

bandwidth wider than that of the interferometer without a SRM. It is then possible

to create a compound mirror with a transmissivity higher (lower reflectivity) than that

of the ITM alone. This reduces the storage time of the signal frequencies of interest,

resulting in an increased bandwidth for the interferometer. Since the interferometer

bandwidth is not limited by the bandwidth of the arm cavities, Fabry-Perot cavities

with high finesse, narrow bandwidth, can be used to maximise the stored energy.

This has also the advantage of reducing the amount of light power which must be

transmitted through the optical substrates of both ITMs and the BS in order to

obtain the same amount of energy stored in the arm cavities. As a consequence the

thermally induced distortion inside the substrates is effectively reduced.

Both modes of operation (SR and RSE) also have the possibility of operating in a

detuned mode. Conventionally the special case of maximum response at zero signal

frequency is termed tuned; all other cases, with peak response at a finite frequency,

are called detuned. This detuning is usually described by either the frequency of peak

response or by the shift of the SRM away from the tuned point, often in terms of an

optical phase shift. This positioning or detuning of the SRM within a wavelength of

the carrier light allows for some narrowing of the detection bandwidth at the expense


of loss of sensitivity outside the bandwidth. This could be valuable in searches for

continuous wave sources of gravitational radiation, like rapidly rotating neutron stars

and fast pulsars or even tracking of a changing periodic signal like a chirp [20].

The configuration of a Michelson interferometer with Fabry - Perot cavities in the

arms and the addition of a PRC and a SRC is referred to as a dual recycled inter-

ferometer. The concept of dual recycling and the design of stable recycling cavities

(PRC and SRC) is reviewed in Chapter 6. Figure 1.5 shows a simple diagram of this

design, which is the one favoured for the next generation of interferometric GW de-

tectors. However this is only part of the story behind the next generation of advanced

GW detectors, which required the development of new techniques to overcome several

noise sources.

To Detector Bench

Fabry-Perot Cavity

ETMY

ETMXBS ITMX

ITMY

Laser

LX

LY

Fabry-Perot Cavity

SRM

PRM

Figure 1.5: Dual recycled interferometer. The addition of a power recycling mirror (PRM)

increases the arms circulating power. The addition of a signal recycling mirror (SRM)

enhances the signal response of the interferometer and could also be used for frequency

tuning and narrow band operation.


1.3.1 Seismic noise

An important source of displacement noise in all ground-based detectors is seismic

noise. Unfortunately it is not possible to control seismic noise, but it is possible to

reduce its effects on the interferometer. Seismic vibration falls sharply with frequency

(f), typically at a level of 10−7/f 2 m/√

Hz, depending on the location of the site in

the world. This makes this source of noise a problem primarily below 100 Hz. At

levels of hundreds of mHz the spectrum of the seismic noise is enhanced by what is

known as micro-seismic peaks. Dominant micro-seismic signals from the oceans are

linked to characteristic ocean swell periods, and occur approximately between 4 to

30 seconds [21].

Techniques for reducing the effects of seismic noise have long been known and yet

new designs and approaches continue to improve vibration isolation. The most basic

vibration isolation system is a simple pendulum which has evolved to the complex

systems of today. The advanced vibration isolation systems planned for the next

generation of interferometric GW detectors have an array of different techniques for

pre-isolation in order to reduce the seismic vibration to levels where the detection

of gravitational waves is feasible. This is only possible however down to a limit of

about 10 Hz. This limit is due to random gravitational forces (or gravity gradient

noise) caused by density fluctuations of the medium surrounding the GW detector;

also including contributions from the motion of isolated bodies in the vicinity nor-

mally related to human activity [22]. Substantial effort has been put into mechanical

vibration isolation systems. A review of the vibration isolation efforts is presented in

Chapter 3, including the performance of an advanced vibration isolation system for

AIGO.

1.3.2 Laser source noise

Even though lasers are commonly thought to be stable sources of light, they have both

frequency and spatial fluctuations. The spatial instability of a laser beam, known

as beam jitter, is due to the mixing of higher order modes with the fundamental

mode TEM00. Amplitude fluctuations are created by beam jitter whenever the beam

interacts with a spatially sensitive element such as an optical cavity. The noise at


the dark fringe of the interferometer output will be affected by such beam jitter

effects. In addition, frequency fluctuations of the laser fundamental mode give rise

to additional noise at the dark fringe. In the case of GW detection these variations

can introduce substantial measurement noise. Filtering the laser light with an optical

filter or mode-cleaner as illustrated in figure 1.6 can solve this problem.

The input mode-cleaner (IMC) acts as a spatial filter. It provides passive stabil-

isation of time dependant higher order spatial modes, transmitting the fundamental

mode TEM00 and attenuating the higher order modes. The concept was first sug-

gested by Rudiger et al in 1981 [23]. As a frequency stability element it can also

suppress frequency fluctuations of the fundamental mode, but without DC stability.

A complete optical design of a high power mode-cleaner is presented in Chapter 4.

M1

M2

M3

Figure 1.6: A classical input mode-cleaner layout showing the laser light path and the

relative position of the mirrors, where M1 (input coupler) and M2 (output coupler) are flat

mirrors and M3 a convex curved mirror.

1.3.3 Optical noise

Optical noise is traditionally divided into two sources of noise. The first is related to

the interaction of light with the test masses, while the second is related to the counting

of photons by a photo-detector. The power that builds up inside the cavity exerts

significant forces on the mirrors due to the momentum of the photons. The more

power circulating in the cavity, the more photons, and a greater force acting on the test

masses. The number of photons ‘bouncing-off’ the test mass at any given time follows

a Poisson distribution and the statistical distribution of the force on the test mass

is known as radiation pressure noise. In interferometric GW detectors this is mainly


a problem at low frequencies (around a few Hz). However radiation pressure can

generate other problems like parametric instabilities [24] and optical spring effects [25],

which can be difficult to control. A discussion of parametric instabilities through the

estimation of diffraction losses is presented in Chapter 5.

Photon shot noise is the second source of optical noise. It is related to the statis-

tical arrival time of photons and also follows a Poisson distribution for the counting

of photons. At the interference fringes the fluctuations due to the random arrival of

photons can look like a GW signal. The more photons we use, the smoother the inter-

ference fringe. Therefore very high power from a continuous wave laser is necessary

to minimise the shot noise.

The source of fluctuations for both shot and radiation pressure noise are the vac-

uum fluctuations, which enter the interferometer from the output port [26]. When

laser light and vacuum fluctuations are injected into optical cavities with suspended

mirrors, the vacuum fluctuations are ponderomotively squeezed by the back action of

its radiation pressure on the suspended mirrors. The shot noise spectral density is flat,

while the radiation pressure amplitude spectral density has a 1/f shape. The stan-

dard quantum limit is then defined by the quadrature sum of the shot and radiation

pressure noise at a given frequency. However the standard quantum limit for the noise

of an optical measurement scheme usually refers to the minimum level of quantum

noise which can be obtained without the use of squeezed states of light [27, 28].

1.3.4 Thermal noise

Another source of fundamental noise that contributes to the displacement noise of

a test mass is due to thermal fluctuations. Thermal noise is caused by mechanical

loss in the system in accordance with the “Fluctuation-Dissipation Theorem” [29]. In

GW interferometers thermal noise is usually divided into two categories, suspension

thermal noise and test mass thermal noise. In both cases the noise comes from the

vibration of atoms which depend on Boltzmann’s constant kB and is proportional to

kBT , where T is the temperature. Suspension thermal noise corresponds to the effect

of thermal fluctuations from the suspension system as seen at the mirror, thus the

strongest effect is from the last stage of the suspension directly in contact with the


test mass. The second component is due to the atoms on the test mass itself, causing

the surface of the mirror to vibrate. However the test mass thermal noise can also be

defined according to the source of the noise, substrate and coating. Coating thermal

noises are defined by differences between the coating material and the substrate mate-

rial, mechanical losses between coating layers contribute to the total Brownian noise

of the test mass. Thermal fluctuations in the coating produce noise via thermo-elastic

and thermo-refractive mechanisms. The study of multilayer dielectric coatings is an

active area of research with new coatings under test for GW interferometers [30, 31].

1.3.5 Readout scheme

An incoming GW signal of frequency ωg will induce sidebands on the carrier light

(frequency of 2.8× 1014 Hz). These sidebands will be located at ±ωg from the carrier

light. To be able to read such a high frequency signal, an optical oscillator is necessary

in order to demodulate the output signal and read the gravitational wave signal.

The first generation of interferometric GW detectors favoured a heterodyne read-

out scheme, where radio-frequency (RF) sidebands are modulated on the carrier light

before entering the interferometer. A large mirror offset (centimetre scale) to create

a macroscopic asymmetry in the Michelson arms of the interferometer (also known as

Schnupp asymmetry [32]) is required to allow leakage of the RF sidebands through

the output port of the interferometer whilst maintaining a dark fringe. This signal

can be used as the local oscillator. The next generation of interferometric GW de-

tectors however will see the addition of a SRC. The operation of this cavity in a

detuned configuration will introduce an imbalance on the control sidebands, reducing

the sensitivity of the interferometer [33]. The sensitivity reduction will depend on the

relative level of imbalance between the sidebands.

A homodyne readout scheme requires a small fraction of carrier light to be used

as the local oscillator. One approach is to obtain a signal from the incoming beam

before it enters the interferometer, and feed this signal as the local oscillator at the

output port. Another approach that also allows a small amount of light to appear at

the output port is to introduce a very small offset (picometre scale) at the Michelson

arms of the interferometer. This technique, known as direct-conversion (DC) readout,


was proposed by Fritschel in 2000 [34] and tested at the 40 m interferometer at the

California Institute of Technology [35]. It also has the advantage of increasing the

signal to shot noise ratio by eliminating the vacuum fluctuations at twice the frequency

of the modulation [36].

1.3.6 Output mode-cleaner

Despite best efforts to keep the arm cavities and recycling cavities perfectly aligned

there will always be some degree of misalignment, for example due to radii of curvature

mismatch and/or surface deformation of the mirrors. These factors can individually

be very small, but they all contribute to light leaking to the dark port, which increases

the noise and reduces the fringe contrast. With the addition of an output mode-

cleaner (OMC) all but the fundamental mode component of the contrast defect will

be rejected [37, 38, 39, 40]. Figure 1.7 shows a dual recycled interferometer including

an IMC and an OMC.

The OMC design depends on selection of either a RF (optical heterodyne detec-

tion) or DC (optical homodyne detection) readout configuration. In a RF readout

scheme the control sidebands are used as local oscillator for signal demodulation, as

a consequence a long OMC with individually isolated mirrors could be used in order

to transmit the sidebands to the output port. However a short OMC with a broad

bandwidth could also be used to reject higher order optical modes. A DC readout

has several technical advantages over the traditional RF readout technique, including

laser and oscillator noise coupling as well as reduced shot noise [41]. Current designs

for the next generation of interferometric GW detectors favour a DC readout scheme

(Advanced LIGO, Advanced VIRGO, and GEO-HF). The DC readout scheme relies

on a small offset (usually on the order of a few ten picometres) that allows a small

amount of carrier light into the otherwise dark port. This carrier light is used as

the local oscillator, and as a consequence the sidebands are not needed at the dark

port. However they will still be present in the interferometer, since they are needed

for auto-alignment error signal. For a fixed OMC cavity length, the transmission of

the sidebands decreases as the finesse increases. Therefore the OMC requirements

include low finesse for reducing the sidebands, the transmission of the carrier funda-


M1 M2

M3

To Detector Bench

Input Mode Cleaner

ETMY

ETMX

SRM

BSPRM ITMX

LIMC

LaserFabry-Perot Cavity

Fabry-Perot Cavity

LY

LX

Output Mode-cleaner

Figure 1.7: Diagram showing a dual recycled interferometer with input mode-cleaner and

output mode-cleaner. Arm cavities, input mode-cleaner and output mode-cleaner are not

at the same scale.

mental mode and low back-scatter light so as to not reduce the sensitivity with added

phase noise. Taking this into account, the favoured configuration for an OMC is a

four mirror cavity in a bow-tie arrangement. This is small enough to be made from

a monolithic piece of fused silica that is then suspended from a vibration isolation

system. In future configurations the OMC could provide an excellent reference for

aligning the injection of squeezed light [42].

1.4 Sensing and control

We have seen that interferometric GW detectors are a collection of optical cavities

coupled together with such precision to allow for the detection of gravitational waves.

However to obtain a working interferometer it is necessary to keep these cavities

locked at all times. This requires a sophisticated sensing and control system, which

1.4. SENSING AND CONTROL 17

has to bring the interferometer from an unlocked state to a configuration appropriate

for collecting science data. This can be divided into three steps. First is the lock

acquisition, where all initially uncontrolled length degrees of freedom are globally

controlled and brought to their operating point. Second is the transition from a

locked interferometer with all degrees of freedom controlled to a configuration where

science data can be collected. Third is to maintain the science mode and the data

collection.

All of the interferometer sensing and control ports (show in figure 1.8) will be

equipped with wavefront sensors (WFS) which also give DC outputs allowing for

fast beam stabilisation. Optical levers are considered for angular control of the core

optics, as well as CCD cameras to locate the beam positions on the high reflectivity

surfaces of all core optics. In addition, the transmitted beam of each optic will be

monitored by a quad-photo-detector (QPD). The main difference between QPD and

WFS is that the QPD are sensitive to the carrier light, whilst the WFS are sensitive

to the RF sidebands [43, 44].

The input signals for the sensing and control systems come from the different

sensors installed around the interferometer. Tuned signals that are band-pass filtered

before being synchronously demodulated are used for detection of the various degrees

of freedom. The outputs of the length and sensing control system, and the alignment

and sensing control system are processed by the global control system before being

distributed to each of the core optics components, through the suspension local control

system, and to the Pre-Stabilised Laser (PSL) for actuation on the laser frequency.

In principle two radio frequency (RF) sidebands are enough to control all 5 length

degrees of freedom of an advanced dual recycled interferometer. These two RF modu-

lation frequencies (f1 and f2) make the use of double and/or differential demodulation

techniques possible. The two RF sidebands allow us to obtain length signals using

single demodulation at f1 and f2 or differential demodulation at f1 ± f2, and double

demodulation at a combination of f1 − f2 and f1 + f2 where the signal is produced

by the beat between the two RF sidebands, equivalent to (f1 × f2).

The signal of each WFS is processed in order to obtain a strong control signal

from the sidebands. The RF signal is split and demodulated at 0o (in-phase) and


Figure 1.8: Distribution of the different sensing apparatus of an advanced interferometric

GW detector. These include WFS and QPD. It also includes the aid of CCD cameras and

optical levers, which are auxiliary lasers and QPD to monitor the optics alignment. (Figure

courtesy of the LIGO Scientific Collaboration (LSC) [43].)

90o (quadrature-phase) giving two signals per detector [45]. Each of these signals

is fed to an input matrix where each length degree of freedom can be calculated.

These are the differential arm length (DARM or L−), Michelson length (MICH or l−),

common arm length (CARM or L+), power recycling cavity length (PRCL or l+), and

signal recycling cavity length (SRCL or lsrc). Figure 1.9 shows their definitions in an

advanced dual recycled interferometer configuration. The signals are digitally filtered

and converted through an output matrix into control signals for the core optics. This

is done by feeding the control signals to the local control of the suspension system,

where the signals are added to the appropriate degrees of freedom of each optic.

In general the length degrees of freedom that need to be controlled and which are

also shown in figure 1.9 are defined as:

DARM Differential arm length: L− = (Lx − Ly)/2

1.4. SENSING AND CONTROL 19

CARM Common arm length: L+ = (Lx + Ly)/2

MICH Michelson length: l− = (lx − ly)/2

PRCL Power recycling cavity length: l+ = lpr + (lx + ly)/2

SRCL Signal recycling cavity length: lsrc = lsr + (lx + ly)/2

ETMy

ETMxBS

PR3

ITMx

ITMy

Fabry-Perot Cavity

Fabry-Perot Cavity

Ly

Lx

L+=(L

x+L

y)/2

L-=(Lx-L

y)/2

l+=l

pr+(l

x+l

y)/2

l-=(lx-l

y)/2

lsrc

=lsr

+(lx+l

y)/2

Degrees of freedom

ly

lx

ls1

lp3

lp2

lp1

lpr

= lp1

+lp2

+lp3

ls3l

sr= l

s1+l

s2+l

s3

SR2

SR1

PR2PR1

M1 M2

M3

Input Mode Cleaner

LIMC

Laser

ls2

To Detector Bench

Output Mode-cleanerL

OMC

SR3

Figure 1.9: Dual-recycled advanced interferometric GW detector. The figure shows stable

recycling cavities and length degrees of freedom, including input mode-cleaner and output

mode-cleaner. In general the input mode-cleaner is part of the input optics sub-system;

while the output mode-cleaner is part of the output optics and they are not part of the

interferometer control system.

Before the interferometer longitudinal degrees of freedom can be controlled it is

necessary to initially align the cavity axes to one another. This includes centring

the beams on the optics as well as aligning the interferometer input beam with the

interferometer optical axes. This is performed with the aid of optical levers installed

on each of the core optics of the interferometer. At this stage there is no need to

operate the laser at full power, which could compromise lock acquisition by radiation


pressure effects. The objective of the initial alignment is to attain a sufficiently high

power build-up in the cavities of the interferometer for locking the longitudinal degrees

of freedom [46].

The next step is to lock the central part of the interferometer comprising the ITMs,

BS, PRM and SRM. This corresponds to three longitudinal degrees of freedom, the

MICH, the PRCL, and the SRCL. With the central interferometer locked it is possible

to lock the arms through the CARM and the DARM signals, locking the whole

interferometer [43]. It is expected that the locked central part of the interferometer

will remain stable during the locking of the arm cavities. If the carrier were used to

lock the central part of the interferometer, the lock could be lost when locking the

arm cavities. The extra phase shift added to the carrier when resonating in the arm

cavities would change the polarity of the control signal, which could drive the control

system unstable. For this reason single demodulation is only used for lock acquisition

of the central part.

Standard Pound-Drever-Hall [47] signals generated by the beating between carrier

and sidebands are strongly dependent on the behaviour of the carrier inside the arm

cavities and on the interferometer losses. Control signals can be obtained from the

beating of the first sideband (f1) with the second sideband (f2) by demodulating

the signal twice. Since double demodulation scheme does not depend on the carrier,

the amplitude and polarity of the control signals obtained from the sidebands are not

affected by the lock or unlock status of the arm cavities. Consequently before the arms

are brought into lock, the control is handed over to double demodulation. However

since double demodulation signals do not work effectively far from the locking point

so that they are not ideal for lock acquisition. Moreover, in the configuration for

broadband detection, there is no detuning of the SRC. In such a configuration the

two sidebands can no longer beat and there is no double demodulation signal at all.

The TAMA group [48] proposed and studied the use of signals demodulated at three

times the modulation frequencies (3f) in order to obtain signals that are independent

from the CARM offset necessary to keep the arm cavities out of lock. These signals are

produced by the beating between 2f and f sidebands and between 3f sidebands with

the carrier. The second contribution is typically smaller than the first, so that the

1.5. A NETWORK OF GRAVITATIONAL WAVE DETECTORS 21

3f signal depends very little on the carrier behaviour. Signals detected in reflection

and demodulated at 3f1 and 3f2 are in fact very good signals for PRCL, MICH and

SRCL.

A combination of the transmitted powers of the two arm cavities is used for the

CARM degree of freedom, (more precisely the square root of the sum). For the

DARM degree of freedom the difference between the two transmitted powers is used.

The DARM degree of freedom is controlled by feedback to the end mirrors, while

the CARM degree of freedom is expected to be controlled by feed-back to the laser

frequency. With the described locking scheme the interferometer is brought to the

operating point by passing through stable states, which allow the activation of a

frequency servo even during lock acquisition.

1.5 A network of gravitational wave detectors

In principle a single GW detector would be enough to detect gravitational waves. An

individual GW detector is almost omni-directional with a wide antenna pattern and

consequently has poor angular resolution. In order to undertake GW astronomy and

astrophysics, a network of detectors is indispensable.

1.5.1 Resonant detectors

In 1997 the Gravitational Wave International Committee (GWIC) was established

[49]. Its main goal is to facilitate international collaboration and cooperation in the

construction, operation and use of the major GW detection facilities world-wide. As

such it is not limited to interferometric GW detectors and also includes the operating

bar detectors (AURIGA [4], EXPLORER, and NAUTILUS [5]), which operate at the

high end of the frequency band between 500 Hz and 5 kHz. Bars are very sensitive to

the direction of the incoming GW, which led to the development of omnidirectional

resonant detectors shaped as a sphere such as the Mario Schenberg [50] and Mini-

GRAIL, with a resonant frequency of 2.9 kHz and a bandwidth around 230 Hz [51].


1.5.2 Interferometric detectors

Operating at a slightly lower frequency band are the ground-based interferometric

GW detectors. The largest of these detectors is LIGO, the Laser Interferomet-

ric Gravitational-wave Observatory, consisting of three kilometre-scale detectors L1,

H1 and H2 [52, 53]. Two of these have 4 km long arms, L1 located in Livingston,

Louisiana, USA and H1 in Hanford, Washington, USA. H2 is a 2 km long interfer-

ometer, which shares the vacuum envelop with H1 at the Hanford facility. In 2007

these detectors completed a two-year data taking run, known as S5 (the fifth science

run for LIGO). At the time of writing only H2 has been left running, while L1 and

H1 are going through an upgrade to increase their sensitivity before starting a new

science run as Enhanced LIGO in the second half of 2009 [54]. Enhanced LIGO will

be an intermediate step between Initial LIGO and Advanced LIGO, which will see H2

upgraded to a 4 km interferometer. On a similar scale with arms stretching 3 km, is

VIRGO near Pisa in Italy [55], built by a consortium of institutions from France and

Italy and operated by the European Gravitational Observatory (EGO) [56]. VIRGO

is also going through an upgrade to become VIRGO + [57] and will join Enhanced

LIGO for its science run. A new set of upgrades will then take place in order to

improve its sensitivity to become Advanced VIRGO [58]. On a smaller scale there

is the German-British GEO600 [59]. Often called a second generation detector since

it already uses some of the ‘advanced’ technology. This project has a slightly differ-

ent topology, being a dual recycled Michelson interferometer with folded arms for its

600 m arms. A series of upgrades are also planned for the GEO600 project, designed

to improve its sensitivity in the high frequency region above 1 kHz, to become GEO-

HF [60]. In Tokyo, Japan has built a 300 m interferometer, TAMA, which was the

first large scale interferometer to achieve continuous operation and has participated

in joint science runs with LIGO [61, 62]. The sensitivity of some of these different

systems are compared in figure 1.10.

In addition to the operating interferometers, GWIC also includes future ground-

based projects such as the Large Cryogenic Gravitational Telescope (LCGT) in Japan,

an interferometer with 3 km arms located 1000 m underground at the Kamioka mine

site [64]. The Japanese group has developed the Cryogenic Laser Interferometric Ob-


Figure 1.10: Measured sensitivity of different interferometric GW detectors. The figure

includes the latest sensitivity curves of LIGO H1 (LHO 4 km), L1 (LLO 2 km), and H2

(LHO 2 km); GEO600 and VIRGO. It also shows the design sensitivity of the VIRGO

interferometer. (Image courtesy of the LIGO Scientific Collaboration (LSC) [63]).

servatory (CLIO) [65], a 100 m interferometer testing technology for the future LCGT.

There are also plans for an advanced GW detector in Australia. The Australian

Consortium for Interferometric Gravitational Astronomy (ACIGA) is developing the

plans for a kilometre-scale interferometric GW detector, the Australian International

Gravitational-wave Observatory (AIGO) [66, 67]. Part of the research and develop-

ment towards this advanced interferometric GW detector is presented throughout this

thesis, including a proposed optical design for AIGO in Chapter 6.

To determine the position of a GW event in the sky, a coherent network analysis

will allow accurate measurement of the different phase fronts of the incident GW on

different detectors, thus obtaining a good angular resolution for an all-sky monitor.

Providing that the orientation of the detectors is different (and therefore the orienta-

tion of their antenna patterns), a network of detectors will provide enough information

about both polarisations components, and allow reconstruction of the polarisation of

the incoming GW. This is partly due to the different geographical location of each

detector over the globe, covering different portions of the sky.


In a network of detectors, broadband stationary noise reduces as the square root of

the number of detectors. While this factor is not large it has a much larger effect on the

number of detectable sources. This depends on the volume accessible by the detector,

which increases as the cube of the detector strain sensitivity. Multiple detectors at

separated sites are essential for rejecting instrumental and local environmental effects

in the data, reducing non-stationary noise by requiring coincident detections in the

analysis. The probability of spurious signals reduces as the power of the number of

detectors.

Figure 1.11 shows the angular uncertainty on the sky when locating a GW event.

With the current network of large detectors (L1, H1 and VIRGO) figure 1.11 (a)

shows the angular resolution corresponding to the red areas, characterised by highly

elongated ellipsoids. Figure 1.11 (b) shows the addition of the planned advanced

GW detectors LCGT and AIGO dramatically reducing the uncertainty areas, and

improving the angular resolution of the GW network. The fact that AIGO will

be located in the southern hemisphere adds an out-of-plane detector (the rest of

the interferometers are in the northern hemisphere), which increases the maximum

baseline and improves the existing angular resolution. This highlights not only the

importance of a southern hemisphere advanced GW detector but the necessity of

operate as a cohesive network. A more complete analysis of the science benefits of

an advanced interferometric GW detector in the southern hemisphere is presented in

Appendix A.

The proposed Einstein gravitational wave Telescope (ET) takes a slightly different

approach. Proposed by eight European research institutes is officially a design study

project supported by the European Commission under the Framework Programme 7.

As a 3rd generation detector in its design stages there are still a few open questions. Its

topology is still under study and could include multiple interferometers co-located [69].

It is most likely that the interferometer (or interferometers) will include cryogenic

technology, will be placed underground and will incorporate an important increase in

laser power, which will require increasing the weight of the mirrors. The main goal of

these efforts is to improve the sensitivity and expand the lower end of the detection

band. With arms 10 km long, the differential length variation will allow the detection


(a)

Dec (

0)

RA cos (Dec) (0)

L+H

VIRGO

0

80

60

40

20

-20

-40

-60

-80

0-150 -100 -50 50 100 150

GEO

(b)

Dec (

0)

RA cos (Dec) (0)

L+H

V

0

80

60

40

20

-20

-40

-60

-80

0-150 -100 -50 50 100 150

G

C

A

Figure 1.11: Angular area maps for a world network of interferometric GW detectors.

(a) shows the antenna pattern for an array including LIGO Livingstone (L), LIGO Hanford

(H), VIRGO (V) and GEO600 (G). (b) shows the antenna pattern when LCGT (C) and

AIGO (A) are included. The angular uncertainty is shown as red ellipsoids in the sky,

with a clear reduction of the uncertainty by adding LCGT and AIGO to the GW network.

(Figures courtesy of L. Wen [68]).

band to be reduced to 1 Hz, only to be limited by gravity gradient noise. Higher

power, requiring heavier test masses, will also expand the higher end of the detection

band to about 10 kHz [70].


1.5.3 Space interferometry

Joining the ground-based efforts are the space detectors, mainly the Laser Interfer-

ometer Space Antenna (LISA), a joint ESA and NASA effort [71]. Three identical

spacecrafts act as an interferometer with an arm length of 5×106 km. The spacecrafts

are joined by a low power laser giving LISA its triangular shape, and creating three

independent arms as shown in figure 1.12. Gravitational waves from distant sources

will warp space-time, stretching and compressing the triangle, which constitutes a

very large GW antenna. Due to its extremely large arms, LISA is expected to detect

gravitational waves at low frequencies (0.1 mHz - 100 mHz). More ambitious plans

to increase the sensitivity of space detectors have been proposed in order to detect

the stochastic background of gravitational waves. The Big Bang Observer (BBO)

will consist of four LISA-like spacecraft constellations [72]. There are plans for an-

other space-based interferometer, the Japanese DECIGO (DECI-hertz interferometer

Gravitational wave Observatory) [73], designed to cover the ‘gap’ between LISA and

the Advanced detectors in the frequency band between 100 mHz and 10 Hz.

The first step for the space detectors is the launch of the LISA Pathfinder [74].

At the time of writing the LISA Pathfinder spacecraft is schedule to be launched in

the first semester of 2010. It will test in flight the very concept of GW detection

by putting two test masses in a near-perfect gravitational free-fall. It will control

and measure their motion with unprecedented accuracy providing invaluable data

regarding the key technologies to be used in the LISA spacecrafts.

1.5.4 Pulsar timing

In order to measure gravitational waves at even lower frequencies a different ap-

proached is required. By monitoring radio beams of distant pulsars for evidence of

gravitational waves interacting with a beam of electromagnetic radiation. Radio-

telescope groups are seeking to detect gravitational waves via precise timing of a

large number of radio pulsars through programs like Parkes Pulsar Timing Array

(PPTA) [75], the Nano-hertz Observatory for Gravitational Waves (NANOGrav) [76],

and the European Pulsar Timing Array (EPTA) [77]. These groups are in the process

of organising themselves as the International Pulsar Timing Array or IPTA. Due to

1.6. CONCLUSIONS 27

Figure 1.12: An artist’s representation of the LISA mission. Three spacecraft, each with

a Y-shaped payload, form an equilateral triangle with sides 5 million km in length. The two

branches of the Y at one corner, together with one branch each from the spacecraft at the

other two corners, form one of up to three Michelson-type interferometers, operated with

infrared laser beams. ( c©NASA images)

the astronomical distances involved these observations are expected to detect gravi-

tational waves at very low frequencies (10−9 – 10−7 Hz). Sources in this regime will

include coalescing super-massive black hole binary systems. At ultra-low frequencies

gravitational waves in the early universe may have left their imprint on the polarisa-

tion of the cosmic microwave background.

1.6 Conclusions

These are exciting times for the GW community. We are getting closer to achieving

direct detection of gravitational waves, which has encouraged continual improvement

of the existing facilities. At the time of writing the improvements to LIGO L1 and

H1 detectors are almost complete and soon a new science run will start (S6) using

the most sensitive instrument built to date, Enhanced LIGO. They will be joined

by VIRGO+ in this new attempt for direct detection of gravitational waves. The

aim of S6 is to collect a year’s worth of data after which the upgrades for Advanced


LIGO and Advanced VIRGO will commence. During this time AIGO and LCGT

groups will push their bids for funding in order to start building their detectors as

soon as possible. Soon after S6 commences GEO600 will begin upgrades to improve

sensitivity on the high end of the detection band, becoming GEO-HF. Studies for the

third generation of ground-based GW detectors are well under way. The imminent

launch of the LISA Pathfinder will mark the start of the space era for gravitational

wave detection. Exiting times indeed.

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Chapter 2

Mode-Cleaner Vibration Isolator

2.1 Preface

With the design of the advanced vibration isolator for the test masses for an inter-

ferometric GW detector well under way, it became evident that a smaller and more

compact version of the isolator would be an excellent system for use with the auxiliary

optics. The first test for this system would be the mode-cleaner optics, which can be

tested as a stand-alone system unlike the recycling cavities optics. For this design

it was considered that in general a vibration isolator system for the auxiliary optics

demands less in terms of seismic isolation than the full scale system. The design

began with an inverse pendulum made of four legs supporting a rectangular structure

used as a table. Weights were used to tune and lower the resonant frequency of the

inverse pendulum, acting as a first stage of horizontal pre-isolation. A pyramidal

Roberts linkage was attached to this structure, providing a second stage of horizontal

pre-isolation. An Euler spring developed for vertical isolation was attached to this

stage and a self-damped pendulum with cantilever springs was in turn used to hang a

bread-board where the optics will be placed. Since the vibration isolator system had

not been completed at the time of writing the presented paper, only the predicted

performance is shown. The system has since been completed and resulting measure-

ments are presented in the postscript of this chapter. The vibration isolators were

built by Andrew Woolley and the technical staff from the UWA School of Physics

workshop.

37

38 CHAPTER 2. MODE-CLEANER VIBRATION ISOLATOR

Application of New Pre-isolation Techniques to

Mode-cleaner Design

Pablo Barriga1, Andrew Woolley1, Chunnong Zhao2,∗, and David G. Blair1

1 School of Physics, University of Western Australia, Crawley, WA6009, Australia2 Computer and Information Science, Edith Cowan University, Mount Lawley, WA 6050,

Australia

Two very low frequency pre-isolation stages can greatly reduce the resid-

ual motion of suspended optical components. In a mode cleaner this can

reduce the control forces required on the mirrors, simplifying lock acqui-

sition and reducing noise injection through control forces. This paper

describes a 12 m triangular suspended mode cleaner under construction

for AIGO high optical power interferometer. A novel and very compact

multistage isolator supports the cavity mirrors. It combines an inverse

pendulum in series with a low mass Roberts Linkage, both with pendu-

lum frequencies below 0.1 Hz. The suspension chain is connected to the

Roberts Linkage via an Euler spring stage and a cantilever spring assem-

bly for vertical isolation. We present an analysis of the mode cleaner,

emphasising the advantage of the improved mode cleaner suspension and

its power handling capability. The effect of seismic noise on the residual

velocity of the mirrors and the predicted frequency stability of the optical

cavity are presented.

2.2 Introduction

A high optical power interferometer is being built at Gingin near Perth to develop

advanced interferometer technology required for advanced gravitational wave (GW)

detectors. The project includes a 12 m triangular ring cavity mode-cleaner to minimise

variations in laser beam geometry. The interferometer is designed to have parameters

∗Now at School of Physics, The University of Western Australia, Crawley, WA6009, Australia

2.2. INTRODUCTION 39

as close as practicable to Advanced LIGO to enable the critical issues of thermal

lensing and radiation pressure effects to be examined.

All interferometer GW detectors include one or more mode-cleaners for laser spa-

tial stabilisation before it is injected into the sensing cavities. All mode-cleaners have

the same layout, a three-mirror triangular ring cavity. Two flat mirrors define the

short side of the triangle, and a concave mirror the acute side of it. Mode-cleaners

generally use a simpler suspension system than those used for interferometer test

masses. For advance interferometers we need mode-cleaners capable of transmitting

at least 100 W of laser power. They all must solve several problems; seismic isolation,

residual velocity, power handling and frequency discrimination. The mirror position-

ing control is based on the well-known Pound-Drever-Hall technique [1]. By reducing

the vibration even at frequencies below the GW detection band (10 Hz − 1 kHz) we

seek to facilitate the locking of high finesse cavities. This not only applies to the main

Fabry-Perot long arms of the interferometer, but to the mode-cleaner and other optical

components like the input optics and the beam-splitter among others. The reduction

of the vibration will reduce the residual motion of the mirrors; as a consequence we

will need less force on the actuators to control them. With lower servo forces required

to maintain the locking noise injection by the actuators will be reduced. All of it

simplifies the locking of the cavities and the control system design.

Project Length (m) Finesse FSR (MHz) End mirror ROC (m)

LIGO (4 km) 12 1550 12.25 17.25

GEO600 4.0, 4.05 2700, 1900 37.48, 37.12 6.72, 6.72

VIRGO 144 986 1 180

TAMA 10 1700 15.24 15

Advanced LIGO 25 2090 6 43.2

AIGO 12 1495 12.39 31.8

Table 2.1: Characteristics of mode-cleaners used on interferometric gravitational wave

detectors around the world, including one proposition for Advanced LIGO (LIGO-G000240-

00-D).


Different configurations have been used for each long base interferometer as sum-

marised in table 2.1. GEO600 has two mode-cleaners in series. LIGO uses a simple

pendulum whilst Advanced LIGO is planning the use of a triple pendulum mirror

suspension system based on the GEO600 suspensions [2]. TAMA300 and GEO600

use double pendulums for the mode-cleaners [3, 4, 5, 6]. At the same time LIGO

and TAMA teams are involved in the implementation of a new suspension system

called Seismic Attenuation System (SAS). VIRGO uses a simpler version of their test

mass superattenuators with only two seismic filters mounted on an inverse pendulum

[7, 8, 9].

In this paper we present the expected behaviour of a mode-cleaner which uses

double pre-isolation stages to obtain very low residual motion of the mirrors. In

order to model the performance we work with seismic data taken at AIGO in Gingin

and with the pre-isolator stages transfer function. We show that very low residual

velocity can be achieved simplifying locking and reducing noise injection.

2.3 Isolation and suspension design

The suspension system for the test masses developed at UWA for the AIGO GW de-

tector consists of different stages of pre-isolation in an effort to achieve the isolation

requirements at low frequencies. The three dimensional (3D) pre-isolator consists

of an inverse pendulum horizontal stage cascaded with a LaCoste vertical stage and

a Roberts Linkage horizontal stage of pre-isolation [10, 11]. A four stage multi-

pendulum system is then mounted, where Euler springs for vertical suspension are

included [12]. Mounting each of the intermediate masses of the pendulum from gim-

bals allowed them to freely rock with respect to a short rigid section of the main

pendulum chain, then viscously coupling these two together with magnetic eddy cur-

rent coupling for self-damping [13].

The mode-cleaner isolation design uses an improved and more compact suspension

for a small optical table where the mirrors are mounted. One of the big differences

between the test masses and the mode-cleaner systems is the available space to fit

them. The mode-cleaner suspension is designed to fit inside a 12 m long and 1 m

2.3. ISOLATION AND SUSPENSION DESIGN 41

(a) (b)

Inverse Pendulum

Roberts Linkage

Euler Spring

Gimbal

Cantilever Spring

Self Damping

Bread-board

Figure 2.1: (a) Mode-cleaner isolator design, where it is possible to see the inverse pen-

dulum with the Roberts Linkage and suspension chain. (b) Design of the Roberts Linkage

and suspension chain. From the Euler spring follow the cantilever springs from where the

breadboard with the optics will hang.

10-2

10-1

100

101

102

-400

-350

-300

-250

-200

-150

-100

-50

0

50

Horizontal and Vertical Isolator Transfer Function

frequency [Hz]

Magnitude [

dB

]

Horizontal

Vertical

Figure 2.2: Frequency space representation of the isolation system transfer function,

where it is possible to identify the resonant frequencies of each of the horizontal and vertical

stages of the pre-isolator. We expect some internal modes (not shown in the figure) of

the order of a few tenths of hertz and Q ∼100 on the vertical transfer function. This

representation was calculated assuming a low pass filter model for each stage.


mm120002

≈l

mm2001

≈l

M1

M2

M3

1222 ll +=L

Pre-Isolator Pre-Isolator

Nd:YAG Laser

r1

= r2

= 0.999 r3

= 0.9999

Figure 2.3: The AIGO mode-cleaner layout formed by two flat mirrors (M1 and M2) in

one end and one curved mirror (M3) at the other end, giving shape to the triangular ring

cavity.

diameter high vacuum pipe

The first stage of horizontal pre-isolation is an inverse pendulum (figure 2.1(a))

with a period of 20 s. Suspended from it is a Roberts Linkage horizontal pendulum

with a resonant frequency less than 0.05 Hz. The configuration of this Roberts Linkage

differs from the one designed for the AIGO test masses in having very low mass and

three attachment points (wires) instead of four, making it easier to tune [11].

Further isolation stages hang from the Roberts Linkage apex (figure 2.1(b)). These

consist of an Euler spring stage and cantilever spring assembly for vertical isolation.

A gimbal is situated in between for a high moment of inertia rocking mass to create

a self-damped pendulum. An array of magnets will provide the damping due to the

induced eddy currents. This provides passive damping for the pendulum modes of

the suspension chain and the vertical modes of the cantilever spring assembly. An

aluminium optical bread-board is suspended from the cantilever spring assembly to

mount the optical components. The mode-cleaner acts as a spatial filter, providing

passive stabilisation of time-dependant higher-order spatial modes and transmitting

the fundamental mode (TEM00). A key factor to reduce the transmission of the

higher-order modes is the radius of curvature of the end mirror (M3 on figure 2.3).

A simplified expression of the suppression factor is given in equation (2.1) [14]. The

suppression depends directly on the ratio between the cavity length (L) and the radius

of curvature (R), the reflectivity of each mirror (r1, r2, r3) and the order of the mode

expressed by the (m+ n) parameter.

2.3. ISOLATION AND SUSPENSION DESIGN 43

Radius (m) 17.4 22.2 26.4 31.8 38.9 43.3

Waist (mm) 1.65 1.94 2.11 2.29 2.47 2.56

Stability (g) 0.31 0.46 0.55 0.62 0.69 0.72

Power density for 586 426 359 305 262 243

100 W (kW cm−2)

Table 2.2: Some of the parameters affected by different radii of curvature and in particular

the effect on the power density on the cavity flat mirrors.

Smn =

(1 +

4

π2

[π2r1r2r3

(1− r1r2r3)2

]sin2

(m+ n)cos−1

[√1− L

R

])1/2

. (2.1)

The suppression of the higher-order modes is not the only consideration when

choosing the curvature of the mirror. The built-up power inside the cavity depends

on the finesse, but the area where this power will concentrate depends on the radius

of curvature of the end mirror M3. This defines the size of the beam waist that

defines the spot size on each mirror and hence the power density on them. Therefore

the curvature of the mirror is limited by the damage threshold of the mirror coating,

normally 1 MW cm−2 and by the stability factor g of the cavity. Due to the high laser

power there is a thermal lensing effect that needs to be taken into account, which for

simplicity and limited space has not been considered.

Figure 2.4 shows the suppression factor for different higher modes. It is clear

from the figure that the smaller the radius of curvature more higher modes will be

transmitted. The larger the radius of curvature the wider is the window from where

we can select a radius that will suppress most of the higher modes. At the same

time the size of the waist will increase. As a consequence the power density on the

flat mirrors will be reduced, but the price is that the cavity becomes less stable as

summarised in table 2.2. Therefore we need to choose a radius of curvature that will

give us a good compromise between power density and cavity stability. A radius of

curvature of 31.8 m will give us a safety margin of 300% on the damage of the mirror

coating with a stability factor of 0.62. Other choices such as 38.9 m and 43.3 m give

higher safety margins but with reduced cavity stability.


10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

M3 Radius of Curvature

Suppre

ssio

n F

acto

r [d

B]

Figure 2.4: Simplified calculations of the suppression factor of the first 15 higher order

modes for a 12 m triangular ring cavity.

2.4 Noise and locking predictions

In order to predict the behaviour of the isolator we used the theoretical transfer

function and seismic data measured at the AIGO site in Gingin. With this data

it is possible to determine the residual motion of the mirrors as shown in figure 2.5.

Here we see that at 10 Hz the residual motion is about 3.4×10−16 m Hz−1/2 (Advanced

LIGO expects a horizontal noise of 3×10−14 m Hz−1/2 at 10 Hz; LIGO-G020151-00-D).

This corresponds to a frequency stability of

δL

L=δf

f, f ≈ 2.82× 1014Hz⇒ δf ≈ 7.9mHz Hz−1/2. (2.2)

The lock acquisition of the cavity will be easier to achieve if the mirror velocities

are below the critical level. This is the velocity at which the frequency shift due to

the Doppler Effect becomes equal to the line-width of the cavity [15]. The velocity

is inversely proportional to the cavity storage time τ , but as can be seen in equation

(2.4) this depends on the length (L) and the finesse (F) of the cavity:

νcr =λ

2τFwhere F =

π√r1r2r3

(1− r1r2r3), (2.3)

2.4. NOISE AND LOCKING PREDICTIONS 45

10-2

10-1

100

101

10-20

10-15

10-10

10-5

100

ModeCleaner Preisolator Horizontal Response to Seismic Noise at Gingin

frequency [Hz]

Magnitude m

/ √H

z

Horizontal Seismic Noise

Vertical Seismic Noise

Horizontal System Response

Coupled Vertical System Response

Figure 2.5: Gingin horizontal and vertical seismic noise and predicted system response.

The vertical response is reduced in three orders of magnitude to simulate the vertical cou-

pling into the horizontal stages.

τ =2(L/c)

|ln(r1r2r3)|≈ 2

L

c

Fπ. (2.4)

The lower the velocity the easier it will be to lock the cavity. Table 2.3 summarises

some of the parameters calculated for AIGO mode-cleaner assuming mirror losses are

only due to transmission.

Using the isolator theoretical transfer function, we obtain the rms residual motion

(indefinite integral) of the mode-cleaner mirrors for the measured seismic noise at

Gingin, as shown in figure 2.6 (a). At 1 Hz the residual motion is about 3.8×10−11 m.

Free spectral range 12.388 MHz

Cavity optical bandwidth 8.285 kHz

Finesse 1495

Storage time 38.42 µs

Critical velocity 9.26 µm s−1

Table 2.3: Theoretical parameters of the AIGO 12 m mode-cleaner.


Figure 2.6 (b) shows the rms mirror velocity where at 1 Hz the mirror velocity is

about 4× 10−11 m sec−1, well below vcr calculated above as 9.26 m sec−1. This means

that in order to lock and control the mirrors we will need to apply forces of about

8.2× 10−6 N at 1 Hz. It can also be seen that above 0.4 Hz the noise is dominated by

the vertical noise coupled into the horizontal stages. The analysis here has ignored

the thermal noise of the test masses which becomes a significant noise source above

10 Hz.

2.5 Conclusions

Using a theoretical transfer function and on-site measured seismic noise we charac-

terise the behaviour of a compact and novel isolation system that will be used for

the mode-cleaner of AIGO high power facility. We have shown that two steps of pre-

isolation allow frequency stability, residual motion and mirror velocities to be kept

well below the critical value. From the Pound-Drever technique only low frequency

corrections are needed to control the mirror position, therefore from these results we

expect to be able to use low servo forces for cavity locking, and hence to minimise

noise injection from the control system.

As part of the design of the mode-cleaner we also calculate the radius of curvature

of the end mirror M3. It is possible to achieve more than 40 dB suppression of the

first 15 modes and to allow the transmission of 100 W of laser power into the main

arms of the detector with a power density of 305 kW cm−2.

Acknowledgements

The authors want to thank Li Ju, John Winterflood, Jerome Degallaix and John Jacob

for helpful discussions. This work was supported by the Australian Research Council,

and is part of the research program of the Australian Consortium for Interferometric

Gravitational Astronomy.

2.5. CONCLUSIONS 47

(a)

10-2

10-1

100

101

10-20

10-15

10-10

10-5

100

105

Mirror Residual Motion

Frequency (Hz)

Mirro

r R

esid

ual M

otion (

m)

Horizontal

Coupled Vertical

(b)

10-2

10-1

100

101

10-20

10-15

10-10

10-5

100

Mirror Velocity

Frequency (Hz)

Mirro

r V

elo

city (

m/s

ec)

Horizontal

Coupled Vertical

Figure 2.6: (a) Mirror rms residual motion as a function of frequency for the mode-cleaner

mirror suspension. (b) Mirror rms velocity as a function of frequency for the mode-cleaner

mirror suspension.


2.6 Postscript

In order to test the mode-cleaner vibration isolator, a full system was assembled on

top of a table. The purpose was to shake the whole system in order to measure

the mechanical transfer function. The driving system comprised a large loudspeaker

specially modified in order to drive the suspension system at low frequencies. The

driving signal was generated by the built-in source signal generator of a spectrum

analyser. This signal was connected to the modified loudspeaker through a high

voltage amplifier. The spectrum analyser was then used in swept sine mode to perform

the measurements at different frequency ranges. Two geophones were mounted on

the suspension system; one on top of the inverse pendulum stage where the vibration

isolator was attached to the driving system and the other on top of the bread-board.

Their differential signal gives the mechanical transfer function of the isolation system.

Measurements were performed for the two horizontal axes X and Y as shown on figure

2.7.

Following several tests and revision of the theoretical model, a new set of mea-

surements was performed using the described experimental setup. Originally the

inverse pendulum stage was modelled as a low pass filter with a corner frequency at

0.1 Hz and a 20 dB/dec roll-off above this corner frequency. Since this roll-off does

not continue indefinitely, a floor at around 30 dB was added to the inverse pendulum

X - Axis

Y - Axis

Figure 2.7: Top view of the mode-cleaner vibration isolation system defining X and Y

axes used for the measurements of the horizontal transfer function.

2.6. POSTSCRIPT 49

theoretical model. The measurements agreed well with the new theoretical model.

Due to operational limitations of the geophones it was not possible to obtain an ac-

curate measurement of the inverse pendulum frequency response for low frequencies.

Therefore the measured curves presented in figure 2.8 show the measurements of the

isolation chain from the Roberts Linkage stage down to the bread-board, with the

theoretical inverse pendulum curve added for comparison.

(a)

10-1

100

101

-200

-150

-100

-50

0

50

Horizontal Transfer Function

Frequency (Hz)

Ma

gn

itu

de

(d

B)

Theoretical Horizontal

Complete Isolator Transfer Function (Y-Axis)

Complete Isolator Transfer Function (X-Axis)

(b)

10-1

100

101

-100

-50

0

50

Vertical Transfer Function

Frequency (Hz)

Ma

gn

itu

de

(d

B)

Theoretic Vertical

Measured Vertical

Figure 2.8: Comparison between theoretical and measured transfer function for both (a)

horizontal axes X and Y and (b) for the vertical axis.

Figure 2.8 (a) shows the preliminary measurement of the horizontal transfer func-


tion. At low frequencies the measurements show a double peak at about 0.1 Hz which

corresponds to the inverse pendulum frequency. The double peak is generated by the

coupling of the X and Y axes of the inverse pendulum, each having slightly different

frequencies. Unfortunately limitations of the geophones do not allow for accurate

measurement at these low frequencies. The inverse pendulum peak is followed by

the Roberts Linkage normal mode at 0.2 Hz, which is higher than the design value.

The normal modes of the pre-isolation stages are followed by two pendulum modes at

1.7 Hz and 3.6 Hz, which correspond to the pendulum that holds the cantilever spring

stage from the Euler spring and the wires that attach the bread-board to the can-

tilever stage. Higher order modes at 7.2 Hz (in the X direction only), 10 Hz (10.2 Hz

in the Y axis and 10.7 Hz in the X axis), and around 17 Hz (16.5 Hz in the X axis

and 18.1 Hz in the Y axis) correspond to resonance modes of the bread-board stage.

These higher order modes were viscously damped using eddy current coupling. This

damping was achieved by attaching a short leg with a neodymium boride permanent

magnet at the end to the tip of each cantilever spring and adding a small copper plate

to each corner of the bread-board just underneath each leg-magnet pair.

Figure 2.8 (b) shows the preliminary measurement of the vertical axis of the

vibration isolator. The first peak at 1.4 Hz corresponds to the Euler spring resonant

mode and the second peak at 3.5 Hz to the cantilever spring stage. In comparison with

the measured results, the theoretical model shows lower Q-factor and a lower resonant

frequency of 0.8 Hz for the Euler spring stage and a lower frequency close to 2 Hz for

the cantilever spring stage. This difference can be partly attributed to the higher

frequency of the Euler spring resonant mode. The measurement also shows a third

resonant mode close to 10 Hz that corresponds to the coupling of a high frequency

mode from the bread-board.

After several sets of measurements, recommendations for system improvements

were made. Including the improvement of the inverse pendulum leg-flexure alignment

control to achieve more consistent tuning at low frequencies. A more robust Roberts

linkage frame with additional tune damping at the top. Replacement and tuning of

the Euler spring blades in order to lower the resonant frequency. A more radical

improvement will be the addition of a LaCoste stage, but utilising an Euler spring

2.7. REFERENCES 51

Mode Cleaner Measurements

Figure 2.9: Mode-cleaner vibration isolator system mounted inside a pipe for testing. At

the far right hand side of the suspension it is possible to see the ports in the pipe that were

intended for use as the main laser port and cable feed-through.

design approach for the spring components. This will allow for a sensing and actuation

scheme similar to the main advanced vibration isolators.

Unfortunately the GW research group suffered a funding shortage. This affected

several projects including the development of the seismic isolation system for the

mode-cleaner. This resulted in postponement of both the planned vibration isolation

system modifications and the completion of measurements using more sensitive de-

vices such as shadow sensors. This forced us to focus on a related system, a suspended

optical cavity utilising an advanced isolation system, which is presented in the next

chapter. The electronics developed for the mode-cleaner isolation system were then

used for the local control of the advanced vibration isolation system.

2.7 References

[1] R. W. P. Drever, J. L. Hall, F. V. Kowalski, et al, “Laser Phase and Frequency

Stabilization Using an Optical Resonator,” Appl. Phys. B 31 (1983) 97–105.

[2] N. A. Robertson, G. Cagnoli, D. R. M. Crooks, et al, “Quadruple suspension

design for Advanced LIGO,” Class. Quantum Grav. 19 (2002) 4043–4058.


[3] S. Goßler, M. M. Casey, A. Freise, et al, “The modecleaner system and suspension

aspects of GEO 600,” Class. Quantum Grav. 19 (2002) 1835–1842.

[4] M. V. Plissi, K. A. Strain, C. I. Torrie, et al, “Aspects of the suspension system

for GEO 600,” Rev. Sci. Instrum. 69 (1998) 3055–3061.

[5] M. Ando, K. Tsubono for the TAMA collaboration, “TAMA project: Design and

current status,” AIP Conf. Proc. 523 (2000) 128–139.

[6] A. Takamori, M. Ando, A. Bertolini, et al, “Mirror suspension system for the

TAMA SAS,” Class. Quantum Grav. 19 (2002) 1615–1621.

[7] F. Bondu, A. Brillet, F. Cleva, et al, “The VIRGO injection system,” Class.

Quantum Grav. 19 (2002) 1829–1833.

[8] G. Losurdo, M. Bernardini, S. Braccini, et al, “An inverted pendulum preisolator

stage for the VIRGO suspension system,” Rev. Sci. Instrum. 70 (1999) 2507–

2515.

[9] A. Bernardini, E. Majorana, P. Puppo, et al, “Suspension last stages for the

mirrors of the VIRGO interferometric gravitational wave antenna,” Rev. Sci.

Instrum. 70 (1999) 3463–3472.

[10] J. Winterflood, “High performance vibration isolation for gravitational wave de-

tection,” PhD Thesis, School of Physics, The University of Western Australia,

Chapter 7, 2001.

[11] F. Garoi, J. Winterflood, L. Ju, et al, “Passive vibration isolation using a Roberts

linkage,” Rev. Sci. Instrum. 74 (2003) 3487–3491.

[12] J. Winterflood, Z. B. Zhou, L. Ju, D. G. Blair, “Tilt suppression for ultra-low

residual motion vibration isolation in gravitational wave detection,” Phys. Lett.

A 277 (2000) 143–155.

[13] J. Winterflood, “High performance vibration isolation for gravitational wave de-

tection,” PhD Thesis, School of Physics, The University of Western Australia,

Chapter 3, 2001.

2.7. REFERENCES 53

[14] M. Barsuglia, “Stabilisation en frequence du laser et controle de cavites optiques

a miroirs suspendus pour le detecteur interferometrique d’ondes gravitationnelles

VIRGO,” PhD Thesis, Universite de Paris-Sud, Orsay, Chapter 7, 1999.

[15] M. Rakhmanov, “Doppler-induced dynamics of fields in Fabry-Perot cavities with

suspended mirrors,” Appl. Opt. 40 (2001) 1942–1949.

Chapter 3

Advanced Vibration Isolator

3.1 Preface

This chapter comprises two papers describing the advanced vibration isolation system

developed at UWA. Both papers have been submitted to the Review of Scientific

Instruments journal. The first paper presents a review of the vibration isolator and

includes the measurements of the frequency response of the cavity, which led to the

characterisation of vibration isolator performance. The second paper introduces the

local control system which allows for feedback control and normal mode damping.

The control electronics and its diagrams are presented in Appendix C.

Development of the mode-cleaner vibration isolation system included design and

development of control electronics. These were developed to amplify and filter the

input signal from the shadow sensors and to amplify the signal that drives the coils

of the magnetic actuators. Since the shadow sensors and actuators installed in the

mode-cleaner are very similar to the ones installed in the full scale vibration isolation

system, the same electronics (with the exception of gain tuning) were used for the

advanced vibration isolator local control system. In addition to development of the

electronic boards, a backplane mounted on the back of a 6U chassis was necessary

in order to distribute the I/O signals between the vibration isolator and the DSP

(Digital Signal Processor) at the core of the local control system.

This required the design of signal distribution and cabling to connect the con-

trol electronics to the different sensors and actuators distributed along the vibration

isolator structure. In order to facilitate these connections, a vacuum-compatible in-

termediate board was designed and installed on the vibration isolators. The author

55

56 CHAPTER 3. ADVANCED VIBRATION ISOLATOR

was not involved in the wiring of the advanced vibration isolators. The intermediate

board for the signal distribution was designed in consultancy with J. C. Dumas. The

built-in libraries provided by the DSP manufacturer allow for the control loops to run

on the DSP board whilst isolated from the local operating system. The user control

and interface was written in LabViewr predominantly by J. C. Dumas.

The second vibration isolator was being assembled and tuned at the AIGO test

facility in Gingin by J. C. Dumas, A. Woolley and technicians. With the first vibration

isolator fully operational in the main lab, several tests were carried out to determine

some of the loop gains to be used in the PID control loops (even though not all the

loops are PID, some only use integral gain and some only derivative gain for damping).

Tests of the mechanical transfer functions were also completed. At the same time the

author prepared an optical table and installed a 300 mW Nd:YAG laser with the

necessary optics for cavity mode-matching, phase-modulation and optical isolation of

the system. With support from C. Zhao, the author tested the possibility of using a

reference cavity for laser stabilisation and an acousto-optic modulator for locking the

main cavity. After several tests the dynamic range of the acousto-optic modulator

(used in double pass mode) was found to be insufficient for following the longitudinal

variations of the cavity. Therefore laser locking was achieved directly to the main

cavity, as presented in the first paper of this chapter.


Compact vibration isolation and suspension for

AIGO: Performance in a 72 m Fabry Perot cavity

P. Barriga, J. C. Dumas, A. A. Woolley, C. Zhao, D. G. Blair

School of Physics, The University of Western Australia, Crawley WA 6009, Australia

This paper describes the first demonstration of vibration isolation and sus-

pension systems which have been developed with view to application in

the proposed Australian International Gravitational Observatory (AIGO).

In order to achieve optimal performance at low frequencies new compo-

nents and techniques have been combined to create a compact advanced

vibration isolator structure. The design includes two stages of horizontal

pre-isolation, and one stage of vertical pre-isolation with resonant fre-

quencies ∼100 mHz. The nested structure facilitates a compact design

and enables horizontal pre-isolation stages to be configured to create a

super-spring configuration, where active feedback can enable performance

close to the limit set by seismic tilt coupling. The pre-isolation stages

are combined with multistage 3-D pendulums. Two isolators suspending

mirror test masses have been developed to form a 72 m optical cavity with

finesse ∼700 in order to test their performance. The suitability of the iso-

lators for use in suspended optical cavities is demonstrated through their

ease of locking, long term stability and low residual motion. An accom-

panying paper presents the local control system and shows how simple

upgrades can substantially improve residual motion performance.

3.2 Introduction

The first vibration isolation systems used multiple stages of mass-spring elements

based on rubber and steel stacks followed by a pendulum stage where individual

mode frequencies were of a few Hertz. This technology was inherited from resonant

bar gravitational wave detectors. In the early 1990’s the University of Western Aus-

tralia (UWA) group introduced cantilever blade springs, both for use on cryogenic


resonant bar detector Niobe [1], and then on an 8 m prototype interferometer [2].

All metal vibration isolation components are required for use in high vacuum envi-

ronments (while rubber is not permissible). However multiple stages of mass-spring

elements suffer from relatively high normal mode Q-factors which comes as strong res-

onant enhancements in the transfer function. To avoid this problem the UWA group

proposed the use of pre-isolators [3]. The idea was to precede a multistage isolator

with a single very low frequency stage, which would reduce the seismic excitation of

the higher frequency normal modes [4]. The VIRGO project was the first to use this

concept in a full scale detector [5], by suspending vibration isolation stages consisting

of long pendulums and cantilever springs by a 6 m inverse pendulum stage with very

low resonant frequency [6].

In the 1980’s a group at JILA developed active vibration isolator based on the

super-spring concept. In this case a two element spring with a sensing transducer and

actuator allows a synthetic spring to be created, with resonant frequency determined

by a control system [7, 8].

During the 1990’s Winterflood et al [9, 10, 11] developed a range of pre-isolation

techniques based on geometric cancellation of elastic spring constants –so called geo-

metric anti-springs. Of these, greatest attention has been given to the Roberts Link-

age [12] which is described in section 3.3. The VIRGO group developed magnetic

anti-springs to lower the vertical frequency of their cantilever spring systems [13]. At

the top of the inverse pendulum is the first vertical filter. It contains a set of maraging

steel triangular blades from where a five stage pendulum chain is suspended. Each

intermediate mass is a ∼100 kg drum-shaped vertical filter. Each one of them also

includes magnetic anti-spring systems in order to reduce resonance frequencies [14].

From the last stage, known as marionetta, the test mass and a reaction mass are

suspended providing three degrees of freedom (translation, pitch and yaw) [15]. This

design will also be used in Advanced VIRGO [16].

A new approach to vertical vibration isolation was developed by Winterflood et

al [17] –the Euler spring. This involves loading a vertical elastic column just beyond

the Euler buckling instability; where it becomes a well behave spring with frequency

equal to 1/√

2 of the frequency of a pendulum of the same length. Methods were found


of adding geometric anti-spring elements to this system to create even lower resonant

frequencies [18]. The Euler spring approach is advantageous because the stored elastic

energy is reduced by a factor of (working range/effective length)2, enabling the spring

to be of much lower mass, and with much higher internal mode frequencies.

A problem with multiple pendulum stages is the difficulty of damping their normal

modes without noise injection. Horizontal isolation at the GEO600 project is provided

by a triple pendulum system. This assembly is suspended from a two layer isolation

stack consisting of an active and a passive stage. A separate reaction pendulum is

included so global control forces can be applied to the test mass from a seismically

isolated platform [19]. At UWA we developed the concept of self-damped pendulum

which provides a passive solution to the problem, through dissipative coupling to an

angular degree of freedom within the pendulum mass [20].

The TAMA project in collaboration with the LIGO laboratory is also testing an

advanced vibration isolation system called Seismic Attenuator System or SAS [21].

TAMA-SAS is conceptually similar to the VIRGO design, with a shorter inverse pen-

dulum (2.5 m compared to VIRGO 6 m), a similar top vertical filter and a suspended

triple pendulum stage [22].

Meanwhile the Advanced LIGO project favoured a ‘stiff’ pre-isolation technique

based on hydraulic systems distributed on each corner of the vacuum chamber [23].

These provide vertical and horizontal pre-isolation for the whole chamber combining

geophones and seismometers with high resonant frequencies in a high bandwidth

control loop. Based on the GEO600 design Advanced LIGO will use a quadruple

pendulum suspension for horizontal isolation [24]. Therefore two stages of active

isolation are followed by a pair of parallel quadruple pendulums, the last being the

interferometer test mass, which is opposed by a reaction mass used for actuation

All of the above concepts have been integrated into a single advanced vibration

isolation system. The system we have created uses double pre-isolation, to enable

the super-spring concept to be used to suppress normal peaks. It is nested so that

the entire structure is less than 3 m in height. Including a multi-pendulum stage

with self-damping. Two systems have been installed at the East arm of the AIGO

research facility in Gingin, Western Australia. Test masses are suspended from each


system to form a 72 m optical cavity. A 300 mW Nd:YAG laser locked to the cavity

through the Pound-Drever-Hall technique [25] provides an error signal to measure the

performance of the suspended cavity at low frequencies.

While the vibration isolation system is mainly a passive design, some feedback

is required for low frequency control, such as alignment and drift corrections, and

damping of normal modes. Each suspension system is integrated with a digital local

control system with feedback to the pre-isolation stages and the penultimate pendu-

lum stage. Details of the local control strategy are discussed in an accompanying

article by Dumas et al [26].

In this article we present the first results of the noise performance in the suspended

cavity. In section 3.3 we review the overall isolator design, introducing individual

isolation concepts and components. The experimental setup is presented in section 3.4

including cavity parameters and the laser control system. In section 3.5 we discuss

the results obtained during the operation of the locked cavity. Finally in section 3.6

we discuss the performance of these isolators and future developments.

3.3 Vibration isolation design

The Australian Consortium for Interferometric Gravitational Astronomy (ACIGA)

has developed a high performance compact vibration isolation system for the pro-

posed AIGO interferometer. Novel isolation elements for all vertical and horizontal

stages were individually developed and tested. First they were designed to include

multiple passive low frequency pre-isolation to minimise the seismic excitation of iso-

lator normal modes. This was achieved by applying simple geometric anti-spring

techniques to achieve very low resonant frequencies. Second, they were designed to

attain passive damping of pendulum modes through the concept of self-damped pen-

dulum. Third, Euler springs were used to obtain vertical normal modes frequencies

well matched to the pendulum frequencies. Fourth, special attention was given to

the materials used for the design and construction of the suspension in order to be

vacuum compatible. Finally centre of percussion tuning was used when possible to

optimise the transfer function of individual stages [27].

3.3. VIBRATION ISOLATION DESIGN 61

Control Mass

Test Mass Mirror

Niobium Suspension

Flexure

Top of pre-

isoltator stand

LaCoste

vertical stage

Inverse

Pendulum

Rigid links

Euler springs

vertical stages

Self-damping

system

Three vibration

Isolation stages

Roberts

Linkage

Figure 3.1: Full vibration isolator system and schematic that show the different stages of

pre-isolation and the multi-pendulum stage with a test mass at the bottom of the chain.

3.3.1 Pre-isolation components

The vibration isolation system consists of multiple cascaded stages as shown in fig-

ure 3.1 applying several different techniques to attenuate seismic noise. At the top

of the suspension four short inverse pendulums legs provide the first stage of pre-

isolation. This first pre-isolation stage is effectively a square table mounted on four

inverse pendulums for legs. This allows for horizontal translation, while being rigid to

tilt. The resonant frequency of these pendulums can be tuned to very low frequency

providing very effective pre-isolation. Its cube shape allows for the integration of a

spring system that provides vertical pre-isolation. The LaCoste linkage consists of

diagonally attached springs between each of the four legs on two structures (the in-

verse pendulum and the LaCoste supporting frame). The four lower pivot arms were

fitted with counter-weights in order to provide centre of percussion tuning for the

eight of them. Pre-tensed springs were used in order to obtain the ‘zero-length’ re-

quirement of the LaCoste geometry [28]. Horizontal springs were added and stretched


more than the separation between the pivoting points creating an inverse pendulum

effect. This effect produces a negative spring constant that counteracts the significant

spring constant of the flexure pivots. By making the width adjustable one can tune

the spring constant in order to obtain very low frequencies and a large dynamic range.

Therefore both pre-isolation stages have resonant frequencies below 100 mHz. Both

the inverse pendulum and the LaCoste stages have a cubic geometry which allows for

their combination into a single 3-D structure as illustrated in figure 3.2.

Figure 3.2: First stage horizontal and vertical pre-isolation. The pre-isolator combines

two very low frequency stages: (a) The horizontal Inverse Pendulum and (b) the vertical

LaCoste linkage. The concept of the anti-spring for flexure spring constant cancellation is

shown.

The rigidity of this structure allows for the suspension of a Roberts linkage stage

nested within the two pre-isolators. It is a relatively simple design consisting of

a cube frame suspended by four wires hung off the LaCoste stage as illustrated in

figure 3.3. Its geometry is tuned to restrict the suspension point of the load to an

almost flat horizontal plane, thus making the gravitational potential energy almost

independent of displacement and minimising the restoring force resulting in a low

resonance frequency [12, 27]. At only 1 m height the whole top section is very compact,

including also the topmost stage of the multi-stage pendulum in the same volume.

The pre-isolation stages are mounted on top of a rigid frame so as to have enough

height to suspend the isolation chain from the top of the Roberts linkage as seen in


figure 3.1. The interweaving of the pre-isolation stages together with the use of large

dynamic range sensors and actuators allows for an active feedback control of the pre-

isolator stages enabling performance close to the limit set by seismic tilt coupling [26].

Figure 3.3: The Roberts linkage. (a) shows a one-dimensional diagram of a Roberts

Linkage with a suspended load from point P, which stays in the same plane for variations in

the position of C and D. (b) shows a diagram of the cube shaped design used in the AIGO

suspension.

3.3.2 Isolation stages

Intermediate masses of 40 kg are suspended to form a self-damped pendulum ar-

rangement as illustrated in figure 3.4. The self-damping concept consists of viscously

coupling different degrees of freedom of the pendulum mass as shown in figure 3.5.

Each intermediate mass is pivoted at its centre of mass, with a light ×-shaped frame

fixed to the pendulum link to provide a reference against tilt. Neodymium boride

magnets in a comb-like distribution are mounted to the frame. These are paired

with intermeshing copper plates attached to each corner of the square rocker mass

to create a viscous damping through eddy current coupling reducing the Q-factor of

the pendulum normal modes [20]. An aluminium arm is attached on each side of the

top rocker mass of the pendulum chain. The purpose is to increase the moment of

inertia to further reduce the Q-factor of the lower resonant mode of the multi-stage

pendulum.


Al web rigidly clamped to

the suspension tube

Copper plates

attached to Al web

Magnets, attached

to rocker mass

Rocker mass

Rotational arm rests on Euler springs

and is suspension point for next stage

Euler springs

Bottom of springs is clamped

to suspension tube

Control mass

Niobium flexures

Test mass

Figure 3.4: The multi-stage pendulum including three intermediate masses showing the

rigid section, rocker mass, eddy current damping and Euler springs for vertical isolation.

At the bottom of the chain the control mass stage provides sensing and control for a test

mass suspended with niobium flexures.

Each intermediate mass in the multi-stage pendulum is attached to the next using

Euler springs tuned for low frequency vertical isolation effectively attenuating the

vertical component of the seismic noise. Euler spring stages can be tuned with anti-

spring geometries to achieve good low frequency performance within a very compact

design [17, 18, 29]. Figure 3.6 shows a diagram of an intermediate mass with Euler

spring attachment, while figure 3.7 illustrate in detail one of the intermediate masses.

3.3.3 Control mass and test mass suspension

At the bottom of the multi-stage pendulum chain shown in figure 3.4 a ∼30 kg control

mass stage provides the interface to the test mass. The test mass consists of a 50 mm,

30 mm thick fused silica mirror supported in an aluminium and stainless steel cylinder.


Viscous damping Pivot

Double wire suspended on

a pivot, free to swing.

Rocker mass, high

moment of inertia.

Figure 3.5: Diagram of one self-damped pendulum stage. Magnets that generate eddy

currents on copper plates create the viscous damping for a high moment of inertia rocker

mass.

Figure 3.6: Schematic of the intermediate mass showing the Euler Spring vertical stage

and the attachment to the rocker mass [29].

The purpose is to closely replicate a ‘real’ test mass and allows for the characterisation

of the high performance vibration isolation chain developed at UWA using an optical

cavity. Four niobium ribbons each of 25µm thick, 3 mm wide and 300 mm long are

used to suspend the test mass from the control mass stage. These are design to

minimise internal modes providing at the same time a high Q-factor [30].

For the initial suspension we have used temporary brass pins clamped to the

suspension ribbons through a high pressure contact tooth instead of the high pressure

contact pins bonded to the end of the ribbons [31]. The ribbon clamping mechanism,

as opposed to a permanent bond, is not ideal. However, the current design provides

high enough contact pressure (approaching the yield strength of niobium) to minimise

slip stick friction, whilst not weakening the suspension ribbon at the point of clamping.


Al web rigidly clamped to

the suspension tube

Copper plates

attached to Al web

Magnets attached

to rocker mass Rocker mass

Rotational arm rests on Euler springs

and is suspension point for next stage

Euler springs

Bottom of springs is clamped

to suspension tube

Integrated 3D isolator stage

Figure 3.7: Intermediate mass showing the integration of the high moment of inertia

rocker mass with the Euler spring vertical stage. The intermediate mass has a hollow tube

in the center to allow for the suspension wire to go through all the stages. The figure also

shows the×-shaped frame that hold the copper plates on top of the rocker mass. Attached

to the rocker mass are the magnets that create the damping through eddy current generation

on the copper plates [20].

Even though the same pin design has been used for the initial brass pins extra thermal

noise may be induced. Currently, priority lies in testing the performance of the

vibration isolation system. Therefore a temporary brass/niobium suspension at the

expense of a possible increase in thermal noise is acceptable.

Figure 3.8 illustrates the control mass stage. A cage attached to the control mass

provides both mechanical safety stops for the test mass, and a low noise reference from

which actuation and local sensing of the test mass can be performed. The control

mass is suspended from the vibration isolation system using a single wire. It also

contains actuators and sensors from which 5 degrees of freedom (translation in all 3

dimensions, yaw and pitch) can be accessed [26].

3.3.4 Integrated system

The isolator structures for AIGO were designed to solve a range of problems in

vibration isolation. This system relies on passive damping design, three stages of

pre-isolation and the use of several novel isolation techniques to achieve nanometer


Actuation arms

Control mass

Suspension

cage

Test mass Niobium

ribbons

Figure 3.8: Control mass stage with test mass suspended. Actuation arms holding per-

manent magnets are attached to the control mass. From the control mass a suspension cage

is attached and a test mass is suspended by four niobium ribbons.

residual motion at low frequencies. The three pre-isolation stages incorporate an in-

verse pendulum as a first horizontal pre-isolation and a LaCoste linkage for vertical

pre-isolation. Nested inside is a second stage of horizontal pre-isolation based on a

Roberts linkage. This compact triple pre-isolation structure supports an isolation

stack which consists of three low frequency three-dimensional isolator stages combin-

ing self-damped pendulums and Euler springs. At the bottom of the chain a control

mass provides the interface with the test mass through niobium flexures.

The local control system is used to compensate for movements induced by the

seismic noise in the local reference system of a single suspension, in particular at the

resonant frequencies of the suspension system. Position readout is done through large

dynamic range shadow sensors. Feedback forces maintain position and alignment of

the pre-isolation stages and the control mass through magnetic actuators attached to

an inertial frame and referred to the ground. An optical lever was added outside of

the vacuum system for test mass readout. The control system is also used to damp

some normal modes through velocity feedback. Details of the local control system


and its performance can be found in an accompanying paper by Dumas et al [26].

3.4 Experimental setup

The experimental setup consists of two vibration isolation systems. Each one with

a mirror as a test mass. A second isolator is necessary in order to have an inertial

reference. In this way we create a 72 m optical cavity. Each system is inside a tank and

under vacuum (10−6 mbar) interconnected through a ∼70 m long 400 mm diameter

pipe. Both systems run their own local control system as described by Dumas et

al [26]. The light source is a continuous wave 300 mW single frequency, non-planar

ring oscillator, Nd:YAG laser (λ = 1.064µm) mode-matched to the suspended cavity.

The laser beam centring onto the input test mass (ITM) and the end test mass (ETM)

is done with the aid of a CCD camera outside each station. The locking of the cavity

is done using a standard Pound-Drever-Hall technique. Neither frequency nor power

stabilisation nor spatial mode-cleaner are used in this experiment.

Figure 3.9: The picture shows the ETM during the assembly of the second suspension

system. A 2 inches mirror is mounted in a stainless steel support that gives the test mass

the same size of a ‘real’ test mass. At the back of the ETM we can see the electrostatic

board.

3.4. EXPERIMENTAL SETUP 69

3.4.1 Cavity parameters

At the bottom of the suspension system a test mass hangs from the control mass stage

using niobium ribbons. An extra optic-modulator was added to the laser control loop

in order to measure the cavity free spectral range (FSR) using the sideband locking

method [32]. The FSR was measured at 2.069 MHz which corresponds to a cavity

length of 72.45 m. Each of these test masses is made of a 50 mm fused silica mirror

mounted in an aluminium and stainless steel mass. The purpose is to facilitate the

mounting of the smaller mirror to the suspension system and give the payload the

size of a ‘real’ test mass. The ITM is flat and the ETM has a radius of curvature of

720 m. The parameters of the cavity formed by these two test masses are summarised

in table 3.1. A picture of the ETM during assembly of the second suspension system

can be seen in figure 3.9.

ITM radius of curvature ∞

ETM radius of curvature 720 m

Cavity g factor 0.8994

Waist size radius 8.565 mm

ITM spot size radius 8.565 mm

ETM spot size radius 9.031 mm

Waist position from ITM 0 m

Free spectral range 2.069 MHz

Table 3.1: Parameters for the 72 m cavity

In order to characterise the cavity we measured the intensity decay time of the

transmitted beam. Figure 3.10 shows an average of the measurements of the cavity

decay time with a value of τc = 56.3± 0.2µs. This value allows us to derive more

parameters for this cavity as shown in table 3.2 [33].

3.4.2 Suspension system transfer function

The mechanical transfer function of the ITM suspension system was measured. The

driving signal was injected at one of the actuators of the inverse pendulum. The


-0.1 0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cavity Decay

time (ms)

Am

plitu

de (

V)

1.626 e(-17.755 t)

Figure 3.10: Average of the cavity decay time. Several measurements were done in order

to determine an average decay time of 56.3± 0.2µs.

Cavity Parameters Formula Value

Finesse 2π (c/L) τc 732

Cavity bandwidth (2πτc)−1 2827 Hz

Cavity Q factor 2πfoτc 9.97× 1010

Total losses L/ (cτc) 8585 ppm

Reflectivity product (1− L/ (2cτc))2 0.9914

Cavity pole 1/ (4πτc) 1413 Hz

Table 3.2: Cavity parameters derived from our measurements. Here L corresponds to

the round trip optical path length, fo the optical frequency, c the speed of light in vacuum,

and τc corresponds to the measured characteristic decay time of the intensity defined as

I(t) = I0 exp (−t/τc).

response was measured at the control mass stage on one of the shadow sensors on the

same axis. As a consequence the higher frequency pendulum modes are too small to

be detected at the control mass shadow sensor. During the measurements there was

no control of any of the stages of the vibration isolator.

Figure 3.11 shows the measured mechanical transfer function of a complete vi-


10-2

10-1

100

101

-100

-80

-60

-40

-20

0

20

40

ITM frequency response

frequency (Hz)

Magnitude (

dB

)

Figure 3.11: ITM horizontal mechanical transfer function. The transfer function was

measured at the control mass level using the shadow sensor for sensing. The driving signal

was injected at the magnetic actuator mounted on the inverse pendulum.

bration isolator under vacuum. Two peaks at ∼ 58 mHz and ∼ 69 mHz correspond to

the X and Y axis respectively of the first pre-isolation stage. This is followed by the

Roberts linkage, which also has different frequencies for X and Y axis. The last peak

before the roll-off corresponds to the main pendulum mode at around 500 mHz [20].

This was confirmed with separate measurements on each axis [34].

The mechanical transfer function has the characteristic features of a ‘soft’ system,

with high peaks at frequencies below 100 mHz. The laser control system will need

high gain at low frequencies in order to follow the cavity length variations due to the

low frequency displacement of the test masses.

3.4.3 Laser control system

The isolation system can greatly reduce residual motion at high frequencies. However

a laser control system is necessary for the low frequencies. Based on a couple of

low noise amplifiers (Stanford Research Systems SR560) we design a simple control

system for the laser as shown in figure 3.12. A phase-modulator with a local oscillator

of 10 MHz is used to generate the required sideband. The error signal is then obtained


by a photo-detector reading the reflected signal which is synchronously demodulated

at 10 MHz. This error signal is send to a low noise voltage pre-amplifier with a cut-off

frequency of 3 Hz, a gain of 5, and a roll-off of 20 dB/dec, which compensates for

the pendulum modes above 1 Hz. The resonant frequencies of the pre-isolation stages

are around 0.1 Hz. Therefore large displacements at low frequencies are generated

by micro-seismic noise which could drive the system unstable. To control these low

frequency oscillations a second SR560 is installed. This instrument is set as a second

order (40 dB/dec roll-off) low pass filter with a corner frequency of 0.3 Hz and a gain

of 5 so as to avoid instabilities due to the phase lag induced by the filter around the

corner frequency. Ideally matching the roll-off of both filters is necessary in order

to obtain a smooth slope between the two filters and the optical cavity frequency

response. However the gains at the SR560 instruments were limited to a maximum of

5 otherwise the second filter (cut-off 0.3 Hz) saturates. The combined signal is added

in parallel to the second signal from the power splitter. The resulting signal goes

through a high voltage amplifier with a gain of 40 before going into the PZT port of

the laser. The voltage applied to the PZT (bond to the laser gain medium) stresses

the laser medium to adjust the laser frequency. The PZT has a frequency response

flat to about 100 kHz, at this frequency the phase is already diverging from zero. The

PZT

East Arm Cavity ~80m FIPM

10 MHz

IFLO

RF

Current

TempLaser

Laser Controller

Power Supply

LNA

0.3 Hz

12 dB/Oct

Gain 5 (10)

LNA

3Hz

6 dB/Oct

Gain 5

Inv Output

HV Amp

Gain 40

+

A

Figure 3.12: Diagram that shows the laser control system. A signal generator is used

to generate the 10 MHz sideband used for cavity locking. PM corresponds to a phase

modulator and FI a faraday isolator. The diagram does not include the optical components

necessary to steer and mode-match the beam into the main cavity.


10-1

100

101

102

103

104

105

0

50

100

Semi-theoretical transfer function

frequency (Hz)

Magnitude (

dB

)

10-1

100

101

102

103

104

105

-150

-100

-50

0

50

100

150

frequency (Hz)

Phase (

deg)

Unity Gain: 14.730 kHz

Phase: 37.5 o

Figure 3.13: The semi-theoretical transfer function is a combination of the measurements

of the electronics in the control loop and the optical cavity theoretical frequency response

and the PZT frequency response.

optical cavity can be modelled as a first order low pass filter with a cut-off frequency

given by the cavity pole at ∼ 1.4 kHz and a roll-off ∼ 20 dB/dec. The resulting semi-

theoretical transfer function of this system can be seen in figure 3.13. As expected the

magnitude curve shows a 20 dB/dec slope between 3 Hz and the cavity pole. However

above the cavity pole the slope increases to a measured value of 33.65 dB/dec around

unity gain; lower than the expected 40 dB/dec, which could have led to an unstable

system. This difference comes from the PZT frequency response and the Q-factor

around its corner frequency effectively ‘lifting’ the slope of the frequency response.

The phase also diverge from the slope around 100 kHz due to the phase-lag introduced

by the PZT response.

Figure 3.14 shows the measurements of the laser control loop around the unity

gain. The measurements were made using a spectrum analyser connected to point

A in the control loop as shown in figure 3.12. Using the swept sine mode we inject

the source signal and add it to the error signal then measure the frequency response

between the resulting signal and the input one (source + error signal). The figure

shows a good agreement between the semi-theoretical and the measurements of the


103

104

105

-20

0

20

40

Measured frequency response

frequency (Hz)

Magnitude (

dB

)

103

104

105

-150

-100

-50

0

50

100

150

frequency (Hz)

Phase (

deg)



Phase: 37.5 o

Phase: 37.0 o

Figure 3.14: Comparison between the semi-theoretical curve and an average of a few

measurements of the loop frequency response at high frequency.

frequency response. The semi-theoretical curve shows a unity gain around 14.7 kHz

and the average of the measurements a unity gain at 14.9 kHz. Correspondingly the

measured phase is in good agreement with the semi-theoretical curve at the unity gain

frequency with an average of 37.0o with a semi-theoretical value of 37.5o. This phase

value also shows the stability of the control loop. A gain of 150 was calculated in order

to match the semi-theoretical curve with the measured closed loop transfer function.

This gain includes the cavity gain, the mixer and the PZT. These components were

not included when measuring the electronics.

3.5 Measurements and results

The experimental set-up presented in the previous section was used to measure the

performance of the advanced vibration isolation system. The main measurements

were the residual motion of the test masses, including longitudinal motion as well as

pitch and yaw measurements.

Figure 3.15 shows the frequency response of the PZT driving signal. This signal

corresponds to the contribution of the two vibration isolator systems to the longitudi-

nal displacement. Around 70 mHz the contributions of the main pre-isolator frequency

3.5. MEASUREMENTS AND RESULTS 75

10-1

100

101

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

frequency (Hz)

Magnitude (

dB

Vrm

s/ √

Hz)

East arm displacement

Cavity displacement

Laser noise

Figure 3.15: Measurement of the frequency response of the laser PZT signal. The top

(blue) line shows the cavity frequency response and the bottom (red) line the laser noise. We

notice that above 1 Hz the main contribution to the cavity displacement frequency response

comes mainly from the laser noise.

10-1

100

101

10-10

10-9

10-8

10-7

10-6

East arm residual motion (PZT)

frequency (Hz)

rms d

ispla

cem

ent

(m)

PZT: 5.8e-009 m @ 1 Hz

Figure 3.16: Residual motion of the east arm cavity derived from the frequency response

measurements. A residual motion of 5.8× 10−9 m can be seen at 1 Hz which is reduced to

1× 10−9 m just below 5 Hz.


can be seen in the figure. The following peaks correspond to the Roberts linkage

resonant frequency (265 mHz) and the main pendulum frequencies at 640 mHz, and

950 mHz respectively. Figure 3.15 also shows the measured laser noise, which shows

that above ∼ 1.2 Hz the main component of the measured cavity displacement sig-

nal corresponds to laser noise. This measurement allows us to calculate the cavity

residual motion shown in figure 3.16. As expected there is an increase in residual

motion at the resonant frequencies. The residual motion of this cavity up to 1 Hz is

5.8× 10−9 m which is then reduced to 1× 10−9 m at just below 5 Hz. However this

mainly corresponds to the laser noise contribution.

10-1

100

101

10-6

10-5

10-4

frequency (Hz)

(ra

d)

Pitch and Yaw angular residual motion

Yaw: 4.62e-5 rad @ 0.1 Hz

Yaw: 4.79e-6 rad @ 0.1 Hz

Pitch: 6.74e-6 rad @ 0.1 Hz

Pitch: 4.14e-6 rad @ 0.1 Hz

Figure 3.17: Pitch and yaw angular residual motion for the ITM. The dotted lines show

the angular residual motion for the control mass when controlled only with the shadow

sensor. The continuous line shows the angular residual motion using the optical lever

control loop.

One of the main problems during the assembly and operation of the isolators and

the main cavity was the pendulum mode of the niobium test mass suspension. With-

out the electrostatic control there were no sensing of the test mass it self. Therefore

only the shadow sensor signal at the control mass stage was available for the signal

readout. The signal to noise ratio at the shadow sensor turned out to be too poor to

be able to detect pitch and yaw pendulum modes of the niobium ribbons at 3.3 Hz

3.6. CONCLUSIONS AND FUTURE WORK 77

and 1.75 Hz respectively, with particularly high Q in the yaw mode.

The addition of an optical lever allowed for the measurement of these modes. This

was installed outside the vacuum tanks in both isolators using a quad photo-detector

to feedback the signal to the control electronics and into the DSP. Due to the limited

area of the quad photo-detector the first stage of control of the test mass was done

using the shadow sensor readout, which allowed for a larger range for the control and

positioning of the test mass. Once the control signal was within the quad photo-

detector range the control of this last stage was handover from the shadow sensor

control loop to the optical lever control loop with a much better signal to noise ratio.

This allowed for the use of band-pass filters and damping loops for each pendulum

mode as part of the optical lever PID control loop. More details are presented in the

accompanying paper by Dumas et al [26].

Figure 3.17 shows a comparison between the residual motion of the pitch and yaw

modes. The dotted lines show the optical lever control loop off whilst the continuous

lines show the optical lever control loop on. Control loop off means that only the

shadow sensor readout and its corresponding PID loop are being use for controlling

the test mass position (pitch and yaw). The optical lever control loop on means that

the shadow sensor control loop is turned off, but the level of the control signal at

the time of switching from shadow sensor to optical lever control is used as an offset

for the optical lever control loop. Figure 3.17 shows at 3.3 Hz a reduction of the

pitch mode residual motion from 3.5µrad to 1.9µrad at the same frequency when the

optical lever control loop is on. The yaw resonant mode at 1.75 Hz is reduced from

26µrad to 2.5µrad when the optical lever control loop is on. The angular residual

motion obtained from these measurements is 4.79× 10−6 radians rms up to 100 mHz

in yaw and 4.145× 10−6 radians rms up to 100 mHz in pitch.

3.6 Conclusions and future work

We have shown for the first time that vibration isolators that combine multiple pre-

isolation stages, self-damping pendulums, Euler springs and niobium ribbons suspen-

sion have no unforeseen difficulties. Using a pair of advanced vibration isolators we


have shown that these systems have long term stability, and are responsive to the

controls required to operate long optical cavities. Two vibration isolator systems

were assembled 72 m apart as to form a suspended optical cavity. This was assembled

and tested on the east arm of the AIGO research facility in Gingin. At the same

time a local control system and its control electronics were developed for each of the

vibration isolators. This allowed us to operate and control the suspended optical

cavity. We have also demonstrated that without direct actuation at the test mass, it

is possible to achieve a residual motion of 1 nm at ∼5 Hz for the suspended cavity.

This includes the addition of an optical lever for improvement of the angular residual

motion by actuation at the control mass and therefore no direct actuation on the test

mass was required.

Future work includes the development of an auto-alignment system. Even though

we were able to lock the cavity and operate for long periods of time, an auto-alignment

system will dramatically improve the duty cycle. This is part of the current devel-

opment of a hierarchical global control scheme, which will allow for the operation of

longer cavities with higher finesse. This will be assembled and tested at the AIGO

research facility. Improvements to the local control system are also planned including

the addition of an electrostatic control for the test mass. The addition of a refer-

ence cavity and/or a pre-mode-cleaner will reduce the laser noise at high frequencies,

allowing isolator performance to be characterised above 10 Hz. However without a

full interferometer it will be difficult to characterise the isolator response in the kHz

range.

Acknowledgements

The authors would like to thanks the technical staff at The University of Western

Australia and Gingin for building each of the thousands of pieces that form the

isolators, in particular, Steve Pople, Peter Wilkinson, Peter Hay and Daniel Stone.

We would also like to thanks Eu-Jeen Chin and Ben Lee for their collaboration. This

work was supported by the Australian Research Council, and is part of the research

program of the Australian Consortium for Interferometric Gravitational Astronomy.

3.7. LOCAL CONTROL 79

3.7 Local control

Compact vibration isolation and suspension for

AIGO: Local control system

J. C. Dumas, P. Barriga, C. Zhao, L. Ju, D. G. Blair

School of Physics, The University of Western Australia, Crawley WA 6009, Australia

High performance vibration isolators are required for ground based grav-

itational wave detectors. To attain very high performance at low fre-

quencies we have developed multi-stage isolators for the proposed AIGO

detector in Australia. New concepts in vibration isolation including self

damping, Euler springs, LaCoste springs, Roberts Linkages, and double

pre-isolation require novel sensors and actuators. Double pre-isolation en-

ables internal feedback to be used to suppress low frequency seismic noise.

Multi-degree of freedom control systems are required to attain high per-

formance. Here we describe the control components and control systems

used to control all degrees of freedom. Feedback forces are injected at the

pre-isolation stages and at the penultimate suspension stage. There is no

direct actuation on test masses. A digital local control system hosted on a

DSP (digital signal processor) maintains alignment and position, corrects

drifts, and damps the low frequency linear and torsional modes without

exciting the very high Q-factor test mass suspension. The control sys-

tem maintains an optical cavity locked to a laser with a high duty cycle

even in the absence of an auto-alignment system. An accompanying pa-

per presents the mechanics of the system, and the optical cavity used to

determine isolation performance. A feedback method is presented which

is expected to improve the residual motion at 1 Hz by more than one order

of magnitude.


3.8 Introduction

In a companion article [35] we describe test mass vibration isolation and suspension

systems developed for the proposed Australian International Gravitational Observa-

tory (AIGO). The performance of an individual isolator system cannot be measured

due to the lack of an inertial reference. For this reason a pair of isolators were con-

figured to suspend mirrors for a 72 m optical cavity. The isolators were developed

to satisfy the sensitivity requirements of advanced interferometric gravitational wave

detectors which require test masses to be isolated from seismic noise at frequencies

down to a few hertz. For a target sensitivity of 10−20 m this typically requires a seismic

attenuation of more than 10 orders of magnitude. In addition to high performance

isolation within the detection bandwidth, it is critical to interferometer operation that

the isolator provides minimal residual motion at low frequencies. This facilitates cav-

ity locking and minimises noise injection through actuation forces. Pendulum systems

inherently have large Q-factors, therefore it is often necessary to damp the normal

modes of the suspension system.

Such requirements can be addressed by a local control system. A feedback loop

injects control forces at various actuation points on the isolator. These forces are

derived by applying appropriate filters to error signals from local transducers such as

position sensors. Typically the local control system is responsible for several tasks,

each requiring different bandwidths and different filters. For example, in the fre-

quency band DC to ∼10 mHz the control system is responsible for drift correction,

positioning and alignment. In the frequency band up to ∼1 Hz the control system

is mainly needed for damping normal mode peaks. For higher frequencies (Hz–kHz)

the control system may be required to suppress high frequency noise, in the form of

active vibration isolation. Despite being conceptually simple, the operation of local

control systems is complicated by resonant modes and mode interaction between iso-

lation components and different degrees of freedom. As a result, the feedback scheme

often requires complex filter designs to avoid noise injection at critical eigenmodes

that would interfere with the noise budget of the test mass.

In the gravitational wave community there have been two broad approaches to the

vibration isolation problem. The first, and most widely adopted, including this work,


has been to create mostly passive vibration isolators based on mass-spring systems.

To stabilise them a local control system is used to control or damp certain normal

modes. The second approach is to invest heavily in very sensitive seismometers to

measure the seismic noise, and then to use active feedback in a rather stiff system to

actively suppress the measured motion. In the first case, one allows the system itself

to provide an inertial reference. In the second, the inertial reference is provided by

the test mass of the seismometer. In the design presented here we extend the idea

of the system itself being the inertial sensor, by using a pair of very low frequency

stages that are designed specifically to allow relative sensing and feedback, to provide

an additional means of active suppression of very low frequency seismic noise.

The VIRGO project [5, 6] and the TAMA project [36, 21] have used the first

approach using multiple passive isolation stages, and relying on the control system

mainly for damping and alignment. The GEO600 project [19] uses a combination of

an active layer and several passive stages. The LIGO project [37, 38] first implemented

an active pre-isolation stage to overcome problems of excess seismic noise. For the

Advanced LIGO project they have constructed a system which combines stages of

stiff active isolation with multiple pendulums [23].

The AIGO vibration isolator is conceptually similar to the VIRGO Superatten-

uator [39], but is more compact and has an extra stage of pre-isolation. The com-

pactness is made possible through the use of novel isolation techniques with multiple

pre-isolation stages [27]. Pendulum normal modes are passively damped through self-

damping [20]. Feedback is applied to the pre-isolation stages and the penultimate

suspension stage to steer the mirrors and maintain alignment and positioning, while

correcting for various sources of drift such as temperature fluctuations. Low frequency

control loops also damp fundamental modes of the pre-isolation stages and angular

modes (yaw) of the suspension chain to minimise low frequency residual noise domi-

nated by the eigenmodes of the isolation chain. These low frequency resonances can

otherwise have large amplitudes as they are located close to the micro-seismic peak of

the seismic background. Therefore careful consideration has to be taken when design-

ing the feedback loop in order to avoid exciting the test mass suspension fundamental

mode, which by design has extremely low loss and hence low Brownian motion. To


facilitate this, the pitch and yaw of the test mass are monitored directly through an

optical lever. Control forces are applied indirectly to through a control mass from

which the test mass is suspended, in similar fashion to the VIRGO marionetta [40].

In order to provide a flexible platform capable of satisfying the various control re-

quirements, a digital control system was implemented on a Sheldon Instrument DSP

board [41], running on a PCI bus. Each vibration isolator is integrated with an inde-

pendent digital control system. The two only differ in minor adjustments of gains and

corner frequencies to match small differences in the mechanical modes of the isolation

stack.

Two vibration isolators were installed at the AIGO test facility in Gingin, Western

Australia. The systems were installed in the East arm of the vacuum envelope to

form a 72 m suspended cavity. A Nd:YAG laser was locked to the cavity using the

Pound-Drever-Hall [42] technique. In a companion article [35] we discuss the vibration

isolator design and the low frequency performance as determined from the error signal

of the optical cavity.

This article presents the control architecture and systems to enable the potential

performance of the isolators to be realised. In section 3.9 we describe the integration of

the isolator components with specially developed sensor and actuator components in-

cluding high dynamic range shadow sensors, high force magnetic actuators and ohmic

position control systems. The control scheme is presented in section 3.10, where we

also show how the novel dual pre-isolation approach is expected to allow major im-

provements in performance through use of so called super-spring techniques. Finally

in section 3.11 we review cavity locking results which confirm the performance of the

control system, and demonstrate the capability of the system for use in interferometric

gravitational wave detectors.

3.9 Experimental setup

3.9.1 Isolator components

The AIGO vibration isolation design is discussed in detail in the companion arti-

cle [35], and only a brief review is presented here. It consists of 9 cascaded stages


as illustrated in figure 3.18, including 3 stages of pre-isolation in a compact nested

structure from which is suspended a triple self-damped pendulum. A vertical stage

based on Euler springs is co-located with each of the 3 pendulums. Anti-Spring

geometries are implemented into various stages to reduce fundamental mode frequen-

cies [29, 43, 27].

LaCoste stage

Inverse pendulum

Self-Damped

pendulums

Control mass

Test mass

Roberts Linkage

Euler stage

a

e

d

c

b

Figure 3.18: Isolation stages of the AIGO suspension chain. The pre-isolation stages

include a, b and c. The isolation stack is defined as the three identical stages of self-

damped pendulums with Euler stages; (a) Inverse pendulum pre-isolator [4] (b) LaCoste

Linkage [4] (c) Roberts Linkage [27] (d) Euler springs [29] (e) Self-damped pendulums [20].

The pre-isolator consists of several Ultra Low Frequency (ULF) stages; the Inverse

Pendulum, the LaCoste linkage [4] and the Roberts linkage [12], each with their

resonant frequencies in the order of 100 mHz. The inverse pendulum stage can be

tuned close to 0.05 Hz to provide low frequency horizontal pre-isolation. It has a

large dynamic range with ±10 mm in all directions, which can be used to buffer


temperature drifts and tidal effects. The inverse pendulum requires minimal force to

displace it and is therefore well adapted to actuation [44]. Vertical pre-isolation is

provided by the LaCoste linkage which is a combination of negative springs to null out

flexure stiffness and zero length coil springs to support a 250 kg load. The LaCoste

linkage, like the inverse pendulum, has a large dynamic range and can be tuned below

0.05 Hz. The Roberts linkage provides the second horizontal pre-isolation stage [27].

The combination of coil and magnet provides the actuation for the positioning

and damping for both the inverse pendulum and the LaCoste stages. Additional

heating of the coil springs at the LaCoste stage provides a means of compensating

for slow temperature drifts in the vertical direction as well as correcting for creep

in the isolation chain. Position control at the Roberts linkage stage is done through

heating of the individual wires. A low frequency isolation stack is suspended from the

Roberts linkage consisting of three almost identical stages (see figure 3.18). A 40 kg

mass load is suspended from each stage in a self-damped pendulum arrangement [20]

and each is combined with an Euler spring for vertical isolation [29].

The test mass is suspended from a control mass which can be actuated in pitch,

yaw, and horizontal translation. The suspension design uses four Niobium ribbons [31]

to form a low loss suspension with pendulum Q-factor ∼ 106. The control mass

itself is suspended from the isolation system by a single suspension wire. All optical

cavity testing has been done without direct sensing or actuation on the final test

mass of the system, beside the use of optical levers. An integrated electrostatic

actuator/RF sensor has been developed [45] as a final stage of low level control. This

will be implemented when the vibration isolators are used in a full interferometer

configuration.

3.9.2 Control hardware

Shadow sensors

The position of several stages of the isolation stack is monitored by the local control

system through optical shadow sensors. A shadow sensor as illustrated in figure 3.19 is

a simple device consisting of an LED shining an infrared beam onto two photodiodes

40 mm away, each photodiode is 10 mm in length. A long and thin flag of the same


width (10 mm), is attached to the stage to be monitored. It is positioned perpendic-

ularly to the two sensors, such that the flag forms a shadow falling roughly equally

on both photodiodes. The resulting current from each photodiode is approximately

proportional to the area illuminated, and as the flag is displaced across the sensor the

difference of the two photodiode signals forms a linear response. Each photodiode

signal is amplified and converted to a signal in the ±10 V range for the ADC module

of the DSP board to be read by the digital control system. The shadow sensor has a

relatively large dynamic range of ±5 mm, and typical sensitivity of 10−10 m/√

Hz [4].

Infrared LED

Photodiodes

Shadow card

Figure 3.19: The shadow sensor is a simple device, where an LED shines a beam onto two

photodiodes, and an intermediate shadow mask is attached to the part to be measured.

Magnet-coil actuator

Magnetic-coil actuators are used to control several stages as described in section 3.9.3.

Each actuator consists of a pair of coils assembled together as illustrated in figure 3.20.

Two designs of magnetic actuators are used in the isolator. Large actuators are used

on the pre-isolation stage, for position control, drift correction, and damping ULF

normal modes. This design uses coils with ∼1600 turns of 0.25 mm wire to form

a coil diameter 65–80 mm, with a resistance of ∼115 Ω. A ∅20×10 mm permanent

neodymium boride magnet is used to result in a force of ∼160 mN with a current

of 100 mA driving the actuator (50 mA each coil, connected in parallel). A smaller

actuator design is used at the control mass, with a coil diameter of 25–30 mm, made

of ∼600 turns of 0.25 mm wire. These coils have a resistance of ∼37 Ω, and are paired


with a ∅10×10 mm magnet. The magnetic field within the actuator coils is nearly

uniform within 1% in the central 10 mm of its range, allowing a large dynamic range

in the control system.

Magnet

CoilCoil

Figure 3.20: The magnet-coil actuator. A magnet mounted on an isolation stage is placed

in the centner of two coils that are mounted on the support frame.

Wire heating

In addition to the magnetic actuators, some stages are controlled by passing a current

through particular suspension elements. The elements warm up and lengthen through

thermal expansion. This control method is relatively effective when the system is un-

der vacuum, as heat does not dissipate through convection, but only by the relatively

slow processes of radiation and conduction. The advantage of this method is that it

removes the complication of added parts, and in some cases provides much greater

dynamic range. This is used to correct for large drifts caused by daily and seasonal

ambient temperature changes. This actuation method responds with a quadratic re-

lationship, but it is linearised in the digital feedback loop. The horizontal control of

the Roberts linkage, and the vertical control of the LaCoste linkage, both employ this

strategy.

The four suspension wires of the Roberts linkage are individually wired to current

power supplies, allowing the length control of each of them by thermal expansion as

they warm up. Since the Roberts linkage is very sensitive to any change of tension

in any of the wires due to it’s carefully tuned folded configuration, a relatively small


change of length is enough to control the stage through it’s entire dynamic range

∼10 mm. The position of the Roberts linkage is controlled via the circulating current

which in turn is controlled by the local control system through integral feedback to

the current power supplies.

The LaCoste linkage has a large dynamic range and can be controlled through

its entirety by magnetic actuators at a fixed ambient temperature. However daily

and seasonal temperature fluctuation cause drifts that would far exceed the capacity

of the actuators. A 1oC change will offset the balance point of the LaCoste linkage

by it’s entire range of 10 mm. For this reason, wire heating is an essential part of

the LaCoste control loop. The coil springs of the LaCoste linkage are electrically

connected in series, and can be heated to change the spring constant of the springs,

which greatly affects the vertical position or balance point of the stage. Low frequency

control (DC–10mHz) of this stage is achieved by regulating this current, while the

magnetic actuators are used at higher frequencies (∼100s mHz).

3.9.3 Control implementation and degrees of freedom

The inverse pendulum can be monitored and actuated in 3 Degrees of Freedom (DoF),

two in the horizontal plane (X and Y ), and one angular (yaw φ), i.e. the rotation

about the vertical axis. These are sensed and actuated through 4 shadow sensors and

4 actuators that are co-located on the inverse pendulum frame as shown in figure 3.21.

The four signals are diagonalised into the 3 DoF X,Y and φ, and each is controlled

independently as a Single Input Single Output (SISO) feedback loop.

The LaCoste stage is a purely 1-dimensional vertical stage (DoF: Z). Two actua-

tors are mounted on opposing sides as illustrated in figure 3.22 and figure 3.21. They

are used to damp the ULF normal mode of the stage. In addition, the LaCoste linkage

can be controlled by heating the coil springs on all four sides of the stage, which are

all connected to a high current supply. A shadow sensor mounted on the side of the

pre-isolation structure monitors the vertical position of the LaCoste linkage.

The Roberts linkage in figure 3.23 is controlled in 2 DoF, X and Y , by passing a

current through its suspension wires. Each of the four suspension wires are electrically

isolated from the rest of the vibration isolator structure and independently connected


X

Y

Horizontal actuator

Shadow sensor

LaCoste frame

Inverse pendulum

Vertical actuator

(LaCoste frame)

Figure 3.21: The inverse pendulum is controlled through shadow sensors and magnetic

actuators.

X

Z

Shadow sensor

Vertical actuator

Inverse Pendulum

Heated suspension

coil spring

Figure 3.22: The LaCoste stage is controlled through a shadow sensor and magnetic

actuator as well as the heating of the suspension coil spring.

to a high current power supply. By controlling the circulating current on each wire

it is possible to control its length, and therefore the position. Since the suspension

system is under vacuum the heat loss by convection is minimal. The X and Y signals

from the control mass shadow sensors are used to feedback to this Roberts linkage

actuation method. This control system provides a low frequency correction of any

drift in the Roberts linkage and ultimately of the multi-stage pendulum and the test

mass.

The control mass can be controlled in 5 DoF, three orthogonal translations X,

Y , Z, the rotation about the vertical axis, yaw (φ) as shown in figure 3.24, and the

rotation about the horizontal axis perpendicular to the laser axis, pitch (θ) shown


X

Y

Heated

suspension wire

LaCoste Frame

Figure 3.23: The Roberts linkage is controlled through shadow sensors and the heating

of the four suspension wires.

in figure 3.25. Three horizontal actuators and shadow sensors are co-located in a

120o arrangement on the horizontal plane as seen in figure 3.24, while two vertical

shadow sensors and actuators are co-located on opposing sides of control mass along

the laser axis as in figure 3.25. The signals are digitalised by a sensing matrix into

five orthogonal DoF, X, Y , Z, φ and θ. These are treated by five separate control

loops as independent SISO systems, before the signals are recombined by a driving

matrix into the appropriate actuator signals.

120°

120°

120°

X

Y

Shadow sensor

Actuator

Figure 3.24: The control mass has three actuators and shadow sensors collocated on the

horizontal plane, in a 120o arrangement. These three signals are converted to an orthogonal

reference frame X, Y , Z and φ by a sensing matrix.


Shadow sensor

Actuator

Y

Z

Figure 3.25: The pitch of the control mass is actuated by two vertical magnetic actuators.

3.9.4 Optical lever

Due to poor coupling of the mirror suspension angular modes to the control mass,

it is necessary to have a direct readout of the mirror angular orientation. This was

achieved by a simple optical lever as illustrated in figure 3.26. A laser outside the

vacuum envelope is reflected off the test mass and is measured by a quadrant photo-

diode, also outside the vacuum envelope. In addition to being a direct measurement

from the mirror surface the optical lever provides better sensitivity to angular motion

as it is placed further away from the centre of rotation of the mirror, such that

the same angular rotation corresponds to a much larger arc-length measured by the

quadrant photodiode ∼5 m away from the mirror, than the shadow sensors which is

only 200 mm away. The drawback is the limited dynamic range, it provides ∼1 mrad

which is greatly exceeded by the test mass suspension oscillation when it is excited.

Therefore the shadow sensor feedback is used for initial damping of the angular modes,

before the optical lever signal can be used for feedback, as discussed in section 3.10.2.

3.9.5 The digital controller

The control system is hosted by a Sheldon Instrument DSP board, forming a flexible

multidimensional digital control platform. The board is a SI-C33DSP on a PCI bus,

based on a 150 MHz Texas Instruments TMS320VC33 DSP using a mezzanine board

SI-MOD6800 to provide 32 input channels (16 bit ADC), 16 output channels (16 bit


Shadow

sensors

Quadrant

photo-detector

Laser

Vacuum envelope

Test mass20

0 m

m

5 m

Figure 3.26: The optical lever setup, using a quadrant photodiode placed outside the

vacuum envelope.

DAC), and digital input/outputs [41]. The input channels are used as described in

table 3.3. Most input channels are used for the shadow sensors which require two

inputs each, one per photodiode. Two more inputs are used for the vertical and

horizontal axis readout of the quadrant photodiode used with the optical lever, and

two more are wired to auxiliary connectors to inject any arbitrary analogue signal.

The output channels are used for the control signals of actuators and current power

supplies.

Stage inputs outputs

Inverse Pendulum 8 4

LaCoste Stage 2 2

Roberts Linkage 4 4

Control Mass 10 5

Optical Lever 2 0

Auxiliary 2 1

Total: 28 16

Table 3.3: I/O channel allocation usage of DSP

An intermediate analogue system amplifies and filters the input and output chan-


nels between the DSP and the control components (shadow sensors, wire heating and

actuators) with the exception of the quadrant photo-detector used in the optical lever

which is integrated on a board with pre-amplification. The analogue electronics con-

sists of 13 boards in a standard 6U rack, each board contains a dual photo-detector

circuit for the pair of photodiodes in one shadow sensor, and a control signal circuit

to drive an actuator. The dual photo-detector circuit contains a transimpedance am-

plifier, anti-aliasing filters and an amplifier. The signal of both photodiodes is then

distributed to two inputs on the DSP board. The control signal is distributed from

one DSP output to the corresponding channel on the control circuit, which contains

anti-aliasing filters and a high speed current amplifier before distribution to the ac-

tuator coils. An additional board in the 6U rack contains five filter circuits for the

wire heating control signals. These five signals are then distributed to five external

voltage controlled current supplies.

The control scheme, algorithms, and the user interface, are written and operated

in LabViewr. The built-in libraries provided by the DSP manufacturer allows for

the control loops to run on the DSP board in real time including ADC and DAC at a

100 Hz sampling rate. The user interface is ran on a host PC to monitor every stage

of the isolation chain and adjust control parameters, such as filters and loop gains as

necessary.

3.10 Control scheme

The control scheme has three main purposes. One is to maintain alignment and posi-

tioning for all stages, against drifts such as caused by ambient temperature changes,

or tidal effects. These effects are extremely low frequency, with timescales from tens

of minutes to days. Therefore the control strategy consists of low gain integration

feedback. The control system has also to maintain the test mass alignment to obtain

a resonant cavity. The alignment of the test mass is controlled indirectly via pitch

and yaw of the control mass stage. However sensing is done with the control mass

shadow sensors and directly from the test mass trough the optical lever. The control

is achieved by both proportional and integration control while aligning the cavity,

3.10. CONTROL SCHEME 93

and integration only when maintaining the cavity locked.

While the isolation design relies on passive isolation, some active damping is re-

quired for some ULF resonant modes of the pre-isolation, as well as low frequency

torsional modes (yaw) of the entire chain. Additionally, the two normal modes of

the test mass suspension (pitch and yaw) must be damped at least initially after any

alignment or positioning offset. The extremely high Q-factor of the Niobium suspen-

sion would otherwise result in several days of oscillations after any large perturbation.

While controlling the alignment the control system must also avoid driving the sus-

pension resonant modes, this is done through carefully placed filters in the feedback

gain.

Figure 3.27: Block diagram of the isolation local control system. The signals from shadow

sensors and a quadrant photodiode are used to feedback to several stages using magnetic

actuators or high current heating.

At each stage the sensor signals are converted into orthogonal DoF by a sensing

matrix, such that the control system consists of independent SISO systems, which

simplifies the control strategy and requirements over a MIMO system (Multiple Input

Multiple Output). The various DoF of each stage relevant to the control system, as


discussed in section 3.9.3, can be summarised in table 3.4. Figure 3.27 illustrate the

physical location for sensing and actuation of the feedback loops.

Stage DoF

Inverse Pendulum X, Y , φ

LaCoste Stage Z

Roberts Linkage X, Y

Control Mass X, Y , Z, φ, θ

Optical Lever φ, θ

Table 3.4: Stages and relevant degrees of freedom in the control scheme

3.10.1 Pre-isolation feedback

Damping of the yaw (φ) mode of the inverse pendulum is done via velocity feedback

(F (s) = ∂∂t

(φIP − φground)) in a bandpass filter 0.3 Hz<f<0.7 Hz. But the two hor-

izontal modes require some consideration to maintain stability. Figure 3.28 shows

the loop gain with feedback loop gain using a relatively high damping gain and a 2nd

order low pass filter at 0.7 Hz. Note that lowering the corner frequency of the low pass

filter would lead to instability as can be seen from the small gain margin. In order

to damp the large resonance at 70 mHz we apply a damping gain using the inverse

pendulum shadow sensor output (which is a measurement of x1−x0, as illustrated in

figure 3.31). This control method is being replaced by the scheme described in sec-

tion 3.10.3 but it was used for initial testing of the cavity. In principle it should also

be possible to damp the second resonant mode (cause by the Roberts linkage) with

phase compensation at the appropriate frequency, but this is difficult to implement

effectively while maintaining stability due to the small frequency separation of the

two pre-isolation stages. However damping of the Roberts linkage can be done with

the method described in section 3.10.3.

The LaCoste Stage feedback method in the vertical direction Z is divided between

the actuator and the coil heating. The shadow sensor signal is fedback to the actuator

with a damping gain and a low pass filter at 0.7 Hz. The vertical signal from the


(a)

10−1

100

101

−60

−40

−20

0

20

40

Frequency

Mag

nitu

de (

dB)

Loop gainClosed loop TF

10−1

100

101

−100

0

100

Frequency

Pha

se

(b)

Figure 3.28: Loop gain G(s) = IP (s)S(s)C(s)A(s) and closed loop transfer function

H(s) = A(s)IP (s)S(s)1+A(s)IP (s)S(s)C(s) of Inverse Pendulum horizontal DoF. The diagonalised signal

from the inverse pendulum shadow sensors is fedback at the inverse pendulum actuators (x1

and F1 in figure 3.31 respectively). The digital compensator C(s) ∝ ∂∂t(Xip−Xground)LPF

with a low pass filter at 0.7 Hz.

control mass is also fedback to the coil heating supply with a low integration gain,

after being linearised by taking the square root of the resulting control signal. The

coil heating method allows to correct for a large range of ambient temperature which

would be impossible with the actuators alone, while the fast response of the magnetic

coils is used to damp the ULF normal mode. It is possible to also use the control

mass vertical signal for damping feedback to the LaCoste actuators, but this is not

currently implemented to avoid coupling to the control mass pitch, pending a thorough

investigation of the sensing matrix digitalisation and coupling of the DOFs.


The Roberts linkage control merely consists of position feedback to the heating

current supplies, with a low gain integral loop, to the two axes, X and Y , at frequen-

cies below 10 mHz. Position sensing of the control mass is fedback to the position

control of the Roberts linkage, to minimise forces injected at the control mass.

3.10.2 Control mass feedback

The control mass horizontal translation X and Y , is fedback to the pre-isolation stages

with a low integration gain, to minimise noise injection at the control mass. Pitch

(θ) and yaw (φ) are controlled to maintain alignment of the optical cavity, as well as

to damp the pitch and yaw resonant modes of the control mass. The control mass is

an order of magnitude heavier than the test mass, hence the suspension resonances

are weakly coupled to the control mass. This results in a poor signal to noise ratio

at the suspension resonances. An optical lever is therefore used to monitor the pitch

and yaw directly from the mirror.

The yaw resonance is damped with the shadow sensor control otherwise its ampli-

tude is too large for the range of the optical lever ∼1 mrad. Although the oscillation

couples weakly to the control mass, there is sufficient signal to noise ratio to damp

the test mass into a range where the optical lever becomes operable, then a much

better level of control can be achieved (∼µrad). Low frequency resonances caused by

the torsion of the suspension wires dominate the residual angular motion with either

control method. The optical lever achieves better performance due to higher angular

sensitivity, and decoupling from translation motion (X, Y , Z).

If the optical lever goes out of range, either due to a large offset, or large amplitude

in the suspension normal mode, the control system will damp the pitch and yaw modes

with a strong velocity feedback gain. Integral and proportional feedback are used in

the loop for cavity alignment, it also includes band-pass filters to damp the suspension

modes. In particular, the phase of the control signal at the suspension resonances must

be reversed, since the control mass coupled oscillation is out of phase to the test mass

oscillation. Once the optical lever is in range, the damping feedback automatically

changes the source of its error signal to the optical lever signal (quadrant photodiode).

This process is automated by the control system though a set of boolean operations


to determine the state of the system according to threshold values on the range of

the optical lever and shadow sensors. The transition from one loop to the other

is ‘smooth’ in the sense that the DC force resulting from integral and proportional

set-points are passed on from one loop to the other.

Since the pitch and yaw damping feedback operate at the highest bandwidth of

the control scheme, they are the most affected by the phase lag due to the 100 Hz

sampling rate. At the pitch resonance of 3.3 Hz, the minimum possible phase lag due

to the 100 Hz operation of the ADC/DAC, is approximately 12o, at the yaw resonance

of 1.75 Hz it is 6o.

Figure 3.29 shows a comparison between the frequency responses of the pitch and

yaw modes with the optical lever control loop off (dotted line) and optical lever control

loop on (continuous line). Optical lever control loop off means that only the shadow

sensor readout and its corresponding control loop are being use for controlling the

test mass position. The optical lever control loop on means that the shadow sensors

control loop is turned off, but the level of the control signal at the time of switching

the loop is used as an offset for the optical lever control loop. Figure 3.29 shows a

pitch mode reduction of 20 dB at 3.3 Hz when using the optical lever control loop.

The reduction on the yaw mode is more significant with about 40 dB at 1.75 Hz.

Figure 3.30 shows the yaw angular motion of the test mass in time using either

shadow sensors or optical lever feedback. The set-point has been removed so as to

centre the curves on zero removing the offset introduce by the control loop set-point.

The total yaw displacement is dominated by the torsional mode of the suspension

wires, causing very low frequency yaw oscillations at ∼20 mHz.

3.10.3 Optimised feedback for pre-isolation

A feedback scheme was devised based on the super-spring concept which takes ad-

vantage of the dual stage of horizontal pre-isolation in our design. This is an active

control method consisting of feeding back the position of the loaded end of a mass-

spring system to the spring suspension, and keep their relative distance constant in

order to synthesise a very-low frequency system [8, 46, 47, 48]. The concept can be

equally applied to pendulum systems. Shadow sensors mounted between the inverse


(a)

10−1

100

10−6

10−5

10−4

10−3

Control mass Pitch frequency response

frequency (Hz)

Mag

nitu

de (

rad/

Ö H

z)

SSOL

(b)

10−1

100

10−6

10−5

10−4

10−3

Control Mass Yaw frequency response

Frequency (Hz)

Mag

nitu

de (

rad/

Ö H

z)

SSOL

Figure 3.29: Measurement of the frequency response of the test mass in it’s two angular

DoF (the resonant modes are highlighted). (a) pitch θ with test mass suspension mode

at 3.3 Hz. Note that the broad peak at 280 mHz is due to the rocking mode of the control

mass. (b) yaw φ with suspension mode at 1.75 Hz. The dotted line shows the measurements

with the optical lever feedback off, using only the shadow sensor signal for feedback. The

solid line shows the measurements with the optical lever control loop on. The optical lever

was used for the measurement of both curves.

pendulum frame and the Roberts linkage as shown in section 3.9.3 could be used for

such a purpose. These sensors provide a signal ∝ (x2−x1) as illustrated in figure 3.31.

Feedback to the inverse pendulum actuators would permit to lower the effective in-


0 200 400 600 800 1000 1200 1400 1600 1800 2000

-600

-400

-200

0

200

400

Yaw

µ ra

d

time (sec)

SS

OL

Figure 3.30: Test mass yaw angular motion using different feedback loop. The optical

lever feedback greatly improves the limit imposed by the low signal to noise ration of the

shadow sensor when sensing the suspension normal modes.

verse pendulum resonance. In addition, velocity feedback ∝ ∂∂t

(x2 − x1) would also

allow effective damping of the Roberts linkage resonant mode and simplify stability

considerations compared to feedback of the inverse pendulum position. Modelling of

this control method shows that the low frequency isolation below a few Hertz can

be improved by an order of magnitude compared to that described in section 3.10.1.

A comparison of the theoretical performance with the initial setup is shown in sec-

tion 3.11.

Figure 3.31: A simple 2 pendulum system illustrating the pre-isolation feedback using

shadow sensors. The inverse pendulum position is referenced to the ground, for low fre-

quency position control (∼∫

(x1 − x0)dt. The Roberts linkage is referenced to the inverse

pendulum, and can be feedback to lower the first resonant mode, and damp the second

(∼ (x2 − x1) + ∂∂t(x2 − x1).


3.11 Initial cavity measurements

Preliminary performance results of the cavity displacement noise have been obtained

from initial trials of cavity locking. The locking scheme employed in these runs

where as described in section 3.10, with simple damping feedback at the inverse

pendulum stage, and did not yet implement the super-spring feedback concept in the

pre-isolation stage. A standard Pound-Drever-Hall feedback system was implemented

to keep the laser locked to the 72 m cavity. The companion paper [35] describes the

optical scheme and laser feedback in more details.

The integrated residual motion per test mass can be calculated from the control

signal and it is shown in figure 3.32. Here we see that above 1 Hz, the residual

motion is ∼3 nm per test mass. The measured curves compares closely with the

predicted performance for the simple feedback at the inverse pendulum. Note that

the theoretical performance without any control is an order of magnitude lower at the

same frequency, however the large resonance of the pre-isolation mode at ∼70 mHz is

too large to practically lock the optical cavity.

The inverse pendulum feedback loop permits to damp the normal mode, at the

sacrifice of high frequency noise injection. This simple feedback is also ineffective in

damping the second resonant mode of the two body system formed by the inverse

pendulum and Roberts linkage. An optimised feedback system was devised using the

super-spring concept as discussed in section 3.10.3, which could achieve an acceptable

residual motion at low frequencies without compromising the isolation performance.

This scheme which requires additional sensors on the Roberts linkage stage, has been

tested on a single isolator, and will be implemented and tested in the cavity in the

near future.

The stability of the cavity, and in particular the angular noise of the test mass,

can be demonstrated by a long term record of the power inside the cavity. This was

achieved by measuring the transmitted light at the end test mass (ETM). In figure 3.33

we plot a histogram of measured power over two hours. As an auto-alignment scheme

has not been implemented at this initial stage, continuous lock could not be achieved

over long periods (24 hours).

3.12. CONCLUSIONS 101

10-1

100

101

10-11

10-10

10-9

10-8

10-7

10-6

10-5

Frequency (Hz)

Resid

ual m

otion (

m)

Cavity residual motion

Measured performance

Model with IP feedback

Model without control

Model with RL feedback

(a)

(b)

(c)

(d)

Figure 3.32: (a) The measured integrated residual motion of the cavity (xrms =√∫∞f xf2df). It is at the nanometre level above 1 Hz. Note that the measurement is

limited by laser noise above 2 Hz [35], due to the free running laser. (b) A model with the

same feedback scheme as used for the measurement. (c) The modelled performance with an

optimised pre-isolation feedback scheme, using the super-spring concept. (d) A modelled

performance if no feedback was implemented.

3.12 Conclusions

A local control system was implemented in a novel isolation and suspension design

for laser interferometer gravitational wave detectors. The system provides feedback

for position control, cavity alignment, and damping of normal modes. Three DoF are

controlled by ohmic thermal tuning of the length of pendulum wires in the Roberts

linkage and by thermal spring constant of the LaCoste linkage. Large dynamic range

shadow sensors and actuators allow more than ±5 mm three dimensional position

control of the test mass. Two suspensions systems and their associated control sys-

tems were installed to form a 72 m optical cavity. Without using direct test mass

control, it was possible to lock the cavity and maintain lock, with a residual motion

of 3 nm per test mass above 1 Hz. Low frequency residual motion at micro-seismic

frequencies is expected to improve by over an order of magnitude once the second

horizontal pre-isolation stage is used for feedback. Angular control of the test mass is


0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3x 10

4 Histogram of transmitted power

Amplitude (V)

count

Figure 3.33: The power inside the cavity is represented in a histogram over a period of

two hours. Note that the X-axis is of arbitrary unit: the voltage of the photo detector

measuring transmitted light.

aided by an optical lever, with an automatic transition between local sensors and the

optical lever. Additionally both sensing and direct actuation of the test mass will be

possible by a capacitive plate mounted on the control cage. The cavity demonstrates

long term stability, indicating that there are no unexpected noise sources or drifts in

the system.

Acknowledgments

We would like to acknowledge the Australian Research Council for their support of

this work. We also thank David Ottaway for his helpful discussions. This project

is part of the research program of the Australian Consortium for Interferometric


3.13. POSTSCRIPT 103

3.13 Postscript

A thorough study of the optical lever was undertaken during measurements of vi-

bration isolator performance described in the presented paper. This section includes

additional information removed from the paper due to page limits recommended by

Review of Scientific Instruments journal.

Figure 3.34 shows the difference between the test mass control using only the

shadow sensors and the reduction on the test mass angular displacement when using

the optical lever. For these graphs the set-point on each axis has been removed to

centre the curves on zero, removing the offset introduced by the control loop set-point.

The distribution of the test mass position in time shows a Gaussian distribution from

which we can determine the standard deviation for each case. Therefore the yaw pen-

dulum mode when using the shadow sensor has a standard deviation of 319.5µrad,

which is reduced to 35.3µrad when using the optical lever. This also improves the me-

dian value and improves the precision and the accuracy when reaching the set-point.

Figure 3.35 (b) shows the reduction in pitch mode to be much smaller, with a stan-

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-600

-400

-200

0

200

400

Yaw

µ ra

d

time (sec)

SS

OL

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-100

-50

0

50

100

Pitch

µ ra

d

time (sec)

SS

OL

Figure 3.34: Test mass angular motion when controlled using only the shadow sensors;

and when controlled using the optical lever. We can clearly see the difference in the yaw

mode, however there is a much smaller difference in the pitch mode.


dard deviation of 60.1µrad when using the shadow sensor and reduced to 57.8µrad

with the optical lever. However there is an improvement in the median value.

(a)

-800 -600 -400 -200 0 200 400 6000

500

1000

1500

2000

2500

Angle (µ rad)

Count

Shadow sensor angular motion distribution: Yaw

Shadow Sensor

Optical Lever

(b)

-150 -100 -50 0 50 100 1500

1000

2000

3000

4000

5000

6000

Shadow sensor angular motion distribution: Pitch

Angle (µ rad)

Count

Shadow Sensor

Optical Lever

Figure 3.35: Histograms showing the gaussian distribution of the angular motion for the

ITM in pitch and yaw. (a) shows a broad distribution when driving the control mass with

the signal from the shadow sensors and a much narrow distribution when using the optical

lever signal. (b) also shows the difference between controlling the control mass with the

shadow sensor signal and the optical lever signal, however the difference is much smaller.

3.14. REFERENCES 105

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Chapter 4

Mode-Cleaner Optical Design

4.1 Preface

Following the work completed with the vibration isolation system for the mode-

cleaner, it was natural to progress to the optical design. A preliminary design was

started during the vibration isolator design presented in the previous chapter. The

initial design assumes that the suppression of the higher order modes is similar to

a two mirror cavity, which is not necessarily correct since the input mode-cleaner

design consists of three mirrors in a triangular distribution. The symmetry of the

suppression of higher order transverse modes is broken due to the odd number of mir-

rors in the mode-cleaner ring cavity. Therefore a new approach for the optical design

was necessary. The optical design is divided into three parts corresponding to three

different publications. The first paper shows a complete analysis of the higher order

mode suppression properties of the triangular ring cavity used as an input mode-

cleaner. This first paper also includes a preliminary analysis of the thermal lensing

problem induced by the high circulating power in the mode-cleaner. This work was

followed by a more complete study of the effects of high circulating power in the input

mode-cleaner. In the third publication a complete simulation of thermal effects are

presented, including the thermal gradients induced in the mode-cleaner optics. The

effects of astigmatism induced by the high circulating power are calculated and a

new optical design for an astigmatism-free mode-cleaner for advanced interferometric

gravitational wave detectors is presented.

111

112 CHAPTER 4. MODE-CLEANER OPTICAL DESIGN

Optical Design of a High Power Mode-Cleaner for

AIGO



Australia

Laser beam geometry variations such as beam jitter and frequency fluc-

tuations are a critical source of noise in the output signal of a laser inter-

ferometer gravitational wave detector. In order to minimise this noise a

resonant vibration isolated optical filter or mode-cleaner is required. For

advanced gravitational wave detectors such a mode-cleaner is required

to be able to handle transmitted power ∼100 W, and an internal circu-

lating power of 45 kW. This paper addresses the design requirements of

such a mode-cleaner. We characterise the mode-cleaner requirements and

the effects of high laser power on the optics and its consequence on the

suppression of higher order modes.

4.2 Introduction

The Australian Consortium for Gravitational Astronomy (ACIGA) has built a high

optical power test facility at the site of the proposed Australian International Gravi-

tational Observatory (AIGO), north of Perth in Western Australia. This facility will

play three vital roles in gravitational wave research. In the short term it will be used

to collaborate in the development of high optical power technologies required for the

next generation of Advanced Gravitational Wave (GW) detectors. The second is to

demonstrate the operation of a very low noise 80 m base line advanced interferome-

ter. Third will be the development of the southern hemisphere long baseline detector,

AIGO.



The 80 m interferometer has been designed to have parameters as close as prac-

ticable to Advanced LIGO to enable the critical issues of thermal lensing, radiation

pressure and optical spring effects to be examined. An essential part of the facility

will be a triangular ring cavity mode-cleaner to minimise variations in laser beam

geometry.

The spatial instability of a laser beam, known as beam jitter, is due to the mixing

of higher order modes with the fundamental mode (TEM00). Amplitude fluctuations

are created by beam jitter whenever the beam interacts with a spatially sensitive

element such as an optical cavity. The noise at the dark fringe at the interferometer

output will be affected by such beam jitter effects. In addition frequency fluctuations

of the laser fundamental mode give rise to additional noise at the dark fringe. All

these noise sources can be minimised by using a mode-cleaner as illustrated in figure

4.1.

2l

M1

M2

M3

1222 ll +=L

Vibration

Isolator

Vibration

Isolator

1l

Figure 4.1: Simplified schematic showing the layout of the AIGO mode-cleaner.

The mode-cleaner acts as a spatial filter. It provides passive stabilisation of time

dependant higher order spatial modes, transmitting the fundamental mode (TEM00)

and attenuating the higher order modes. The concept was first suggested by Rudiger

et al in 1981 [1]. As a frequency stability element it can also suppress frequency

fluctuations of the fundamental mode, but without DC stability.

The designer is able to vary cavity length, mirror radius of curvature (ROC) and

number of mode-cleaners in series. The choice of parameters can allow optimisation of

frequency stability, power handling capacity, and choice of free spectral range (FSR).

However these factors are not independent and there is no unique optimum solution,


Project Length [m] FSR [MHz] ROC [m] Stability

GEO600 [2] 4 37.48 6.7 0.403

4.05 37.12 6.7 0.396

VIRGO [3] 144 1 180 0.200

TAMA [4] 9.85 15.24 16 0.384

LIGO Caltech [5] 13.54 11.07 21.2 0.362

Advanced LIGO [6] 25 6 43.2 0.421

Table 4.1: Summary of the mode-cleaner configurations used by other gravitational wave

interferometers.

as can be seen in table 4.1 where the various mode-cleaner solutions adopted by the

other GW interferometric projects are presented. The FSR needs to match desired

interferometer modulation frequencies (to allow sideband transmission). Large ROC

leads to increased spot size and higher power handling capacity, but with loss of

angular stability.

In the following sections, we first present a summary of the key design aspects and

theoretical background to a mode-cleaner and use this to derive suitable parameters

for the AIGO mode-cleaner. Then we address the issue of thermal lensing for the

AIGO high power mode-cleaner. We show that design choices depend on the highest

higher order mode for which suppression is desired, and that issues remain concerning

astigmatism due to thermal lensing in the 45o mirrors.

4.3 The mode-cleaner

A mode-cleaner is a cavity used in transmission, with a geometry chosen such that the

fundamental mode is non-degenerate. This is a cavity where the fundamental mode

is resonant and the higher order modes, having different cavity eigen-frequencies, are

attenuated or suppressed. How effective the cavity will be for filtering the higher

order modes is then given by the suppression factor Smn of the cavity [7], given by:

4.3. THE MODE-CLEANER 115

Smn =

[1 +

4F2

π2sin2

(2π∆νmn

cL

)]1/2

, (4.1)

where

F = π

√r1r2r3

(1− r1r2r3)(4.2)

∆νmn =c

2L(m+ n)

1

πarccos

(√1− L

R

). (4.3)

Equation (4.1) shows the suppression factor of any higher order mode TEMmn.

Here F corresponds to the finesse of the cavity as shown in equation (4.2), where r1,

r2, and r3 are the reflectivity of each mirror. L is the length of the cavity and c is the

speed of light in vacuum. Equation (4.3) corresponds to the difference in frequency

between any higher order mode TEMmn and the fundamental mode TEM00. This

frequency difference not only depends of the order values m and n, but also depends

on the length of the cavity and the ROC of the end mirror R. The relation between

the length of the cavity and the ROC is commonly known as the stability g-factor of

the cavity as shown in equation (4.4). In the case of the mode-cleaner, and in order

to have a stable cavity this value has to be 0 < g < 1.

g = 1− L

R. (4.4)

Clearly the ROC of the end mirror M3 is one of the essential variables when

designing a mode-cleaner. By equation (4.4), the ROC and the length of the cavity

define the stability g-factor. Through equations (4.1) and (4.3), the g-factor defines

the suppression factor of the higher order modes. The power that builds up inside

the cavity depends on its finesse, but the area of the mirrors where this power will

concentrate depends on the ROC of the end mirror M3. Therefore the ROC defines

the size of the beam waist that defines the spot size on each mirror and hence the

power density. Consequently the curvature of the mirror is limited by the damage

threshold of the mirror coating and by the stability g-factor of the cavity. Typical

damage threshold is ∼1 MW cm−2 [8].

The transmission factor of higher order modes, Tmn is given by the ratio of the

transmission factor of the fundamental mode, T00, and the suppression factor of the


higher order modes as shown in equation (4.5). The transmission of the fundamental

mode will depend on the transmission of the input and output couplers M1 and M2

(t1 and t2), and the mirrors reflectivity as shown in equation (4.6),

Tmn = T001[

1 +(

2πFsin

(2πLc

∆νmn))2]1/2

(4.5)

where

T00 =t1t2

(r1r2r3). (4.6)

The general rule for the resonant condition of a cavity says that this one way

phase shift must be an integer number of half cycles [9]. Since the total round trip

phase shift must be an integer multiple of 2, it must satisfy:

2kL− 2(m+ n+ 1) arccos (√g) = 2πq. (4.7)

Where k = ω/c corresponds to the wave number, g is the cavity stability factor

and q an integer that represents the axial mode number.

All of the above is true for a two mirror cavity and in general for any cavity with an

even number of mirrors. It has been shown that symmetry with respect to the vertical

axis implies that the spatial dependence of the field with the plane of incidence is an

even function. For this analysis the vertical axis is defined perpendicular to the plane

of incidence. In this case a ring cavity formed by three or four mirrors is completely

equivalent. If the field distribution is anti-symmetric with respect to the vertical axis,

then there is a difference of half a wavelength between an even and an odd number

of mirrors [10].

As a consequence a ring cavity with an odd number of mirrors like the mode-

cleaner of figure 4.1 will have the same frequency shift for the higher order modes

with similar (m + n) value, depending on the symmetry of the horizontal modes m

with the vertical axis.

We can formalise this by writing the following general expression:

2kL− 2(m+ n+ 1) arccos (√g)− π (1− (−1)m)

2= 2πq. (4.8)


The FSR is defined by equation (4.9). The frequency for any higher order mode

at any axial mode q is then given by equation (4.10):

ν0 =c

2L, (4.9)

νmnq = qν0 +ν0

π(m+ n+ 1) arccos (

√g) +

ν0

2

(1− (−1)m)

2. (4.10)

The frequency difference between the fundamental mode and any higher order

mode associated with the same q axial mode is then given by:

∆νmn =ν0

π(m+ n) arccos (

√g) +

ν0

2

(1− (−1)m)

2. (4.11)

In this frequency difference expression two factors can be distinguished:

arccos(√

g)

π, (4.12)

1

2

(1− (−1)m)

2. (4.13)

The first shown in (4.12) corresponds to the Gouy phase shift factor [9]. However

the limiting values for this factor will also depend on the geometrical design of the

optical cavity. In our case there are two possible limiting values for the Gouy phase

shift factor: g close to 1 and g = 0 a.

When g is close to 1 the Gouy phase shift factor becomes close to 0. Therefore the

frequency of all transverse modes associated with a given axial mode q are clustered

on the high frequency side. This is the case where the end mirror M3 corresponds

to a flat mirror which gives rise to a minimum ∆νmn. This implies that most of the

higher order modes will be transmitted together with the fundamental mode. Hence

a mode-cleaner will not be effective if g is very close to 1. Such a cavity also has

extreme angular sensitivity and is almost unstable.

If g is equal to 0 the Gouy phase shift factor becomes 0.5 and the TEM01 associated

with the q-th axial mode will fall exactly half way between the q and q+1 axial modes,

although the TEM10 associated with the same q-th axial mode will fall exactly at the

aA third possible limit is g close to −1 but this is not applicable to a stable mode-cleaner, where

0 < g < 1.


q + 1 axial mode. In our case this situation occurs if the end mirror M3 has a ROC

equal to the length of the cavity. Therefore if m is even and (m+n) is even the mode

will be transmitted. Moreover if m is odd and (m + n) is odd the mode will also be

transmitted. This case can be extended to every M3 ROC that is a multiple of the

length of the cavity L, where different multiples of the modes will be transmitted.

The second factor shown in (4.13) corresponds to the frequency shift between

modes of the same order, but present only if the horizontal mode m is odd. This

introduces an extra shift of half the FSR.

Combining equations (4.5) and (4.11) the transmission factor is given by:

Tmn = T001[

1 + 4F2

π2 sin2(

(m+ n) arccos(√

g)

+ π2

(1−(−1)m)2

)](1/2). (4.14)

The phase shift introduced by the fact that the mode-cleaner is formed by an odd

number of mirrors is normally not taken into consideration. If this is not considered

there will be no difference between m odd or even for any TEMmn mode. For example

TEM31 is not the same as TEM22, even though in both cases (m+n) = 4. As can be

deduced from equation (4.11) each of these modes will have a different frequency. In

figure 4.2 it is possible to see that the transmission factor does depend on this value,

as suggested in equation (4.14).

Using expression (4.14) it is possible to simulate the transmission factor of the

higher order modes for the mode-cleaner as a function of the g-factor. Figure 4.2

shows the distribution of the higher order modes. Here for example, even though

TEM13 and TEM22 have the same value of (m+ n), because they will have different

eigen-frequencies they need different g-factor to resonate in the mode-cleaner.

For the same cavity length a higher g-factor (closer to 1) means a larger ROC,

therefore less power density and less thermal effects on the mirrors.

Using the first 20 modes (corresponding to a total of 231 modes with (m+n) ≤ 20)

we calculate a g-factor that minimise the transmission of the higher order modes as

shown in figure 4.3. Choosing an end mirror with a ROC close to the length of the

cavity many higher order modes will be transmitted. Also we have to consider the

damage threshold of the mirror coatings. In our case with an input power of ∼100 W


Transmission of Higher Order Modes

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

g factor

Tra

nsm

issi

on fa

ctor

TEM01

TEM02

TEM03

TEM04

TEM10

TEM11

TEM12

TEM13

TEM20

TEM21

TEM22

TEM30

TEM31

TEM40

TEM13TEM31

TEM03TEM21

TEM04TEM11TEM22TEM40

TEM02TEM04TEM10TEM12TEM20TEM30TEM40

TEM12TEM30

TEM01TEM20TEM02TEM03TEM04TEM21TEM40

TEM13TEM31

Figure 4.2: Transmission of the higher order modes as function of the stability g-factor

of a three mirror ring cavity to be used as a mode-cleaner for the AIGO interferometer.

by choosing a ROC larger than 20 m (g-factor higher than 0.5) we assume a safety

margin higher than 50% with just 0.43 MW cm−2 on the mirrors M1 and M2.

It is important to notice that by including more higher order modes to the calcu-

lations we are reducing the suppression of the lower higher order modes by 6 dB to

10 dB. This due to the fact that we need to move away from what could be a minimum

of a lower mode in order to avoid the transmission of a higher one. Also important

is to give higher priority to the lower modes when choosing a g-factor or a radius of

curvature, since these modes are more likely to appear in the incident beam. This is

the main reason not to choose a radius of curvature around 21.5 m where the highest

transmission (or lowest suppression) corresponds to TEM11 mode.

All of the above lead us to choose a ROC of 22.5 m, which means a g-factor of

0.5556. This results in a waist radius of 1.9460 mm and power density of 378 kW cm−2

M1 and M2.

For this simulation a circular Gaussian beam was assumed inside the mode-cleaner.


The purpose is to select the best ROC for M3 that maximises the mode-cleaner

rejection of higher order modes. Therefore the elliptical profile that appears due to

the fact that the input and output couplers are at ∼45o in relation to the input beam

has not being considered here.

20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25

10−2

10−1

100

Higher Order Modes Sum

M3 Radius of Curvature [m]

Mag

nitu

de

Figure 4.3: Detail of transmission of the higher order modes as function of the radius of

curvature of the end mirror M3. The graph shows the transmission factor for modes with

(m+ n) ≤ 20 for radius of curvature between 20 m and 25 m.

4.4 Mode-cleaner thermal lensing

In Advanced GW Interferometers high power lasers will be used [11]. This will intro-

duce thermal lensing effects in the main cavities of the interferometer where the power

builds up to the order of hundreds of kilowatts [12]. This mode-cleaner is designed

with a finesse of ∼1500. For an input laser power of ∼100 W, the expected power

inside the cavity will be ∼45 kW.

Considering fused silica substrates for the mirrors that form the mode-cleaner

cavity it is possible to calculate the thermal effect produced in them. Due to the high

4.4. MODE-CLEANER THERMAL LENSING 121

laser power circulating inside the cavity the mechanical parameters of the mirrors

will suffer a small but significant change. This change in the curvature of the mirror

will alter some of the cavity parameters, like the g-factor and the waist size. Among

other effects these changes will affect the suppression (transmission) factor of the

cavity [13].

The change in the ROC of the mirrors is really a change in sagitta, which corre-

sponds to the curvature depth of the mirror measured across the beam diameter [14].

The sagitta so of the mirror surface with a radius of curvature R over a spot size ω

is given by:

so = R−√R2 − ω2. (4.15)

The change in sagitta caused by heating can be expressed as:

δs ≈ αPa4πk

. (4.16)

Here Pa corresponds to the absorbed light power, k the heat conductivity of the

substrate, and α the thermal expansion. Coating absorption losses of 1 ppm, a thermal

expansion of 0.51 × 10−6 K−1, and heat conductivity of 1.38 W m−1 K−1 have been

assumed for the fused silica substrates.

Due to the geometric distribution of the mirrors to form the mode-cleaner the

laser beam will impinge upon the flat mirrors at an angle of ∼45o. In our case with

a distance of 20 cm between the input and output couplers the angle is 44.71o. As

a result the spot size on the mirrors can not be considered as circular anymore but

elliptical.

The elliptical spot at the mirrors means different sagitta values for the horizontal

and vertical planes causing an elliptical deformation of the mirror. An elliptical profile

will lead to differences in wavefront curvature between the two transverse directions

or astigmatism.

After the high power laser has been switched on and the cavity enters a steady

state, some of the parameters of the cavity will change due to thermal effects in

the mirrors. This steady state situation is what is known as hot cavity. Table 4.2

summarise some of these changes, where for comparison the cold cavity status still


assumes a circular beam profile. The end mirror M3 is expected to increase its ROC

and consequently M1 and M2 will become slightly convex. Hence the minus sign in

the hot cavity ROC of M1 and M2.

Due to the elliptical profile of the beam we now see that the deformation of the

mirrors due to thermal effects it is not even in both axes but elliptical as well. In fact

on the flat mirrors the effect will be stronger on the Y-axis (tangential plane) than

on the X-axis (sagittal plane) producing differences in wavefront curvature between

the two transverse directions.

As seen before the frequency of the higher order modes depends on the g-factor

of the cavity. The change in ROC due to thermal effects alters the cavity g-factor,

changing the frequency spacing between the higher order modes and the fundamental

mode. Within addition to this frequency shift there is a change in the suppression

(transmission) factor of each mode. The increase in the ROC of the mirrors makes the

cavity g-factor also increase, therefore the Gouy phase shift decreases. The frequency

Definition unit Cold Hot Cavity

Cavity X-axis Y-axis

Mode-cleaner length m 10

M1 radius of curvature m flat -2838 -1433

M2 radius of curvature m flat -2838 -1433

M3 radius of curvature m 22.5 22.65 22.80

Cavity g-factor 0.5556 0.5624 0.5692

Mode-cleaner FSR MHz 14.9896

Finesse 1495

Mode-cleaner waist mm 1.9460 1.9529 1.9598

Rayleigh range m 11.18 11.26 11.34

Input power W 100

Stored MC power kW 45.381

Table 4.2: Shows the difference between hot and cold parameters for the AIGO mode-

cleaner (MC). Considering that all mirrors substrates are made of fused silica.

4.4. MODE-CLEANER THERMAL LENSING 123

difference between the fundamental mode and the higher order modes also decreases,

ultimately shifting the higher order modes closer to the fundamental one.

By doing separate analysis for tangential and sagittal planes it is possible to deter-

mine the level of astigmatism introduced by the mirror deformation due to thermal

absorption. We calculate the difference in the frequency shift of the fundamental

mode for each axis as shown in equation (4.10). The difference between X and Y

axis for the fundamental mode is about 65 kHz corresponding to a 0.17%. Also there

is a difference in the frequency shift of the higher order modes suggesting that the

coupling of higher order modes will be stronger on the horizontal axis than in the

vertical one. The worst case is in the X-axis, where the mode TEM(12)(1) will have

a transmission of the ∼4.3% due to the astigmatism present inside the cavity. This

effect could be compensated using radiant heat to vary the temperature profile on

the mirror using compensating methods already developed for thermal lensing in test

masses. This problem is an area of actual study by LIGO [15] as well as ACIGA.

However further study is required to address this issue.

The level of the higher order modes transmitted by the mode-cleaner will strongly

depend on the quality of the incident beam. Moreover the excellence of this beam

will not only depend on the quality of the laser, but also on the performance of the

Pre-mode-cleaner.

In order to achieve these levels of suppression of the higher order modes it is

necessary to specify the ROC of the end mirror quite carefully for a fixed cavity.

More realistic a careful control of the cavity length is needed, in order to match the

length of the cavity to the mirror ROC to obtain the adequate g-factor.

Having different values for the transmission of the higher order modes for the

horizontal and vertical axes shows the effect of the astigmatism inside the cavity. As

a result the beam inside the cavity and therefore the transmitted one will not be

a pure Hermite-Gaussian beam. It will be the fundamental mode plus some small

components from the higher order modes coupled to the fundamental one.


4.5 Conclusions

We have seen some of the effects caused by the high power stored in the mode-

cleaner, including mechanical changes in the optics and how they alter some of the

cavity parameters such as the g-factor and cavity waist. The frequency shift on the

higher order modes and the change produced in the suppression (transmission) factor

were also presented.

The high power stored in the cavity creates a thermal lens effect in the mode-

cleaner optics, which will introduce some astigmatism in the cavity. This effect should

be taken into account when performing the mode matching for the power recycling

cavity and main arms of the future AIGO interferometer. It will also be necessary to

estimate the effect of the thermal lensing in the sidebands that need to be transmitted

through the mode-cleaner in order to control the other cavities in the interferometer.

High power mode-cleaner design presents a challenge due to astigmatic thermal

lensing. Design performance represents a trade off between low order and high order

normal modes. If the injection beam quality is such that mode numbers of order 20

need to be strongly suppressed, either the cavity length or the mirror radii of curvature

need to be tuneable to a precision ∼0.1%. Astigmatic thermal compensation is also

required to achieve sufficient rejection of certain modes with mode number > 10.

However, we note that the fine tuning required can be achieved through the sus-

pension system design previously reported [16], which has an ultra-low frequency

stage capable of fine translation over ∼0.1% of the cavity length. Astigmatic thermal

compensation can be provided using astigmatic heating, based on small modifications

of existing thermal compensation techniques [12].

We have shown that a mode-cleaner with a g-factor ∼0.55 and 10 m length with a

ROC of 22.5 m allows the transmission of the fundamental mode reducing the coupling

of the higher order modes. In addition, this design has good immunity to thermal

lensing effects, with the high order modes frequency offset from the carrier changing

by less than 350 kHz from cavity switch on to high power operation in the worse case

(Y-axis). The mode-cleaner is designed to support 100 W of input power with a safety

margin of more than 60% on the coating power threshold for standard high power

mirrors. Future work include a more detailed study of the effect of the astigmatism


due to high power stored in the cavity and the coupling of higher order modes into

the fundamental mode.

Acknowledgements

The authors would like to thank Jerome Degallaix and Bram Slagmolen for helpful

discussions and Andrew Woolley for his construction of the isolator stage for the

mode-cleaner. We thank David Reitze, Amber L. Bullington and Ken Y. Frazen from

the LIGO group for useful discussions. This work was supported by the Australian

Research Council, and is part of the research program of the Australian Consortium

for Interferometric Gravitational Astronomy.


Astigmatism compensation in mode-cleaner

cavities for the next generation of gravitational

wave interferometric detectors



Australia

Interferometric gravitational wave detectors use triangular ring cavities

to filter spatial and frequency instabilities from the input laser beam.

The next generation of interferometric detectors will use high laser power

and greatly increased circulating power inside the cavities. The increased

power inside the cavities increases thermal effects in their mirrors. The

triangular configuration of conventional mode-cleaners creates an intrinsic

astigmatism that can be corrected by using the thermal effects to advan-

tage. In this paper we show that an astigmatism free output beam can be

created if the design parameters are correctly chosen.

4.6 Introduction

The configuration of interferometric Gravitational Wave (GW) detectors includes

at least one input mode-cleaner [2, 3, 17, 18]. Current detectors have all similar

configuration, consisting of two flat mirrors (M1, M2) used as input and output

couplers and a concave end mirror (M3) as show in figure 4.4. This configuration is

preferred since the reflected light at the input mirror will not be reflected back to

the laser increasing the noise, but used to control the cavity locking. The triangular

configuration presents an intrinsic astigmatism inside the cavity which increases when

the input power is increased.



2l

M1

M2

M3

1222 ll +=L

Vibration

Isolator

Vibration

Isolator

1l

Figure 4.4: Triangular ring cavity layout used as mode-cleaner for the interferometric

gravitational wave detector. This configuration has the advantage of low optical feedback.

The next generation of interferometric GW detectors will also include input mode-

cleaners. Designs have been proposed for Advanced Laser Interferometer Gravita-

tional-wave Observatory (Advanced LIGO) [6], the Large-scale Cryogenic Gravita-

tional-wave Telescope (LCGT) [19], and for the Australian International Gravitational

Observatory (AIGO) being developed in Gingin 90 km north of Perth in Western

Australia.

The results presented below are applied to the mode-cleaner design being devel-

oped for AIGO. The proposed design consists of a 10 m long triangular ring cavity.

Two flat mirrors are used as input and output couplers. At 9.9 m a concave end

mirror with a radius of curvature of 22.5 m is installed. This means cavity stability

or g-factor of 0.5556. The design is chosen to minimise transmission of higher order

modes, and to enable spot size large enough to keep the power density below the

coating damage threshold. The details of this design are given in ref [20].

With a finesse of 1495 a circulating power of ∼45 kW for ∼100 W of input power

is expected. Some of this power will be absorbed by the mirrors causing thermal

effects in the substrate and their deformation as a consequence [14]. The amount

of power absorbed by the mirrors and the substrate greatly depends on the quality

of the coating. The magnitude of the thermal deformation depends on the thermo-

mechanical properties of the substrate, usually fused silica.

It has been previously shown that triangular ring cavities present an intrinsic

astigmatism [21, 22, 23]. In this paper we show that mode-cleaners can use the thermal

effects due to the high circulating power to advantage to compensate and greatly


reduce the astigmatism, producing a mode-cleaner with near ideal performance and

astigmatism free output. We show that there are various possible solutions. We

present different alternatives varying radius of curvature, substrate materials and

coating absorptions in order to obtain an output beam free of astigmatism.

4.7 Mode-cleaner intrinsic astigmatism

The actual input mode-cleaners used by different GW detectors are based on a de-

sign proposed by Rudiger et al in 1981 [1]. The design basically consists of vibration

isolation suspension and suspended mirrors as shown in figure 4.4. Since the beam

impinges onto the flat mirrors at an angle ∼ 45o the spot on the surface of these mir-

rors is strongly elliptical with an eccentricity ∼0.7, which as a consequence produces

an elliptical cavity waist. It has been shown that a cavity with these characteristics

will always have an astigmatic output [23].

An elliptical spot means different sagitta values for the x-axis (parallel to the plane

of incidence) and y-axis (perpendicular to the plane of incidence) causing an elliptical

deformation of the mirror. In fact on the flat mirrors the effect will be stronger on

the y-axis (tangential plane) than on the x-axis (sagittal plane) producing differences

in wavefront curvature between the two transverse directions or astigmatism.

Previously we have shown the astigmatism level of a high power mode-cleaner

for AIGO [20]. In principle the mode-cleaner design consists of a 10 m mode-cleaner

with two flat mirrors and a 22.5 m radius of curvature end mirror. In cold cavity

conditions (very low or no input power) the waist will have an eccentricity of 0.0068

Cold Cavity Hot Cavity

Parameter X-axis Y-axis X-axis Y-axis

Waist [mm] 1.94604 1.94599 1.95264 1.95599

M3 Spot [mm] 2.61073 2.61073 2.61175 2.61228

Table 4.3: Comparison of waist and M3 spot sizes in sagittal and tangential planes between

cold and hot cavity.

4.7. MODE-CLEANER INTRINSIC ASTIGMATISM 129

0 10 20 30 40 50 60 70 80 90 1000.7

0.701

0.702

0.703

0.704

0.705Eccentricity variation with Power

Ecc

entr

icity

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06

Input Power [W]

Ecc

entr

icity

M1 Spot

M3 SpotWaist Size

Figure 4.5: Eccentricity of the spots M1/M2, M3 and the waist for the AIGO high power

mode-cleaner. The upper graph shows the reduction of the spot eccentricity of M1 and M2

when the input power increase. The lower graph shows the eccentricity variation of the M3

spot and cavity waist for different input power values.

compare to an eccentricity of 0.0585 under the hot cavity conditions (∼100 W in-

put power). Increasing an order of magnitude the waist eccentricity, which implies

stronger astigmatism, to levels in which mode matching into the interferometer is

significantly degraded. The intrinsic astigmatism also produces higher order modes

that will degrade the interferometer operation.

Table 4.3 present the values for the x-axis and the y-axis under cold and hot

cavity conditions. The calculations were made using a Matlabr code written by

the authors based on matrix simulations [24, 25]. Together with the enlargement

of the waist and the spot size at M3 due to the thermal effects we note that both

eccentricities also increase. Note that not only the eccentricity increases, but the

major axis of the ellipse changes from the x-axis to the y-axis. This implies that the

waist must cross a circular profile for a certain level of input power.

Fused silica substrate was assumed for the thermal simulation presented in figure 4.5

with thermal expansion of 0.51× 10−6 K−1 and heat conductivity of 1.38 W m−1 K−1.

Coating absorption losses of 1 ppm has also been assumed.


So far we have just considered the thermal effects on the mechanical properties

of the substrate, but we also need to consider the change in curvature of the wave

front of the transmitted beam. This change mainly depends on the effect of a heated

hemisphere due to the absorption at the dielectric coating. There is also a smaller

contribution from the absorption that occurs inside the substrate along the transmit-

ted path. Contributions to the path changes due to the variation of the refractive

index to stress (photo-elastic effect) can be neglected [26].

The following equations have been proposed as good approximations to calculate

the path difference between the centre and the outer parts of the beam when trans-

mitted through a substrate [14, 27]. Equation (4.17) refers to the effect due to the

power absorbed by the dielectric coating, and equation (4.18) to the change due to

the power absorbed by the substrate.

δs ≈ ∂n

∂T

Pa4πk

(4.17)

δs = 1.32∂n

∂T

pa4πk

d, (4.18)

where ∂n/∂T corresponds to the temperature dependence of the refractive index, Pa

is the absorbed light power, k is the thermal conductivity, pa is the power absorbed

per unit length and d is the substrate thickness.

With a value of 8.7× 10−6K−1 for ∂n/∂T , we assume an absorption of 2 ppm/cm

for fused silica substrate and that 90.7% of the input power will be transmitted into the

power recycling cavity. Therefore the light path difference for the transmitted beam

is 22.8 nm due to the coating absorption and 8.1 nm due to the substrate absorption

for a 25 mm thick substrate, which corresponds to an error of ∼ λ/35.

Reduction of the coating absorption losses can be sufficient to correct the astig-

matism at low power levels. For high circulating power this needs to be combined

with changes to the mirrors’ radius of curvature in order to reduce the astigmatism.

4.8. ASTIGMATISM FREE MODE-CLEANER 131

0 20 40 60 80 100 120 140 160 180 2000.7

0.701

0.702

0.703

0.704

0.705Eccentricity variation with Power

Ecc

entr

icity

0 20 40 60 80 100 120 140 160 180 2000

0.01

0.02

0.03

0.04

0.05

0.06

Input Power [W]

Ecc

entr

icity

M1 Spot

M3 SpotWaist Size

Figure 4.6: Eccentricity variation with input power for fused silica substrate with 0.6 ppm

coating absorption losses. The upper graph shows the reduction of the M1 and M2 spot

eccentricity for different levels of input power. The lower graph shows the variation of the

M3 spot and waist eccentricity variation for different levels of input power.

4.8 Astigmatism free mode-cleaner

The results presented suggest that with the right combination of parameters we can

obtain a mode-cleaner with an astigmatism free output. One possible solution is

to create a circular waist inside the cavity. In order to obtain a circular waist the

following configuration is proposed: Mirrors substrate will be made of the same type

of fused silica, but instead of flat input/output couplers we will have 500 m radius

of curvature ones. In order to keep the transmission properties of the mode-cleaner

similar to the original design we change the radius of curvature of the end mirror M3

to 24 m. In the previous design it was suggested a coating with 1 ppm absorption

losses that needs to be improved down to 0.6 ppm.

Figure 4.6 shows that by choosing the parameters mentioned before we will create

a waist with a radius of 2.00021 mm across both axes. This means a circular cavity

waist for 99.76 W of input power. The beam will also have a Rayleigh range of 11.81 m

in both axes and a beam radius of curvature of 23.96 m in both axes.

We still have the problem of the thermal lensing when the beam is transmitted


through the substrate. In this new configuration the values for the path difference

are of 13.6 nm due to coating absorption and 8.1 nm due to transmission through

the substrate for a 25 mm thick substrate, which corresponds to an error of ∼ λ/50.

This result shows a 57% improvement when compared to the previous configuration,

mainly due to less absorbed power in the coating.

Good results with a circular waist of 1.98297 mm can also be obtained using input

and output couplers made of fused silica with a radius of curvature of 750 m. The

best results for this case are obtained for 97.24 W of input power. In order to keep

the higher order modes suppression levels an end mirror with a radius of curvature of

23.7 m is chosen. For this configuration it is necessary to reduce the absorption losses

of the coating down to 0.4 ppm, reducing also the path difference for the transmitted

light to ∼ λ/64. This result can be improved just by using M3 as the output coupler.

Under this configuration not only the astigmatism is reduced also the light path

difference is improved to ∼ λ/74.

It is possible to obtain similar results using sapphire as substrate. In this case the

radius of curvature of the input and output couplers needs to be 1000 m and 22.8 m

for the end mirror. In such case 0.75 ppm coating absorption losses are needed.

The circular waist obtained with an input power of 94.25 W which has a radius of

1.95726 mm in both axes. Rayleigh range of 11.31 m and a beam radius of curvature

of 22.79 m are also achieved. Given that mirror coating parameters are difficult to

precisely specify, it will in practice be necessary to adjust the input power to tune

the mode-cleaner near to the zero astigmatism condition.

Due to its homogeneity it is assume that in fused silica there is no problem in

transmitting the light through the substrate at ∼ 45o, and that the difference between

a normal incidence of the beam and a 45o one it is only at the amount of substrate

that the light will have to go through. In the case of sapphire this assumption is not

valid anymore, and we will have to consider the substrate axis orientation. Therefore

if we want to use sapphire as a substrate for the mode-cleaner mirrors we will need to

consider the use of the end mirror (M3 in figure 4.4) as the output coupler in order

to minimise the incident angle of the beam.


4.9 Conclusions

Triangular mode-cleaners have a mild astigmatism when operating under low power

conditions. High power lasers under development for the next generation of interfero-

metric GW detectors will introduce significant thermal effects in the optical cavities.

In the mode-cleaners thermal deformation worsens the astigmatism of the output

beam. We have shown that by carefully choosing the parameters for the mode-cleaner

it is possible to use those thermal effects to our advantage and dramatically reduce

the astigmatism within the mode-cleaner. Even if we remove the astigmatism inside

the mode-cleaner the use of M2 as an output coupler has the disadvantage of cross-

ing the substrate at nearly 45o. This means longer light-path through the substrate,

therefore more power absorbed and a larger change in the light path between the

centre of the beam and its edge. Moreover the spot at this mirror is elliptical and

as a consequence the wave-front path changes are different for the x-axis and for the

y-axis. In order to avoid this situation we can use the end mirror M3 as an output

coupler. Due to the long distance between M1, M2 and M3 the laser beam crosses the

substrate nearly perpendicular minimising the astigmatism due to thermal lensing.

Future work on thermal effects due to high circulating power in mode-cleaners will

address this problem.

We have shown that by improving the coating absorption losses for fused silica

mirrors it is possible to considerably reduce the astigmatism for ∼100 W of input

power. The right combination between the mirrors’ radius of curvature and coating

reduces the astigmatism while keeping the cavity stability and the higher order modes

suppression levels. A solution using sapphire substrates was also presented. Similar

solutions can be obtained for any high power mode-cleaner.

Acknowledgements

The authors would like to thank Jerome Degallaix, Li Ju and Bram Slagmolen for

helpful discussions. This work was supported by the Australian Research Council,

and is part of the research program of the Australian Consortium for Interferometric



Self-Compensation of Astigmatism in

Mode-Cleaners for Advanced Interferometers

Pablo Barriga, Chunnong Zhao, Li Ju, David G. Blair

School of Physics, University of Western Australia, Crawley, WA6009, Australia

Using a conventional mode-cleaner with the output beam taken through

a diagonal mirror it is impossible to achieve a non-astigmatic output.

The geometrical astigmatism of triangular mode-cleaners for gravitational

wave detectors can be self-compensated by thermally induced astigmatism

in the mirrors substrates. We present results from finite element modelling

of the temperature distribution of the suspended mode-cleaner mirrors

and the associated beam profiles. We use these results to demonstrate

and present a self-compensated mode-cleaner design. We show that the

total astigmatism of the output beam can be reduced to 5×10−3 for ±10%

variation of input power about a nominal value when using the end mirror

of the cavity as output coupler.

4.10 Introduction

Input mode-cleaners are used in gravitational wave interferometers presently in op-

eration. One on each LIGO detector in the USA [17], one in the French–Italian

VIRGO [7], two in series at the British–German GEO600 [2] and one in TAMA300 in

Japan [18]. As an important part of the input optics system mode-cleaners are used

to reduce any spatial or frequency instability of the laser beam. In addition frequency

fluctuations of the laser fundamental mode give rise to additional noise at the dark

fringe. It provides passive stabilisation of time dependant higher order spatial modes,

transmitting the fundamental mode TEM00 and attenuating the higher order modes.

Plans for the next generation of advanced interferometers (Advanced LIGO [6],

LCGT [19] and AIGO) will also include at least one input mode-cleaner. In order to

overcome photon shot noise high power (>100 W) single frequency, continuous wave

Nd:YAG lasers are needed [28]. A small portion of the circulating power will remain

4.11. SUBSTRATE DEFORMATION 135

in the mirrors due to substrate and coating absorptions. This energy will increase the

temperature on the mirror causing a thermal expansion and a change in its radius of

curvature [14]. Therefore the higher the circulating power the stronger the thermal

effects.

It has been shown that triangular ring cavities like the mode-cleaner will always

have a certain level of astigmatism due to the angles at which the beam impinges on

the mirrors [23]. This creates a mild astigmatism in the beam that circulates inside

the cavity, which is worsen when the input (or circulating) power is increased. This

geometrical astigmatism is a consequence of the geometrical distribution of the mirrors

in the triangular ring cavity. Whenever the beam crosses a piece of optics there is

a change in path length between the centre and the outer beam [14]. This thermal

lensing effect is induced by the power absorbed by the coating and the substrate of

the optics. If the beam crosses the output coupler at a relatively large angle then this

effect will be stronger and even worse at high power. Similar effects were reported at

the beam-splitter of the Phase Noise Interferometer at the Massachusetts Institute of

Technology (MIT), where the beam crosses at ∼ 45o as well [29].

In this paper we first analyse the substrate deformation due to the thermal effects

inside the mode-cleaner. This deformation introduces a thermally induced astigmatism

inside the cavity. We propose a solution to this problem using the thermally induced

astigmatism to advantage in order to compensate the geometrical astigmatism present

in the mode-cleaner output beam. We also show the temperature distribution in the

substrate when the mirror is used as an output coupler. This will be used to determine

the thermal lensing effect of the output beam due to the different path between the

centre and the outer parts of the beam. This will help us to quantify the astigmatism

of the output beam and to select the best solution that minimise this effect.

4.11 Substrate deformation

Input mode-cleaners actually in use consist mainly of two flat mirrors that define

the short side of the triangle and a concave mirror that forms the acute angle of an

isosceles triangle as seen in figure 4.7. Due to this configuration and depending on


2l

M1

M2

M3

1222 ll +=L

Vibration

Isolator

Vibration

Isolator

1l

Figure 4.7: Due to its low optical feedback a triangular ring cavity using three suspended

mirrors is preferred as the input mode-cleaner for interferometric gravitational wave detec-

tors.

the distance that separates the two flat mirrors from the concave end mirror (usually

several metres) the laser spot at the flat mirrors will be strongly elliptical while at

the end mirror will be closer to a circumference as can be see in figure 4.8. With an

elliptical spot on the surface of the flat mirrors the thermal effects will be different

for each axis.

(a) (b)

Figure 4.8: (a) Elliptical spot at the M2 mirror inside the cavity. The elliptical spot is

the result of the 45o of the laser beam incident angle. (b) Due to the long distance between

M1/M2 to M3 the spot at the end mirror is nearly circular under the cold cavity conditions.

4.11. SUBSTRATE DEFORMATION 137

We define a coordinate system where the x-axis is parallel to the plane of incidence

on the cavity mirrors and the y-axis is perpendicular to it, leaving z-axis along the

direction of propagation. The thermal effects due to the absorption at the dielectric

coating and the deformation of the substrate as a consequence are stronger in the

y-axis (tangential plane) than in the x-axis (sagittal plane). This causes an elliptical

deformation of the mirror substrate producing differences in wave-front curvature

between the two transverse directions or astigmatism. This effect needs to be carefully

calculated since it will change the g-factor of the cavity, changing the transmission

(or suppression) factor of the higher order modes.

We have shown that the astigmatism inside a triangular mode-cleaner is strongly

dependant on the circulating power, which is defined by the input power and the cavity

finesse [30]. Figure 4.9 shows the eccentricity variation of the M3 spot and cavity

waist with input power. The eccentricity values depend on the radius of curvature

of the mirrors that form the mode-cleaner and the cavity finesse. They also depend

on the mirrors substrate and coating absorption. In our case we assume that all

substrates are of fused silica with substrate absorption of 2 ppm/cm, heat conductivity

k = 1.38 Wm−1 K−1, thermal expansion coefficient α = 0.51×10−6 K−1 and refractive

index temperature dependence β = 8.7× 10−6 K−1.

Input mode-cleaners at the operational gravitational wave interferometers are all

designed with two flat mirrors and one end concave mirror at the acute end of it. In

this case when operating at low power the cavity waist will have an almost circular

profile. As soon as the circulating power is increased the waist gets strongly elliptical

and therefore with high eccentricity, as shown in figure 4.9 (a). This figure also shows

that operating a standard mode-cleaner at a 100 W of input power we have a relatively

high eccentricity of ∼0.06 at the waist, even if we use M3 as an output coupler we

still get an eccentricity of ∼0.02.

When using the design proposed in figure 4.9 (b), we obtain a beam free of astig-

matism at mirror M3. This result is very difficult to obtain in reality due to power

variations and the fact that coating and substrates are never exactly the ones pre-

dicted. However we notice that a variation of ±10% in the input power only increases

the eccentricity from nearly 0 to 0.005 when using M3 as an output coupler. A similar


0 20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

0.08

0.1

Eccentr

icity

Eccentricity variation with Power

0 20 40 60 80 100 120 140 160 180 2000

0.02

0.04

0.06

Eccentr

icity

0 20 40 60 80 100 120 140 160 180 2000

0.01

0.02

0.03

0.04

0.05

Eccentr

icity

Input Power [W]

M3 Spot

Waist Size

M1/M2 = Flat

M3 = 22.5 m

M1/M2 = 1000m

M3 = 23.3 m

M1/M2 = 470 m

M3 = 23.3 m

(a)

(b)

(c)

Figure 4.9: Waist and M3 spot eccentricity variation with input power. The upper

graph shows two flat mirrors and an end concave mirror assuming coating absorption of

1 ppm (best result for M3 at 28.5 W). Figure 4.9 (b) shows the cavity tuned for 100 W input

power, while 4.9 (c) shows an Advanced LIGO type of mode-cleaner tuned for 160 W of

input power. For the last two cases we assumed coating absorption of 0.5 ppm.

case is presented in figure 4.9 (c). With a different design, this time for 160 W of input

power. In this case a variation of ±10% in the input the input power increases the

eccentricity up to 0.006 when using M3 as the output coupler.

By choosing different radius of curvature for the mirrors and combining them with

lower absorption coatings it is possible to design a mode cleaner free of astigmatism.

When doing the design it is important to maintain the intended cavity g-factor, in or-

der to keep the higher order modes non-degenerated in the cavity. Graphs at figure 4.9

only consider the thermal deformation of the substrate. However we note that with

the right combination it is possible to design a mode-cleaner free of astigmatism for

different input power levels. It is interesting to see that at the waist and at M3 there

is also a change in the ellipse major axis. The thermal effects inside the cavity will

stretch the y-axis to the point that crosses a circular profile. It even gets larger than

4.12. THERMAL LENSING 139

the x-axis when the input power (and therefore the circulating power) is increased.

The geometrical astigmatism has the opposite sign from the thermally induced astig-

matism from the mirror absorption. Using the parameters for the mirrors we show

this thermally induced astigmatism can be used to correct the intrinsic geometrical

astigmatism of an isosceles triangular ring cavity mode-cleaner. The balance of these

two effects can lead to a self compensated mode-cleaner.

For the simulations here presented it was assumed a perfectly mode matched

gaussian beam, however the effects of the substrate in the input beam when entering

the cavity were considered. In reality the input beam quality is such that the input

performance of the mode-cleaner is dominated by the poor quality of the input beam.

Therefore mode matching losses are generally significantly smaller than the mode

cleaning losses, and as a consequence negligible.

Figure 4.10: Steady state solution for the bulk absorption case. If M2 is used as out-

put coupler the diagonally transmitted beam produces strong astigmatic thermal lensing.

(90.5 W transmitted power, 0.6 ppm coating absorption, fused silica substrate absorption of

2 ppm/cm).


4.12 Thermal lensing

The thermal lensing effects as described by Hello and Vinet will cause a deformation

in the curvature of the wave-front of the transmitted beam [31]. This change mainly

depends on the effect produced by the power absorbed by the dielectric coating due

to the high power circulating inside the cavity. In our case with a finesse of ∼1500

and ∼100 W of input power there will be ∼45 kW of circulating power. There is also a

small contribution from the power absorbed by the substrate during the transmission

of the output beam.

In most cases the two flat mirrors are used as input and output couplers for the

mode-cleaner cavity. This can be a problem in high optical power mode-cleaners, not

only due to the deformation of the mirror’s substrate, but mainly due to the thermal

lensing effects when transmitting the beam through the output coupler.

Figure 4.10 shows the steady state temperature distribution of a flat mirror when

Figure 4.11: Steady state temperature distributions for coating absorption due to the

circulating power inside the mode-cleaner. M2 high reflectivity coating absorption produces

astigmatic thermal lensing. The spot ellipticity produces different distribution between X

and Y axis. (45 kW of circulating power, 0.6 ppm coating absorption, fused silica substrate

absorption of 2 ppm/cm).

4.12. THERMAL LENSING 141

used as the output coupler. By crossing the substrate at almost 45o the laser beam

will cross a larger section of substrate, which will increase the thermal lensing effect

produced by the temperature distribution compared to a perpendicular transmission.

The main contribution to the thermal lensing from the power absorbed by the coating,

is even larger as can be inferred from the higher temperatures in figure 4.11.

The wave-front change in the light path can be quantified as a change in the

focal length of the beam. In this case there is higher temperature along the y-axis

(vertical cross-section) due to a smaller spot size along the y-axis. This will produce

a stronger deformation of the wave-front that will in consequence produce a stronger

astigmatism in the output beam. Since the distortion is different for each axis we will

study both separately.

By calculating the wave-front deformation including bulk and coating absorptions

it is possible to determine the best lens fit for it. Using a Matlabr code written

by the authors it was possible to calculate the new focal length for the output beam.

This method was later compared to FFT simulations obtaining very close agreement.

The main contributors to the thermally induced optical path change are the depen-

dence of the refractive index on temperature, the strain and the thermal expansion.

Different authors [27, 26] have already examined these effects, which are summarised

in equation (4.19).

δs = 1.32pa

4πk

(∂n

∂T− n3

2ρ12α + 2αn

ω

d

)d. (4.19)

Here δs corresponds to the path change, pa to the power absorbed per unit length,

k to the heat conductivity, ∂n/∂T = β to the refractive index change with tempera-

ture, n the fused silica refractive index, ρ12 the fused silica photo elastic coefficient, α

the thermo-elastic coefficient, ω the beam radius of the intensity profile and d is the

substrate thickness.

If we use M2 as the output coupler the output beam will be strongly astigmatic.

This astigmatism according to our simulations presented in table 4.4 will lead to a

focal length difference between the x-axis and the y-axis of ∼133 m.

If instead of using M2 we use M3 as the output coupler the astigmatism will be

much smaller due to the small ellipticity of the spot, producing similar thermal effects


in both axes. Even though at table 4.4 the results for M3 look the same they have a

difference in focal length of about 1µm.

X - axis (m) Y - axis (m)

M2 output coupler 269.61 136.28 Strong astigmatism

M3 output coupler 253.82 253.82 Mild astigmatism

Table 4.4: Output beam thermal lensing focal length. Best result is obtained when using

M3 as output coupler. Input power of 99.6 W gives a focal length difference between X and

Y axis of ∼ 10−6 m.

4.13 Conclusions

The current mode-cleaner designs in use for interferometric gravitational wave detec-

tors have mild astigmatism at low input power due only to the geometrical distribu-

tion of the mirrors. The same design used with higher laser power leads to significant

astigmatism of the output beam due to thermal effects in the mode-cleaner mirrors.

We have shown that the balance of the two effects can lead to a self-compensated high

optical power mode-cleaner. Our proposed solution for the AIGO mode-cleaner is to

use fused silica substrates, M1 and M2 with a radius of curvature of 1000 m and 23.3 m

for the end mirror, assuming coating absorption losses of 0.5 ppm. The need for an

adaptive optic element for astigmatism control and mode-matching can be avoided,

since using M3 as output coupler contributes negligible additional astigmatism in the

output beam. The design presented keeps the astigmatism below 0.5% over a ±10%

power range for 100 W of input power. Self compensation can be adjusted to any

input power level using an additional single fixed astigmatic corrector at the output.

Acknowledgements

The authors wish to thanks Bram Slagmolen and Stefan Goßler for useful discussions

and Jerome Degallaix for helpful hints in writing the code. This work is supported by

the Australian Research Council, the Department of Education, Science and Training


(DEST), and is part of the research programme of the Australian Consortium for

Interferometric Gravitational Astronomy.

4.14 References

[1] A. Rudiger, R. Schilling, L. Schnupp, et al, “A mode selector to suppress fluctu-

ations in laser beam geometry,” J. Mod. Opt. 28 (1981) 641–658.

[2] S. Goßler, M. M. Casey, A. Freise, et al, “Mode-cleaning and injection optics of

the gravitational-wave detector GEO600,” Rev. Sci. Instrum. 74 (2003) 3787–

3796.

[3] F. Bondu, A. Brillet, F. Cleva, et al, “The VIRGO injection system,” Class.

Quantum Grav. 19 (2002) 1829–1833.

[4] G. Heinzel, K. Arai and N. Mitaka, “TAMA Modecleaner alignment,” Technical

Report, Version 1.6, TAMA, (2001).

[5] B. Bonfield, B. Abbott, O. Miyakawa, et al, “Chracterizing the length sensing

and control system of the mode cleaner in the LIGO 40 m lab,” Technical Report,

T020147-00-R, LIGO, (2002).

[6] G. Mueller, D. Reitze, D. Tanner and J. Camp, “Reference Design for the LIGO

II Input Optics,” LSC Presentation, G000240-00-D, LIGO, (2000).

[7] M. Barsuglia, “Stabilisation en frequence du laser et controle de cavites optiques

a miroirs suspendus pour le detecteur interferometrique d’ondes gravitationnelles

VIRGO,” PhD Thesis, Universite de Paris-Sud, Orsay, Chapter 7, 1999.

[8] Melles Griot, “The Practical Application of Light,” 14.1–14.24 (1999).

[9] A. E. Siegman, “Lasers”, University Science Books, Sausalito, California, Chap-

ter 19, (1986).

[10] C. Mathis, C. Taggiasco, L. Bertarelli, et al, “Resonances and instabilities in a

bidirectional ring laser,” Physica D 96 (1996) 242–250.


[11] LIGO II Conceptual Project Book, M990288-A-M, LIGO, (1999).

[12] J. Degallaix, B. Slagmolen, C. Zhao, et al, “Thermal lensing compensation prin-

ciple for the ACIGA’s High Optical Power Test Facility Test 1,” Gen. Relat.

Gravit. 37 (2005) 1581–1589.

[13] R. Amin, G. Mueller, M. Rakhmanov, et al, “Input Optics Subsystem Conceptual

Design Document,” Technical Report, T020027-00-D, LIGO, (2002).

[14] W. Winkler, K. Danzmann, A. Rudiger and R. Schilling, “Heating by optical ab-

sorption and performance of interferometric gravitational-wave detectors,” Phys.

Rev. A 44 (1991) 7022–7036.

[15] A. L. Bullington, D. Reitze, K. Y. Franzen, Internal communication (2004).

[16] P. Barriga, A. Woolley, C. Zhao and D. G. Blair, “Application of new pre-

isolation techniques to mode-cleaner design,” Class. Quantum Grav. 21 (2004)

S951–S958.

[17] R. Adhikari, A. Bengston, Y. Buchler, et al, “Input optics final design,” Technical

Report, T980009-01-D, LIGO, (1998).

[18] S. Nagano, M. A. Barton, H. Ishizuka, et al, “Development of a light source with

an injection-locked Nd:YAG laser and a ring mode cleaner for the TAMA 300

gravitational-wave detector,” Rev. Sci. Instrum. 73 (2002) 2136–2142.

[19] K. Kuroda, M. Ohashi, S. Miyoki, et al, “Large-scale cryogenic gravitational

wave telescope,” Int. J. Mod. Phys. D 8 (1999) 557–579.

[20] P. Barriga, C. Zhao, and D. G. Blair, “Optical design of a high power mode-

cleaner for AIGO,” Gral. Relat. Gravit. 37 (2005) 1609–1619.

[21] F. J. Raab and S. E. Whitcomb, “Estimation of special optical properties of a

triangular ring cavity,” Technical Report, T920004-00-R, LIGO, (1992).

[22] J. Betzwieser, K. Kawabe, and L. Matone, “Study of the output mode cleaner

prototype using the phasecamera,” Technical Report, T040156-00-D, LIGO,

(2004).


[23] T. Skettrup, T. Meelby, K. Færch, et al, “Triangular laser resonators with astig-

matic compensation,” Appl. Opt. 39 (2000) 4306–4312.

[24] A. E. Siegman, “Lasers”, University Science Books, Sausalito, California, (1986).

[25] H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5 (1966)

1550–1567.

[26] J. D. Mansell, J. Hennawi, E. K. Gustafson,et al, “Evaluating the effect of trans-

missive optic thermal lensing on laser beam quality with Schack-Hartmann wave-

front sensor,” Appl. Opt. 40 (2001) 366–374.

[27] K. A. Strain, K. Danzmann, J. Mizuno, et al, “Thermal lensing in recycling

interferometric gravitational wave detectors,” Phys. Lett. A 194 (1994) 124–132.

[28] D. Mudge, M. Ostermeyer, D. J. Ottaway, et al, “High-power Nd:YAG lasers

using stable-unstable resonators,” Class. Quantum Grav. 19 (2000) 1783–1792.

[29] B. T. Lantz, “Quantum limited optical phase detection in a high power suspended

interferometer,” PhD Thesis, Department of Physics, Massachusetts Institute of

Technology, Chapter 4, (1999).

[30] P. J. Barriga, C. Zhao, and D. G. Blair, “Astigmatism compensation in mode-

cleaner cavities for the next generation of gravitational wave interferometric de-

tectors,” Phys. Lett. A 340 (2005) 1–6.

[31] P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive

mirrors heated by high power laser beams”, J. Phys. I France 51 (1990) 1267–

1282.

Chapter 5

Diffraction Losses and Parametric Instabilities

5.1 Preface

The previous chapter presented the optical design of an input mode-cleaner for ad-

vanced interferometric GW detectors. The improved understanding of higher order

optical modes obtained through this work led to the study of their behaviour in the

arm cavities of the interferometer. Due to the high circulating power, higher order

optical modes interact with the test mass acoustic modes in the arm cavities. If

this three-mode opto-acoustic parametric interaction has enough gain, the arm cav-

ities could potentially be driven out of lock, leaving the interferometer inoperable.

It was therefore very important to understand this phenomenon and determine the

parametric gain as accurately as possible. However there are several parameters that

contribute to the parametric gain and in most cases they are not independent from

each other. One is diffraction losses, which depend on the design parameters of the

arm cavities. Normally they are treated with simple approximations that, as pre-

sented in this chapter, do not give a realistic result. Diffraction losses affect not only

the power build-up of each mode, but also the frequency shift, the mode gain and its

Q-factor, all of which contribute to the possibility of parametric instabilities. This

research was started in collaboration with Biplab Bhawal from the California Insti-

tute of Technology in Pasadena, USA, and also coincided with the author’s visit to

the LIGO Hanford site as part of the science monitors’ program during the LIGO

S5 science run. This visit allowed the author to further discuss parametric instabil-

ities with the wider scientific community. In particular Phil Willems, who hosted

the author’s visit to the Caltech laboratory, gave the author several ideas on how to

147

148CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES

improve the code developed for the simulations. Riccardo DeSalvo in Pasadena also

encouraged the author to continue this research, suggesting study of the apodising

coating and the possible benefits in reducing the parametric gain. At the core of this

chapter is the paper that condenses these studies. This is followed by an extract of a

technical report published for the LSC as a LIGO technical report T060273-00-Z and

an extract of an internal technical report which was presented at the “International

Parametric Instabilities Workshop” held in Gingin in 2007. This presentation can

be found at http://www.gravity.uwa.edu.au/docs/PIworkshopJul07/Pablo.pdf. The

introductions of both reports have been reduced due to similarities with the main

introduction of the published paper. A paper published by the UWA group is pre-

sented in Appendix B for better understanding of parametric instabilities and related

control strategies.

http://www.ligo.caltech.edu/docs/T/T060273-00.pdf

http://www.gravity.uwa.edu.au/docs/PIworkshopJul07/Pablo.pdf


Numerical calculations of diffraction losses and

their influence in parametric instabilities in

advanced interferometric gravitational wave

detectors

Pablo Barriga1, Biplab Bhawal2,∗, Li Ju1, David G. Blair1

1 School of Physics, The University of Western Australia, Crawley, WA 6009, Australia.2 LIGO Laboratory , California Institute of Technology, Pasadena, CA 91125, USA.

Knowledge of the diffraction losses in higher order modes of large optical

cavities is essential for predicting three-mode parametric photon-phonon

scattering, which can lead to mechanical instabilities in long baseline grav-

itational wave detectors. In this paper we explore different numerical

methods in order to determine the diffraction losses of the higher order

optical modes. Diffraction losses not only affect the power build up inside

the cavity but also influence the shape and frequency of the mode, which

ultimately affect the parametric instability gain. Results depend on both

the optical mode shape (order) and mirror diameter. We also present a

physical interpretation of these results.

5.2 Introduction

In order to detect gravitational waves large laser interferometers have been built with

arms formed by Fabry-Perot cavities stretching up to 4 km. The actual interferom-

eters are very close to their design sensitivity, but this may still not be enough to

detect gravitational waves. In order to increase their sensitivity, advanced laser in-

terferometer gravitational wave detectors will require much higher circulating optical

power. The high power increases the high frequency (>100 Hz) sensitivity, but also

enhances undesired effects including the possibility of parametric instability, which

∗Now at Google Inc., CA 94043, USA


were first predicted by Braginsky et al [1] with further study by Kells et al [2] and

Zhao et al [3].

Parametric instabilities in advanced gravitational wave interferometers are pre-

dicted to arise due to a 3-mode opto-acoustic resonant scattering process in which

the cavity fundamental mode w0 scatters with test mass acoustic modes wm and op-

tical cavity modes w1, which satisfy the condition w0 ∼ wm + w1. The parametric

gain R0 for this process determines whether the system is stable (R0 < 1) or unstable

(R0 ≥ 1). Three mode opto-acoustic parametric processes have not yet been ob-

served. However 2-mode processes which also couple optical and mechanical degrees

of freedom via radiation pressure have been observed in resonant bar gravitational

wave detectors with microwave resonators readouts [4]. More recently they have been

observed in optical micro-cavities with very high Q [5], and low frequencies in short

(∼1 m) suspended optical cavity [6]. In all these cases the mechanical mode frequency

is within the electromagnetic mode linewidth.

It has been shown that for an Advanced LIGO type of interferometer with fused

silica test masses the parametric gain R0 will typically have a value of ∼ 10 [7]. The

parametric gain scales directly as the mechanical Q factor of the test masses and the

optical Q factor of higher order modes. Hence errors in the Q factor of higher order

modes directly affect the estimation of R0. Thus it is very important to have an

accurate estimation of the diffraction losses of the modes. Equation (5.1) shows the

parametric gain in a power recycled interferometer.

R0 ≈2PQm

McLw2m

(Q1Λ1

1 + ∆w21/δ

21

− Q1aΛ1a

1 + ∆w21a/δ

21a

). (5.1)

Here P is the total power inside the cavity, M is the mass of the test mass, Q1(a)

are the quality factors of the Stokes (anti-Stokes) modes, Qm is the quality factor

of the acoustic mode, δ1(a) = w1(a)/2Q1(a) corresponds to the relaxation rate, L is

the cavity length, ∆w1(a) = w0 − w1(a) − wm is the possible detuning from the ideal

resonance case, and Λ1 and Λ1a are the overlap factors between optical and acoustic

modes. The overlap factor is defined as [1]:

Λ1(a) =V(∫

f0(~r⊥)f1(a)(~r⊥)uzd~r⊥)2∫

|f0|2d~r⊥∫|f1(a)|2d~r⊥

∫|~u|2dV

. (5.2)


Here f0 and f1(a) describe the optical field distribution over the mirror surface

for the fundamental and Stokes (anti-Stokes) modes, respectively, ~u is the spatial

displacement vector for the mechanical mode, uz is the component normal to the

mirror surface. The integrals∫d~r⊥ and

∫dV correspond to integration over the

mirror surface and mirror volume respectively.

Traditionally it has been assumed that diffraction losses can be estimated by the

clipping approximation. In this approximation it is assumed that the mode shape

is not altered by the finite mirror geometry, and that the diffraction loss is simply

determined by the fraction of the mode that overlaps the mirror.

It has been demonstrated, for the fundamental mode of a two mirror symmetric

cavity, that the geometry of the system determines whether the clipping approxima-

tion overestimates or underestimates the diffraction losses [8, 9]. Preliminary work

by D’Ambrosio et al [10] indicates that the clipping approximation is not valid for

this particular case. For this reason we have undertaken careful numerical modelling

of the mode losses based on the free propagation of the beams inside a long base-

line interferometer. Using Fast Fourier Transform (FFT) simulations we determine

the diffraction losses inside the main arms of a proposed advanced interferometer

configuration. We present results for typical arm cavity design for advanced inter-

ferometers, based on the proposed design of Advanced LIGO (Laser Interferometer

Gravitational-wave Observatory).

We show that for these very large cavities the mode frequencies are shifted by a

significant amount; and that the size of the mirror not only affects the diffraction

losses but also the cavity gain, mode frequency, Q-factor and the mode shape. All

these parameters are necessary to determine the overlap factor Λ and greatly affect

the predicted parametric gain R0.

First we introduce the FFT simulation method to simulate the behaviour of a

gaussian beam inside a cavity. In the next section this is applied to calculate the

diffraction losses from Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes

in an advanced gravitational wave detector arm cavity, analysing also the limits of

the method. In section 4 these results are compared with results from eigenvalues

simulations. We also analyse the change in diffraction losses when using finite mirrors


of different sizes, including their impact on the optical gain and mode frequency, the

optical Q-factor and mode shape.

5.3 Diffraction losses

Diffraction losses occur in any optical cavity with finite size optics, even if the mirrors

are large compare to the Gaussian spot size. The larger the Fresnel number, the

weaker is the field intensity at the edge of the mirror, and smaller the power loss due

to diffraction loss [11, 12].

If we consider a Fabry-Perot cavity (like the proposed Advanced LIGO arm cav-

ities) with mirrors of diameter 2a and spot size radius we can say that only those

modes of order less than the ratio of the two areas will oscillate inside the cavity with

relatively low losses [13].

Nmax ≈(πa2

πw2

). (5.3)

We use the proposed design of Advanced LIGO as a case in point. It has mirrors

of 34 cm diameter with a spot size of approx 6 cm, which gives us Nmax = 8.03.

Therefore we will analyse modes up the 8th order. We can expect that the cavity

losses for higher order modes will rapidly increase with mode number.

The clipping approximation determines the part of the higher order mode spot

size that will fall outside the mirror’s surface when the mode shape itself is due to

that of an infinite diameter mirror. The diffraction loss in each reflection of a cavity

mode off a mirror is given by:

Dclip =

∫ ∞a

|U(r)|22πdr. (5.4)

Here U(r) is the normalised field of a HG or LG mode with infinite size mirrors

integrated outside a mirror of diameter 2a.

It is already known that the clipping approximation yields a smaller loss than the

calculations based in FFT methods for the TEM00 mode in a long optical cavity.

The FFT method enables the mode shape changes due to the finite mirror sizes to be

estimated and hence enables a much better approximation of the diffraction losses.

5.3. DIFFRACTION LOSSES 153

A good explanation of this method can be found in ref [14], and a more general

explanation in ref [15].

The following calculation relates the internal power with the diffraction losses

due to finite size mirrors. Let Ti, Di and Li be the transmission, diffraction and

dielectric losses respectively for the Input Test Mass (ITM) and Te, De and Le be the

corresponding values for the End Test Mass (ETM). The finesse of such cavity can

then be calculated as [16]:

F ∼=2π

Ti +Di + Li + Te +De + Le. (5.5)

Since we are interested in the effects of the diffraction losses over the circulating

power inside the cavity in steady state, we assume perfect mode matching for the

input beam. Therefore the peak value for the circulating intensity in a purely passive

cavity at resonance can be written as [17]:

Icirc ≈4(Ti + Li)

(Ti +Di + Li + Te +De + Le)2Iinc. (5.6)

In the case of infinite size mirrors there are no diffraction losses, then Di = De = 0.

For our simulations we use the following values, which coincide with proposed

parameters for Advanced LIGO interferometers, where Ti = 5000 ppm, Te = 1 ppm

and Li = Le = 15 ppm [18]. In this case it is clear that the major loss contribution

comes from the transmission losses of the ITM.

Note that equation (5.6) does not contain a mode matching parameter because

of the assumption of perfect mode matching. This assumption allows us to refer the

incident beam to the cavity waist, this simplifies the analysis.

In order to calculate the diffraction losses we use a lossless cavity in parallel with

the cavity under study. After each round trip the resulting beam is normalised and

propagated in to a lossless cavity, this cavity has the same characteristics of the cavity

under study but with lossless mirrors. In such cavity diffraction losses are the only

cause of power loss. Therefore we can calculate the diffraction losses per round trip

using equation (5.7), where PNormj corresponds to the normalised total circulating

power per round trip. In the case of infinite sized mirror PNormj is always 1, hence no

diffraction losses.


Dclip = 1− PNormj . (5.7)

This method proved to be more accurate than measuring the diffraction losses on

every round trip using the incremental power of every round trip as shown in equation

(5.8). This method is more susceptible to numerical errors due to the digitisation of

the beam.

D =

(1− pj

pj−1

)− (Ti + Li + Te + Le) . (5.8)

Here pj corresponds to the power contribution of round trip j. Therefore pj−1

corresponds to the power contribution of the previous round trip.

A third way to estimate the diffraction losses is to calculate the eigenvalues for

the cavity. At the end of section 5.5.1 we comment on the verification of our results

based on comparison with eigenvalues calculations. This eigenvalues γmn are such that

after one round trip the eigenmodes will satisfy the simplified round trip propagation

expression [19]:

γmnUmn(x, y) =

∫∫K(x, y, x0, y0)Umn(x0, y0)dx0dy0. (5.9)

The magnitude of the eigenvalues due to the round trip losses will be less than

unity. Therefore we can calculate the power loss per round trip as:

Power loss per round trip = 1− |γmn|2. (5.10)

We now go on to present the details of our FFT simulation methods.

5.4 FFT simulation

Since each mode has a different resonant frequency depending on the order of the

mode we must analyse each mode separately. The first step is to generate the mode

we are interested in. Part of the study also includes the comparison between HG and

LG modes. For simplicity the mode is generated at the waist of the cavity. HG modes

are given by:

5.4. FFT SIMULATION 155

Um,n(x, y, z) =

(1

2π

) 12

√exp j(2m+ 2n+ 1)ψ(z)

2m2nm!n!w(z)2×

Hm

(√2x

w(z)

)Hn

(√2y

w(z)

)

exp

−j2kz − jk

(x2 + y2

2R(z)

)− x2 + y2

w(z)2

. (5.11)

Here m and n correspond to the order of the transverse modes, w(z) is the spot

size radius, R(z) is the beam radius of curvature, ψ(z) is the Gouy phase shift, k is

the wave number and Hm() is the mth order Hermite polynomial.

LG modes are also a valid representation of the higher order modes, this time in

cylindrical coordinates rather than rectangular, and given by:

Ul,m(r, φ, z) =

√4l!

(1 + δ0,m)π(l +m)!

(exp j(2l +m+ 1)ψ(z)

w(z)

)cos(mφ)×(√

2r

w(z)

)m

Ll,m

(2r2

w(z)2

)exp

−jkz − jz r2

2R(z)− r2

w(z)2

.(5.12)

Here l corresponds to the radial index and m to the azimuthal mode index, Ll,m()

are the generalised Laguerre polynomials, δ0,m = 1 if m = 0 and δ0,m = 0 if m > 0.

The rest of the variables are the same as in the HG modes.

The Fourier transform corresponds to the transformation of the beam profile into a

spatial frequency domain. We can create a propagation matrix based on an expansion

of the optical beam in a set of infinite plane waves travelling in slightly different

directions [20] given by

A(p, q, zL) = exp−jkzL + jπλ(p2 + q2)zL

. (5.13)

Where zL corresponds to the distance which we will propagate the beam, p and

q are the coordinates in the Fourier space or the spatial frequencies. To apply this

propagation matrix it is also necessary to transform the field of the input beam by

using a two dimensional FFT. The Fourier transform of a gaussian function is always

another gaussian transform of the same order: i.e.


F U(x, y, z) =

∫∫U(x, y, z)e−jpxe−jqydxdy. (5.14)

and the inverse Fourier transform is then written as:

F−1 U(p, q, z) =

(1

2π

)2 ∫∫U(p, q, z)ejpxejqydpdq. (5.15)

Once the beam has been propagated by multiplying point to point both the field

matrix and the propagation matrix we can transform the field back to the time domain

using the two dimensional inverse FFT. If z0 is the propagation starting point then

the final field corresponds to:

U(x, y, zL) = F−1 FU(x, y, z0) × A(p, q, zL) . (5.16)

Thus, this method basically consists of transforming the input field into the Fourier

domain in two dimensions and propagation of this field along the z axis. Then

the resulting field is transformed back to the time domain using the inverse FFT.

Based on this principle we developed our own code in Matlabr, which allowed us to

propagate the field and to reflect it off the mirrors surface. Two independent codes

were developed, one at the University of Western Australia (UWA) and the other

at the California Institute of Technology (Caltech), for the purpose of verification of

results. For the simulation we assume a perfect surface for the test masses, but any

imperfection can easily be added. Starting from the waist of the cavity we propagate

the beam down to the ETM where it is reflected from the mirror surface and propagate

it back through the waist to the ITM. Here part of the beam is transmitted out of the

cavity through the substrate, and the rest is reflected back to the waist of the cavity

completing the round trip. This is then iterated until the power inside the cavity has

reached the steady state.

In order to calculate the diffraction losses for different higher order modes it was

necessary first to make the modes resonant inside the cavity. Starting from a nominal

value of 4 km for the cavity length we move the ETM away from the ITM up to a

maximum of half the laser wavelength (λ = 1.064µm) until we find the cavity length

that maximises the circulating power for a particular mode. For each small step that

the ETM is moved several round trips are done to calculate the power built-up inside

5.5. RESULTS 157

the cavity. Once the cavity is set at the resonance length we propagate the mode

inside the cavity for several round trips until the circulating power reaches the steady

state, which corresponds to the maximum circulating power.

Most of the calculations presented here were done using a 128×128 elements grid.

This proved to be good for the calculations of the parameters we are interested in,

for the different modes. However, it was observed that for higher order modes (order

higher than 6) a grid of 128 × 128 is not enough when using mirrors of infinite size

due to aliasing. In those cases we increased the size of the grid to a 256× 256, using

also finer elements. Another option could have been the use of anti-aliasing filters or

an adaptive grid.

5.5 Results

In order to determine the parametric instability gain we need to know more than

just the diffraction losses of the higher order modes. Therefore we also calculated

the optical gain, the frequency, and the optical Q-factor of each mode. We also

examined the mode shapes, and how they change with the size of the mirrors. In

each case the calculations were done separately for each mode. We explored the

variation of the size of the mirrors that form the cavity, while keeping the mirror

radius of curvature and losses constant. The results were crossed checked between

UWA and Caltech finding very good agreement between both simulations. A cavity

formed by two mirrors with the losses previously mentioned and a constant radius of

curvature of 2076 m at a nominal length of 4000 m was assumed for these simulations.

On each simulation a pair of mirrors of the same diameter was used. For all our

simulations it was assumed that the substrate of the test masses will have two flat

sides proportional to the substrate diameter for suspension attachment as shown in

figure 5.1. We also assumed that the mirror coating covers the whole front surface of

the test mass (surface 1 in figure 5.1).


Figure 5.1: Advanced LIGO substrate dimensions for ITM and ETM test masses showing

the flat sides of the test mass for suspension attachment. From LIGO drawing D040431-B.

5.5.1 Diffraction losses

The diffraction losses for different modes for varying mirror size are presented in

figure 5.2. As expected the diffraction losses are the same for modes of order 1 like

HG01, HG10 and LG01. For HG modes the order number is given by (m+ n) while

for LG modes the order number is given by (2l +m).

It is clear (as previously showed by Fox and Li in ref [11]) that the higher the

order of the mode the higher is the loss, especially for smaller mirrors (smaller Fresnel

number). Diffraction losses can be separately analysed by the order of the mode, but

when the order of the mode increases we need to take in to account the symmetry of

the modes. For example mode HG60 is mainly distributed along one axis compared

to HG33, which is of the same order but is evenly distributed on the mirror’s surface.

In such case and due to the energy distribution of each mode the losses for mode

HG33 are much smaller than mode HG60.

From figure 5.2 we can deduce that when the order of the mode is increased the

grouping of the diffraction losses spreads out due to the different symmetries of the

modes. If we compare the diffraction losses of modes of order 6 and 7 (figures 5.2 (c)

and 5.2 (d) we notice that there is an overlap of some modes. Therefore the low loss

modes of order 7 like LG07 have lower diffraction losses than the high loss modes of

order 6, namely LG22 and LG30.

Even though it is difficult to distinguish in figure 5.2, modes HG11 and LG02 have

5.5. RESULTS 159

(a) (b)

14 14.5 15 15.5 16 16.5 17 17.5 1810

−2

10−1

100

101

102

103

104

105

106

Diffraction losses for modes up to order 4

Mirror radius size [cm]

Diff

ract

ion

loss

es [p

pm]

HG00HG01HG10LG01LG02HG11HG20LG10LG03HG12HG30LG11LG04HG22HG13HG40LG12LG20

14 14.5 15 15.5 16 16.5 17 17.5 1810

3

104

105

106

Diffraction losses for 5th order


Diff

ract

ion

loss

es [p

pm]

LG05HG23HG14LG13HG50LG21

(c) (d)

14 14.5 15 15.5 16 16.5 17 17.5 1810

3

104

105

106

Diffraction Losses for 6th order


Diff

ract

ion

loss

es [p

pm]

LG06HG33HG24HG15LG14HG60LG22LG30

14 14.5 15 15.5 16 16.5 17 17.5 1810

3

104

105

106

Diffraction Losses for 7th order


Diff

ract

ion

loss

es [p

pm]

LG07HG43HG52LG15HG16LG23HG70LG31

Figure 5.2: Diffraction losses for different higher order modes. Starting from the top left

corner we present modes up to order 4 in ascendant orders from HG00 up to LG20. For

clarity we separately present modes of order 5, 6 and 7. HG and LG modes are plotted

together for comparison.

almost the same losses. This is easy to explain since we can see in figure 5.3 that

LG02 corresponds to HG11 twisted by 45o (or vice-versa) and therefore orthogonal to

each other. Therefore the difference in diffraction losses comes from the flat sides of

the test mass affecting differently each mode. It is also interesting to notice that the

highest loss is in mode LG10, which is a more symmetric one, but has more energy

at the edge of the mirror, compared with other 2nd order modes.

We have compared our results with calculations of the diffraction losses using the

cavity eigenvalues by Juri Agresti from Caltech [21]. He calculated the diffraction

losses for several LG modes. His results are in very close agreement with the results


Figure 5.3: Intensity profile at the ITM in a mirror of diameter 34 cm. On the left hand

side we can see mode HG11 compared to mode LG02 in the middle. Also for comparison

the energy distribution of mode LG10 is presented on the right hand side.

here presented. The average difference between the FFT simulation and the eigen-

value approximation is of 1%, while the biggest difference is less than 5% for lower

order modes. However his calculations were done for a previous design of Advanced

LIGO test masses with smaller mirrors of 31.4 cm in diameter, which we also use for

comparison.

In parallel with the FFT simulations and the eigenvalues calculations done at

Caltech we did our own eigenvalues calculations based in an eigenvector method

proposed by C. Yuanying et al [22]. The results obtained through this method showed

that in a perfect aligned cavity with cylindrical test masses (circular mirrors) only

LG modes and their rotated orthogonal modes will resonate. However the need of

suspend the mirrors requires the test masses to have two flat sides as can be seen in

figure 5.1 (LIGO technical document D040431-B). This breaks the symmetry.

As the circular symmetry is broken when solving with the eigenvector method

for this cavity it shows that HG modes are now part of the eigenvectors solution of

this cavity. Therefore HG modes are partially supported by the cavity even if it is

perfectly aligned. However these modes are mainly distributed along the horizontal

axis aligned with the flat sides of the test mass. The symmetry break not only changes

the eigenvectors and eigenvalues solution for this system, but also induces a particular

orientation of the higher order mode that minimises the diffraction losses.

The main difference with the eigenvector method is that in the FFT simulation

5.5. RESULTS 161

Diffraction losses comparison between FFT and Eigenvalues

HG

00

HG

01

LG

01

HG

11

HG

02

LG

02

LG

10 H

G12

HG

03

LG

03 LG

11

HG

22

HG

13

LG

04

LG

12

LG

20

HG

14

HG

05

HG

23

LG

05 LG

13

LG

21

HG

15

HG

24

HG

33

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

LG

07 LG

15

LG

23

LG

31

HG

08

HG

17

HG

26

HG

35

HG

44

LG

08

LG

16

LG

24

LG

32

LG

40

HG

07

HG

04

HG

06

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 5 10 15 20 25 30 35 40 45 50

Optical modes

Diffr

actio

n lo

sse

s (

pp

m)

FFT Results Eigenvalues

Figure 5.4: Comparison of diffraction losses obtain with FFT simulations and the eigen-

vector method for the proposed Advanced LIGO type cavity.

we can choose which mode we are going to propagate inside the cavity. The FFT sim-

ulation allows the mode to change its shape while it propagates inside the cavity. In

order to determine when the mode shape is stable we calculate the non-orthogonality

between the input and the circulating beam, this subject to the finesse of the mode

and the power to build up in the cavity. Therefore the mode labels in the graphs cor-

respond to the input mode used in the FFT simulation and not necessarily correspond

to that of the final mode shape.

Figure 5.4 show the diffraction losses results obtained with both methods. The

results are in close agreement, but we noticed that some of the modes that we injected

are not supported by the cavity. These modes in fact do not appear in the eigenvector

method; moreover we can see that those modes are the ones that their mode shape

changes in to a lower loss mode of the same order. This explains why modes HG15,

HG24, HG33 have all similar losses as mode LG06. We further analyse the mode

shape changes in section 5.5.4.


5.5.2 Optical gain

Considering the parameters here presented for Advanced LIGO type optical cavities,

the optical gain of the main arms is about 800. Assuming infinite size mirrors and the

losses already presented for this cavity, the gain that we obtain from the simulations is

close to 793. Using mirrors of finite size increases the diffraction losses, thus reducing

the gain. This can also be deduced from equation 5.6.

14 14.5 15 15.5 16 16.5 17 17.5 1810

−5

10−4

10−3

10−2

10−1

100

Optical Gain for higher order modes

Mirror size radius [cm]

Gai

n

HG00HG01LG02HG20LG10LG03HG30LG04HG40LG05LG20LG06HG50LG07HG60LG30HG70

Figure 5.5: Cavity optical gain for some HG modes of different orders. The modes have

been plotted in descendant order, starting from the top with HG00 to finish with HG70 at

the bottom. HG and LG modes are plotted together for comparison.

For a given finite mirror size the higher the order of the mode, the higher the

diffraction losses. This in turn means lower optical gain and lower finesse for the

higher order mode, which as a consequence implies a reduction of the circulating

power. Figure 5.5 shows the optical gain for some of the HG and LG modes. The

figure shows how the gain of each order changes with the mirror size (or Fresnel

number). The fundamental mode doesn’t change much, but as the order of the mode

increases the diffraction losses increase reducing the gain. With infinite size mirrors

all the modes have the same gain since there are no diffraction losses.

Using the nominal Advanced LIGO mirror size (34 cm diameter) we can plot the

5.5. RESULTS 163

Higher Order Modes Optical Gain

HG

00

HG

01

LG

01

HG

11

HG

20

LG

02

LG

10

HG

12

HG

30

LG

03

LG

11

HG

22

HG

13

HG

40 L

G04

LG

12

LG

20

HG

14

HG

50

HG

23

LG

05

LG

13

LG

21

HG

15

HG

24

HG

33

HG

60

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

HG

70

LG

07

LG

15

LG

23

LG

31

HG

17

HG

26

HG

35

HG

44

LG

08

LG

16

LG

24

LG

32

LG

40

HG

08

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0 5 10 15 20 25 30 35 40 45 50

Optical Modes

Ga

in

Figure 5.6: Optical gain variation for higher order modes for a mirror diameter of 34 cm.

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.20

500

1000

1500

2000

2500

3000

3500

Mirror size [m]

Inte

nsity

TEM40

intensity profile for different mirror size

Inf

36 cm

34 cm

32 cm

31.4 cm

30 cm

28 cm

Figure 5.7: Intensity profile variation of mode HG40 due to the different mirror size.

optical gain versus mode number as shown in figure 5.6. Here we notice how the gain

is reduced for a particular mirror size when the order of the mode increases. Again

we can see the dependency on the energy distribution and symmetry of the mode.


The gain reduction due to the finite size of the mirrors can also be appreciated

in figure 5.7, which shows the effect in the intensity profile of mode HG40. We can

see how the optical gain is reduced with the size of the mirrors. The use of smaller

mirrors increases the diffraction losses leading to smaller gain as shown in table 5.1.

This effect is even stronger in higher order modes.

5.5.3 Mode frequency

As previously mentioned the calculation of the resonance length was done by moving

the ETM away from the ITM until the circulating power is maximised. The resonance

length is different for each mode, but when the mirror size was changed a minor

variation in resonance length was noticed. This suggests that mirrors of different

size will also alter the resonance conditions of the cavity, thus changing the mode

frequency. It is well known that the frequency shift for higher order modes is given

by [23]:

υ0

π(m+ n) arccos(

√g1g2) for HG modes, (5.17)

υ0

π(2l +m) arccos(

√g1g2) for LG modes. (5.18)

Here υ0 corresponds to the Free Spectral Range (FSR) in Hz, g1 and g2 correspond

to the stability factor of each mirror, define as g = (1 − L/R), L being the cavity

Mirror Diameter Diffraction Losses Optical Gain

(cm) (ppm)

28 116022 1.37

30 52437 6.07

32 19256 34.01

34 5315 187.40

36 1506 469.50

Table 5.1: Diffraction losses and cavity optical gain of mode HG40 for different mirror

size.

5.5. RESULTS 165

length, and R the radius of curvature of the mirror. Thus to calculate the frequency

separation of a higher order mode with respect to the fundamental mode we use the

following relation:

∣∣∣∣∆f∆l

∣∣∣∣ =1

LfY AG +

N

π

c

2L2

1

(2LR− L2)1/2. (5.19)

Where ∆f corresponds to the frequency variation of the mode, ∆l to the cavity

length variation and fY AG to the Nd : Y AG laser fundamental mode frequency. Here

N = (m+ n+ 1) for HG modes and N = (2l +m+ 1) for LG modes, c corresponds

to the speed of light in vacuum and R the radius of curvature of the mirrors. We

note that in reality the laser is locked to the TEM00 cavity mode. Equation (5.19)

is special case for cavities with two mirrors of the same radius of curvature and as a

consequence with g1 = g2.

Using equations (5.17) and (5.18) the mode separation per mode order using the

proposed Advanced LIGO parameters is 4.593 kHz. According to our simulations

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 5.8: Modes of order 7 as they would appear in an infinite sized mirror. Therefore

no diffraction losses or mode shape changes affect these modes. Figures (a), (b), (c) and (d)

correspond to modes HG07, HG16, HG25 and HG34. Figures (e), (f) (g) and (h) correspond

to modes LG07, LG15, LG23 and LG31.


this is true only when using mirrors of infinite size. The use of mirrors of finite size

introduces diffraction losses which not only reduce the circulating power but also shift

the frequency of the higher order modes.

First we consider the fundamental mode where the frequency deviations are negli-

gible, of the order of millihertz. The smallest mirror in our simulations is of 28 cm in

diameter. In this case the frequency variation from the infinite mirror case is about

24 × 10−3 Hz, which compared to the laser fundamental mode frequency, is a varia-

tion of 10−17. For a mirror of 34 cm in diameter the frequency variation is reduced to

1.9× 10−3 Hz.

These results show a clear agreement with the diffraction losses for modes of order

7 shown in figure 5.2 (d). The highest diffraction losses are from mode LG31, while the

lowest losses are from mode LG07, which also has the smallest frequency variations

from the infinite mirror case. These results show that the frequency not only depends

on the mode order, but also on the symmetry and energy distribution of the mode

subject to the size of the mirror. The frequency variation will also have an impact

on the possible parametric instabilities calculations. Figure 5.9 show the frequency

variations for the higher order modes in the proposed cavity with mirrors of 34 cm in

diameter.

Cavity losses define the optical Q-factor, which for any given mode is given by:

Mirror ∆freq ∆freq ∆freq ∆freq ∆freq ∆freq ∆freq ∆freq

Diameter HG70 HG61 HG52 HG43 LG07 LG15 LG23 LG31

(cm) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz)

28 3466 1658 1345 1277 1251 2715 3735 4213

30 2144 799 686 659 649 1584 2324 2698

32 897 282 248 237 233 688 1127 1371

34 317 103 70 64 61 215 400 515

36 73 59 34 25 21 76 145 189

Table 5.2: Frequency shift of modes of order 7 for different size mirrors compared to the

frequency of the same mode when using infinite sized mirrors.

5.5. RESULTS 167

Higher Order Modes Frequency Shift

HG

00

HG

01

LG

01 H

G11

HG

20

LG

02 L

G10

HG

12

HG

30

LG

03

LG

11

HG

22

HG

13

HG

40

LG

04

LG

12 L

G20

HG

14

HG

50

HG

23

LG

05

LG

13

LG

21

HG

15

HG

24

HG

33

HG

60

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

HG

70

LG

07

LG

15

LG

23

LG

31

HG

08

HG

17

HG

26

HG

35

HG

44

LG

08 L

G16

LG

24

LG

32

LG

40

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0 5 10 15 20 25 30 35 40 45 50

Optical Modes

De

lta

Fre

qu

en

cy (

Hz)

Figure 5.9: Frequency variations from the theoretical value (infinite size mirror) for each

higher order mode when using finite size mirrors of diameter 31.4 cm.

Q =w

2δT. (5.20)

Here w corresponds to the frequency of the mode and δT to the relaxation rate of

that particular mode [1]. Therefore it is expected that the higher the diffraction losses

the lower the optical Q of that mode. The Q-factor of the optical modes has a direct

effect on the parametric gain R0 therefore a reduction of this factor will also reduce

the parametric gain. The Q-factor of the optical modes will also have a dependency

on the size of the mirrors, since both the frequency and the losses depend on the size

of the mirror as well. Figure 5.10 shows the optical Q-factor for the higher order

modes for the 34 cm diameter mirrors. Here we can see that the Q-factor follows the

same trend as in the optical gain, which also is inversely proportional to the total

losses of the cavity.

5.5.4 Mode shape

Higher order modes have a more spread intensity profile. As a consequence depending

on the symmetry of the mode, it will cover a larger area of the mirror’s surface. This

not only causes high diffraction losses due to the energy loss per round trip, but in


Optical Q factor for higher order modes

HG

00

HG

01

LG

01

HG

11

HG

20

LG

02

LG

10

HG

12

HG

30

LG

03

LG

11

HG

22

HG

13

HG

40

LG

04

LG

12

LG

20

HG

14

HG

50

HG

23

LG

05

LG

13

LG

21 HG

15

HG

24

HG

33

HG

60 LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

HG

70

LG

07

LG

15

LG

23

LG

31

HG

08

HG

17

HG

26

HG

35

HG

44

LG

08

LG

16

LG

24

LG

32

LG

40

1.E+10

1.E+11

1.E+12

1.E+13

0 5 10 15 20 25 30 35 40 45 50

Optical Modes

Q f

acto

r

Figure 5.10: Optical Q-factor for the higher order modes in the proposed cavity.

some cases it also distorts the mode. The overlap parameter Λ used to calculate the

parametric gain R0 depends on the mode distribution over the mirror’s surface [3].

An interesting case is mode HG33. This mode resonates and keeps its shape inside

the cavity if the mirrors are of infinite size (figure 5.11 (a)), but when using finite size

mirrors the mode is completely distorted and after a few round trips it doesn’t look

like a mode HG33 anymore, but more like mode LG06, although twisted by 30o and

thus orthogonal to LG06 as can be seen in figure 5.11. The distorted mode HG33

still show some features from the original mode like the small energy distribution at

the centre of the intensity profile. However it is easy to see the similarity and why

the diffraction losses are higher for mode HG33 due to the energy loss at the edge of

the mirror.

The interpretation is that even if we forced mode HG33 in to a perfectly aligned

cavity it will not resonate inside the cavity and will give rise to a ‘twisted’ LG06 mode

(figure 5.11 (b)). This mode will also be one of the eigenvectors of the cavity since

it is orthogonal to the original LG06 shown in figure 5.11 (c). Similar effects were

observed for modes HG43 and HG52.


(a) (b) (c)

Figure 5.11: (a) Intensity profile of mode HG33 in a infinite sized mirrors. The next

two intensity profiles are for finite sized mirrors of diameter 34 cm. (b) Shows the intensity

profile of mode HG33 on the ITM at end of the simulations. (c) Shows the intensity profile

of mode LG06 also for finite size mirrors.

5.6 Conclusions

We have shown how the diffraction losses of various modes in large optical cavities

depend on the diameter of the mirrors. Moreover they also depend on the shape

of the mirror. We show that the predicted mode frequencies are also offset from

the infinite mirror case by up to a few kHz. The diffraction losses are needed to

determine the optical Q-factor of each mode, and this combined with mode frequency

data is necessary to estimate the possibility of parametric instabilities, through the

calculation of the parametric gain. We have shown that finite size mirrors significantly

alters the shape of the higher order modes, and due to high diffraction losses on

each round trip also the optical gain is reduced. The mode shape variations affect

the overlap integral calculation which determines the opto-acoustic coupling in the

parametric instabilities calculations.

We wish to point out that in a power recycled interferometer, the design of the

power recycling cavity can vary the coupling losses so as to increase the high order

mode losses. The high order losses can never be less than the diffraction losses

predicted here. However it is also true in the case of coupled cavities the mode shape

could be significantly altered (compared with the single cavity modes considered here),

and this could vary the diffraction losses. The presence of a signal recycling cavity


can also affect the parametric instability.

Acknowledgements

The authors would like to thanks Erika D’Ambrosio, Bill Kells and Hiro Yamamoto

for useful discussions. Also thanks to Jerome Degallaix for his help in developing the

FFT code at UWA, and Juri Agresti for his eigenvalues calculations. David G. Blair

and Li Ju would like to thanks the LIGO Laboratory for their hospitality. Biplab

Bhawal is supported by National Science Foundation under Cooperative Agreement

PHY-0107417. This work was supported by the Australian Research Council, and

is part of the research program of the Australian Consortium for Interferometric


5.7. APODISING COATING 171

5.7 Apodising coating

Study of the possible reduction of parametric

instability gain using apodising coating in test

masses

P. Barriga1 and R. DeSalvo2

1 School of Physics, University of Western Australia, Crawley, WA6009, Australia2 LIGO Laboratory , California Institute of Technology, Pasadena, CA 91125, USA.

5.7.1 Introduction

As part of the study of the possibility of the occurrence of parametric instabilities

in advanced gravitational wave interferometers, it was suggested the use of an apo-

dising coating for the test masses. The main goal is to reduce the parametric gain

R0 by increasing the diffraction losses of the high order optical modes keeping the

fundamental mode diffraction loss below 1 ppm.

This section concentrates on the effect of an apodising coating over the diffraction

losses for an Advanced LIGO type of cavity. Diffraction losses changes will affect

the frequency of the mode, the Q-factor and the total losses, also affecting the mode

shape and the overlapping parameter as a consequence. Therefore it is not straight

forward to determine the effect of diffraction losses changes over the parametric gain,

but in general we need higher diffraction losses in order to reduce the parametric

instability gain.

The diffraction losses for an Advanced LIGO type of cavity have been previously

calculated [16, 24]. The Fast Fourier Transform (FFT) method developed at The

University of Western Australia allows us to inject any optical mode in to the cavity.

Inside the cavity the mode is propagated using a FFT and thus the beam is free to

change according to the resonance conditions imposed by the cavity parameters. Pre-

vious simulations show that a mode which is not supported by the cavity will morph

in to a different mode of the same order but with lower diffraction losses. A change in


the mode shape is accompanied by the corresponding frequency shift. In such cases it

is not possible to say that the nominal mode resonates inside the cavity. As a result

the simulations presented here show the high order modes diffraction losses of the in-

jected mode. Figure 5.12 shows a comparison of the diffraction losses obtained using

a FFT simulation and the results obtained using the eigenvalues calculations for this

cavity using an eigenvector method [22]. For these calculations a test mass of 34 cm

in diameter was assumed according to LIGO drawing D-040431-B. A homogeneous

coating with 50 ppm losses was also assumed.

Diffraction losses comparison between FFT and Eigenvalues

HG

00

HG

01

LG

01

HG

11

HG

02

LG

02

LG

10

HG

12

HG

03

LG

03 LG

11

HG

22

HG

13

LG

04

LG

12

LG

20

HG

14

HG

23

LG

05 LG

13

LG

21

HG

15

HG

24

HG

33

LG

06

LG

14

LG

30

HG

16

HG

43

HG

52

LG

07 LG

15

LG

23

LG

31

HG

05

LG

22

HG

06

HG

04

HG

07

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 5 10 15 20 25 30 35 40

Optical modes

Diffr

actio

n lo

sse

s (

pp

m)

FFT Results Eigenvalues

Figure 5.12: Diffraction losses for an Advanced LIGO type cavity, comparison between

FFT and eigenvalues calculations.

5.7.2 Apodising coating

In order to compare the effect that an apodising coating will have on the diffraction

losses the designs presented in figure 5.13 were tested using different values for L1,

L2 and L3.

Several simulations were done in order to determine the best coating absorption

combination by comparing results based on the diffraction losses of the fundamen-

tal mode. In all simulations the same substrate was used according to the LIGO

document E060001-00, which corresponds to a 34 cm diameter test mass including


the chamfer and the flat sides for suspension attachment. Let Ti = 5000 ppm and

Li = 15 ppm be respectively the transmission and dielectric losses for the Input Test

Mass (ITM) and Te = 1 ppm and Le = 15 ppm the corresponding values for the End

Test Mass (ETM) [18]. In this case it is clear that the major loss contribution comes

from the transmission losses of the ITM. The results presented in the next section

correspond to the more relevant ones.

The simulations also show that the minimum coating size for the fundamental

mode to have diffraction losses of 1 ppm corresponds to a circular coating with a

diameter of 33.1 cm. This is assuming that outside the coating all photons will be

loss. However by reducing the losses outside the central coating we are able to reduce

the size of this central coating proportional to the reduction of the outer ring losses.

For these simulations the same coating is assumed for both ITM and ETM.

L1

L2

17 cm

11.33 cm L312.59 cm

17 cm

L1

Figure 5.13: Proposed apodising coatings design for the ITM and ETM of an Advance

LIGO type cavity. The main purpose is to simulate their influence on diffraction losses and

ultimately their effect in the parametric gain R0.

5.7.3 Results

Figure 5.14 shows the diffraction losses for a standard homogeneous coating with

L = 50 ppm, also other configurations with homogeneous coatings with losses of 1000

ppm and 10000 ppm have been considered. The figure also include two apodising

coatings, one with losses given by L1 = 50 ppm and L2 = 25000 ppm (coating mean

value of 18714 ppm) and one with losses given by L1 = 50 ppm and L3 = 100000 ppm


(coating mean value 68961 ppm). However due to the different energy distribution of

the higher order modes the absorption losses of the apodising coatings will be different

for each mode. However the simulation results presented in figure 5.14 show no big

difference in terms of diffraction losses when using different coatings.

Normal DiffLoss DS DiffLoss DiffLoss DiffLoss DS2

(L = 50) (L1 = 50 (L1 = 1000) (L = 10000) (L1 = 50

L2 = 25000) L3 = 105)

mean L = 18714 mean L = 68961

0.6899 0.8388 0.6882 0.6817 0.8745

Table 5.3: Diffraction losses for the fundamental mode HG00 using the different coatings.

All values in ppm.

Table 5.3 shows the diffraction losses for different coating losses on the test masses.

We can infer from the table that when using a homogeneous coating the higher the

losses the lower the diffraction losses. When using an apodising coating the diffraction

losses for the fundamental mode are increased, but always keeping them below 1 ppm.

Diffraction Losses Coating Comparison (mirror = 34 cm)

HG

00

HG

01

LG

01

HG

20

HG

11

LG

02

LG

10

HG

30

HG

12

LG

03 LG

11

HG

40

HG

13

HG

22

LG

04 L

G12

LG

20 H

G50

HG

14

HG

23

LG

05 L

G13

LG

21

HG

60

HG

15

HG

24

HG

33

LG

06 LG

14

LG

22

LG

30

HG

70

HG

16

HG

52

HG

43

LG

07 LG

15

LG

23

LG

31

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 5 10 15 20 25 30 35 40

Optical modes

Diffraction losses (ppm

)

(L = 50) (L1 = 50, L2 = 25000) (L = 1000) (L = 10000) (L1 = 50 L3 = 100000)

Figure 5.14: Diffraction losses of higher order modes for an Advanced LIGO type of cavity

using different combinations of homogeneous and apodising coatings.

For higher order modes the diffraction losses comparison between the different


coatings will depend of the energy distribution on the beam profile. Therefore modes

with higher energy distribution closer to the edge of the test mass will be more affected

by the apodising coating. A similar effect can be seen in the shape of the mode which

is also affected by the coating losses, which in consequence will affect the overlapping

parameter and therefore the parametric gain R0. Also due to the higher losses the

gain of the cavity is reduced, thus the circulating power inside the cavity also drops.

Diffraction Losses Ratio (mirror = 34 cm)

HG

00

HG

01

LG

01

HG

20

HG

11

LG

02

LG

10

LG

03

LG

11

LG

04

LG

12

LG

20

HG

50

HG

14

HG

23

LG

05

LG

21

HG

15

HG

24

HG

33

LG

06

LG

14

LG

22

LG

30

HG

70

HG

16

HG

52

HG

43

LG

07

LG

15

LG

23

LG

31

HG

30

HG

40

HG

22

HG

12

HG

13

LG

13

HG

60

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 5 10 15 20 25 30 35 40

Optical modes

Ratio

AP1/Normal (L=1000)/Normal (L=10000)/Normal AP2/Normal

Figure 5.15: Ratio between the different coatings and a normal coating, assuming that

the normal coating has homogeneous losses of 50 ppm.

Figure 5.15 shows the ratio between the different coatings when compared to the

standard coating, which is assumed to have homogeneous losses of 50 ppm. It is inter-

esting to see that the biggest effect of the differential coatings is on the fundamental

mode, with an increase of losses of 27%, which is reduced for the higher order modes.

Most of the higher order modes have more or less similar diffraction losses except for

the more symmetric HG modes.

Figure 5.16 shows the comparison between the three different coatings, where

DiffLoss 50 ppm corresponds to the original calculations with a homogeneous coating

with dielectric losses of 50 ppm. DiffLoss AP1 corresponds to the first test with a

central area of 50 ppm and an external ring of ∼5.7 cm with a loss of 25000 ppm.

DiffLoss AP2 corresponds to a central area of 50 ppm as well, but with an external

ring of ∼4.1 cm with a loss of a 100000 ppm. As we can see from figure 5.16 there is


Diffraction Losses Comparison for Different Coatings

HG

01

HG

11

HG

20

LG

02

LG

10 H

G1

2

HG

30

LG

03 LG

11

HG

22

HG

13

HG

40

LG

04

LG

12

LG

20

HG

14

HG

50

HG

23

LG

05 L

G1

3

LG

21

HG

15

HG

24

HG

33

HG

60

LG

06

LG

14

LG

22

LG

30

HG

43

HG

52

HG

70

LG

07 LG

15

LG

23

LG

31

HG

00

LG

01

HG

16

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 5 10 15 20 25 30 35 40

Optical Modes

Diffr

ac

tio

n L

os

se

s (

pp

m)

DiffLoss 50ppm DiffLoss AP1 DiffLoss AP2

Figure 5.16: Diffraction losses comparison between the original homogeneous coating and

the two apodising coatings.

not much difference between the different coatings, however we can still see that the

biggest difference in terms of diffraction losses is for the lower modes, in particular

the fundamental mode HG00.

The main difference however is in the coating absorption for each mode, while

having a homogeneous coating shows homogeneous absorption it is not the case for

the apodising coatings. In the case of an apodising coating the absorption will depend

on the energy distribution of the mode, therefore it is also important the change in

shape of the mode since it will also affect its absorption. This calculation was done by

normalising the circulating power inside the cavity and integrating the field of each

mode over the absorption map over the test mass surface.

Figure 5.17 shows the results for the two different apodising coatings. Not sur-

prisingly the coating absorption goes up with the mode order. As in the diffraction

losses case this is cause by the mode shape changing inside the cavity. This is caused

by the circular symmetry, which favours the resonance of LG modes more than HG

modes. We have to remember that in this case no external means of exciting higher

order modes have been considered, no mirror tilt, no mechanical resonance and no

suspension residual noise for example.

We can also notice in figure 5.17 that for the lower modes there is not much


Comparison of coating absorption

HG

11

HG

20

LG

02

LG

10 H

G1

2

HG

30

LG

03 L

G1

1

HG

22

HG

13

HG

40

LG

04

LG

12

LG

20

HG

14

HG

50

HG

23

LG

05

LG

13

LG

21

HG

15

HG

24

HG

33

HG

60

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

HG

70

LG

07

LG

15

LG

23

LG

31

HG

00

HG

01

LG

01

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 5 10 15 20 25 30 35 40

Optical Modes

Co

atin

g A

bso

rptio

n (

pp

m)

Absorption AP1 Absorption AP2 50 ppm

Figure 5.17: Shows the different dielectric absorption for higher order modes. We notice

that there is not much difference between the two coatings when analysing the lower modes,

but there is a clear difference for the higher order ones.

difference between the two coatings absorptions. But for the higher order modes,

starting from order 3, the two curves start to show some difference. This is due to

the energy distribution of the higher order modes and the fact that the outer ring of

the second coating has higher losses even though it is slightly narrower.

5.7.4 Conclusions

Based on these results the use of an apodising coating will increase the diffraction

losses for several modes. Somewhat unanticipated was to see that the effect is bigger

in the lower order modes. The diffraction loss change comes with the corresponding

frequency shift and a Q-factor change for the optical mode. Consequently there is

also a change in the mode shape due to the different coating losses. In general these

changes are too small and any effect on the overlapping parameter will be negligible.

As a result the effect of an apodising coating on the parametric instabilities gain R0

for an advanced gravitational wave interferometer it is also expected to be negligible.


Acknowledgements

The authors would like to thank Phil Wilems and Erika D’Ambrosio for useful discus-

sions. Pablo Barriga would like to thank the LIGO Laboratory for their hospitality.

This work was supported by the Australian Research Council, and is part of the

research program of the Australian Consortium for Interferometric Gravitational As-

tronomy.

5.8. MIRROR TILT 179

5.8 Mirror tilt

Effect of mirror tilt in higher order optical modes

5.8.1 Introduction

The physical parameters of the cavity like distance between mirrors, their radius of

curvature, size and losses (substrate and coating) define the characteristics of the

cavity. These include the beam waist and spot size at the input and end test mass

(ITM and ETM), the final circulating power, the free spectral range, diffraction losses

and the higher optical modes frequency separation. All these parameters are necessary

to determine the possibility of parametric instability in the main arms of advanced

gravitational wave detectors by the calculation of the parametric gain R0 (shown in

equation 5.1).

As seen in section 5.2 in order to estimate the parametric gain R0 accurately it

is very important to correctly estimate the frequency, the diffraction losses and the

Q-factor of the higher optical modes. First we review the effects of the geometry of

the mirror in the higher order modes and the frequency separation. This defines the

reference for the analysis on the effect that the tilt of the mirror has on the frequency

of the higher order modes for an Advanced LIGO type of cavity. In general a mirror

tilt will increase the cavity diffraction losses, depending on the ratio between the size

of the mirror and the laser spot size. The simulations show that the tilt of the mirrors

increases the diffraction losses changing the frequency separation of the higher order

modes. As a consequence the Q-factor and the gain of the optical modes is also

affected, which ultimately affects the power inside the cavity.

5.8.2 Mirror geometry

An important parameter that needs to be taken in to account is the geometry of the

test mass. The frequency of the transverse modes depends on the radius of curvature

of the mirror and the cavity length. Normally infinite size mirrors and therefore

symmetric are considered for the calculations of the frequency mode separation. It

has been shown that the size of the mirror introduces a frequency shift from the


theoretical value expected for the frequency of the transverse modes, and this shift

increases with the order of the modes [24]. In laser interferometric gravitational wave

detectors it is necessary to suspended the optics in order to isolate the test mass from

seismic noise. The proposed solution is then to cut two flat sides in opposite sides of

the substrate as shown in figure 5.1 which allows for the bonding of the fused silica

suspension elements.

Assuming that the coating covers all of surface 1 we calculate the eigenvalues

for an Advanced LIGO type cavity using an eigenvector method proposed by C.

Yuanying et al [22]. The results obtained through this method showed that in a

perfectly aligned cavity with symmetric cylindrical test masses (circular mirrors) only

Laguerre-Gaussian (LG) modes and their rotated orthogonal modes will form the

eigen-solution. Due to the cylindrical symmetry of this system they are able to freely

rotate and diffraction losses will be independent of the mode orientation, hence no

preferred orientation for the optical modes on the test mass. In reality, we know that

due to misalignments of the optics Hermite-Gaussian (HG) modes will also appear,

but these can still be represented as a linear combination of LG modes. The current

design for the suspension of the test masses requires them to have two flat sides for

suspension attachment as shown in figure 5.1 (LIGO technical document D040431-B).

This breaks the cylindrical symmetry of the system and therefore of the mirror under

the assumption that the coating covers the whole front surface of the substrate. This

break of symmetry affects the eigenvectors of this cavity. There is a small increase in

diffraction losses due to the smaller area of the high reflective coating, but there is also

a preferred orientation for the eigenvectors, which minimise the loss for that particular

mode, but in turn affects the orthogonal optical mode as shown in figure 5.19 using

mode LG21 as an example. A similar case with mode LG03 can be seen in figure 5.20.

Figures 5.18, 5.19 and 5.20 show the effect of the flat sides of the test masses on

the resonant optical modes inside the cavity. Using mode LG21 as an example we

notice that in circular mirrors the mode can freely rotate on the mirror surface with

no effect in diffraction losses. With the flat sides needed for the attachment of the

suspension the symmetry is broken introducing an orientation of the optical mode.

This orientation corresponds to the one that minimises the diffraction losses for that


LG21LG21

Figure 5.18: Optical mode LG 21 in a perfectly aligned cavity with circular mirrors.

LG21 HG05?

Figure 5.19: Optical mode LG21 in a perfectly aligned cavity with flat sides. The shape

of the mirrors induce an orientation on the mode which minimises the diffraction losses, but

worsens the orthogonal mode.

particular mode. However this strongly affects the corresponding orthogonal mode

now resembling a mode HG05, which is the result of having a square area within the

surface of the mirrors. This can only occur if the spot size is big enough so the higher

order modes intensity pattern is constrained by the edge of the mirror. In advanced

interferometric gravitational wave detectors large spot size are required in order to

minimise the test mass thermal noise [25].

Table 5.4 shows the diffraction losses and the shift in frequency from the theoretical

value of 22.964 kHz (or infinite sized mirror case) for an Advanced LIGO type cavity.

The same effect can be seen in all higher order modes with the exception of the modes


(a)

LG03 LG03

(b)

LG03 HG21?

Figure 5.20: The figure shows another example of the effects of the flat sides for suspension

attachment in the cavity eigenmodes and their orientation. (a) shows the case of a perfectly

aligned cavity with circular mirrors where mode LG03 and its orthogonal mode (also LG03)

are part of the eigen-solutions for this cavity, (b) shows the case for the mirror with flat sides

for mirror suspension where mode LG03 also resonates, but with a different orientation and

the orthogonal mode resembles an HG21 optical mode.

with circular symmetry like LG10, LG20, LG30,...etc. With the break of symmetry

these modes do not form part of the eigen-solutions for this cavity, but they could

still resonate in the cavity if the conditions are right.

As seen in figures 5.18 and 5.20 (a) in a cavity with circular mirrors an optical

mode and its orthogonal mode are coupled. As a consequence they will have the

same diffraction losses, mode frequency separation, optical gain and Q-factor. The


Diffraction losses Frequency shift

(ppm) (Hz)

LG21 circular 31377 43.581

LG21 flat sides 40621 57.308

HG05? flat sides 43286 65.894

Table 5.4: Comparison for LG21 mode between circular mirrors and mirrors with flat

sides for suspension attachment. Only one of the circular mirror cases is presented since

the orthogonal mode has similar values.

flat sides in the mirrors break this symmetry as seen in figures 5.19 and 5.20 (b).

With the break in symmetry an optical mode will not have the same diffraction losses

as its orthogonal mode. Figure 5.21 shows that the higher the order of the mode

the larger the diffraction losses difference between orthogonal modes. This difference

in diffraction losses also implies different frequency separation, different optical gain

and optical Q-factor. Therefore the higher the order of the optical modes the larger

these differences will be, and as a consequence a decoupling of the higher order optical

modes doublets.

The mirror size also plays an important role in the diffraction losses and the fre-

quency shift of the higher order modes as seen in sections 5.5.1 and 5.5.3. Figures 5.22

and 5.23 show the influence of the mirror size on the frequency separation between

the fundamental mode and the higher order modes. We notice that the higher the

order of the mode the stronger the effect of the mirror size. This occurs due to the

more spread pattern of the higher order modes and as a consequence the larger area

of the energy distribution which increases the diffraction losses for these modes. The

figures show the results for simulations using the Advanced LIGO design. However

the secondary horizontal axis shows the ratio between the spot size and the mirror

size and therefore can be applied to any advanced interferometer design.


Mirror geometry and diffraction losses

LG

01

LG

02

LG

03

LG

11

LG

04

LG

12

LG

05

LG

13

LG

06

LG

21

LG

07

LG

14

LG

22

LG

15

LG

23 L

G1

6

1

10

100

1000

10000

100000

0 2 4 6 8 10 12 14 16 18

Optical modes

Diffr

action

losses (p

pm

)

Figure 5.21: The graph shows the difference between orthogonal modes in diffraction

losses introduced by the mirror geometry. The difference appears due to the break in

symmetry of the mirrors.

5.8.3 Mirror tilt

We have defined the conditions of a perfectly aligned cavity with finite size mirrors.

Therefore we can proceed to misalign the cavity by tilting one the mirrors. This will

allow us to study the effect of the mirror tilt in higher order modes diffraction losses

and frequency shift.

For our simulation we use the FFT based code previously presented in section 5.4.

This time it also includes a mirror tilt angle for the ETM as shown in figure 5.24.

It has been shown that the tilt perturbation shifts the mode pattern to one side of

the cavity which results in additional coupling loss [26]. The increase in diffraction

power loss due to misalignment is proportional to the square of the mirror tilt angle

and therefore a quadratic increase of power loss with the tilt angle will occur.

Figure 5.25 shows a simplification of the problem, where a small tilt in one test

mass changes the optical path of the laser beam. Now the tilted optical path crosses

the horizontal (or perfectly aligned) optical path at an angle θ. This path length

difference (∆L) can be calculated with some simple trigonometry.

The optical path difference is define by ∆L = L∗ − L, where L∗ = La + Lb.


Frequency Change with Mirror Size

15000

20000

25000

30000

35000

40000

0.14 0.15 0.16 0.17 0.18 0.19 0.2

Mirror size radius (m)

Fre

qu

en

cy

[H

z]

HG16

HG43

HG52

HG70

LG07

LG15

LG23

LG31

HG15

HG33

HG24

HG60

LG06

LG14

LG22

LG30

HG14

HG23

HG50

LG05

LG13

LG21

HG13

HG22

HG40

LG04

Order 7

Order 6

Order 5

Order 4

0.429 0.3000.3160.3340.3530.3750.400

(Spot size/Mirror size) ratio

Frequency change with mirror size

Fre

qu

en

cy [

Hz]

Figure 5.22: The graph shows for higher order modes the variation in frequency from the

fundamental mode TEM00. For modes of order lower than 4 the variation is too small for

the scale of this graph and are not shown. We notice that for a given spot size the smaller

the mirror the higher the frequency shift and therefore the higher the divergence from the

theoretical value. This can also be seen on the top secondary x axis which shows the ratio

between the spot size and the mirror size.

tan(θ) =∆x1

L1

=∆x2

L2

. (5.21)

The tangent of the angle θ can be determined by the displacement of the spot size

(∆x1,2) and the distance between the mirror and the centre of rotation of the beam

path (L1,2) as shown in equation (5.21).

L1 =L(

1 +∣∣∣∆x2

∆x1

∣∣∣) , (5.22)

La =√L2

1 + ∆x22, (5.23)

Lb =√L2

2 + ∆x21. (5.24)

Equations (5.22), (5.23) and (5.24) show the relations between the beam path and

the displacement of the spot on the mirrors in order to calculate the length difference


Frequency Change with Mirror Size for modes of order 7

32000

32500

33000

33500

34000

34500

35000

35500

36000

36500

37000

0.14 0.15 0.16 0.17 0.18 0.19 0.2

Mirror size radius (m)

Fre

qu

en

cy [H

z]

HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31

0.3750.400

Advanced LIGOSpot size: 60 mmTest Mass radius: 170 mm

0.429 0.3000.3160.3340.353

(Spot size/Mirror size) ratio

Frequency change with mirror size for modes order 7

Figure 5.23: The graph shows for modes of order 7 the effect of the mirror size in the

frequency gap with the fundamental mode. We use the Advanced LIGO design for our

study, where the spot size has a diameter of 0.6 cm and the mirror a diameter of 17 cm. The

simulation does include the flat sides for suspension attachment.

α

L = 4000 m

R1 R2

Figure 5.24: Considering only one of the mirrors to be tilted there is a change in the

optical path for the laser beam. As a consequence there is a change in the resonance

conditions of the cavity.

between the perfectly align situation and the tilt one. For these calculations also the

Sagitta S has been considered, and is defined by:


∆x1

∆x2

θ

La

Lb

θL

2

L1

baLLL

LLL

+=

+=*

21

Figure 5.25: With some simple geometry it is possible to calculate the change in the path

length of the laser beam and also the displacement of the spot from the centre of both test

masses.

S1 = R−√R2 −∆x2

1 (5.25)

tan(θ) =∆x1√

R2 + ∆x21

tan(θ) =Rα

2R− L

∆x1 =Rθ√1 + θ2

. (5.26)

Analogous calculations for the end test mass show that the spot displacement is

given by:

∆x2 = R (θ − α) . (5.27)

It is well known that the frequency variation (∆f) of the fundamental modes for

variations of the cavity length is given by:

∆f =∆L

LfY AG. (5.28)

Where ∆L corresponds to the optical path length variation, L is the total optical

path length and fY AG is the laser frequency. In our case we are using an Nd:YAG laser

with a wavelength of 1.064µm, which corresponds to a frequency of 2.82 × 1014 Hz.

However for higher order modes this frequency variation has an extra component. For

higher order modes the frequency is then given by:

f =c

2L

(q +

N

πarccos(

√g1g2)

). (5.29)


Where c corresponds to the speed of light in vacuum, q to the axial mode, g1 and

g2 corresponds to the stability factor of each mirror. N corresponds to the mode

number given by (m + n + 1) for HG modes or (2m + n + 1) for LG modes. In this

case the radius of curvature of both mirrors is the same with a value of R, therefore

g = g1 = g2. This means that the frequency variation for higher order modes due to

a small change in the optical path length is given by:

∆f =∆L

LfY AG + ∆L

(N

π

c

2L2

1√2LR− L2

). (5.30)

The extra term in this equation is comparatively small, only 11.4 mHz for high

order optical modes of order 2.

It has also been shown that cavities with stability g-factor close to 1 are more

efficient [27]. However the resonator becomes very sensitive to misalignment of the

mirrors and diffraction losses will increase rapidly with mirror tilt. The losses in-

crease with the square of the tilt angle and can be characterised by the misalignment

sensitivity Di as proposed by Hauck et al [28]:

D2i =

πL

λ

(gjgi

)(1/2)1 + g1g2

(1− g1g2)(3/2). (5.31)

Where Di corresponds to the misalignment sensitivity of mirror i referred to mirror

j, with L the cavity length, λ the laser wavelength and g1 and g2 the stability factors

for mirrors 1 and 2. The reciprocal value of Di is the tilt angle for that mirror which

increases the losses by ∼ 10%. If both mirrors are misaligned, the losses proportional

to D2i are summed up. Therefore the misalignment of the cavity is defined as:

D =√D2

1 +D22. (5.32)

We are analysing a case with a cavity length of 4 km and mirrors with 2076 m

radius of curvature the stability factor is given by g = g1 = g2 = −0.9268, and with

λ = 1.064µm we then obtained a Di value of 6.44 × 105 which in turn means that

diffraction losses will increase by ∼ 10% for a tilt angle of 1.6× 10−6 radians.

The use of Fast Fourier Transform (FFT) to simulate the propagation of the beam

inside the cavity allows the mode shape to change due to the finite mirror size and

hence enables a much better approximation of the diffraction losses, and therefore


of the other parameters of interest [14]. A simulation program based on the beam

propagation using FFT was developed at UWA which was previously used for the

calculation of diffraction losses in advanced interferometers [24]. The same program

allows us to introduce mirror tilts and estimate their effect in the diffraction losses

and frequency of the higher order optical modes.

Due to the tilt of the mirror there will be a displacement of the spot proportional

to the mirror tilt angle. In a cavity with finite mirrors this mode pattern shift closer

to the edge of the mirror results in an additional diffraction loss. The FFT code

not only allowed us to estimate the diffraction losses and frequency changes for each

mode, but also to check the spot displacement and therefore the change in geometry

of the cavity.

The increase in diffraction losses is proportional to the square of the tilt angle,

but the constant of proportionality decrease with increasing mirror size (or Fresnel

number). In the case of an infinite concave mirror the diffraction losses will always

be zero for any small tilt angle. However even in the case of an infinite size mirror

there will be an increase in power losses due to mismatch between the input beam

and the circulating beam. The loss of circulating power is proportional to the square

of the tilt angle and its constant decrease with mirror size.

Figure 5.26 shows the shift in frequency when the ratio between the spot size and

mirror size varies. The figure shows where the Advanced LIGO design is in terms of

this ratio and shows that for higher order modes (which are more affected) a change

of 1 % in the spot size/mirror size ratio produces a change between 7 Hz and 50 Hz

in frequency for modes of order 6, depending on the mode energy distribution. For

example, LG14 will have a frequency shift of 7 Hz for a 1 % ratio change, while LG30

will have a frequency change of 47 Hz. This variation is larger for higher high order

modes. This however it also implies that thermal tuning can be use to advantage for

frequency tuning of the higher order modes.

With the mirror tilted, i.e. the cavity is misaligned, the mode is no longer concen-

tric on the test mass and there is an asymmetric edge effect. This is due to the fact

that the mode moves closer to one edge, and because we are on the exponential tail of

the mode the differential effect is different on the near side leading to a quadratic tilt


Frequency change with spot/mirror size ratio for modes of order 6

0

500

1000

1500

2000

2500

3000

0.32 0.34 0.36 0.38 0.4 0.42 0.44

Spot size/Mirror size ratio

∆ F

req

ue

ncy (

Hz)

LG06 LG14 LG22 LG30

Advanced LIGO

Figure 5.26: The graph shows the variation in frequency of modes of order 6. Effects

like thermal lensing will change the (spot size/mirror size) ratio changing the frequency gap

between the fundamental mode and higher order modes. This frequency shift will change

the resonant conditions for parametric instabilities.

sensitivity. This explains why higher order modes are more affected. It also shows

that the effects will depend on the orientation of the mode profile in relation to the

tilt. For example, for a certain tilt direction, mode HG05 will be more affected than

HG50 which comes as a consequence of the flat sides on the test mass substrate as

shown in figure 5.28.

The frequency change will depend on the orientation of the mode profile in relation

to the tilt. This can also be interpreted as the frequency change for a given higher

order mode will depend on the tilt direction in relation with the energy distribution

of a particular mode. Therefore the frequency gap between the fundamental mode

and the higher order modes will also depend on this relative orientation.

5.8.4 Conclusions

We have shown the effect of the mirror geometry and the mirror tilt in an align cavity

in terms of diffraction losses and the frequency separation of the higher order modes.


Frequency change with ETM mirror tilt for modes of order 4

18370

18375

18380

18385

18390

18395

18400

0 200 400 600 800 1000 1200

Mirror tilt (nrad)

Fre

quency (

Hz)

LG04 LG12 LG20 HG40 HG31 HG22

LG04

LG12

LG20

HG40

HG31

HG22

Figure 5.27: The graph shows the variation of the frequency gap between the fundamental

mode and the modes of order 4. We notice that the effects of the mirror tilt in the frequency

gap depends on the energy distribution of the mode. As a consequence the frequency

variation for a given tilt of the mirror not only depends on the mode order but also on the

energy distribution of the optical mode.

This study forms part of a broader understanding of the possibility of parametric

instabilities in advanced interferometric gravitational wave detectors.

Using an FFT-based code developed at UWA we have quantified the effects of

mirror tilt in terms of the frequency shift. In reality the cavity length effect due to

displaced spots is generally a common effect on the TEM00 and higher order modes

which should be removed by the control system.

We have also shown that changes in the spot size/mirror size ratio can affect the

frequency separation between higher order modes. This can be use to advantage by

thermally tuning the spot size on the test masses as a way to control the diffraction

losses. In turn it is possible to control the mode gain, optical Q-factor and frequency

separation, ultimately controlling the parametric gain R0.


Frequency change with mirror tilt

0

10

20

30

40

50

60

70

80

0 200 400 600 800 1000 1200

Mirror tilt (nrad)

∆ F

req

ue

nc

y (

Hz

)

HG04 HG40 HG05 HG50

HG04

HG50

HG40

HG05

Figure 5.28: The graph shows the frequency variation for a few modes of order 4 and 5.

It shows that the frequency change will depend on the orientation of the mode profile.

5.9 Postscript

The calculations made in the publication of “Numerical calculations of diffraction

losses in advanced interferometric gravitational wave detectors” were based on the

advanced LIGO design at that time. Since publication in June 2007, the design of

Advanced LIGO has changed. In this particular case our main concern is the effect

that the new arms configuration will have on the diffraction losses and therefore on

the parametric gain. The changes to the radius of curvature of the test masses seek

to reduce the spot size at the beam-splitter, effectively reducing the diffraction losses

in the recycling cavities [29]. There is also a reduction of the reflectivities of the

mirrors, reducing the finesse of the arm cavities. This reduction does not affect the

quantum noise in an advanced dual recycling interferometer such as Advanced LIGO.

The main reason for the reduction of finesse for the arm cavities is the use of fused

silica as the substrate of choice for Advanced LIGO. By selecting fused silica as the

substrate for the test masses, absorption is less of a problem when compared to the

original sapphire option [30]. Table 5.5 shows the parameters of the arm cavities

5.9. POSTSCRIPT 193

comparing the previous design with the current design of Advanced LIGO.

Parameter Previous AdvLIGO Current AdvLIGO

ITM radius of curvature 2076 m 1971 m

ETM radius of curvature 2076 m 2191 m

ITM spot size radius 60 mm 55 mm

ETM spot size radius 60 mm 62 mm

Waist size radius 11.6 mm 11.8 mm

Waist position from ITM 2000 m 1885 m

Cavity g-factor 0.8589 0.8499

Mode spacing 32.846 kHz 32.696 kHz

Table 5.5: Comparison of design parameters between the previous Advanced LIGO design

and the current one.

Figure 5.29 shows the diffraction losses for both configurations. It shows that some

modes are more affected by the change in radius of curvature of the test masses, but

in general the difference in the round trip diffraction losses is very small. This was

expected since the increase in spot size at the ETM is compensated by the reduction

at the ITM, leaving the round trip total roughly the same. The most affected modes

are those with more energy spread over the mirror coating. Leading to bigger spot

size on the ETM which has a greater impact on round-trip losses.

Conversely the optical gain and Q-factor of the higher order modes depend on

the total losses, including diffraction losses and coupling losses. As a consequence we

can see a reduction of the gain in figure 5.30, in particular for the fundamental mode

and the lower of the higher order modes. This was expected, since the reduction

of cavity finesse implies less circulating power in the arm cavities. Interestingly the

higher order modes have somewhat higher gain in the current design, even though it

is only a small fraction. The gain for the fundamental mode was reduced from 790

to 285, which implies a considerable reduction in circulating power. The case of the

Q-factor shown in figure 5.31 is slightly different. Here the higher order modes are

always higher in the previous design, with a Q-factor of the fundamental mode of


Comparison of diffraction losses between original and current Advanced LIGO design

HG

00

HG

01

LG

01

HG

11

HG

02

LG

02

LG

10

HG

12

HG

03

LG

03 LG

11 HG

22

HG

13

HG

04

LG

04 LG

12

LG

20

HG

14

HG

05

HG

23

LG

05 LG

13

LG

21

HG

15

HG

24

HG

33

HG

06

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

HG

07

LG

07 LG

15

LG

23

LG

31

HG

08

HG

17

HG

26

HG

35

HG

44

LG

08 LG

16

LG

24

LG

32

LG

40

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0 5 10 15 20 25 30 35 40 45 50

Optical modes

Diffr

actio

n lo

sse

s (

pp

m)

Original AdvLIGO Current AdvLIGO

Figure 5.29: Comparison of diffraction losses between the previous design for Advanced

LIGO and the current design.

4.7× 1012, which is reduced to 1.7× 1012 in the current configuration.

The reduction of the optical gain of the higher order modes and of Q-factor in

particular for the lower of the higher order modes, points to a lower parametric gain

and therefore a lower chance of parametric instabilities in the current Advanced LIGO

design. Since this is only in comparison with the previous design it does not guarantee

that parametric instabilities will not occur.

High Order Modes Optical Gain

HG

00

HG

01

LG

01

HG

11

HG

02

LG

02

LG

10

HG

12

HG

03

LG

03

LG

11

HG

22

HG

13

HG

04

LG

04

LG

12

LG

20

HG

14

HG

05

HG

23

LG

05

HG

07

LG

13

LG

21

HG

15

HG

24

HG

33

HG

06

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

LG

07

LG

15

LG

23

LG

31

LG

16

HG

17

HG

26

HG

35

HG

44

LG

08

LG

32

LG

40

LG

24

HG

08

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0 5 10 15 20 25 30 35 40 45 50

Optical modes

Ga

in


Figure 5.30: Comparison between the gain of the higher order modes in the previous

Advanced LIGO design and the current design.


Optical Q factor for higher order modes

HG

00

HG

01

LG

01

HG

11

HG

02

LG

02

LG

10

HG

12

HG

03

LG

03

LG

11

HG

22

HG

13

HG

04

LG

04

LG

12

LG

20

HG

14

HG

05

HG

23

LG

05

LG

13

LG

21

HG

15

HG

24

HG

33

HG

06

LG

06

LG

14

LG

22

LG

30

HG

16

HG

43

HG

52

HG

07

LG

07

LG

15

LG

23

LG

31

HG

17

HG

26

HG

35

HG

44

LG

08

LG

16

LG

24

LG

32

LG

40

HG

08

1.E+10

1.E+11

1.E+12

1.E+13

0 5 10 15 20 25 30 35 40 45 50

Optical modes

Q fa

cto

r


Figure 5.31: Comparison between the Q-factor of the higher order modes in the previous

Advanced LIGO design and the current design.

Frequency Comparison

HG

00

HG

01

LG

01 H

G11

HG

02

LG

02

LG

10

HG

12

HG

03

LG

03

LG

11

HG

04

LG

04 LG

12

LG

20

HG

16

HG

07

LG

15

LG

23

LG

31 H

G08

HG

17

HG

26

HG

35

HG

44

LG

08

LG

16

LG

24

LG

32

LG

40

HG

06

HG

33

HG

24

HG

15

LG

21

LG

13

LG

05

HG

23

HG

14

HG

05

LG

07

HG

52

HG

43LG

30

LG

22

LG

14

LG

06

HG

22

HG

13

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0 5 10 15 20 25 30 35 40 45 50

Optical modes

De

lta

Fre

qu

en

cy (

Hz)

Original AdvLIGO New AdvLIGO

Figure 5.32: Comparison between the frequency shift of higher order modes in the

previous Advanced LIGO design and the current design.

5.10 References

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instability in FabryPerot interferometer,” Phys. Lett. A 287 (2001) 331–338.

[2] W. Kells and E. D’Ambrosio, “Considerations on parametric instabilities in

Fabry–Perot interferometer,” Phys. Lett. A 299 (2002) 326–330.

[3] C. Zhao, L. Ju, J. Degallaix, et al, “Parametric instabilities and their control


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[4] B. D. Cuthbertson, M. E. Tobar, E. N. Ivanov and D. G. Blair, “Paramet-

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[5] H. Rokhsari, T. Kippenberg, T. Carmon, K. J. Vahala, “Radiation-pressure-

driven micro-mechanical oscillator,” Opt. Express 13 (2005) 5293–5301.

[6] T. Corbitt, D. Ottaway, E. Innerhofer, et al, “Measurement of radiation-

pressure-induced optomechanical dynamics in a suspended Fabry-Perot cavity,”

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[7] L. Ju, S. Gras, C. Zhao, et al, “Multiple modes contributions to parametric in-

stabilities in advanced laser interferometer gravitational wave detectors,” Phys.

Lett. A 354 (2006) 360–365.

[8] R. E. Spero, “Diffraction loss and minimum mirror size,” Technical Report,

T920002-00-D, LIGO, (1992).

[9] R. E. Spero, “Update on diffraction loss and minimum mirror size,” Technical

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[10] E. D’Ambrosio, “Nonspherical mirrors to reduce thermoelastic noise in ad-

vanced gravitational wave interferometers,” Phys. Rev. D 67 (2003) 102004

(16pp).

[11] A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys.

Tech. J. 40 (1961) 453–488.

[12] G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter

through optical wavelength masers,” Bell Syst. Tech. J. 40 (1961) 489–508.

[13] M. Born and E. Wolf, “Principles of optics”, 7th edition, (Cambridge University

Press, 2003), 412–516.


[14] J. Y. Vinet, P. Hello, C. N. Man, and A. Brillet, “A high accuracy method

for the simulation of non-ideal optical cavities,” J. Phys. I France 2 (1992)

1287–1303.

[15] E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier

transform methods,” Proc IEEE March (1974) 410–412.

[16] B. Bhawal, “Diffraction losses of various modes in advanced LIGO arm cavity,”

Technical Document, T050234-00-E, LIGO, (2005).

[17] A. E. Siegman, “Lasers”, University Science Books, Sausalito, California,

(1986) 398–456.

[18] R. Lawrence, “Active wavefront correction in laser interferometric gravitational

wave detectors,” PhD Thesis, Department of Physics, Massachusetts Institute

of Technology, 2003.

[19] B. E. A. Saleh, M. C. Teich, “Fundamentals of photonics”, John Wiley & Sons,

Inc., (1991), 310–341.

[20] A. E. Siegman, “Lasers”, University Science Books, Sausalito, California,

(1986), 626–662.

[21] J. Agresti, LIGO Laboratory, California Institute of Technology, Pasadena, CA

91125, USA, (private communication, 2005).

[22] C. Yuanying, W. Youqing, H. Jin, and L. Jiarong, “An eigenvector method for

optical field simulation,” Opt. Commun. 234 (2004) 1–6.

[23] J. P. Goldsborough, “Beat frequencies between modes of a concave-mirror op-

tical resonator,” Appl. Optics 3 (1964) 267–275.

[24] P. Barriga, B. Bhawal, L. Ju, and D. G. Blair, “Numerical calculations of

diffraction losses in advanced interferometric gravitational wave detectors,” J.

Opt. Soc. Am. A 24 (2007) 1731–1741.

[25] Y. Levin, “Internal thermal noise in the LIGO test masses: A direct approach,”

Phys. Rev. D 57 (1998) 659–663.


[26] J. L. Remo, “Diffraction losses for symmetrically perturbed curved reflectors

in open resonators,” Appl. Opt. 20 (1981) 2997–3002.

[27] R. B. Chesler and D. Maydam, “Convex-Concave Resonators for TEM00 Op-

eration of Solid-State Ion Lasers”, J. Appl. Phys. 43 (1972) 2254–2257.

[28] R. Hauck, H. P. Kortz, and H. Weber, “Misalignment sensitivity of optical

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Chapter 6

Stable Recycling Cavities

6.1 Preface

The author presented a preliminary design for the Australian International Gravita-

tional wave Observatory (AIGO) during an international workshop in Gingin, West-

ern Australia. At this workshop Volker Quetschke from the University of Florida

presented the concept study for a stable power recycling cavity proposed for the Ad-

vanced LIGO design. Following several discussions on the possible benefits of stable

recycling cavities compared with marginally stable recycling cavities used by the op-

erating interferometric GW detectors, the author started to research the possibility

of incorporating a stable power recycling cavity in the AIGO design. The main de-

sign constraint was the requirement to use the available infrastructure to fit the extra

mirrors and vibration isolation systems necessary for the stable recycling cavity. A

stable signal recycling cavity was successfully added to the design and discussed with

Guido Mueller (also from the University of Florida) during the Amaldi 7 conference on

gravitational waves in Sydney. A more complete presentation by G. Mueller followed

at a Parametric Instabilities workshop in Gingin. The original concept of incorporat-

ing a stable power recycling cavity was extended to include a stable signal recycling

cavity. Following the selection of the RF sidebands for the length sensing control

system the design of AIGO was then extended to form a complete optical design for

an advanced dual recycled interferometric gravitational wave detector. This extended

study was completed in collaboration with the University of Florida, in particular with

the co-authors of the paper “Optical design of the proposed Australian International

Gravitational Observatory”. The calculations were done by the author in consulta-

199

200 CHAPTER 6. STABLE RECYCLING CAVITIES

tion with the group at the University of Florida. Muzammil Arain did the mode

matching calculations and the tolerances of the mirrors. Additional simulations were

completed during the evolution of this design, some results of which are presented in

the paper at the core of this chapter. A later simulation of the beam-splitter thermal

effects was recorded as an internal technical report and added as a postscript to this

chapter.


Optical design of the proposed Australian

International Gravitational Observatory

Pablo Barriga1, Muzammil A. Arain2, Guido Mueller2, Chunnong Zhao1

and David G. Blair1

1 School of Physics, The University of Western Australia, Crawley, WA 60092 Department of Physics, University of Florida, Gainesville, FL32611, USA

Marginally stable power recycling cavities are being used by nearly all

interferometric gravitational wave detectors. With stability factors very

close to unity the frequency separation of the higher order optical modes

is smaller than the cavity bandwidth. As a consequence these higher or-

der modes will resonate inside the cavity distorting the spatial mode of

the interferometer control sidebands. Without losing generality we study

and compare two designs of stable power recycling cavities for the pro-

posed 5 kilometre long Australian International Gravitational Observatory

(AIGO), a high power advanced interferometric gravitational wave detec-

tor. The length of various optical cavities that form the interferometer

and the modulation frequencies that generate the control sidebands are

also selected.

6.2 Introduction

The addition of a southern hemisphere interferometric gravitational wave (GW) de-

tector in Gingin (∼80 km north of Perth in Western Australia) will greatly improve the

angular resolution of the existing network of GW detectors [1, 2], while also provid-

ing resolution for both polarisations, thereby allowing measurement of the luminosity

distance of sources. Thus the addition of a single southern hemisphere GW detector

of sensitivity comparable to the proposed northern hemisphere detectors (Advanced

Laser Interferometric Gravitational-wave Observatory (LIGO) [3], Large-scale Cryo-

genic Gravitational-wave Telescope (LCGT) [4], and Advanced VIRGO [5]) allows

the number of detectable sources to be doubled [6].


Similar to the 4 km long LIGO detector or the 3 km VIRGO detector, the proposed

Australian detector (AIGO) will be a Michelson interferometer with 5 km long optical

cavities in each arm. The interferometer will be kept at a dark fringe such that all

the light reflects back towards the laser. A mirror placed between the laser and

the Michelson Interferometer (MI) is then used to form the power recycling cavity

(PRC) that further increases the circulating power in the arm cavities [7]. Future

detectors such as Advanced LIGO and AIGO will also employ signal recycling where

an additional mirror at the output port of the MI is used to build-up the signal in a

tuneable frequency band.

The spatial eigenmodes of both recycling cavities have to match the spatial eigen-

mode of the arm cavities. This ensures an efficient extraction of the signal or GW

induced sidebands and a good mode matching between the spatial modes of the carrier

and the spatial modes of phase modulation radio frequency (RF) sidebands used to

control the longitudinal and angular degrees of freedom. It also improves the coupling

of the carrier field into the arm cavities. The first generation of large scale interfer-

ometric GW detectors, LIGO and VIRGO, used marginally stable power recycling

cavities in an essentially flat/flat configuration. These cavities did not confine the

spatial modes of the RF sidebands which led to significant spatial mode-mismatch

between them and the carrier [8]. A sophisticated thermal correction system was

necessary in order to overcome these problems during the commissioning phase of the

interferometer [9].

Stable PRC for Advanced LIGO to confine the spatial modes of the RF sidebands

have first been proposed by the University of Florida [10, 11] as part of their input

optics work [12]. A group at the California Institute of Technology discovered that

marginally stable signal recycling cavities will also reduce the amplitude of the GW-

sidebands by resonantly enhancing the scattering of light into higher order spatial

modes [13]. Since then the concept of stable recycling cavities has been intensively

discussed for all next generation interferometric gravitational wave detectors and they

are now part of the baseline design for Advanced LIGO [14, 15].

This paper analyses the concept of stable recycling cavities within the design

scenario of the proposed AIGO interferometer. In Section 6.3 we will review dual-


recycled Michelson interferometer and define key parameters of the proposed AIGO

interferometer. The suppression of higher order modes (HOM) as a function of the

Gouy phase will be reviewed in the Section 6.4. In Section 6.5 we will discuss and

compare possible designs of stable power recycling cavities given the current vacuum

envelope of the Gingin test facility. The current vacuum envelope and the specific

design of the PRC will then be used to select the RF used to generate the control

sidebands. In the remainder of the paper we will apply the same design principles

discussed in Section 6.5 to design the signal recycling cavity (SRC).

M1 M2

M3

To Detector Bench

South Fabry-Perot Cavity

East Fabry-Perot CavityInput Mode Cleaner

100W PSLNd:YAG laserλ = 1064nm

ETM1

ETM2

SRM

BSPRM ITM2

ITM1

L1

L2

l1

l2

lsr

lpr

LIMC

L+=(L

2+L

1)/2

L-=(L2-L

1)/2

l+=l

pr+(l

2+l

1)/2

l-=(l2-l

1)/2

lsrc

=lsr

+(l2+l

1)/2

Degrees of freedom

Figure 6.1: Configuration of the proposed AIGO interferometer as a dual recycling inter-

ferometer with marginally stable recycling cavities. The figure shows a pre-stabilized 100

W Nd:YAG laser and the input mode cleaner (IMC). The power recycling mirror (PRM),

beamsplitter (BS), both input test masses (ITM1 and ITM2), and both end test masses

(ETM1 and ETM2). It shows the power recycling cavity (PRC) formed by the power recy-

cling mirror (PRM) and both input test masses (ITM1 and ITM2) passing through the BS.

Also the signal recycling cavity (SRC) formed by the signal recycling mirror (SRM) and

both ITMs. It also shows the different degrees of freedom that need to be controlled.


6.3 Dual recycling interferometers

An interferometric GW detector consists of a set of coupled optical cavities in a MI

as shown in figure 6.1. Long km-scale cavities are used to enhance the displacement

sensitivity of the MI arms. The power recycling mirror (PRM) creates a composite

cavity (PRC) with the common mode of the arm cavities, while the signal recycling

mirror (SRM) creates another composite cavity (SRC) with the interferometer differ-

ential mode. This configuration operated in signal recycling mode was first proposed

by the Glasgow group [16] while the resonant sideband extraction (RSE) mode was

later described by Mizuno et al [17].

The lengths of the two long Fabry-Perot cavities that form the arms of the inter-

ferometric GW detector need to be controlled. In addition, the MI has to be kept dark

such that all the light is send back towards the laser. The position of the PRM has to

be controlled to keep the PRC on resonance for the laser field. The SRM is controlled

to resonantly enhance or extract the signal fields. Using the notation described in

figure 6.1, it involves controlling five length degrees of freedom (L+, L−, l+, l−, lsrc) of

the seven suspended optics PRM, BS, ITM1, ETM1, ITM2, ETM2 and SRM. Length

sensing and control schemes for this interferometer configuration were first developed

and tested by [18, 19, 20, 21]. These schemes are still evolving as the interferometer

designs evolve [22]. All utilise phase modulation sidebands and/or phase locked lasers

to generate the necessary control signals. However, it is currently assumed that two

pairs of phase modulation sidebands are sufficient to control all longitudinal degrees

of freedom. The lengths of both recycling cavities are directly linked to the RF used

to generate the sidebands. This will be discussed in more detail in Section 6.6.

In order to calculate the parameters for the stable PRC we need to define the

Fabry-Perot arms of the interferometer. A large spot size is needed in order to max-

imise the averaging over the thermal fluctuations in the test masses and thereby

reduce the test mass thermal noise [23]. At the same time it is necessary to keep

the fundamental mode diffraction losses sufficiently low (normally below 1 ppm per

round trip). AIGO current design uses sapphire as the substrate of choice for the test

masses. The parameters of the arm cavities for the proposed AIGO interferometer

are presented in table 6.1.

6.4. HIGHER ORDER MODES SUPPRESSION 205

Cavity Length 5000 m

ITM (ETM) radius of curvature 2734 m

ITM (ETM) diameter 32 cm

Cavity g-factor 0.687

Waist size radius 16.095 mm

Spot size radius 55.014 mm

Free spectral range 29.979 kHz

High order modes separation 24.312 kHz

Cavity finesse 1220

Cavity pole 12.29 Hz

Table 6.1: Parameters of the arm cavities for the AIGO interferometer. The proposed

design assumes sapphire test masses with a diameter of 32 cm and a spot size radius of

55 mm in order to keep diffraction losses of the fundamental mode below 1 ppm.

6.4 Higher order modes suppression

The transversal spatial eigenmodes of optical cavities formed between mirrors with

spherical radii of curvatures can be approximated by a set of Hermite-Gaussian eigen-

modes. For a given wavelength λ this set is defined by a waist size w0 and its location

z0 along the optical axis [24]. Each eigenmode is described by a pair of mode numbers

m, n. Each number is associated with one of the transversal directions defined by our

coordinate system and corresponds to the order of the Hermite mode which is used

to describe the field distribution along this coordinate axis. Current interferometric

GW detectors operate with the 00-mode being resonant in the arm cavities and the

PRC. Imperfections in the mirrors, such as radii of curvature mismatches, misalign-

ments, or other spatial variations of the mirror surfaces will cause scatter between the

eigenmodes. This will reduce the power in the 00-mode, increase the stray light, and

create spurious error signals. The scatter is resonantly enhanced into specific HOM if

they are close to resonance and encounter only very small diffraction losses inside the

interferometer. Modes with large mode numbers have larger effective cross sections

and finite apertures will increase their interferometer internal losses. In addition, the


scatter efficiency between the 00-mode and HOM with large mode numbers is usually

much lower than the scatter efficiency into low order HOM. Consequently, the res-

onant enhancement of scattered light into HOM is mainly a problem for the lowest

order HOM. The goal of the stable recycling cavities is to avoid incidental resonances

of any of the low order HOM.

Transversal eigenmodes are resonant inside a cavity if their roundtrip phase shift

is a multiple of 2π. The roundtrip phase shift of each Hermite-Gaussian mode has two

contributions. The main contribution is the phase shift associated with plane-wave

propagation:

φ = 2kL (6.1)

where L is the length of the cavity. This phase shift is common to all modes. The

second contribution is associated with the Gouy phase [25], which can be calculated

from the generalised cavity g-factor:

ΨG = arccos√±g (6.2)

with

g =A+D + 2

4(6.3)

where A and D are the diagonal matrix elements of the ABCD matrix which describes

the round-trip through the cavity.

Each Hermite-Gaussian mode is acquiring an additional phase which depends on

the mode number:

Ψmn = (m+ n+ 1)ΨG (6.4)

This additional phase breaks the degeneracy between the modes and causes the

HOM to have different resonance frequencies or length [24].

Figure 6.2 shows the normalised build-up of the first ten HOM (mode number =

m+ n) as a function of the Gouy phase ΨG of the cavity. Any significant build-up of

a HOM could significantly reduce the stability of the optical interferometer with the

6.4. HIGHER ORDER MODES SUPPRESSION 207

0 0.05π 0.1π 0.15π 0.2π 0.25π 0.3π 0.35π 0.4π 0.45π 0.5π10

-2

10-1

100

High Order Modes Transmission

Tra

nsm

issio

n [

a.u

.]

10 9 8 7 6 5 10 9 4 8 7 10 3 6 9 8 5 10 7 9

12

0

1

2

3

4

5

6

7

8

9

10

Ψ: Gouy phase shift (rad)

Figure 6.2: Transmission of HOM as function of the Gouy phase shift. All HOM resonate

at ΨG = 0, however only even modes resonate at ΨG = 0.5π. By selecting a Gouy phase

around 0.18π the highest transmission is for modes of order 5 and 6, while at ΨG close to

0.15π it will be orders 6 and 7, but with a higher transmission for order 1. From 0.5π to π

the transmission of HOM mirrors the transmission here presented.

lower modes being the most critical ones. The graph shows which of the lower order

modes are the least critical ones and can be used to optimise the Gouy phase inside

the cavity. For example a Gouy phase of ΨG ≈ 0.18πrad or a cavity g-factor of 0.7

for the recycling cavities would allow the modes with mode numbers 5 and 6 to have

some build-up while all other modes are well suppressed.

Figure 6.3 shows the suppression of HOM compared to the 00-mode for different

cavity designs with different stability g-factors. All shown designs have a finesse of

≈ 95. The first three designs do not use any focusing elements inside the recycling

cavities. The first design (L = 12 m) corresponds to a marginally stable recycling

cavity; this is about the maximum length for a non-folded recycling cavity which

would fit into the Gingin central building. The second (L = 500 m) and third designs

(L = 1 km) would require to fold the recycling cavity through one of the long vacuum


tubes which would also house the arm cavities. Although the suppression of HOM

increases for longer recycling cavities, it is still rather small compared to a recycling

cavity with a g ∼ 0.7. In the remainder of the paper we will use this g-factor for

the recycling cavities. This g-factor can be obtained with two different Gouy phases:

ΨG = 0.18π rad = 0.58 rad and ΨG = 0.82π rad = 2.57 rad. At these Gouy phase

values the first HOM (TEM01 and TEM10) have lower build-up when compared to a

marginally stable design. However the lower build-up of these modes in the recycling

cavity is somewhat compensated by the higher overlap between the carrier and the

RF sidebands in the stable PRC. This allows for error signals for alignment controls.

Moreover, the alignment sensing matrix for the stable cavity design allows for a better

decoupling of various alignment signals. This is a trade-off which will require further

studies.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

High order modes suppression for recycling cavities

Ma

gn

itu

de

(d

B)

Mode order

Marginally stable (g = 0.999998) 500 m stable (g = 0.998207) 1 Km stable (g = 0.995366) Short stable (g = 0.707643)

Figure 6.3: Comparison of the intensity suppression of HOM between a 12 m long

marginally stable recycling cavity, a 500 m long inline PRC, a 1 km long inline PRC, and

the proposed design for a stable PRC. The graph shows HOM up to order 10.

6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES 209

6.5 Possible solutions for stable

recycling cavities

In the previous section we showed that without any additional focusing elements very

long recycling cavities are needed to gain enough Gouy phase in order to achieve a

moderately good suppression of HOM. Since this option is impractical we evaluate

some alternative solutions that allow us to obtain the required Gouy phase and thus

the necessary HOM suppression without extending the recycling cavities by a large

amount. To be more specific we use the current vacuum envelope at the Gingin test

facility to constrain the parameter space. The vertex of the two arms restricts the

distance between the BS and the recycling mirrors to about 3 m while the dimensions

of the central building restricts the distance between the ITMs and the recycling

mirror to about 12 m. Currently the end stations are ∼80 m from the central tank.

For a future 5 km detector, the ITMs could be relocated into these ’end’ stations while

the BS, the recycling cavity mirrors, and the auxiliary optical components could all

be left in the central station.

The ABCD matrix method [26] is used to simulate the propagation of the laser

beam inside the interferometer. The spatial mode inside the recycling cavity has to

match the circulating beam of the main arm cavities. Therefore we start our analysis

from the waist of the arm cavities moving backwards towards the recycling cavity

input mirror. It is assumed that the test mass substrates are made of sapphire which

has an index of refraction of 1.75. Consequently, the ITM has an effective focal length

of −3645 m for both recycling cavities.

In the following two sections we will discuss two designs for stable recycling cavities

for AIGO. Both designs would fit into the current vacuum envelope at Gingin. They

will be compared in the last section of this chapter.

6.5.1 Straight stable recycling cavity

A first option for a stable recycling cavity will be the addition of a lens inside the

common arm of the MI. As the distance between the PRM and the BS is limited

to 3 m, the focusing element would have to have a focal length in the order of or


PR1PR2

South

PR2East

ITM

South

ITM

East

BS

From MC

Figure 6.4: Schematic diagram for the proposed stable PRC design for AIGO advanced

interferometer (figure not to scale). This solution includes a lens inside the recycling cavity

in order to achieve the required Gouy phase.

smaller than 3 m to accumulate any appreciable Gouy phase. This will create a

beam size of well below ∼ 100µm on the PRM. This small beam size has multiple

disadvantages such as an intensity of at least 10 MW/cm2 for typical input powers

and power recycling gains and very stringent requirements on the focal length of the

lens and the radius of curvature (ROC) of the recycling mirror [10].

These problems can be significantly reduced when moving the ITMs to the current

end stations and place one lens (PR2) in each arm of the MI as shown in figure 6.4.

This lens could be combined with the compensation plate which will be installed in

most high power interferometers to compensate thermal lenses generated in the ITM

substrates [27, 28] or it could even be polished into the backside of the ITM substrate.

The focal length of this lens would be roughly similar to the distance between the

ITM and the central building. This creates a waist close to PR1, which then would

be placed at the position where the acquired Gouy phase is equal to the design Gouy

phase. Its ROC needs to match the ROC of the Gaussian eigenmode at that location

in order to mode-match the recycling cavity to the arm cavities. Here and throughout

this document the mode-matching refers to mode-matching in power.


(a)

0.997

0.997

0.9975

0.9975

0.998

0.998

0.9985

0.9985

0.999

0.999

0.9995

0.9995

PR1 radius of curvature normalized error

PR

2 focal le

ngth

norm

alize

d e

rror

Mode matching as function of the optics tolerances

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-3

0.02

(b)

0.9

99

9

0.9

99

9

0.9

99

92

0.9

99

92

0.9

99

94

0.9

99

94

0.9

99

96

0.9

99

96

0.9

99

98

0.9

99

98


PR

2 focal le

ngth

norm

alize

d e

rror

Mode matching as function of the optics tolerances

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-3

Figure 6.5: (a) Mode-matching drop as a function of PR1 radius of curvature and PR2

focal length. Note that mode-matching drops mainly due to PR2 radius of curvature error.

(b) The optimised mode-matching after repositioning PR1.

An important consideration for the design of the stable recycling cavity is the

tolerance of the mirrors and lenses. Here we consider mode-mismatch, manufacturing

tolerances based on the current glass-manufacturing technology and the ability to

correct them by adjusting the distance between PR1 and PR2. This will be discussed

using a specific set of parameters (see tables 6.2 and 6.3) for the PR mirrors which


fix the length and generate a Gouy phase of ∼ 0.58 rad. The nominal ROC of PR1

in this design is −1.630 m while the focal length of PR2 is 77.775 m.

Any changes from the design values of PR1 and PR2 will decrease the mode-

matching into the arm cavity. Figure 6.5(a) shows the mode-matching drop as a

function of the normalised error in ROC of the mirrors. Figure 6.5(a) shows that the

mode-matching depends mainly on the normalised error in PR2. This is also easy to

understand since the mode-matching in a simple two element telescope depends on

the absolute errors in the focal lengths. However, as shown in figure 6.5(b) this can

be compensated by changing the distance between PR2 and PR1 to match up with

the as-build focal lengths. Note that the Gouy phase after re-optimising the distance

is again very close to its design value. For example, with these tolerances the Gouy

phase change is about 0.07 rad in the worse case. This can be recovered by distance

re-optimisation to about ±0.02 rad.

6.5.2 Folded stable recycling cavity

A different approach is the possibility of a folded recycling cavity allowing a stable

PRC to be constructed within the existing facilities. In this case we add two mirrors

to the recycling cavities to attain the necessary Gouy phase for the HOM suppression.

Figure 6.6 shows a schematic of the proposed design for a folded stable PRC.

The original PRM from figure 6.1 is replaced by three power recycling mirrors PR1,

PR2 and PR3. This creates a mode-matching telescope that also works as a PRC.

By carefully choosing the ROC and the distance between these mirrors it is possible

to obtain any g-factor. As a consequence the distance lpr shown in figure 6.1 now

corresponds to the distance between PR1 to the BS passing by the secondary and

tertiary mirrors PR2 and PR3.

As seen before the distance between the input mirror of the recycling cavity and

the BS is very short. This restriction requires us to install the PRC mirrors before the

BS (to the left of the BS in figure 6.6). Because of the strong focusing power needed

to achieve the required Gouy phase the power density at PR1 is around 3 MW/cm2.

It is a similar case to the one presented in the previous section where a short distance

cavity will have a very steep change in Gouy phase (again due to a very short Raleigh


PR1 PR2

PR3

ITM

South

ITM

East

BS

From MC

Figure 6.6: Schematic diagram for the proposed stable PRC design for AIGO advanced

interferometer (figure not to scale).

range) making it very difficult to control and with very high power density on the

input mirror.

As a consequence we need to put PR2 ‘behind’ the BS as in figure 6.6. This set

some constraints to the geometry of the recycling cavity. First PR1 and PR2 will

need to be offset from the PR3–BS–ITM line. The larger the offset the larger the

angle at which the beam will impinge PR2 and PR3 increasing the astigmatism in

the PRC. However a larger offset will allow for larger distances between the mirrors

which will ease the mirror tolerance for the Gouy phase control. It is also necessary to

keep a minimum clearance between the BS and the PR2–PR3 circulating beam which

sets a maximum for the distance between PR1 and PR2 and the angle of incidence at

PR3. The parameters for the proposed designs which take all these constraints into

account are listed in tables 6.2 and 6.3. The smallest spot size radius is 1.887 mm at

PR2 sustaining a power density of 17.9 kW/cm2; well below the damage threshold.

A potential problem with the folded stable cavity design is the astigmatism due

to the angles at which the beam will impinge the mirrors. Given the constraints

discussed above the angles at which the beam will impinge PR2 and PR3 are very

similar and close to 2.38o. The combination of a relatively short distance between


the mirrors and the even shorter focal length creates some astigmatism. The worse

case is at PR1 with a vertical spot radius of 2.021 mm and a horizontal spot radius of

3.360 mm, corresponding to an ellipticity of 0.8. This is interesting when compared to

the case with a Gouy phase of ∼2.57 radians, where the ellipticity is the same, but the

horizontal spot size radius is reduced to 1.216 mm, hence changing the ellipse major

axis and the power density. In both cases the difference in focal length between the

horizontal and vertical axis is close to 1.5 m. The difference in Gouy phase between

the two axes makes the suppression of the HOM by means of a stable recycling cavity

ineffective without the addition of extra optical components. Using off-axis parabolic

mirrors it is possible to compensate for the astigmatism due to the large angles of

incidence. Assuming the use of off-axis parabolic mirrors, next we present an optical

layout for the mode-matching telescope for an intermediate optical mode between the

horizontal and the vertical directions.

Following a similar criterion as in the straight PRC we assigned the tolerances for

the ROC of the mirrors that form the folded stable PRC. The designed value of PR2

ROC is−0.407 m, while the designed value of PR3 ROC is 12.4 m. Figure 6.7(a) shows

the decrease in mode-matching due to ROC errors. It shows the mode-matching as

a function of the normalised error in ROC of PR2 and PR3. As seen in the straight

cavity case we can improve the mode-matching by optimising the distance between

PR2 and PR3. After optimising the distance the mode-matching improves to better

than 99.9%. Note that moving PR2 will require repositioning PR1 to keep the length

of the recycling cavity constant.

Adaptive heating on PR2 and PR3 can mode-match to a significant modal space

in the arm cavity. The mode-matching can drop by as much as 0.5% over the range

of expected values for the arm cavities test masses. This by itself shows that the

proposed cavity is quite tolerant to errors in ROC. However, it is desirable to improve

the mode-matching because of errors in the ROC of the ITM can change the Gouy

phase of the recycling cavity. Since this static error is a one time only process, the

mode-matching can be improved by repositioning the PR2 mirror also restoring the

Gouy phase. The improved mode-matching is shown in figure 6.7(b).


(a)

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9


PR

3 r

adiu

s o

f curv

atu

re n

orm

alized e

rror

Mode matching as function of mirror tolerances

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-3

0.5

(b)

0.99

0.992

0.994

0.996

0.998

0.998

ITM radius of curvature (m)

ET

M r

adiu

s o

f curv

atu

re (

m)

Mode matching as function of the ITM and ETM

2640 2660 2680 2700 2720 2740 2760 2780 2800 2820

2640

2660

2680

2700

2720

2740

2760

2780

2800

2820

0.9880.986

Figure 6.7: (a) Mode-matching as a function of proposed radius of curvature tolerance

limits on PR2 and PR3 for a fixed folded PRC design. The contour lines are lines of constant

mode-matching. (b) Improved mode-matching as a function of expected values of ITM and

ETM radius of curvature after optimising PR2 position.

6.5.3 Comparison between designs

Due to the long distance between PR1 and the lens PR2 in the straight design, there

is a smooth transition of Gouy phase from 0 radians near the ITM to ∼0.58 radians


at PR1 (stability g-factor of ∼0.7). In the folded design there is a Gouy phase shift

of only 1.6 × 10−4 radians from ITM to PR3. From PR3 to PR2 the Gouy phase

shift is ∼0.58 radians (stability g-factor of ∼0.7). Since PR3 is in the far field of

the arm cavities a very small Gouy phase shift is induced between ITM and PR3.

As a consequence the distance between them has a very small effect in this stable

PRC design. For this simulation we assumed that the ITM is at the end station ∼80

m from the BS. These results show that for the folded design the ITM could also

be inside the main lab. The small change in Gouy phase can be compensated by

adjusting the position or the ROC of PR2. As a consequence the effectiveness of the

stable PRC will be mostly affected by errors in the ROC of PR2.

Figure 6.8(a) shows the tolerance of the PR2 lens in the straight cavity design

while figure 6.8(b) shows the tolerance of the PR2 mirror in the folded cavity design.

Both figures show the accumulated Gouy phase variation and the spot size radius

at PR1 as a function of the PR2 normalised error. We notice that for both designs

similar percentage errors in PR2 have a similar effect in the spot size variation and

the accumulated Gouy phase shift at PR1. The spot size has a steeper variation with

the normalised error in the straight cavity design, with the accumulated Gouy phase

also showing a steeper slope around the design value. This makes the tolerances for

manufacturing the optics very demanding, but we have shown that these errors can

be overcome by repositioning the optics.

The major contributor to the thermal effects in both designs is the substrate

thermal lens in the ITM. With high circulating power inside the arm cavities this

substrate thermal lens is mainly generated by the power absorbed by the coating

of the ITMs. For almost 1 MW of circulating power, the absorbed power is about

0.5 W. The required amount of compensation depends upon the material chosen. For

a lens made of fused silica, the required compensation will be in the shape of an

annulus pattern and the required compensating power would be about ten times

the absorbed power. A CO2 laser can be used to create the required pattern, in

such case about 5 W would be sufficient for the compensation [29]. Note that two

optical components would be necessary to compensate the thermal effects inside the

interferometer. For the folded design, this could be the substrate side of the ITMs


(a) (b)

-0.1 -0.05 0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

Gouy phase and spot size at PR1 as function of PR2

PR2 ROC Normalized Error

One

way

Gou

y ph

ase

(rad

)

-0.1 -0.05 0 0.05 0.1 0.150

5

10

15

Spo

t siz

e at

PR

1 (m

m)

Gouy phase Spot size

-0.1 -0.05 0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

Gouy phase and PR1 spot size as function of PR2

PR2 Normalized Error

One

way

Gou

y ph

ase

(rad

)

-0.1 -0.05 0 0.05 0.1 0.150

5

10

15

Spo

t siz

e at

PR

1 (m

m)

Gouy phase Spot size

Figure 6.8: (a) Variation of the accumulated Gouy phase in the straight PRC design and

spot size radius at PR1 as a function of the PR2 lens normalised focal length. (b) Variation

of the accumulated Gouy phase in the folded PRC design and spot size radius at PR1 as a

function of the normalised PR2 ROC.

or independent compensation plates. In the straight design PR2 can be used for

compensation. With one optical element in each arm, it is possible to compensate

both common mode as well as differential mode distortions in the recycling cavities.

A major concern is the spot size at the BS. With a spot size radius of 57.85 mm in

the folded design the temperature rise due to the circulating beam is less than 0.01o

K from room temperature. This assumes a substrate like Suprasil 3001 for the BS,

with substrate absorption of 0.25 ppm/cm [30]. As a consequence the thermal effects

are quite small and the mode-mismatch due to these effects negligible. In the straight

cavity design the spot size radius at the BS is only 2.30 mm and therefore the thermal

effects stronger. The temperature rise at the substrate is close to 0.01o K, while at

the coating almost 0.06o K. As a consequence the strongest mode-mismatch will occur

in the inline arm, caused by the beam crossing the substrate at 45o inducing some

astigmatism. The mode-mismatch can be reduce to less than 0.1 % by repositioning

PR2 also reducing the astigmatism. By using the PR2 lens as a compensation plate it

is also possible to make some corrections through variations in the lens focal length.

Since each MI arm of the PRC has its own lens it is possible to tune the mode-

matching more effectively since each PRC arm will have a different mismatch with


the main arm cavities.

The bigger spot size in the folded cavity design will also increase the diffraction

losses by at least an order of magnitude. This however can be overcome by reducing

the spot size at the vertex using different ROC for the ITM and ETM [15]. While

in the case of the straight cavity design the much smaller spot size radius at the BS

means that its contribution to diffraction losses is negligible.

A smaller spot size generates tighter alignment restrictions. Any off-centring of the

beam at the BS will increase the leakage of power from the PRC into the dark port.

The amount of power that leaks into the dark port depends on this contrast defect,

the circulating power in the PRC and the transmissivity of the SRM. The restrictions

are achievable with the advanced vibration isolation system under development for

AIGO [31], but will require further revision.

In the straight cavity design we have two optical elements, one inside each arm,

which can be used for thermal compensation. The thermal effects on the BS can be

compensated by operating on these two optical elements. The repositioning of the

PR2 elements and the control of their ROCs via thermal compemsation provides the

four degrees of freedom required for mode matching into four cavities; namely PRM

and inline arm, PRM and off-line arm, SRM and inline arm and SRM and off-line

arm.

The smaller beam size at the BS will also increase the coating noise contribution

of the BS to the overall noise of the interferometer. Thermo-optic noise can be a

main source of noise in advanced interferometers, but it has been suggested that it

has been overestimated in the past [32]. It is one of the main fields of research for the

next generation of interferometric GW detectors. Therefore the entire design would

have to be further refined looking into other effects such as thermo-optic noise.

Table 6.2 shows the distances between the different optical elements of each stable

recycling cavity design. The focal length of the optical elements selected for these

simulations are presented in table 6.3. In both designs PR1 is a mirror. PR2 is a lens

in the straight cavity solution while for the folded case is a mirror. PR3 is a mirror

in the folded cavity solution and it is not required in the straight design. Table 6.4

shows the spot size radius on the different optical elements that form each of the


Straight + lens PRC Folded stable PRC

Elements Distance (m) Elements Distance (m)

ITM - PR2 1.50 ITM - BS 80.51

PR2 - BS 78.50 BS - PR3 2.20

BS - PR1 2.20 PR3 - PR2 6.02

PR2 - PR1 6.00

Total 82.20 Total 94.73

Table 6.2: Distance between the different optical components in both cavity recycling

designs.


Optics Focal Length (m) Focal Length (m)

PR1 -0.815 13.726

PR2 77.775 -0.203

PR3 - 6.2

ITM -3645 -3645

Table 6.3: Focal length of the different components that form the inline PRC with a lens

to obtain the necessary Gouy phase and the folded PRC, which uses three mirrors to obtain

the necessary Gouy phase.



Position Spot size (mm) Position Spot size (mm)

Arm waist 16.095 Arm waist 16.095

ITM 55.014 ITM 55.014

PR2 55.067 BS 57.850

- - PR3 57.929

BS 2.302 PR2 1.887

PR1 0.917 PR1 2.058

Table 6.4: Development of the spot size radius on different optical components starting

from the arm waist through the ITM back to the PRC input mirror PR1. In the inline

design the lens PR2 is between the ITM and the BS, while in the folded design the mirrors

are between the input mirror PR1 and the BS. Note the difference in spot size radius at the

BS.

stable PRC proposed designs.

Based on the results of our analysis a straight stable PRC seems a better choice.

The mode-mismatch due to ROC errors and the Gouy phase variations are lower than

for the folded stable PRC. The mode-mismatch due to the thermal gradient at the

BS can be corrected in both designs. A folded stable PRC will introduce some degree

of astigmatism. However this can be greatly reduced by properly selecting the angle

of incidence at two off-axis parabolic mirrors, but this will further complicate the

optical design and the control system of the interferometer.

6.6 Sidebands and the stable recycling cavities

6.6.1 Modulation frequencies calculations

In the previous section we have selected a Gouy phase for the suppression of HOM

in the PRC. Therefore we have a known length for a stable PRC. The resonance

condition of this cavity will determine the modulation frequencies (fm) for the RF

sidebands. In this case the carrier sees an over-coupled arm reflectivity. Therefore if

6.6. SIDEBANDS AND THE STABLE RECYCLING CAVITIES 221

the carrier is resonant in the PRC the sidebands need an extra phase shift, thus the

following relation applies to the modulation frequencies of the sidebands.

fm =

(n1 +

1

2

)c

2LPRC. (6.5)

Here n1 is an integer and LPRC the average PRC length, which is defined by

lpr + (l1 + l2)/2. With l1 and l2 the distance from the BS to the ITM, l1 on the

perpendicular arm and l2 on the inline arm as seen in figure 6.1, and c the speed of

light in vacuum.

The selected modulation frequencies and the carrier frequency will all have to go

through the IMC. Therefore the length of the IMC (LIMC) has to be such that its

FSR, defined as FSRIMC = c/2LIMC , allows the transmission of the carrier and both

modulation frequency sidebands. Therefore the sideband frequencies need to satisfy

the following relation as well:

fm = n2c

2LIMC

. (6.6)

Here n2 is an integer equal or bigger than 1. Therefore the two sideband modula-

tion frequencies will be an integer multiple of the PRC FSR (FSRPRC).

In the previous section we have seen that the length of the straight PRC is about

82.2 m, while for the folded cavity solution the length is 94.7 m. This means that

each stable cavity solution has a different FSR; 1.82 MHz for the straight cavity and

1.58 MHz for the folded cavity solution. As a consequence each configuration will

require different modulation frequencies. In both designs the modulation frequencies

must not resonate inside the arms of the interferometer.

Using equations (6.5) and (6.6) we can then calculate the maximum length of

the IMC subject to the vacuum constraints. At the Gingin main lab in the current

configuration the maximum IMC length is 10 m. This implies that for a straight

cavity the IMC length will be 9.67 m with a FSR of 15.5 MHz. For the folded design

the IMC length is 9.97 m with a FSR of 15.0 MHz. The maximum modulation

frequency is determined by the bandwidth of the high efficiency photodiodes used in

the readout, thus we choose to demodulate at fm1 + fm2 < 100 MHz.

By introducing a difference in the arm lengths in the recycling cavities, we can


arrange for the output of the interferometer to be dark for the carrier but not dark for

one of the sidebands [33]. This asymmetry is required in order to have some signal

in the SRC that allows for the control of that degree of freedom. This Michelson

arm length difference known as Schnupp asymmetry [34] is defined as l− ≡ |l1 − l2|

which corresponds to the difference between BS–ITMSOUTH (inline) and BS–ITMEAST

(perpendicular) lengths. We can define this asymmetry as l = c/(4fm2) since we

assume that the reflectivities of the mirrors in the PRC and SRC are the same.

Therefore the asymmetry will be 967 mm for the straight PRC and 997 mm for the

folded cavity.

The lower modulation frequency (fm1) needs to be resonant in the PRC and non-

resonant in the arm cavities. This allows the control of the common mode signal

and the PRC length. The higher modulation frequency (fm2) needs to be resonant

in the PRC and in the SRC as well in order to be able to control the SRC length

and the Michelson arms. Under these conditions the higher modulation frequency

will be 77.5 MHz for the straight PRC, with fm1 + fm2 = 93.0 MHz, and 75.2 MHz

for the folded cavity solution with fm1 + fm2 = 90.2 MHz. There is however an

ongoing investigation for a different selection criteria for the modulation frequencies

that could allow for a lower high frequency to be selected without extending the

Schnupp asymmetry [35].

6.6.2 Signal recycling cavity

The length for the SRC will depend on the operation scheme selected for the interfer-

ometer. For example for a narrow band detection of black hole - black hole inspiral or

neutron star - neutron star inspiral we can select a peak frequency around 300 Hz [36].

This has been extensively studied and it is not the purpose of this paper to analyse

in detail the theory behind the detuning of the SRC in order to obtain the desired

resonance peak frequency [37, 38]. We will study the case where no detuning (φs = 0)

is required, which corresponds to a RSE configuration for the interferometer. Both

modulation frequencies need to fulfil equations (6.4) and (6.5), but only the higher

frequency (fm2) will be use to determine the length of the SRC as per the following

relation [20]:

6.6. SIDEBANDS AND THE STABLE RECYCLING CAVITIES 223

LSRC + ∆LSRC =c

2πfm2

(n3π + φs) . (6.7)

Here LSRC + ∆LSRC corresponds to the length of the SRC plus its detuning, φs

corresponds to the signal recycling detuning in radians and n3 an integer. In order

to operate the interferometer in a broadband RSE configuration we need to select n3

carefully. Since the modulation frequencies are multiples of each other and only one

of the sideband frequencies (fm2) needs to be resonant in the SRC. When operating

in a detuned configuration all frequencies will be off-resonance.

The peak frequency response of the interferometer will depend on the SRC de-

tuning. However the peak frequency not only depends on the length of the SRC. It

also depends on the coupling of the cavities, which in turn depends on the trans-

mission and losses of the mirrors that shape the interferometer. How these cavities

are coupled (under-coupled, over-coupled or matched) will play an important role in

the interferometer frequency response. At the same time the coupling combined with

the length of the cavities and in particular the ratio between the lengths of the SRC

and the arm cavities will define the slope of the detuning. The bandwidth of the fre-

quency response and the highest possible frequency at which will be possible to tune

the interferometer will be defined by the transmission and losses of the mirrors. This

is not a simple matter and has been studied in greater detail in references [37, 38].

With a folded PRC a folded SRC of ∼96.73 m is needed in order to obtain a

broadband RSE interferometer. This implies a distance of ∼16.2 m between the BS

and SR1, which can be easily accommodated at the Gingin test facility (there is a

possible maximum of 7 m between the BS and the SRM). This corresponds to a folded

stable SRC that follows a similar design as the PRC. The ROC of the SR1 mirror

will need to be of 1.52 m. The SRC will then have a Gouy phase of 2.65 radians or a

g-factor of 0.78. For simplicity we have assumed that the mirrors for the SRC are of

the same ROC as the one used in the PRC, but this will not necessary be the case.

A folded SRC allows us to select the ROC of the mirrors to obtain an optimum Gouy

phase. This is still under intense investigation [13]. If required, a marginally stable

SRC could be easily accommodated, instead of the SRC extra mirrors (SR1, SR2 and

SR3), one SRM at the output of the interferometer can be installed in order to obtain


a marginally stable SRC. However special attention will be needed to select the right

reflectivity for the mirror according to the combined reflectivities of the PRC mirrors.

The straight PRC solution needs a slightly different approach since the lens PR2

is sitting inside the MI. To obtain a stable SRC we still need to make the higher

modulation frequency (fm2) resonant in the SRC. This means a SRC of 84.13 m long

with a SR1 of 1.5 m ROC at 4.13 m from the BS. With the mirror at this position the

Gouy phase shift for the SRC cavity will be 2.4 radians (g-factor ∼0.52). However

if a marginally stable SRC is required it will be necessary to increase the cavity

Gouy phase to π radians. Since the Gouy phase transition is rather slow under this

configuration it is not possible to obtain the necessary extra Gouy phase shift within

the available space inside the Gingin main lab. As a result it will be necessary to add

another lens in front of SR1 to obtain the extra Gouy phase. We can install this lens

1.9 m in front of the SR1 and with a focal length of 1.88 m obtain a Gouy phase of

nearly π radians at the SR1 making the SRC a marginally stable cavity. In such case

the SRC will have the same length as the previous solution, but the SR1 will need to

be replaced by a 1.2 m ROC mirror. This extra lens could also be used for tuning the

SRC in order to obtain a more suitable Gouy phase.

6.6.3 Summary

A summary of the calculations for the different cavity lengths is presented in ta-

ble 6.5 for the proposed AIGO interferometer. Table 6.6 shows the details of the

recycling cavities for both proposed solutions. The results are for a double recycling

interferometer with stable PRC and stable SRC to be operated as a broadband RSE

interferometer.

6.7 Discussion

We studied two alternative solutions for stable recycling cavities included in the op-

tical design of an advanced interferometric GW detector. The operation as a dual

recycled interferometer with a broadband RSE scheme was also presented. This can

be extended to a detuned narrow band interferometer by selecting the appropriate

6.7. DISCUSSION 225

Parameter Straight PRC Folded PRC

LIMC 9.671 m 9.972 m

LPRC 82.200 m 94.731 m

LArms 5000 m 5000 m

LSRC 84.134 m 96.725 m

Asymmetry 0.967 m 0.997 m

fm1 15.500 MHz 15.032 MHz

fm2 77.501 MHz 75.161 MHz

Table 6.5: Optical length of the different cavities proposed for the AIGO dual recycled

interferometer.

Parameter Straight PRC Folded PRC

PR1 - BS 2.200 14.221

BS - ITMInline 80.484 81.009

BS - ITMPerp 79.516 80.011

L PRCInline 82.684 95.230

L PRCPerp 81.716 94.232

SR1 - BS 4.134 16.215

Table 6.6: Distance (in meters) between the different mirrors that form the proposed

stable recycling cavities for the AIGO interferometer.


detuning for the SRC. The proposed design includes arm cavities of 5 km long and

stable recycling cavities. This will further increase the sensitivity of the interferom-

eter by increasing the power in the recycling cavities and in the main Fabry-Perot

arms. We have shown that, without the addition of extra optical elements, kilometre

long inline recycling cavities are not efficient in terms of HOM suppression. However,

the addition of a lens in the recycling cavity allows for the adjustment of the accumu-

lated Gouy phase within the PRC and thus to select a suitable level of suppression

of the HOM. The study of an alternative solution with a folded stable PRC was also

presented. This can be accommodated in probably every GW design depending on

the constraints on each particular case. The latter design introduces some level of

astigmatism in the stable recycling cavities which will depend on the geometry of

the cavity. This is especially adverse in the SRC since it will contain the informa-

tion of the GW signal coming from the arm cavities. We have also shown that the

straight design is less susceptible to errors in the accumulated Gouy phase due to

errors in the focal length of the optics which can be corrected by small changes in the

position of the lens also recovering the mode-matching into the main arm cavities.

The mode-mismatch introduced by the thermal gradient induced in the BS as well

as any mismatch due to ROC errors can be compensated by re-optimising the optics

position and/or the use of the PR2 lens as a thermal compensation plate. These are

the main reasons to favour a straight recycling cavity with the addition of a lens in

order to obtain the required Gouy phase and HOM suppression. Even though, this

design will impose more stringent alignment requirement for the optical elements. It

is expected that a reduction of HOM of about 20 dB will help to reduce the possibility

of parametric instabilities in advanced interferometers to a level where they can be

further reduced using passive techniques [39]. Future work will address in more detail

the effect of stable recycling cavities in the possibility of parametric instabilities. A

further refined look into the noise sources, alignment issues, vibration isolation, and

thermo-optic noise among others will also be necessary. This will help to outline the

final design of future advanced interferometric GW detectors.

6.7. DISCUSSION 227

Acknowledgments

The authors would like to thanks Slawomir Gras from The University of Western

Australia, Jerome Degallaix from the Albert Einstein Institute Hannover and Volker

Quetschke from the University of Florida for useful discussions. This work is sup-

ported by grant 0653582 of the National Science Foundation, the Australian Research

Council, and is part of the research program of the Australian Consortium for Grav-

itational Astronomy.


6.8 Beam-splitter thermal effects

Beam-splitter thermal effects in the proposed

AIGO stable recycling cavity

6.8.1 Introduction

As part of the ongoing design of AIGO, the future Australian advanced interferometric

gravitational wave detector we include the study of two options for a stable power

recycling cavity (PRC). The aim is to use the current location of the East and South

end stations ∼80 m away from the central tank at the vertex of the two arms, which

will contain the beam-splitter (BS).

The two possible solutions under study are a folded mode-matching telescope

cavity, similar to the current design for Advanced LIGO, and a straight cavity which

instead uses an additional lens to obtain the necessary Gouy phase that will allow for

a higher order mode frequency gap larger than the cavity bandwidth. Both proposed

designs consider the installation of the ITM at the location of the current end stations.

PR1 PR2

PR3 ITM

South

ITM

East

BS

From MC

PR1 PR2South

PR2East

ITM

South

ITM

East

BS

From MC

Figure 6.9: Schematics showing the two possible solutions for the AIGO stable recycling

cavity, a folded stable recycling cavity on the left and a straight recycling cavity with the

addition of a lens on the right.

6.8. BEAM-SPLITTER THERMAL EFFECTS 229

6.8.2 Astigmatism in folded design

One of the main problems of the folded cavity solution is the astigmatism created

by the angles at which the beam impinges the mirrors that form the folded recycling

cavity. The elliptical spots at the mirrors due to the beam impinging at an angle

means different sagitta values for the horizontal and vertical planes. With a circu-

lating power of 2 kW the PRC astigmatism is enhanced by the small thermal effects

that this circulating power will induce in the mirrors.

A similar situation occurs in the current Advanced LIGO design. Even though

it is a milder effect due to the smaller angles at which the beams hit the mirrors

that form the folded PRC. The distance between the vacuum tanks in the current

Advanced LIGO design allows for the mirrors PR1 and PR2 to be placed before the

BS. Distances of more than 16 m between the mirrors that form the PRC allows them

to achieve small angles. In the proposed AIGO design we can only achieve ∼12 m, but

with PR2 behind the BS and only ∼3 m from PR3 to the BS. PR2 is then behind the

BS as shown in figure 6.9 the clearance between the BS and the PR2–PR3 circulating

beam creates an extra constraint to the minimum angle we can achieve in this design.

Figure 6.10 shows the astigmatism on both proposed folded designs (Advanced

LIGO and AIGO). The figure shows the different beam size in the horizontal and

vertical axis as a result of the beam impinging the PRC mirrors at an angle. A

measure of this is shown as the ellipticity of the beam at the PR1 mirror. Table 6.7

presents a summary of these values.

Interferometer AIGO Folded PRC Advanced LIGO Folded PRC

Axis X Y X Y

Waist position (m) 11.44 4.33 4.22 4.65

Waist size radius (m) 1.24 1.86 1.21 1.16

Table 6.7: Comparison of the astigmatism induced in the stable folded cavity designs for

AIGO and Advanced LIGO. The table shows the waist spot size radius and its position in

both stable folded PRC as a way of comparison.


-10 -5 0 5 10 15 20 25 300

10

20

30

40

50

60

70

Gaussian Beam in Power Recyling Cavity

beam

siz

e (

mm

)

distance (m)

-10 -5 0 5 10 15 20 25 300

10

20

30

40

50

60

70

Gaussian Beam in Power Recyling Cavity

beam

siz

e (

mm

)

distance (m)

x

y

Spot size radius

X = 3.360 mm

Y = 2.021 mm

e ~ 0.80

Spot size radius

X = 1.787 mm

Y = 1.692 mm

e ~ 0.32

x

y

Figure 6.10: Astigmatism comparison between the AIGO design (left) and latest Ad-

vanced LIGO PRC design (right). Both designs are folded stable recycling cavities confined

within their own vacuum constraints. They have different parameters and thus different

spot size at PR1, but it can be seen that the ellipticity at PR1 for AIGO is ∼0.8 while for

the Advanced LIGO design is only ∼0.3.

6.8.3 Thermal effects

The main problem in the current straight cavity design is the smaller spot size at

the BS. In the folded cavity design the BS spot size radius is 57.85 mm (similar to

the 55.69 mm of the current Advanced LIGO design). Since the BS transmits 50% of

the incoming beam it only has a few layers of coating, hence low coating absorption.

For these simulations a coating absorption of 0.1 ppm was assumed. The substrate

absorption of the proposed substrate (Suprasil 3001) [30] for the main optics is only

0.25 ppm/cm, and thus the thermal effects induced by a large spot size are negligible.

In the straight cavity design the spot size radius is reduced to 2.29 mm at the BS,

which means higher power density.

Thermal lensing is mainly caused by the power absorbed by the coating and the

power absorbed by the substrate. The power absorbed by the coating will cause

a mechanical deformation of the optics changing its sagitta, while the temperature

gradient induced by the beam going through the substrate will induce a path difference

between the central and the external part of the transmitted beam. This is mainly

caused by the refractive index change with temperature. In the case of the BS the

thermal effect due to substrate absorption will only affect the in-line beam, since the


perpendicular beam doesn’t cross the substrate of the BS. An ideal BS will transmit

only 50% of the beam reflecting the other 50%. However the beam is reflected back by

the arm cavities. Therefore the beam circulating through the substrate and reflected

in the coating will be close to the full circulating power in the PRC. Therefore both

arms will have slightly different beam profiles at each ITM, which in turn will have

different mode mismatch at the waist of the cavity. For all the simulations presented

here a circulating power in the PRC of 2 kW was assumed.

Figure 6.11 shows the different temperature gradient at the BS under both configu-

rations. The simulations include the effects of coating absorption and bulk absorption.

In the folded configuration the maximum temperature rise is only 0.006o K above room

temperature (300o K) for 2 kW of circulating power. In the straight configuration the

highest temperature is 0.029o K degrees above room temperature mainly due to the

coating absorption, which can be seen in figure 6.11 (d). This difference is caused by

the different spot size at the BS, with a radius of 57.9 mm in the folded configuration

and 2.3 mm in the straight configuration.

Figure 6.12 shows the thermal lensing effect in the position and size of the waist

in the in-line arm cavity. In these graphs only the thermal effects of the BS are

considered, the thermal effects at the ITM and recycling cavity optics have been

omitted in order to compare only the contribution of the BS to the mode mismatch.

The right hand side plot in the figure corresponds to the waist size and position after

the lens PR2 has been repositioned in order to compensate for a much larger error.

Originally the waist radius spot size is around 8.4 mm and the position almost 1.1 km

closer to the ITM! A similar effect could be seen in the perpendicular arm, which

didn’t come as a surprise since the major temperature effect comes from the power

absorbed at the coating. Figure 6.13 shows the waist size radius and its position in

the perpendicular arm cavity.

Even though the mode mismatch seems to be quite large the total effect in terms

of power loss it is not as bad as originally thought. Based on the calculations of

the couplings of the fundamental mode into higher order modes proposed by D. Z.

Anderson [40] we obtained that the power loss to higher order modes due to the mode

mismatch is given by:


(a) (b)

(c) (d)

Figure 6.11: Comparison of the temperature profile in the BS due to the PRC circulating

power in both proposed AIGO designs. The left hand side shows the temperature gradient

induced in the folded PRC, while the right hand side shows the case with a straight PRC.

All figures are set to the same temperature scale. We notice the high peak in temperature

due to the small spot size at the BS under the straight stable cavity design caused by coating

absorption.

∆P

P=

(∆w0

w0

)2

+

(λ (∆z)

2πw20

)2

. (6.8)

Where, ∆w0 corresponds to the difference between the size of the intrinsic waist of

the cavity and the actual size of the waist of the incoming beam, with w0 the intrinsic

waist of the cavity. ∆z corresponds to the difference in the position of the waist of

the cavity; this is between the intrinsic position and the actual position of the waist,

with λ the wavelength of the laser, in our case 1.064µm.


-200 -150 -100 -50 0 50 100 150 20014

14.5

15

15.5

16

16.5

17

17.5

18

Spot size and waist position variation with BS thermal lensing

Distance (m)

Beam

radiu

s (

mm

)

-200 -150 -100 -50 0 50 100 150 20014

14.5

15

15.5

16

16.5

17

17.5

18


Distance (m)

Beam

radiu

s (

mm

)

Waist (mm) Pos (m)

W 16.095 0

X 16.241 -21.345

Y 16.214 -17.395

Waist (mm) Pos (m)

W 16.095 0

X 16.666 -5.523

Y 16.616 6.795x

y

x

y

Figure 6.12: Comparison of the waist position and size between the two PRC designs.

The graphs only show the effect of the BS thermal lensing in the in-line arm cavity. It does

not include thermal effects in the ITM or PRC mirrors.

-200 -150 -100 -50 0 50 100 150 20014

14.5

15

15.5

16

16.5

17

17.5

18


Distance (m)

Beam

radiu

s (

mm

)

-200 -150 -100 -50 0 50 100 150 20014

14.5

15

15.5

16

16.5

17

17.5

18


Distance (m)

Beam

radiu

s (

mm

)

Waist (mm) Pos (m)

W 16.095 0

X 16.096 -0.110

Y 16.095 -0.001

Waist (mm) Pos (m)

W 16.095 0

X 16.577 -0.053

Y 16.579 -0.419x

y

x

y

Figure 6.13: Comparison of the waist position and size between the two PRC designs.

The graphs only show the effect of the BS thermal lensing in the perpendicular arm cavity.

It does not include thermal effects in the ITM or PRC mirrors.


In-line Arm Perpendicular Arm

X Y X Y

Waist size radius (mm) 16.666 16.616 16.577 16.579

∆ Waist position (m) -4.306 8.011 1.163 0.797

Power loss(%) 0.127 0.108 0.090 0.091

Table 6.8: Comparison between thermal effects in both arms of the interferometer.

Therefore we have four different values for the power loss, since we have different

values for the x and y axis position and size of the waist on each arm. The results

are presented in table 6.8.

6.8.4 Discussion and conclusions

After this brief study of the thermal effects at the BS due to the circulating power and

some of its consequences, the conclusion would be to favour the straight cavity design

over the folded cavity design. The thermal lensing at the BS will make the mode

mismatch between the PRC and arm cavities larger for the straight cavity design

even after repositioning the lens. However the astigmatism induced in the folded

design outweighs the mode mismatch from the straight cavity. The ideal solution

will be to increase the distance between the BS and the recycling mirrors, but also

increasing the distance between the recycling mirrors in order to obtain a gentler

slope for the Gouy phase transition.

A major concern is the spot size at the BS. With a spot size radius of 57.85 mm

in the folded design the temperature rise due to the circulating beam is less than

0.01o K from room temperature assuming a substrate like Suprasil 3001 with substrate

absorption of 0.25 ppm/cm. As a consequence the thermal effects are quite small and

the mode mismatch due to these effects is almost negligible. However in the straight

cavity design the spot size radius at the BS is only 2.29 mm and therefore the thermal

effects much stronger. In this case the main temperature rise is at the coating of the

BS. The temperature rise at the substrate is close to 0.02o K, while at the coating

almost 0.03o K. The strongest mode mismatch will occur in the in-line arm, caused by

6.9. REFERENCES 235

the beam crossing the substrate at ∼ 30o also inducing some astigmatism. This can

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fact that each arm of the PRC has its own lens helps to tune the mode matching

more effectively since each PRC arm will have a different mismatch with the main

arm cavities.

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http://www.optics.heraeus-quarzglas.com/

Chapter 7

Summary and conclusions

This thesis has presented the author’s contributions to the field of interferometric

gravitational wave detection including: vibration isolation system design for auxiliary

optics and test masses; advanced vibration isolation performance and local control

system design; high power mode-cleaner optical design; and advanced interferometric

gravitational wave detector arm cavities and stable recycling cavities modelling and

design, which included the selection of RF sidebands for length sensing and control.

The compact prototype vibration isolation system design for the mode-cleaner

optics uses a four leg inverse pendulum and a pyramidal Roberts linkage as a double

stage horizontal pre-isolation and includes Euler springs and cantilever spring blades

for vertical isolation. A gimbal between the vertical stages provides a high moment of

inertia rocking mass, including permanent magnets and copper plates for self damping

of the pendulum modes. This design has shown good performance which will further

improve once the described modifications are in place. The auxiliary vibration isolator

is not only designed to be used for the mode-cleaner optics, but also for the optics

of the recycling cavities. The development and testing of control electronics for the

mode-cleaner vibration isolator (as part of the original design) was started in parallel

with the mechanical design. The electronics were tested with the advanced vibration

isolator soon after testing with the mode-cleaner vibration isolator was completed.

A more complex, advanced vibration isolator for the test masses was developed at

UWA. The combination of an inverse pendulum and Roberts linkage provide a double

stage of horizontal pre-isolation, while a LaCoste spring configuration is used for

vertical pre-isolation. Ultra-low frequency control was provided for the LaCoste stage

and the Roberts linkage stage through an ohmic thermal position control which, when

239

240 CHAPTER 7. SUMMARY AND CONCLUSIONS

combined with large dynamic range magnetic actuators, provides complete control

of the positioning and damping of the different stages of vibration isolation. The

position sensing was provided by shadow sensors with large dynamic range. The

addition of magnetic actuators to the Roberts linkage stage in combination with the

local control system upgrade will allow for a super-spring configuration in the near

future, improving the low frequency performance of the vibration isolator.

A passive multistage pendulum combines Euler springs for vertical isolation with

rocker masses. The masses were combined with copper plates and permanent magnets

to create passive viscous damping, using eddy currents to reduce the Q-factor of the

pendulum normal modes. A control mass at the end of the pendulum chain provides

the last stage of control and an intermediate stage for the test mass suspension made

of Niobium ribbons. Since the installed electrostatic control system was not ready for

use, the test mass was controlled through the control mass stage rather than directly.

Due to the low signal to noise ratio of the control mass signal, the sensing of the test

mass angular displacement was arranged through an optical lever placed outside the

vacuum tank. The readout signal from a quad photo-detector allowed for the test

mass readout signal to be used in a control loop for damping of the high Q pendulum

modes of the test mass suspension.

The local control system is based on an off-the-shelf DSP board by Sheldon In-

struments. The remainder of the control electronics were developed by the author

with the support of C. Zhao at UWA. This included the development of dual chan-

nel electronic boards and filter boards. All the necessary boards were mounted on

a purposely developed backplane required for the distribution of the I/O signals for

the local control system. The control electronics for the electrostatic board will be

mounted in a separate chassis (installation pending at time of writing).

The control electronics are connected to the vibration isolator through DB–25

cables which are fed into the vacuum tank through vacuum compatible connectors.

Since all components inside the vacuum tank have to be ultra high vacuum compatible,

thin polyimide copper wires are used for signal distribution. It was necessary to

bundle the signals together in DB–25 connectors in order to pass through a six-

way cross feed-through and reach inside the vacuum tank. Once inside the vacuum

241

tank the signals first reach an intermediate board, from where they are distributed

around the vibration isolator to the different sensors and actuators. This excludes

the high current signals used for the ohmic thermal position control, which are fed

into the vacuum tank through a separate high current connector with thicker vacuum

compatible wiring.

Long term stability of the vibration isolator will improve with the addition of

an auto-alignment system. Even though the cavity could be locked and operated

for long periods of time, an auto-alignment system will dramatically improve the

duty cycle. The auto-alignment system forms part of the current development of a

hierarchical global control system, which will allow for the operation of longer cavities

with higher finesse. The local DSP architecture will be replaced with a centralised

system, where a single multi-CPU digital controller running real-time Linux OS will

take over the control of the vibration isolators. Multiple PCIe-PCIX extension chassis

with shared memory at each isolator will be connected to the controller using fibre

optics, facilitating data exchange between each local control and the central processing

unit.

An optical design for the mode-cleaner was started in parallel with the auxiliary

optics vibration isolator. The main objective of this design was to suppress the

optical higher order modes and stabilise the laser beam entering the main cavities of

the interferometer. Due to the high laser power required in advanced interferometric

gravitational wave detectors, this design also considered the thermal effects of the

high circulating power inside the mode-cleaner. These effects will change the cavity

g-factor and as a consequence the suppression level of the higher order modes and

more importantly will induce astigmatism in the outgoing beam. A new design, which

could substantially reduce the astigmatism, was proposed taking into consideration

the thermal effects induced by the high circulating power inside the mode-cleaner

cavity.

The analyses of optical mode suppression and thermal effects in the mode-cleaner

cavity were then translated into the main arm cavities. The simulations undertaken

for the behaviour of the higher order modes in the arm cavities were based on an FFT

code developed by the author, with the main purpose being to analyse the different

242 CHAPTER 7. SUMMARY AND CONCLUSIONS

effects of the mirrors on the higher order optical modes. Of particular interest were

the effects with direct influence on parametric instabilities and the parametric gain

R0, such as diffraction losses and the frequency separation of the higher order modes

from the fundamental mode. Analysis of a stand-alone cavity showed that the optical

mode parameters were usually underestimated and that degrees of freedom such as

mirror size, mirror tilt, optical mode orientation and energy distribution can have

a large effect on the estimation of the parametric gain. Further analysis went on

to add stable recycling cavities, with the initial intent of converting the marginally

stable power recycling cavity into a stable power recycling cavity for dual recycled

interferometric gravitational wave detectors. A stable power recycling cavity will help

to improve the control signals and the power build-up in advanced interferometers.

The stable design has also been extended to the signal recycling cavity. Following

recent Advanced LIGO approval of the addition of extra auxiliary suspensions for

the extra mirrors necessary for a stable recycling cavities, it will be necessary to

analyse the impact of these cavities on parametric instabilities and the parametric

gain. This will help to improve the design of advanced interferometric gravitational

wave detectors in general and AIGO in particular.

Appendices

243

Appendix A

Science Benefits of AIGO

The Science benefits and Preliminary Design of

the Southern hemisphere Gravitational Wave

Detector AIGO

D. G. Blair1, P. Barriga1 A. F. Brooks2, P. Charlton7, D. Coward1, J-C. Dumas1, Y. Fan1,

D. Galloway6, S. Gras1, D. J. Hosken2, E. Howell1, S. Hughes8, L. Ju1, D. E. McClelland3,

A. Melatos4, H. Miao1, J. Munch2, S. M. Scott3, B. J. J. Slagmolen3, P. J. Veitch2,

L. Wen1, J. K. Webb5, A. Wolley1, Z. Yan1, C. Zhao1

1School of Physics, University of Western Australia, Crawley, Perth, WA 6009, Australia2Department of Physics, The University of Adelaide, Adelaide, SA, 5005 Australia3Department of Physics, Australian National University, Canberra, ACT 0200, Australia4School of Physics University of Melbourne, Parkville, Vic 3010, Australia5School of Physics, The University of New South Wales, Sydney 2052, Australia6School of Mathematical Sciences, Monash University, Vic 3800, Australia7School of Computing and Mathematics, Charles Sturt University, NSW 2678, Australia8Department of Physics, Massachusetts Institute of Technology, Cambridge, MA

02139-4307, USA

The proposed southern hemisphere gravitational wave detector AIGO in-

creases the projected average baseline of the global array of ground based

gravitational wave detectors by a factor ∼4. This allows the world ar-

ray to be substantially improved. The orientation of AIGO allows much

better resolution of both wave polarisations. This enables better distance

245

246 APPENDIX A. SCIENCE BENEFITS OF AIGO

estimates for inspiral events, allowing unambiguous optical identification

of host galaxies for about 25% of neutron star binary inspiral events. This

can allow Hubble Law estimation without optical identification of an out-

burst, and can also allow deep exposure imaging with electromagnetic

telescopes to search for weak afterglows. This allows independent esti-

mates of cosmological acceleration and dark energy as well as improved

understanding of the physics of neutron star and black hole coalescence.

This paper reviews and summarises the science benefits of AIGO and

presents a preliminary conceptual design.

A.1 Introduction

Currently there are four kilometre scale gravitational wave detectors in the world

(3 LIGO detectors in the USL1 in Livingston, Louisiana, H1 and H2 co-located at

Hanford, Washington) and the VIRGO detector in Pisa, Italy. There are also smaller

detectors in Europe and Asia: GEO600 in Hannover, Germany, and TAMA in Tokyo,

Japan. In the coming decade, advanced detectors will be built, either as upgrades

to existing facilities (Advanced LIGO and Advanced VIRGO), or as new detectors:

LCGT in Japan and AIGO in Australia. The advanced detectors are designed to have

improved low frequency performance and lower shot noise, leading to an amplitude

sensitivity about 10 times better than existing detectors, enabling them to monitor

a volume of the universe 1000 times larger than current detectors. Frequent neutron

star inspiral events should be detectable as well as more distant binary black hole

coalescences. Coalescing neutron star binary systems will be able to be observed to

about 200 Mpc, while black hole binaries will be able to be observed to distances

∼1 Gpc (see for example E E Flanagan and S A Hughes [1]). Future improvements

using third generation detectors will improve this capability even further.

The correlation of electromagnetic events with gravitational wave signals provides

enormous science benefits. First it allows the velocity of gravitational waves to be

estimated. Second, if the source is a binary inspiral, it allows the luminosity distance

to be determined from the gravitational wave inspiral event itself, independent of

A.2. SCIENTIFIC BENEFITS OF THE AIGO OBSERVATORY 247

the red shift determined from observation of the host galaxy. This allows a powerful

independent probe of the Hubble law, cosmological acceleration and the equation of

state of dark energy [2]. However, this requires the identification of the gravitational

waves source locations to find the electromagnetic counter-part of the event.

Individual gravitational wave detectors have poor angular resolution with a beam

width of ∼120 degrees, so they are good all sky monitors but are completely inade-

quate for directional searches. This situation is greatly altered if an array of detectors

is used. Then the coherent analysis of signals from the array allows the network to

have diffraction limited resolution, where, as with VLBI radio astronomy, the angular

resolution is set by the ratio of the signal wavelength to the product of the projected

detector spacing and the signal to noise ratio. A world wide array of detectors can

achieve an angular resolution of ∼10 arc minutes for signals in the audio frequency

terrestrial detection band as discussed further below. However, the two dimensional

projected detector spacing can only be large for all directions in the sky if the array

contains a southern hemisphere detector. Here we summarise the scientific benefits of

the AIGO detector and the then go on to summarise a preliminary conceptual design

for this detector.

A.2 Scientific benefits of the AIGO observatory

It has long been recognised that an Australian detector disproportionately improves

the science return of the existing international network of gravitational wave detectors.

This disproportionate impact comes about for several reasons.

First, an Australian detector would greatly improve our ability to determine, from

gravitational waves alone, the location a gravitational wave event on the sky. Gravita-

tional wave detectors largely determine position by triangulation using time of arrival

information between different detectors-phase fronts of an incident gravitational wave

interact with different detectors at different times. Coherent network analysis effec-

tively resolves these differing times of arrival, enabling the detector array to be an all

sky monitor with good angular resolution over all source directions Detailed calcula-

tions by Wen et al [3, 4, 5] indicate that inclusion of AIGO would on average improve


the international network’s ability to localise sources from about 12 square degrees

to a fraction of a square degree as shown in figure A.1. Without AIGO, the error

ellipses are typically about 1.5 deg×8 deg. With AIGO, they are significantly smaller

than 1 deg× 1 deg. The error ellipse would then be well matched to the field of view

of most sensitive optical, X-ray and radio telescopes, so that it becomes possible to

conduct very long exposure searches for electromagnetic signatures of gravitational

wave events.

The second major impact of an Australian detector would be to improve greatly

our ability to measure both polarisations of a signal. At least two detectors with sub-

stantial different orientations are needed to fully reconstruct both wave polarisations

from the data. The LIGO detectors in the USA are oriented in such a way that they

do not provide information about both polarisation components. This was a delib-

erate choice-by orienting both detectors such that they each measure the same wave

polarisation, the statistical confidence in any given detection is greatly increased. For

the initial goal-unambiguous first detection of gravitational waves-this is a natural

and appropriate choice of detector orientations. Unfortunately, this is not such a

good choice of detector orientations once direct detection has occurred.

The primary goal of developing the science of gravitational-wave astronomy re-

quires the measurement of the polarisations, as this greatly increases our ability to

infer astronomically important information. Consider, for example, waves from bi-

nary coalescence, of which advanced detectors expect to detect more than 20 events

per year. In this case, the amplitude ratio of the two gravitational wave polarisations

encodes the inclination of the plane of the binary orbit with respect to the line of

sight from the Earth. Once the orbital inclination is defined, the frequency evolution

of the gravity wave signal contains a complete description of the system, and the

observed amplitude therefore encodes the distance of the source. This remarkable

property of gravitational wave signals enables them to be very powerful cosmological

probes. Adding an Australian detector to the network immensely augments its capa-

bility to measure polarisations simply due to its orientation on the nearly spherical

surface of the Earth. Thus with AIGO the network can obtain quite precise distance

information to sources [2].

A.2. SCIENTIFIC BENEFITS OF THE AIGO OBSERVATORY 249

The third benefit from the Australian detector relates to the noise performance of

the network. For broadband stationary noise, the noise of a network of detectors is

reduced as the square root of the number of detectors. While this factor is not large

(only ∼25%), it has a much larger effect on the number of detectable sources, since

the number of detectable sources depends on the volume of the accessible universe,

which increases as the cube of the detector strain sensitivity. Thus the global array

can be expected to detect almost double the number of signals with the addition of a

single southern hemisphere detector of sensitivity comparable to the northern hemi-

sphere detectors. For non-stationary noise, a larger network has the benefit of being

much better at rejecting spurious signals. Such signals must mimic a gravitational

wave passing through the network by arriving at each detector at a time and with

appropriate amplitude to be consistent with a real gravitational wave signal. The

probability of such a signal reduces as the power of the number of detectors, so the

addition of a single detector greatly reduces this probability. This reduction can be

by a factor of 10–100 depending on the types of signal.

The above three improvements provided by adding AIGO to the world array vastly

enhances the knowledge we can gain about the gravitational wave sources. For some

sources, such as core collapse supernovae, the waves are likely to be poorly understood

prior to gravitational wave observations. For others such as waves from black hole

binary mergers, the signals are likely to be only moderately well understood, while

waves from coalescing binaries prior to merger are well understood. In the not well

understood regime one must use the observed waves to solve an inverse problem and

obtain an understanding of the dynamics of the source. Gursel and Tinto [6] developed

methods for performing the “inverse” problem, and examined how well it could be

implemented using detectors located in North America, Germany, and Australia.

Their work demonstrates that the addition of AIGO to the network greatly improves

the reconstruction of such waves. In the other extreme of a well understood system,

the signal can be used to define the source distance and location on the sky.

Figure A.1 demonstrates the advantage of increasing the number of detectors in

the array and also of obtaining maximum out-of-plane volume in the array by placing

one detector in the southern hemisphere. AIGO significantly improves the angular


resolution and also eliminates the ambiguity problem which arises if all the detectors

are close to a common plane. The out-of-plane response also increases the maximum

baseline significantly thereby obtaining good angular resolution in almost all sky

directions. The array is even further improved if LCGT is added.

(a) (b)

Figure A.1: Angular area maps for world array. The angular uncertainty for each geocen-

tric sky direction is indicated as an ellipse on the sky. These are normally highly elongated.

(a) LIGO and VIRGO. (b) LIGO, VIRGO and AIGO. A further improvement is obtained

if LCGT is added to the array as shown in figure A.2.

To quantify the problem of host galaxy determination, we need an estimate of the

number of galaxies within the detector array angular resolution. Figure A.2 shows

the average number of galaxies per 1σ error ellipse for different gravitational wave

detector arrays, as reported in Wen et al in 2007 [5]. We see that the average number

of galaxies at 200 Mpc varies from in excess of 200 for LIGO–VIRGO array (LHV)

to about 4 if AIGO an LCGT are added to the array. Taking the galaxy distribution

into account, Wen et al showed that about 25% of sources can be unambiguously

identified with a galaxy. This allows the Hubble Law to be tested without actual

identification of an optical outburst.

A.3. PRELIMINARY CONCEPTUAL DESIGN FOR AIGO 251

200 400 600 800 1000

distance (Mpc)

num

ber

of

ga

laxie

s

within

a 1

–σ

err

or

circle

104

102

100

10-2

Figure A.2: The average number of galaxies expected within a 1–σ error ellipse for

different gravitational wave detector arrays, based on the angular resolution of each array

for each sky direction. The figure shows that the number of galaxies per error ellipse is

reduced from almost 200 for the LIGO–VIRGO array, to less than 10 for LIGO–VIRGO–

AIGO. For a single additional detector, AIGO gives the greatest benefit, but the best array

contains AIGO and LCGT. The symbols in the figures are: C–LCGT, A–AIGO, V–VIRGO,

LH–LIGO.

A.3 Preliminary conceptual design for AIGO

A preliminary conceptual design for AIGO utilises the maximum arm length possible

on our site of about 5 km. The extra arm length has the advantage of diluting local

noise sources such as thermal noise and control system noise, allowing slightly less

demanding specification for coating acoustic loss and control systems. With Advanced

LIGO (AdvLIGO) test mass and control specifications, the maximum inspiral range

is increased from about 200 Mpc to 250 Mpc, corresponding to a doubling of the

accessible volume of the universe.

Our design uses slightly smaller beam spots than AdvLIGO, (55 mm) to maintain

1 ppm arm cavity diffraction loss. This slightly increases the test mass thermal noise

but reduces the overlap factor for parametric instability. The vacuum arms are chosen

to be the same diameter as LIGOs. The vacuum design [7] has been shown to allow


LIGO vacuum specifications to be exceeded. A passive solar thermal bakeout system

has been demonstrated experimentally [8]. The vacuum system will include future

provision for ion pumps.

Vibration isolation will use multistage passive isolators developed at UWA [9],

subject to successful evaluation. Two are currently being evaluated on an 80 m optical

cavity. The isolators have an extra pre-isolation stage compared with the VIRGO

design. Otherwise they are conceptually similar to those of VIRGO but are much

more compact, occupying about 10% of the volume (and are correspondingly cheaper).

We propose to use sapphire for the test masses, because this material has a low

acoustic mode density compared with fused silica, which would reduce the problem of

parametric instability [10]. Sapphire also has the advantage of fast thermal response,

which means that the system comes into thermal quasi-steady state much more rapidly

than fused silica, further aiding parametric instability control [11].

The absorption requirement of the two inner test masses is 50 ppm/cm. This is

typical of good quality material currently available. The end test masses can use

sapphire of lower optical quality. Auxiliary optics will be made from fused silica:

mirrors for a quasi-stable power recycling cavity, the 10 m input mode-cleaner and

1 m output mode cleaner.

The test mass suspension system follows the design developed by Lee at UWA [12],

utilising four thin niobium ribbons with micro-cantilever suspension from equatorial

holes in the test masses. The test masses are controlled electrostatically using annular

bifilar comb capacitors incorporating an RF local control readout which serves as an

auxiliary local sensor. They are supported by a control mass which plays the role of

a VIRGO marionetta. Main test mass control is by actuators on the control mass.

The laser, injection locked to the successful Adelaide 10 W laser now in use [13],

will initially have a power of 100 W. This may need to be augmented to 200 W, subject

to performance of the vacuum squeezing. Vacuum squeezing is currently not part of

the AIGO baseline design but could be implemented if the technology is available.

Two prototype auxiliary optics suspensions have already been developed [10].

These use two stages of horizontal pre-isolation (one inverse pendulum, one Roberts

linkage), one Euler spring and one pair of blade springs for vertical isolation. These

A.4. DISCUSSION AND CONCLUSIONS 253

will be used to support the mode cleaner mirrors as well as the power recycling and

signal recycling mirrors.

Hartmann sensor technology for monitoring wavefront distortion has been devel-

oped at Adelaide and recently demonstrated in Gingin [15]. The AIGO design in-

cludes Hartmann sensors with CO2 laser thermal control to monitor and compensate

wavefront distortion in all test masses as well as the beam-splitter and compensa-

tion plates. The control specification is to better than 1 nm. Closed loop thermal

compensation control has been successfully implemented at the Gingin facility.

The proposed interferometer configuration will be detuned Resonant Sideband

Extraction. The control strategy is likely to be based on an ANU concept in which

control signals are injected into both beam-splitter ports. We propose to use an 80 m

stable power recycling cavity containing a thermal compensator lens.

The input mode-cleaner design for AIGO will use a triangular 10 m cavity with

apex mirror output for reduced astigmatism [16]. An output mode cleaner will be

employed to reject light in higher order modes and control modulation sidebands from

reaching the photo-detectors. This is likely to be based on the AdvLIGO concept of

a 4 mirror ring cavity silicate bonded onto a glass breadboard.

A.4 Discussion and conclusions

We have shown that a global array of gravitational wave detectors that contains

AIGO is substantially improved. Better polarisation resolution allows improved dis-

tance estimates for inspiral events. Roughly 25% of detected inspiral events can be

identified with a particular galaxy. This enables independent measurements of the

Hubble constant with or without detection of an optical outburst. The addition of

AIGO doubles the number of detectable sources and reduces non-stationary noise by

more than an order of magnitude. A 5 km interferometer has significant advantages,

particularly in reducing thermal noise, and in principle can detect double the number

of sources compared with a single AdvLIGO interferometer.


A.5 References

[1] E. E. Flanagan and S. A. Hughes, “Measuring gravitational waves from binary

black hole coalescences. I. Signal to noise for inspiral, merger, and ringdown,”

Phys. Rev. D 57 (1998) 4535–4565.

[2] N. Dalal, D. E. Holz, S. A. Hughes, and B. Jain, “Short GRB and binary black

hole standard sirens as a probe of dark energy,” Phys. Rev. D 74 (2006) 063006

(9pp).

[3] L. Wen and B. Schutz, “Coherent data analysis strategies using a network of

gravitational wave detectors,” Technical Report, G050043-00, LIGO, (2005).

[4] L. Wen, “Network analysis of gravitational waves,” Technical Report, G060336-

00, LIGO, (2006).

[5] L. Wen, E. Howell, D. Coward, and D. Blair, “Host galaxy discrimination us-

ing world network of gravitational wave detectors,” in Proceedings of the XLI-

Ind Rencontres de Moriond on Gravitational Waves and Experimental Gravity

(J. Dumarchez and J. T. T. Van, eds.), 123–131, The Gioi Publishers, 2007.

[6] Y. Gursel and M. Tinto, “Near optimal solution to the inverse problem for

gravitational-wave bursts,” Phys. Rev. D 40 (1989) 3884–3938.

[7] S. Sunil and D. G. Blair, “Investigation of vacuum system requirements for a

5 km baseline gravitational-wave detector,” J. Vac. Sci. Technol. A 25 (2007)

763–768.

[8] D. Berinson, D. Blair, P. Turner, and T. Simaile, “Test of a model solar bakeout

system for laser interferometer gravitational wave detectors,” Vacuum 44 (1993)

151–154.

[9] E. J. Chin, J. C. Dumas, C. Zhao, et al, “AIGO high performance compact

vibration isolation system,” J. Phys. Conf. Ser. 32 (2006) 111–116.

[10] L. Ju, S. Gras, C. Zhao, J. Degallaix, and D. Blair, “Multiple modes contributions

A.5. REFERENCES 255

to parametric instabilities in advanced laser interferometer gravitational wave

detectors,” Phys. Lett. A 354 (2006) 360–365.

[11] J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal tuning of optical cavities

for parametric instability control,” J. Opt. Soc. Am. B 24 (2007) 1336–1343.

[12] B. H. Lee, “Advanced Test Mass Suspensions and Electgronstatic Control for

AIGO”. PhD Thesis, School of Physics, The University of Western Australia,

2007.

[13] D. Mudge, M. Ostermeyer, D. J. Ottaway, et al, “High-power Nd:YAG lasers

using stable–unstable resonators,” Class. Quantum Grav. 19 (2002) 1783–1792.

[14] P. Barriga, A. Woolley, C. Zhao, and D. G. Blair, “Application of new pre-

isolation techniques to mode cleaner design,” Class. Quantum Grav. 21 (2004)

S951–S958.

[15] A. F. Brooks, T.-L. Kelly, P. J. Veitch, and J. Munch, “Ultra-sensitive wavefront

measurement using a Hartmann sensor,” Opt. Express 15 (2007) 10370–10375.

[16] P. J. Barriga, C. Zhao, and D. G. Blair, “Astigmatism compensation in mode-

cleaner cavities for the next generation of gravitational wave interferometric de-

tectors,” Phys. Lett. A 340 (2005) 1–6.

Appendix B

Control of Parametric Instabilities

Strategies for the control of parametric instability

in advanced gravitational wave detectors

L. Ju, D. G. Blair, C. Zhao, S. Gras, Z. Zhang, P. Barriga, H. X. Miao, Y. Fan and

L. Merrill

School of Physics, University of Western Australia, Crawley, Perth, WA 6009, Australia

Parametric instabilities have been predicted to occur in all advanced high

optical power gravitational wave detectors. In this paper we review the

problem of parametric instabilities, summarise latest findings, and assess

various schemes proposed for their control. We show that non-resonant

passive damping of test masses reduces parametric instability but has a

noise penalty, and fails to suppress the Q-factor of many modes. Reso-

nant passive damping is shown to have significant advantages but requires

detailed modelling. An optical feedback mode suppression interferometer

is proposed which is capable of suppressing of all instabilities but requires

experimental development.

B.1 Introduction

Ground-based gravitational wave detectors are designed to make extremely high pre-

cision measurements of the motion of test masses with perturbations limited by quan-

tum measurement theory. To obtain high sensitivity, high laser power is required. If

257

258 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES

there is any mechanism that a small amount of this power can be coupled directly into

oscillatory vibration of the test masses, this can lead to their uncontrolled mechanical

oscillation. A particular means by which this can occur is called parametric insta-

bility (PI). The possibility of PI in advanced laser interferometer gravitational wave

detectors was first shown to be a potential problem by Braginsky et al [1, 2]. Later,

three-dimensional modelling showed that parametric instability was likely for current

proposed interferometer configurations [3]. If the high frequency acoustic modes in

test masses have significant spatial overlap with high order optical cavity modes that

satisfy a resonance condition, then parametric instability could occur. It appears that

parametric instability is likely to be a potential threat to any interferometer which

uses high optical power and low acoustic loss test masses. Instability causes test mass

acoustic modes to ring up to a large amplitude in a time that could be in the range

50 ms to hundreds of seconds. The large amplitude destroys the interferometer fringe

contrast. The phenomenon cannot be filtered or reduced through any post-processing

of data.

Since the original predictions, parametric instability has been extensively mod-

elled. It has been undertaken by groups at The University of Western Australia

(UWA), Moscow State University, The Japanese Large Scale Cryogenic Gravita-

tional Wave Telescope (LCGT) project and members of the Laser Interferometer

Gravitational Observatory (LIGO) laboratory at the California Institute of Technol-

ogy [3, 4, 5, 6, 7, 8]. Modelling requires detailed knowledge of optical modes and

acoustic modes in test masses. Results are dependent on small changes in the test

mass geometry (which change the acoustic mode spectrum) and on the mirror di-

ameter and shape, which affect the optical mode spectrum. Small features such as

small wedge angles in the test masses and the placing of flats on the test mass sides

have very strong effects because they break both the acoustic mode and optical mode

degeneracy, acting in general to increase the number of potentially unstable modes.

There is no significant disagreement between estimates of instability. However, the

precise detail of instability is extremely sensitive to system parameters. For example,

a change in mirror radius of curvature of ∼1 part in 104 is sufficient to modulate

individual mode parametric gain by a large factor.

B.1. INTRODUCTION 259

Such changes are within the uncertainties of test mass material parameters and

1–2 orders of magnitude larger than the mirror radius of curvature tolerance. Because

of the high sensitivity of the resonant interactions to small parameter changes, only

models which use the same finite element modelling (FEM) test mass meshing, the

same acoustic losses of test masses including the contributions from coatings, the same

model for optical cavity diffraction losses and the same material parameters will give

strictly identical results. Modelling results also contain significant uncertainties due to

the limited precision of FEM [9]. Thus modelling is unlikely to be able to predict the

detailed instability spectrum of a real large-scale interferometer. However, modelling

results are likely to present an accurate statistical picture. Best estimates currently

predict an average of ∼10 unstable modes per test mass in an Advanced LIGO1 type

configuration.

The most realistic modelling to date has been applied to a proposed Advanced

LIGO configuration. Modelling takes into account the test mass shape and acoustic

losses due to mirror coatings. Results predict that between 0.2% and 1% of acoustic

modes in the frequency range 10–100 kHz are likely to be unstable depending on the

precise instantaneous value of the effective mirror radius of curvature. The parametric

gain for unstable modes varies between 1 and 103. Over a radius of curvature range

of 30 m (a range chosen to take into account thermally induced changes as well

as manufacturing tolerances), current estimates indicate a positive parametric gain

greater than unity for ∼20–40 modes spread across four test masses. The number of

modes and their gain fluctuate as the radius of curvature is thermally tuned. Over

the 30 m radius of curvature thermal tuning range, more than 700 acoustic modes

are parametrically excited in each test mass (but at any one radius of curvature, the

number is only about 5–10 per test mass).

Parametric instability estimates were first obtained for interferometer arm cavities

alone [1]. However, the real gravitational wave detector configuration is more compli-

cated with nested cavities of power recycling and signal recycling cavities [10]. The

effects of degenerate power and signal recycling cavities were then considered [2, 5],

including the realistic case of non-matched arm cavities [11]. By selectively suppress-

1http://www.ligo.caltech.edu/advLIGO/scripts/summary.shtml

http://www.ligo.caltech.edu/advLIGO/scripts/summary.shtml


ing arm cavity modes, the effect of using stable (non-degenerate) power recycling

cavities can be estimated. To date, all configurations analysed show instability.

The magnitude of parametric instability gain scales as the product of three Q-

factors: those of the optical cavity main mode, the relevant cavity high order mode

and the test mass acoustic mode. Unstable modes are generally in the frequency

range of 10–100 kHz. Most suggestions of methods for controlling instability have

focused on changing the value of the Q-factors of the acoustic modes.

In any room temperature interferometer in which thermal lensing is significant,

parametric instability will be tuned by time-varying thermal lensing. Thus, changes

will occur over a thermal lensing timescale, which is seconds to minutes in single

crystal test masses (such as sapphire or silicon) and ∼1 h for fused silica test masses.

In 2007, the UWA group observed three-mode parametric interactions for the first

time [3]. The parametric gain was shown to be tunable through variation in the test

mass radius of curvature in agreement with predictions.

In a cryogenic interferometer, parametric instability is likely to be frozen into

a particular configuration since the temperature coefficient for thermal lensing falls

effectively to zero at cryogenic temperatures. However, the precise configuration is

unlikely to be able to be predicted in advance.

Since advanced gravitational wave detectors are already under construction, it is

critical to focus on strategies for the control of parametric instability, which is the

focus of this paper. Instability control should be achievable without noise penalty

because the instabilities are all narrow band and all occur outside the measurement

band. However for many reasons, noise free PI control is not simple. Section B.2

summaries the theory of parametric instability and the status of modelling results

and explains why PI has not already been seen in LIGO and VIRGO. Section B.3

discusses various approaches to PI control and shows that simple methods have noise

penalties, disadvantages and risks. Optical feedback control is also discussed, which

although complex, can be implemented as a fully automatic external suppression

system.

B.2. PARAMETRIC INSTABILITIES THEORY AND MODELLING 261

B.2 Parametric instabilities theory and modelling

B.2.1 Summary of theory

Parametric interactions can be considered as classical ponderomotive interactions

of optical and acoustic fields or as simple scattering processes [13], as indicated in

figure B.1. In (a), a photon of frequency ω0 is scattered, creating a lower frequency

(Stokes) photon of frequency ωs and a phonon of frequency ωm, which increases

the occupation number of the acoustic mode. In (b), a photon of frequency ω0 is

scattered from a phonon creating a higher frequency (anti-Stokes) photon of frequency

ωa, which requires that the acoustic mode is a source of phonons, thus reducing its

occupation number. The scattering could create entangled pairs of phonons and

photons [14].

(a) (b)

ωm

ω0

ω1

= ω0

- ωm

ωm

ω0 ωa

= ω0

+ ωm

Figure B.1: Parametric scattering of a photon of frequency ω0 (a) into a lower frequency

Stokes photon, ωs, and a phonon of frequency ωm, and (b) into a higher frequency anti-

Stokes photon ωa, which require destruction of a phonon.

Figure B.2 illustrates three-mode parametric interactions in an optical cavity from

a classical viewpoint. In this example, stored energy in the form of a TEM00 mode is

shown scattering into a TEM11 mode by a particular acoustic mode. The interactions

can only occur strongly if two conditions are met simultaneously. First, the optical

cavity must support eigenmodes that have a frequency difference approximately equal

to the acoustic frequency |ω0 − ω1| ≈ ωm. Here, ω0 is the cavity fundamental mode

frequency while ω1 represents either Stokes or anti-Stokes high order mode. Second,


the optical and acoustic modes must have a suitable spatial overlap. Parametric

interaction effect can be expressed by parametric gain R. If R > 1, instability will

occur.

Acoustic

mode ωm

Cavity fundamental mode ωo

(Stored energy)

Radiation pressure force

Input

frequency ωo

Stimulated high order

optical mode ω1

Figure B.2: A cartoon of three-mode interactions in an optical cavity from a classical point

of view, showing an acoustic mode scattering stored energy from the cavity fundamental

mode into a high order mode while acting back on the test mass by radiation pressure.

The parametric gain in a simple cavity is given by [2]

R = ±4P0Q1Qm

LMcω2m

Λ

1 + (∆ω/δ1)2 (B.1)

Here, P0 is the power stored in the TEM00 mode, which is the fundamental mode

in the cavity; M is the mass of the acoustic resonator; L is the length of the cavity;

ωm is the acoustic mode frequency; ∆ω = |ω0 − ω1| − ωm; δ1 = ω1/2Q1 is the half

linewidth (or damping rate) of the high order optical mode; ω1 is the frequencies of

the Stokes (anti-Stokes) high order optical modes; and Q1 and Qm are the quality

factors of the high order optical mode and the acoustic mode respectively. The factor

Λ measures the spatial overlap between the electromagnetic field pattern and the

acoustic displacement pattern defined by [2]


Λ =V(∫

ψ0 (−→r⊥)ψ1 (−→r⊥)uz−→dr⊥

)2

∫|ψ0|2−→dr⊥

∫|ψ1|2−→dr⊥

∫|−→u |2dV

, (B.2)

where ψ0 and ψ1 describe the optical field distribution over the mirror surface for

the TEM00 mode and higher-order modes respectively, −→u is the spatial displacement

vector for the mechanical mode and uz is the component of −→u normal to the mirror

surface. The integrals∫ −→dr⊥ and

∫dV correspond to integration over the mirror

surface and the mirror volume V , respectively.

It should be pointed out that there are often multiple-mode interactions, and the

above equation should include a summation over all the possible modes (both Stokes

and anti-Stokes modes) [15]. The positive and negative signs in R correspond to

Stokes and anti-Stokes modes, respectively.

From equation (B.1), it can be seen that parametric gain is a product of three

parameters: P0, Qm and Q1, corresponding to the arm cavity power, acoustic mode Q-

factor, and high order optical mode Q-factor respectively. Compared with the baseline

parameters for Advanced LIGO, the initial LIGO arm cavity power (P0 ∼ 20 kW) is

∼40 times lower, the acoustic Q (∼ 106) is ∼10 times lower and optical Q1 (finesse

of ∼200) is ∼2 times lower than those of Advanced LIGO (P0 ∼ 800 kW, Qm ∼ 107,

finesse of ∼450). In addition, the test masses are smaller so that the acoustic mode

density is lower at high frequency. The parametric gain in initial LIGO should be

more than ∼800 times lower than Advanced LIGO, and the risk of instability is very

low. Hence, it is no surprise that LIGO has not observed PI.

It is important to point out that the derivation of equation (B.1) is based on

a model that does not consider diffraction losses of the optical cavities. We found

that in the case of a simple cavity, the addition of diffraction loss does not change

the formulation and equation (B.1) is still valid. To incorporate diffraction losses

in the analysis of parametric instabilities in interferometers with recycling cavities

is mathematically difficult and has not yet been accomplished. However, in practice

for optical cavities with relatively low finesse, the diffraction losses of the low spatial

order modes are small compared with the coupling loss. Only very high order modes

have strong diffraction losses. The very high order mode contribution to PI is usually

small; thus, diffraction losses can often be ignored. So, the results presented here do


not strongly depend on this issue.

B.2.2 Modelling approach

We have conducted many simulations of various interferometer configurations. In

general, we use finite element modelling to simulate several thousand acoustic modes

of the test masses and their diffraction losses [3]. We have used both modal analysis

and FFT codes to model cavity optical modes. We have simulated interferometers

with both signal recycling mirror (SRM) and power recycling mirror (PRM) using

Advanced LIGO parameters. This involves an elaborated form of equation (B.1)

which takes into account both power and signal recycling [11]. We normally assume

identical main cavities of 4 km length and a circulating power of 830 kW. We also as-

sume that a power recycling cavity operates in a marginally stable scheme. For stable

power recycling cavities, one expects that the finesse of some high order modes will

increase, thereby increasing Q1 and the associated parametric gain. Unfortunately,

we do not have the formalism available at present to analyse PI in an interferometer

with stable power recycling cavities.

Stokes and anti-Stokes modes are constructed using a set of transverse optical

modes up to eleventh order. In a typical analysis, about 60 optical higher order modes

(HOM) and 5500 elastic modes of the test mass are taken into account. The HOM

mode shapes and resonant frequency are obtained numerically using the eigenvalue

method whereas elastic modes are calculated using finite element modelling. The

test mass model takes into account most of the detailed structure. For our most

precise model (see results below), test masses are modelled as 20 cm thick 17 cm

radius cylinders with flats on the circumference, including chamfers and back face

wedge of 0.5o. In addition, the optical modes frequency detuning due to the finite

mirror geometry [3] is also taken into account. The quality factor of optical modes

is determined by coupling losses, while the Q-factor of elastic modes is based on

substrate losses [17] and coating losses [18]. Diffraction losses of the high order modes

are not significant for low finesse design presented below.

Our analysis usually includes up to the fifth longitudinal mode number from the

main cavity mode, which ensures that all possible interactions of optical modes with


an elastic mode are taken into account. However, usually only the first three mode

numbers contribute significantly to the parametric gain. The PI analysis is carried

out for different radii of curvature (RoC) of the end test mass (ETM) mirror. In

the analysis presented below, the ETM RoC is allowed to vary from 2.171 km to

2.201 km with 0.1 m steps. The radius of curvature for input test mass (ITM) is set

as a constant of 1.971 km. For each ETM RoC data point, the resonance condition

∆ω and the overlapping parameter Λ are calculated. Estimating PI for different RoC

of the ETM enables us to simulate the thermal tuning of the interferometer and thus

probe changes of the resonant conditions.

B.2.3 Modelling results

Our simulations of the Advanced LIGO detector with power and signal recycling cav-

ities reveal that the instability can occur over the whole RoC range considered. For

this configuration, there is a very strong dependence of the mirror radius of curvature

on the parametric gain R of unstable modes. Figure B.3 shows the number of un-

stable modes for different RoC. At certain values of RoC, there is up to 20 unstable

modes while for other values, only a few unstable modes are present. The number

of unstable mode is on average 8± 3 for each test mass. However, the total number

of unstable modes over the whole range is as high as 777. Figure B.4 shows the

highest parametric gain of the unstable modes at different radius of curvature. It

can be seen that parametric gain can vary from ∼2 up to ∼1000. It should be noted

that the instability condition for certain mechanical modes is very sensitive to the

change of radius of curvature, in the range ∼0.1 m. Therefore, very small changes of

the effective mirror curvature result in strong changing of the instability condition

from one mode to another. This can be visualised by examining the movies located

at http://www.gravity.physics.uwa.edu.au/docs/PI Movies.htm. These movies show

parametric gain for all modes during a sweep through the test mass radius of curva-

ture. We recommend playing the sweep manually by dragging the tracker button, to

enable careful examination of the gain-frequency structure.

Figure B.5 shows a statistical result of gain values of unstable modes. It can be

seen that the majority of unstable modes have R < 5. However there are a substantial

http://www.gravity.physics.uwa.edu.au/docs/PI_Movies.htm


number of modes with gain R > 100.

Figure B.3: Number of unstable modes as a function of ETM radius of curvature for

an interferometer with advanced detector parameters. Both substrate and coating acoustic

losses were taken into account for the elastic modes Q-factor. The transmissivity of the

SRM, PRM, ITM, ETM are set to proposed Advanced LIGO values 20%, 2.5%, 1.4%, and

5 ppm, respectively. The detuning of the signal recycling cavity was δ=20 deg.

B.3 Possible approaches to PI control

B.3.1 Power reduction and thermal radius of curvature

control

The simplest approach to PI control is to reduce the input power by a factor equal

to the peak instability gain that might be encountered. To ensure stable operation in

a reasonable range of mirror radii of curvature, the power would have to be reduced,

say, to 1% of nominal power, thereby increasing the detector shot noise by an order

of magnitude.

B.3. POSSIBLE APPROACHES TO PI CONTROL 267

Figure B.4: Maximum parametric gain for an interferometer with different ETM radius

of curvature using advanced detector parameters.

Figure B.5: The parametric gain distribution of unstable modes, for all of the unstable

modes in a 30 meter radius of curvature range. The majority unstable modes have R

value< 5.


A much better option would be to use CO2 laser heating to stabilise the interfer-

ometer test masses at radii of curvature where the peak parametric gain is minimum

as indicated in figure B.4. In the absence of any other damping scheme, a power level

of ∼10% of the proposed peak power could be achievable.

Since power causes thermal radius of curvature tuning, the process of powering up

sweeps the cavity through different parametric instability gain regimes on a test mass

thermal timescale (∼1000 s for fused silica test masses). The ring-up time constant

of an unstable mode can be written as [15]

τ =2Q

ωm(R− 1). (B.3)

For a test mass to ring-up from thermal amplitude of 10−14 m to 10−9m (the

assumed breaking lock amplitude), the break lock time will be

tB =23Q

ωm(R− 1). (B.4)

For example, a 30 kHz mode with Q = 107 and R = 20 would cause the inter-

ferometer to break lock in about 1 min. While thermal actuation with CO2 lasers

may compensate for this, the process is complicated by the long time constants in

fused silica test masses. It is not clear to us how to simultaneously control the in-

stantaneous radius of curvature (which determines the parametric instability) and the

thermal lensing environment (which defines the optical mode matching) with a single

CO2 laser actuation system. Because of the long thermal time constants, interferom-

eter lock acquisition would need to use thermal-lens-dependent power level control

to navigate past instabilities without losing lock. Once operation is stabilised to a

radius of curvature where the PI gain is low, the power level could be ramped up to

just below the PI threshold. The ramping up of power would need to be slow enough

so that the radius of curvature stayed within a tight range. It could be possible to

operate proposed interferometers at between 10% and 30% of the proposed power

in the absence of other control schemes. After losing lock, it might be a very slow

process to regain stable operation because of the long thermal memory of the test

masses.


B.3.2 PI control by using ring dampers or resonant acoustic

dampers

Ring dampers

The idea of ring dampers is to apply lossy strips on the circumference of test masses to

suppress the Q-factors of the acoustic modes. This Q-factor reduction will not greatly

increase the thermal noise of the test mass if the lossy parts are far from the laser

spot at the test mass [19, 20]. By carefully choosing the position of the lossy strips,

it is possible to reduced the test mass mode Q-factor without greatly degrading the

thermal noise performance of the interferometer. The lossy strips could be an optical

coating or a layer of Al2O3, Au or Cu applied by conventional ion-assisted deposition

techniques. We have analysed the use of ring dampers. This work has been published

elsewhere [6] and here we summarise their performance.

The ring damper method can effectively suppress many acoustic mode Q-factors

by a factor of ∼50. As shown in figure B.5, most of the unstable modes have R values

< 5. Thus, by applying ring dampers to the test masses most of the unstable modes

can be suppressed. Unfortunately the effect of the ring damper is not uniform: some

modes are only weakly damped and few unstable modes are very difficult to suppress.

Figure B.6 shows the reduction of unstable modes as a function of RoC for a typical

ring damper design. Here we assume a simple cavity interferometer without recycling

cavities but with Advanced LIGO parameters. The ring damper is a 2 cm wide, 20 m

thick strip with loss angle of 10−2. The exact number of unstable modes at certain

RoC may differ from those when recycling cavities are considered but the statistical

results of the simulation are not altered. There is always a thermal noise penalty by

applying lossy ring dampers on the test masses. The model shown here contributes a

Brownian thermal noise penalty of 5% [6]. If the noise penalty is allowed to increase

to 20%, substantial instability free windows appear in the RoC tuning range.

Resonant dampers

The use of resonant dampers, widely used in vibration isolation systems, was proposed

by Evans et al [21], and DeSalvo [22]. Preliminary analysis shows that in principle,


(a)

(b)

Figure B.6: Example of unstable mode suppression using a 2 cm wide, 20 m thick rind

damper on the test masses in a simple cavity interferometer with Advanced LIGO param-

eters. (a) Unstable mode number without ring damper on the test masses. (b) Unstable

mode number with ring dampers on the test masses. There is substantial reduction of

unstable modes due to the applied lossy ring damper.

it is possible to attach a small lossy spring-mass resonator to a test mass to damp

the resonant modes. If we choose the mass of the damper to be 1 g and the resonant

frequency of the damper to be 20 kHz, it is easy to show that the Q-factor of the

resonances of a 10 kg effective mass mirror, in the range of 30–100 kHz, can be reduced


from 107 to < 5 × 105 as shown in figure B.7. This result assumes perfect coupling

to the test mass modes.

To be usable in practice, there are several factors that must be considered. For

the damping to be effective, the damper should be placed sufficiently close to the

antinodes of the resonant mode to be damped. Therefore, many dampers may be

necessary to obtain good coupling to the large number of potentially unstable modes.

Also from the thermal noise point of view, it is desirable that the dampers should

be placed far away from the laser spot. This limits the effectiveness of the damping

for some modes that have high amplitude near the laser spot. Detailed modelling is

required to determine an optical configuration of dampers and their thermal noise

contribution.

103

104

105

106

107

103

104

105

Dam

ped m

ode Q

Frequency (Hz)

Figure B.7: Internal modesQ-factor of a test mass with a small resonant damper attached.

Parameter used: effective mode mass of the test mass ∼10 kg; original Q-factors of the

internal modes ∼ 107; small resonator damper mass ∼1 g; damper Q-factor ∼1 and the

frequency of the damper ∼20 kHz. Here we assume perfect coupling between the damper

resonance and the test mass internal modes.

However, it is clear from figure B.6 that strong damping can be achieved in the 10-

100 kHz band where instabilities occur, with weak damping within the gravitational

wave signal band. Thus, this method is particularly worthy of further investigation.


B.3.3 Local control of PI by acoustic excitation sensing and

feedback

Local feedback is a potentially simple solution to PI. The idea is to sense the acoustic

excitation of a test mass and provide a derivative feedback force using standard control

theory. Motion sensing could use optical, radio-frequency or electrostatic sensing,

while feedback forces could be applied electrostatically. However, there are several

issues with this method.

a) First, the sensor must have sensitivity sufficient to maintain amplitudes small

compared with the position bandwidth of the interferometer (position bandwidth is

the linewidth expressed in displacements units). The position bandwidth of Advanced

LIGO is ∼1 nm. The sensing noise must be small compared with this distance or else

the feedback controller will excite all mechanical modes within its bandwidth (say 15–

100 kHz) to an amplitude equal to the sensor noise floor. As long as the sensor noise

floor is less than about 10−12 m/√

Hz, this will not represent a severe degradation

of fringe contrast. However, if the sensor noise was 10−10 m/√

Hz (typical of current

local control sensors), this would represent a significant loss of fringe contrast. It is

also worth pointing out that the requirement of test mass motion < 10−20 m/√

Hz in

the interferometer detection band (below a few kHz) means that the force noise from

feedback actuation on the test mass should be sufficiently low to avoid injecting extra

noise in the detection band. This imposes even tighter requirements on the sensor

noise floor.

b) The second problem is the large number of test mass acoustic modes that are

potentially excited, as shown in section B.2.3. Many of the acoustic modes have a

complex mode structure. The overlap integral between each acoustic mode and an

electrostatic sensor must be large to be able to get good signal coupling. It is often

very difficult to excite test mass modes using electrostatic exciters because the exciter

applies forces across a node so that the positive and negative displacements partially

cancel, leading to small electromechanical coupling. In practice, we have found it

very difficult to achieve strong coupling to high Q modes with high frequencies in our

high Q test masses in an 80 m cavity. To accommodate high coupling to a large set of

modes, the sensor and actuator would have to be broken up into a set of separately


addressable elements. Each mode would require a different combination of actuator

elements. Sensing suffers from the same problem, so that high signal-to-noise sensing

will require sensing at different locations on the test mass.

c) The next problem is one of gain. The parametric gain we wish to suppress

is typically 10, sometimes 100 and if unlucky more than 1000. Let us assume that

we choose a gain of 100 as the maximum we want to control. This means that the

actuator must be able to excite the test mass mode at a rate 100 times faster than the

ring-down time. For example, if the acoustic mode ring-down time is 10 s, the ring-up

time will be 0.1 s. Such strong excitation is possible for small gap spacings and high

excitation voltages, but in our experience such strong coupling is difficult to achieve.

It would require a very high voltage small gap spacing exciter, and it would have to

be designed so that it did not supply residual in-band noise above the detector noise

floor.

B.3.4 PI control using global optical sensing and electrostatic

actuation

PI will be very easily detected in the dark fringe signal from the interferometer, which

is a much better probe of instability than any local sensor. The signal could be applied

to all test masses simultaneously even though the unstable acoustic modes will be

shared across four test masses. There should be no problem with driving test masses

at frequencies where there is no instability, except under the unlucky circumstance

that the frequency coincided with another test mass acoustic mode. If this occurred,

the system should also be able to automatically damp such a self-induced instability.

Using the global PI signal overcomes the signal-to-noise ratio (SNR) problem

associated with local sensing, but there still remains the need to apply quite large

forces to the test masses in the case of high gain instabilities. This is still limited

by the problem of achieving a large overlap integral with the electrostatic driver as

discussed in section B.3.2. Control of the end test masses will need to take account of

the time delay phase shift. To evaluate the active acoustic damping solution requires

the testing of suitable actuators to confirm adequate coupling to all of the predicted

unstable test mass acoustic modes.


B.3.5 PI control using global optical sensing and direct

radiation pressure

The force required to reduce the Q-factor of an acoustic mode is directly proportional

to its amplitude. If a suppression system is required to act on large acoustic am-

plitudes (say 10−10 m), the force requirements are too large to use direct radiation

pressure actuation. However, direct radiation pressure actuation is practical if insta-

bility is caught before the amplitude has grown too large. If we apply direct radiation

pressure feedback control when the oscillation increases to ten times the equilibrium

thermal amplitude, the maximum force F0 required to reduce the acoustic mode Q-

factor is

F0 = −10×√

2mω2kBT

Qf

, (B.5)

corresponding to a laser power of

P =cF0

2= −5c×

√2mω2kBT

Qf

, (B.6)

where kB is Boltzmann’s constant, c is the speed of light and Qf is the desired Q-

factor. Assuming an effective mass m = 10 kg, ωm = 30 kHz, Qf = 105, T = 300 K,

the required power is P = 0.8 W. Here, we have used point mass approximation which

assumes 100% overlap between the acoustic mode and the actuation force. For non-

ideal overlap, corresponding to actuation far from an antinode of the acoustic mode,

more power will be required.

In 2005, we used a laser walk-off delay line as a radiation pressure actuator [23] and

proposed to use direct radiation pressure as a means of PI control. Multiple reflections

can reduce the power requirements or increase the force by ∼30 times. In all cases, the

main problem is to obtain large overlap with all acoustic modes. The actuation phase

depends on the location of the actuator and must be well defined for each relevant

mode. If instability was not suppressed early enough, such an actuator would be

unable to achieve suppression. This threshold effect imposes strong requirements on

the reliability and completeness of the suppression system.

Braginsky and Vyatchanin [24] suggested using an external short optical ‘tran-


quiliser’ cavity to control instability. This method is also limited by the requirement

of overlap between the actuator and the test mass acoustic modes. In [25], the method

was shown to be technically difficult but viable in principle. Again, in practice several

such systems would be required for each test mass to obtain adequate overlap with

the acoustic modes, greatly increasing the complexity of the interferometer system.

B.3.6 PI control by global optical sensing and optical

feedback

The parametric interaction provides forces to the test mass via the high order optical

modes which are excited in the cavity. It should be possible to suppress the high order

mode in the cavity by introducing an anti-phase high order mode. Here, we summarise

the analysis of such a high order mode interference system. It has the advantage that

it could act very rapidly and even suppress very high gain instabilities. If it was

practical and robust, it could eliminate the need for any other instability suppression

scheme.

To model the system with optical feedback, Zhang et al [26] used a classical model

of a cavity with fields as shown in figure B.8 including an injected field f0(t). The

fundamental mode (Ei with frequency ω0) and the high order mode (fi with frequency

ω1) contribute to create the radiation pressure force at the differential frequency of

ω0 − ω1. This force acts back on the test masses which vibrates at its internal mode

frequency ωm. The parametric instability comes from the interaction between the

radiation pressure force and the mechanical mode vibration.

If no other external field is injected into the cavity apart from the fundamental

mode, the high order mode field fi is produced due to the scattering of the funda-

mental mode into higher order modes and their resonant build-up inside the cavity.

The back action force is determined by the product of fi and Ei. As Ei is a constant

depending on the input power and cavity parameters, the strength of the instability

is solely determined by fi and consequently by the parameters in equation (B.1)the

frequency difference between ω0 − ω1 and ωm, the overlap between the high order

mode and the test mass internal mode, and the quality factors of the cavity and the

test mass internal mode. If we inject another high order mode optical field f0 out


ELaser

3(t)

Ein

0(t)

4(t)

L

E1

E4

E3

2(t)

(t)

1(t)

Figure B.8: The fields of the fundamental mode, high order mode and the feedback of

the cavity. The fundamental mode fields with frequency ω0 is denoted Ei (i = 1, 2, 3, 4),

and the high order mode with frequency ω1 is denoted fi (i = 1, 2, 3, 4). The feedback field

is f0, L is the length of the cavity and x(t) is the perturbation of the cavity length due to

mirror vibration.

of phase with f1 into the cavity, it will destructively interfere with f1 to suppress

the instability. By detecting the optical signal at the interferometer dark port, it is

possible to determine the amplitude, frequency and phase of f1. One can then inject

field f0 with an appropriate phase and amplitude to suppress the instability.

Figure B.9 is a simplified schematic diagram for this optical feedback control

system. The beam pick-off mirror (BS1) diverts a small part of the main laser beam to

a PZT mirror which is used for locking a beam injection Mach-Zehnder interferometer.

After passing through two phase modulators and a phase mask, the pick-off beam

recombines with the main beam at BS2 and then is injected to the interferometer. The

phase modulator EOM1 together with photo-detector PD1, mixer and the amplifier

are used to phase lock the pick-off beam to the main beam. The modulator EOM2 is

used to create the sideband at the test mass internal mode frequency ωm. The phase

mask converts the fundamental mode to the high order mode. The photo-detector

PD2 detects the high order mode amplitude and frequency that is fed back to control

the driver of EOM2 to create an appropriate level sideband signal. The sideband

signal is injected into the cavity to suppress the high order mode created by the

test mass internal mode scattering. This prevents the build-up of the frequency ω1,

thereby suppressing the instability.

B.4. CONCLUSIONS 277

Figure B.9: Schematic diagram of the PI optical feedback control setup. The beam

pick-off mirror (BS1) diverts a small part of the main laser beam to the PZT mirror. After

passing through two phase modulators and a phase mask the pick-off beam recombines with

the main beam at BS2 and then is injected to the interferometer.

Detailed analysis and numerical results [26] show that in principle, it should be

possible to suppress parametric instability in the next generation detector such as

Advanced LIGO. This would only require external optical and electronic components

at the corner station and would not require modification of the test masses or the local

control systems. It has the advantage that it could act very rapidly and even suppress

very high gain instabilities. The disadvantage of this method is that each high order

mode to be suppressed required a MachZehnder arrangement and a phase mask for

adding the high order mode. This will increase the complexity if many modes were

to be suppressed. However, it would be possible to use this technique combined with

other methods for PI suppression. If the injection optics in advanced interferometers

included the basic beam splitters required, particular problematic instabilities could

be suppressed without internal modification of the interferometer. An experiment at

the high optical power facility [9] at Gingin is planned to investigate this solution.

B.4 Conclusions

We have shown that without taking active control measures, advanced gravitational

wave interferometers may suffer from parametric instabilities. These will render the


devices inoperable at high power.

The stochastic nature of instability has been emphasised, with instability some-

times occurring suddenly, as the thermal conditions of test masses vary with time.

The difficulty of making precise predictions has also been emphasised. Control of

instability using simple active damping is shown to be difficult.

We have shown that passive ring dampers can significantly reduce parametric

gain but not eliminate instability. Simple modelling shows that a number of passive

resonant dampers could be attached to test masses to reduce instability and probably

control it, but this requires detailed modelling.

A less invasive technique based on optical feedback has been proposed. This

technique allows instability control by selectively suppressing all large amplitude high

order modes. This technique enables all instabilities to be controlled but requires

knowledge of the high order mode, and a separate phase mask for each high order

mode that requires suppression. This is a most attractive back-up solution which

could be implemented during commissioning if required for particular high order

modes without requiring mechanical changes to the core optics.

Acknowledgment

We would like to thank the LIGO laboratory and the International Advisory Com-

mittee of the Gingin Facility for their encouragement and advice. This research was

supported by the Australian Research Council and the Department of Education,

Science and Training. This paper has been assigned LIGO Document Number LIGO-

P080081.

B.5 References

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Fabry-Perot interferometer,” Phys. Lett. A 293 (2002) 228–234.

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[4] L. Ju, S. Gras, C. Zhao, et al, “Multiple modes contributions to parametric

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[5] A. Gurkovsky, S. Strigin, and S. Vyatchanin, “Analysis of parametric oscillatory

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[6] S. Gras, D. Blair, and L. Ju, “Test mass ring dampers with minimum thermal

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[7] H. S. Bantilan and W. P. Kells, “Investigating a Parametric Instability in the

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[8] K. Yamamoto, T. Uchiyama, S. Miyoki, et al, “Parametric instabilities in the

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Phys. Rev. D 38 (1988) 2317–2326.

[11] S. Strigin and S. Vyatchanin, “Analysis of parametric oscillatory instability in

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[12] C. Zhao, L. Ju, Y. Fan, et al, “Observation of three-mode parametric interactions

in long optical cavities,” Phys. Rev. A 78 (2008) 023807.

[13] J. Manley and H. Rowe, “Some General Properties of Nonlinear Elements-Part

I. General Energy Relations,” Proc. of the IRE 44 (1956) 904–913.


[14] S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable en-

tanglement and quantum-state teleportation between optical and macroscopic

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[15] L. Ju, C. Zhao, S. Gras, et al, “Comparison of parametric instabilities for different

test mass materials in advanced gravitational wave interferometers,” Phys. Lett.

A 355 (2006) 419–426.

[16] P. Barriga, B. Bhawal, L. Ju, and D. G. Blair, “Numerical calculations of diffrac-

tion losses in advanced interferometric gravitational wave detectors,” J. Opt. Soc.

Am. A 24 (2007) 1731–1741.

[17] S. D. Penn, A. Ageev, D. Busby, et al, “Frequency and surface dependence of

the mechanical loss in fused silica,” Phys. Lett. A 352 (2006) 3–6.

[18] I. Martin, H. Armandula, C. Comtet, et al, “Measurements of a low-temperature

mechanical dissipation peak in a single layer of ta2o5 doped with tio2,” Class.

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[20] K. Yamamoto, M. Ando, K. Kawabe, and K. Tsubono, “Thermal noise caused by

an inhomogeneous loss in the mirrors used in the gravitational wave detector,”

Phys. Lett. A 305 (2002) 18–25.

[21] M. Evans, J. S. Gaviard, D. Coyne, and P. Fritschel, “Mechanical Mode Damping

for Parametric Instability Control,” Technical Report, G080541-00, LIGO, 2008.

[22] R. DeSalvo, 2008, LIGO Laboratory, California Institute of Technology,

Pasadena, CA 91125, USA, (private communication, 2005).

[23] M. Feat, C. Zhao, L. Ju, and D. G. Blair, “Demonstration of low power radia-

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[25] S. W. Schediwy, C. Zhao, L. Ju, et al, “Observation of enhanced optical spring

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Appendix C

Vibration Isolator Control Electronics

C.1 Introduction

The local control system is a multiple input-multiple output (MIMO) system, which is

then converted via a sensing matrix into a single input-single output (SISO) system.

Each input passes through a dedicated digital filter and PID control loop before

being converted by a driving matrix into signals to drive each of the actuators. As

a consequence different matrices, filters and control loops have been created for the

different stages depending on their resonant frequencies, the number of actuators, and

their distribution on that particular stage. The control electronics provides means

to interconnect actuators and sensors to the DSP on which the local control system

is based. The user control and interface was written in LabViewr. The built-in

libraries provided by the DSP manufacturer allow for the control loops to run on

the DSP board isolated from the local operating system. Through the interface the

operator can remotely monitor the performance of the different stages of the isolator,

and adjust the various loop gains and settings if necessary. The operator interface is

a series of graphical screens showing the current status of each stage of the isolators.

Figure C.1 shows a block diagram of the local control system and figure C.2 shows a

general schematic for its wiring.

C.2 Control electronics

The control electronics were developed in standard 6U height Eurocard format at

UWA. Each board in the control electronics contains two channels. Each channel

283

284 APPENDIX C. VIBRATION ISOLATOR CONTROL ELECTRONICS

4CS

4SS

5SS

5ACT

4ACT

La Coste Vertical Stage1HC + 2ACT + 1SS

Roberts Linkage4HC + 2SS

Pre-Isolator

Control Mass5ACT + 5SS

Test Mass Control

High Current

Power Supply

High Current

Power Supply

High Current

Power Supply

High Current

Power Supply

1SS

2ACT

1HC

4HC

High Current

Power Supply

1CS

Optical Lever1QPD

2SS

Laser

λ = 1064 nm1OL

1QPD

Multi-pendulum stage

Inverse Pendulum4ACT + 4SS

DSPSheldon MOD-6800

Figure C.1: Block diagram of the local control system for the advanced vibration isolator.

provides an infrared LED control circuit, a dual photo-detector circuit, and a control

signal circuit. The LED control circuit (shown in figure C.8) provides power to the

infrared LED and monitors its connectivity, which can be seen in the front-panel of

the board. The photo-detector circuit (shown in figure C.9) filters and amplifies the

signal from both photo-detectors that comprise the shadow sensor. Both signals are

independently fed to the DSP board, and can also be accessed from the front panel

for external measurements or monitoring. The photo-detectors differential signal can

also be accessed from the front panel. The control signal from the DSP is fed to the

corresponding control channel circuit (shown in figure C.10) through the backplane.

The control signal goes through a filter and a transimpedance circuit before going

through a high speed amplifier. This circuit converts the voltage control signal into

a corresponding current level that drives the magnetic field produced by the coils

which in turn acts on the permanent magnet and thus on the corresponding stage

of the vibration isolator. The control signal circuit includes a potentiometer (POT2

in figure C.10) which allows for the adjustment of the gain in order to maximise the

dynamic range of the actuators. Small differences on each actuator impedance due to

distance and/or wiring, including smaller actuators for the control mass stage, makes

it necessary to tune each circuit in order to maximise the control signal dynamic

C.2. CONTROL ELECTRONICS 285

LaCoste Heating

Roberts Linkage

Heating

Intermediate

Board Inverse

Pendulum

Roberts Linkage

LaCoste

Control Mass

Horizontal

Control Mass

Vertical

PC/DSPControl

Electronics

Current

Power

Supplies

Vacuum

10-6 mbar

Figure C.2: Control electronics and wiring strategy. A 100 pin Sub–D connector is used

to connect the PC/DSP which host the local control software and processor to the control

electronics chassis through its backplane. From here the I/O signals are distributed to

the high current power supplies or to the magnetic actuators. While the shadow sensors

provide the sensing, magnetic actuators control the positioning of each stage. The signals

are distributed through an intermediate board which distributes them to each axis on the

isolator. Low frequency position control is also obtained using current power supplies. These

signals are distributed separately through high current wires to each stage.

range. An external signal can be connected to this circuit through a connector in the

front panel. The external signal is added to the control signal from the local control

system. The filter board (shown in figures C.13 and C.14) contains five independent

filter circuits connecting the control signals to the current power supplies that drive

each of the degrees of freedom of the ohmic position control. Four are used for each

Roberts linkage axis and one for the LaCoste stage which connects all the springs

of this stage in series. Figure C.3 shows the distribution of the different boards and

channels that populate the control electronics.

Figure C.4 shows the location of the different connection points in the isolator

frame, using the east arm ITM vibration isolator as an example. The connectors

with only shadow sensors (ss) are installed for the Roberts linkage sensing, and the

horizontal actuators and shadow sensors (as) are for the inverse pendulum stage.

High current connections (h) are used for the ohmic position control, which provides

ultra low frequency position control for the Roberts linkage and LaCoste stages, while


(a)

Invers

e P

endu

lum

A

Invers

e P

endu

lum

D

Invers

e P

endu

lum

C

Invers

e P

endu

lum

B

La

Coste

Vert

ical C

La

Coste

Vert

ical A

Ro

bert

s lin

kage D

Ro

bert

s lin

kage A

Hig

h C

urr

ent

Filte

rs 1

Hig

h C

urr

ent

Filte

rs 0

Co

ntr

ol M

ass H

orizo

nta

l A

Co

ntr

ol M

ass S

pa

re

Co

ntr

ol M

ass H

orizo

nta

l C

-DC

on

tro

l M

ass H

orizo

nta

l B

-C

Co

ntr

ol M

ass T

ilt

DC

ontr

ol M

ass T

ilt

B

Optical Lever

Input

Au

xilia

ry I/O

(b)

Figure C.3: Control electronics board distribution. Each board has two channels which

filter and amplify the signals from shadow sensors and magnetic actuators. The heating

positioning control signal is connected directly from the DSP to the high current power

supply through the filters board.

vertical actuation (va) is only for the LaCoste stage. In addition, five shadow sensors

and actuators pairs are necessary for the control mass stage. These connectors are

not visible from the top view of the vibration isolator. The central tube connections

(c) are for the electrostatic control and its wiring is fed through the central tube of

the multistage pendulum in order to reach the suspension cage at the bottom where

the electrostatic board is installed next to the test mass.

Since each vibration isolator is orientated according to the test mass orientation,

and therefore with respect to the incoming laser beam, they will have different ge-

ographical orientation. For this reason A, B, C and D are defined as the cardinal

C.2. CONTROL ELECTRONICS 287

North

East ITM Isolator

x

y

z

+V

A

B

C

D

h1

c3

as3op2c2

va1

as2

h1

ss1

va2

as4

h4

c4ss2

op1

c1

h2

h3

ss – shadow sensors

as – horz. actuator +

shad. sens.

c – central tube

h – high current/heating

op – options/auxiliary

va – vertical actuator

Intermediate

Board

High current

board

Figure C.4: The figure shows the distribution of the different connection points in the

isolator. It also shows the distribution actuators for the inverse pendulum and LaCoste

stages and the direction of the actuators response to a positive control signal.

points for the vibration isolators. Each vibration isolator will have different geo-

graphical orientation but the same relative orientation within its components. This

facilitates the local control system set-up since it does not need to be tailored for each

isolator and as a consequence the wiring and channel distribution remains the same.

Inside the vacuum tank each shadow sensor and actuator is ultra high vacuum

compatible and each has to be wired to a connection port requiring ultra high vac-

uum compatible boards, connectors, pins and wires. Polyimide insulated copper

wires of 0.25 mm diameter were used in order to connect the shadow sensors and

actuators around the vibration isolator. The wiring was arranged so as to minimise

the transmission of vibration. It was also designed to be modular, with several iden-

tical extensions and connectors fabricated in parallel. This strategy simplified the

replacement of faulty pieces.

The intermediate board shown in figure C.4 distributes the I/O signals within the

vibration isolator. This board was developed in order to collect the multiple signals


from the shadow sensors and actuators in the isolator and group them in a smaller

number of DB–25 terminals that could be easily connected through a six-way cross,

used as feed-through connection port to the “outside world” as shown in figure C.5.

From the six-way cross, 25 vias cables with standard DB–25 connectors were used to

connect the signals to the control electronics. The east board in figure C.4 distributes

the high current signals for the heating stages of the LaCoste and Roberts linkage. A

picture of the intermediate board as installed on the ITM vibration isolator are shown

in figure C.6, while the diagrams for the signal distribution are shown in figures C.15,

C.16, C.17, C.18, and C.19.

(a) (b)

Figure C.5: The picture shows a 6-way cross used as wires feed-through. Each of the

remaining sides has a vacuum compatible DB–25 that connects the shadow sensors and

magnetic actuators signals.

A backplane was developed in order to distribute the I/O signals between the

different control boards, the vibration isolator, and the DSP. A block diagram of the

backplane is shown in figure C.20 which is followed by the signal distribution through

the backplane connectors. Figure C.21 shows the different I/O signals for the hori-

zontal pre-isolator which correspond to the inverse pendulum actuators and shadow

sensors. Figure C.22 shows the I/O signals for the vertical axis which correspond to

the LaCoste actuators and shadow sensors. Figure C.23 shows the Roberts linkage

C.3. CONCLUSIONS 289

shadow sensors. Figure C.24 shows the signals for the ohmic position control. Fig-

ure C.25 shows the horizontal control mass I/O signals. Figure C.26 shows the tilt (or

pitch) control mass I/O signals. Figure C.27 shows the distribution of the I/O signals

of the different connectors at the backplane. Figure C.28 shows the signal distribution

of the 100 pin Sub-D connector which connects the DSP to the backplane.

(a) (b)

Figure C.6: (a) Intermediate board installed at the ITM vibration isolator. (b) shows

the intermediate board and to the right the high current board.

C.3 Conclusions

The current DSP based local control system will be replaced with a multi-CPU digital

controller server running real-time Linux. The server will be configured to allow

separate control loops to run on each of the CPU cores. A PCI-Express expander

chassis will be used to allow fibre-optic cables to be run to each of the isolators (ITM

and ETM) on each arm of the interferometer. Analog to Digital and Digital to Analog

converters, as well as Anti-Aliasing and Anti-Imaging filters will be installed at each

station. This will connect to the control electronics currently in use for the signal

distribution on each of the vibration isolators.


11

22

33

44

55

66

DD

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Figure C.7: Control board block diagram.


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Date

:

File:

Revis

ion

:

Sheet

of

B

Dra

wn

by:

Pablo

Barr

iga

R36

10

0K

32

6

7 4

8

1

5

U8

OP

90G

P

R35

5K

R34

5K

R39

10

0

R38

68

R37

10

0

10

uF

C32

Tan

t

1u

F

C33

Tan

t

1u

F

C31

Tan

t

3

2

1Q1

PN

2222

A

D3

BZ

X7

9C

2V

4

LE

D1

Fro

nt

LE

D

+15

VC

C

-15

VC

C

LE

D+

con

necte

d to

+15

Vcc

TP

5 Test

Poin

t

TP

6 Test

Poin

t

20

K

1 3

2P

OT

1

25

Tu

rns

Con

necte

d to

P-3

2

Con

necte

d to

P-3

1L

ED

-L

ED

-

Figure C.8: LED circuit diagram.


11

22

33

44

55

66

DD

CC

BB

AA

46

Du

al

Ph

oto

det

ect

or C

ircu

it

19

/06/2

00

9

Pho

toD

iode V

51.S

CH

DO

C

Tit

le

Siz

e:

Nu

mb

er:

Date

:

File:

Revis

ion

:

Sheet

of

B

Dra

wn

by:

Pablo

Barr

iga

-15

VC

C

R20

22

K

C12

48

nF

C11

1u

F R18

16

K

R19

16

K

+15

VC

C

-15

VC

C

C10

1u

F

+15

VC

C

-15

VC

C

R17

22

K

R16

22

K

C9

1u

F

C8

.47uF

+15

VC

C

-15

VC

C

C7

1u

F

R15

15

.8K

R14

10

0K

+15

VC

C

-15

VC

C

-15

VC

C

+15

VC

C

R1

22

K

C1

48

nF

C2

1u

F R3

16

K

R2

16

K

+15

VC

C

-15

VC

C

C3

1u

F

+15

VC

C

-15

VC

C

R4

22

K

R5

22

K

C4

1u

F

C5

.47uF

+15

VC

C

-15

VC

C

C6

1u

F

R6

15

.8K

R8

1K

R7

10

0K

+15

VC

C

-15

VC

C

R9

10

K

+15

VC

C

-15

VC

C

R11

10

K

R10

10

K

R12

10

K

PD

1 O

UT

PD

2 O

UT

+15

VC

C

C30

.1uF

C29

.1uF C

27

.1uF

C28

.1uF

C25

.1uF

C26

.1uF

P3

TP

1

P1

TP

3

P2

TP

2

109

8C

4 11

U2

C

LT

112

5C

N

109

8C

4 11

U4

C

LT

112

5C

N

56

7B

4 11

U2

B

LT

112

5C

N

2 3

1

4 11

A

U2

A

LT

112

5C

N

12

13

14

D

4 11

U2

D

LT

112

5C

N

2 3

1

4 11

A

U4

A

LT

112

5C

N

12

13

14

D

4 11

U1

D

LT

112

5C

N

2 3

1

4 11

A

U1

A

LT

112

5C

N

109

8C

4 11

U1

C

LT

112

5C

N

12

13

14

D

4 11

U4

D

LT

112

5C

N

56

7B

4 11

U1

B

LT

112

5C

N

R13

1K

Con

necte

d to

P-2

3

Con

necte

d to

P-24

Con

necte

d to

P-2

7

Con

necte

d to

P-2

8

PD

1-B

ack

PD

1-B

ack

PD

2-B

ack

PD

2-B

ack

PD

2 O

UT

PD

1 O

UT

Figure C.9: Photo-detectors circuit diagram.


11

22

33

44

55

66

DD

CC

BB

AA

36

Co

ntr

ol

Sig

na

l C

ircu

it

19

/06/2

00

9

Con

trolS

ignal

V5

1.S

CH

DO

C

Tit

le

Siz

e:

Nu

mb

er:

Date

:

File:

Revis

ion

:

Sheet

of

B

Dra

wn

by:

Pablo

Barr

iga

R24

16

K

R25

16

K

R27

22

K

R28

15

.8K

R26

22

KR

29

10

0K

C14

1u

F

C16

.47uF

C17

1u

F

C13

1u

F

C15

1u

F

-15

VC

C

+15

VC

C

-15

VC

C

-15

VC

C

+15

VC

C

5

2 3

4

6

7

1

8

U6

LT

102

8A

CN

8

R31

10

R32

10

0

R33

10

K

-15

VC

C

+15

VC

C

22

uF

C19

Tan

t

22

uF

C18

Tan

t

24

5 3

1U

7

BU

F63

4T

-15

VC

C

+15

VC

C

C20

47

pF

+15

VC

C

CO

IL+

CO

IL-

R23

10

K

+15

VC

C

-15

VC

C

R22

10

K

R21

10

K

C23

.1uF C

24

.1uF

P5

CS_

Fro

nt

TP

4

Test

Poin

t

109

8C

4 11

U3

C

LT

112

5C

N

12

13

14

D

4 11

U3

D

LT

112

5C

N

56

7B

4 11

U3

B

LT

112

5C

N

2 3

1

4 11

A

U3

A

LT

112

5C

N

1K

13

2

PO

T2

25

Tu

rns

Con

necte

d to

P-2

9

Con

necte

d to

P-3

0

Con

necte

d to

P-2

5

CS B

ack

CS B

ack

CO

IL+

CO

IL-

R22

b

10

K

+15

VC

C

-15

VC

C

P4

CSM

_Fro

nt

56

7B

4 11

U4

B

LT

112

5C

N

R40

10

K

R41

10

K

Figure C.10: Control signal circuit diagram.


11

22

33

44

55

66

DD

CC

BB

AA

26

Co

ntr

ol

Boa

rd P

ow

er S

up

ply

19

/06/2

00

9

Pow

erS

up

ply

V5

1.S

CH

DO

C

Tit

le

Siz

e:

Nu

mb

er:

Date

:

File:

Revis

ion

:

Sheet

of

B

Dra

wn

by:

Pablo

Barr

iga

C34

.33uF

+15

VC

C

-15

VC

C

Volt

ag

e R

egu

lato

rs a

nd F

ilte

rs

C36

.1uF

IN3

1

OU

T2

GN

D

U1

0

L7

915

CV

IN1

3

OU

T2

GN

D

U9

L7

815

CV

D1

1N

400

1

D2

1N

400

1

Con

necte

d to

P-1

Con

necte

d to

P-3

GN

D C

on

necte

d to

P-2

2.2

uF

C35

Tan

t

1u

F

C37

Tan

t

-20

Vcc

20

Vcc

20

Vcc

-20

Vcc

Figure C.11: Power supply circuit diagram.


11

22

33

44

55

66

DD

CC

BB

AA

66

Co

ntr

ol

Boa

rd P

Con

nec

tor

19

/06/2

00

9

P_C

onnecto

rs V

51

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

File:

Revis

ion

:

Sheet

of

B

Dra

wn

by:

Pablo

Barr

iga

1B

2A

2B

1A

2C

4A

4B

4C

6A

6B

6C

8A

8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

P1

Con

trol

Board

PD

1 O

ut

PD

2 O

ut

PD

1 I

N

Coil

-

LE

D -

LE

D +

Coil

+

PD

2 I

N

LE

D-

+15

VC

C

CS B

ack

CO

IL+

CO

IL-

CS B

ack

LE

D-

CO

IL+

CO

IL-

PD

1-B

ack

PD

1-B

ack

PD

2-B

ack

PD

2-B

ack

PD

2 O

UT

PD

1 O

UT

PD

2 O

UT

PD

1 O

UT

-20

Vcc

20

Vcc

-20

Vcc

20

Vcc

Figure C.12: P connector signal distribution diagram.


11

22

33

44

DD

CC

BB

AA

12

Fil

ters

Boa

rd

19

/06/2

00

9

HC

_B

oard

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

Repeat (F

C,1

,5)

Fil

ter

V5

0.S

ch

Doc

+20

VC

C

-20

VC

C

C34

.33uF

+15

VC

C

-15

VC

C

C36

.1uF

IN3

1

OU

T2

GN

D

U1

0

L7

91

5C

V

IN1

3

OU

T2

GN

D

U9

L7

81

5C

V

D1

1N

40

01

D2

1N

40

01

Con

necte

d t

o P

-1

Con

necte

d t

o P

-3

GN

D C

on

necte

d t

o P

-2

2.2

uF

C35

Tan

t

1u

F

C37

Tan

t

Figure C.13: Filters board and power supply circuit diagram.


11

22

33

44

DD

CC

BB

AA

22

Fil

ter

Cir

cuit

19/06

/200

9

Filte

r V

50

.SchD

oc

Title

Siz

e:

Nu

mber:

Date:

File:

Revis

ion

:

Sheet

of

A4

Dra

wn b

y:

Pablo

Barri

ga

16K

R1

16K

R2

22K

R3

22K

R4

60K

R5

1uF

C1

1uF

C2

1uF

C3

.47u

F

C4

INP

UT

0.1

uF

C3

7

0.1

uF

C3

6

+1

5V

-15

V

10K

R7

60K

R6

10K

R8

1uF

C5

68n

F

C6

1uF

C7

OU

TP

UT

+1

5V

-15

V

+1

5V

-15

V

-15

V

+1

5V

2 3

1

4 11

A

U1

A

LT

11

25C

N

4 11

56

7B

U1

B

LT

11

25C

N

4 11

12

13

14

D

U1

D

LT

11

25C

N

4 11

109

8C

U1

C

LT

11

25C

N

Figure C.14: High current signal filters circuit diagram.


11

22

33

44

DD

CC

BB

AA

15

Vib

rati

on I

sola

tor

Inte

rmed

iate

Boa

rd

19

/06/2

00

9

Inte

rBoard

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

Vert

2

Vert

3

Vert

6

Vert

7

CMT6

CMT7

CMT8

CMT9

CMT10

CMT11

AU

X1

2

AU

X1

3

AU

X1

5

AU

X1

4

AU

X1

6

AU

X1

7

AU

X1

8

AU

X1

9

AU

X1

1

AU

X1

0

Vert

ical

& R

ob

erts

Lin

kage

Vert

ical

_R

L.S

chD

oc

CMT6

CMT7

CMT8

CMT9

CMT10

CMT11

Con

trol

Mas

s H

ori

zo

nta

l

Con

trolM

assH

.Sch

Doc

Vert

2

Vert

3

Vert

6

Vert

7

Ho

rizon

tal

Ho

rizon

tal.

Sch

Doc

AU

X1

0

AU

X1

1

AU

X1

2

AU

X1

3

AU

X1

4

AU

X1

5

AU

X1

6

AU

X1

7

AU

X1

8

AU

X1

9

Ele

ctro

static C

ontr

ol

Ele

ctro

static.S

ch

Doc

Figure C.15: Intermediate board block diagram.


11

22

33

44

DD

CC

BB

AA

25

Ho

rizo

nta

l S

ign

al

Dis

trib

uti

on

19

/06/2

00

9

Ho

rizon

tal.

Sch

Doc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J4 Ho

rizon

tal

B +

Vert

ical

A

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

Coil+

Coil-

Coil+

Coil-

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J3 Ho

rizon

tal

A +

C

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J2 Ho

rizon

tal

D +

Vert

ical C

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

Coil+

Coil-

Coil+

Coil-

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J1 Ho

rizon

tal

Horz0

Horz1

Horz2

Horz3

Horz4

Horz5

Horz6

Horz7

Horz8

Horz9

Horz10

Horz11

Horz18

Horz19

Horz20

Horz21

Horz22

Horz23

Horz12

Horz13

Horz14

Horz15

Horz16

Horz17V

ert

2V

ert

3

Vert

2

Vert

6V

ert

6

Vert

7

Ho

rizon

tal

D

Ho

rizon

tal

B

Ho

rizon

tal

A

Ho

rizon

tal

C

Vert

ical

Actu

ato

r A

Vert

ical

Actu

ato

r C

Vert

7

Vert

3

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Male

DB

-25 p

in d

istr

ibuti

on

Figure C.16: Horizontal axis connections circuit diagram.


11

22

33

44

DD

CC

BB

AA

35

Ver

tica

l a

nd

Ro

ber

ts L

ink

age

Sig

na

ls

19

/06/2

00

9

Vert

ical

_R

L.S

chD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J5 Vert

ical

& R

L

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J6 LaC

oste

Vert

ical

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J7 Rob

erts

Lin

kage

Vert0

Vert1

Vert4

Vert5

CMT6

CMT7

CMT8

CMT9

CMT10

CMT11

RL0

RL1

RL2

RL3

RL4

RL5

RL6

RL7

Vert2

Vert3

Vert6

Vert7

PD1IN

PD2IN

LED-

LED+

PD1IN

PD2IN

LED-

LED+

PD1IN

PD2IN

LED-

LED+

Con

trol

Mas

s T

ilt D

La C

ost

e V

erti

cal C

Rob

erts

Lin

kage A

Rob

erts

Lin

kage D

Vert

ical

Actu

ato

r A

Vert

ical

Actu

ato

r C

Vert

7

Vert

6

Vert

3

Vert

2

CM

T6

CM

T7

CM

T8

CM

T9

CM

T1

0

CM

T1

1

PD

1IN

PD

2IN

Coil

-

Coil

+

LE

D-

LE

D+

Coil

+

Coil

-

Coil

+

Coil

-

AU

X1

0

AU

X1

1

AU

X1

2

AU

X1

3

AU

X1

4

AU

X1

5

AU

X1

6

AU

X1

7

AU

X1

8

AU

X1

9

AU

X1

0

AU

X1

1

AU

X1

2

AU

X1

3

AU

X1

4

AU

X1

5

AU

X1

6

AU

X1

7

AU

X1

8

AU

X1

9

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Male

DB

-25 p

in d

istr

ibuti

on

Figure C.17: Vertical axis connections circuit diagram.


11

22

33

44

DD

CC

BB

AA

45

Co

ntr

ol

Mas

s S

ign

al

Dis

trib

uti

on

19

/06/2

00

9

Con

trolM

assH

.Sch

Doc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J8 Con

trol

Mas

s

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J10

CM

Ho

rz A

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J9 CM

Ho

rz C

-D T

ilt

D

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J11

CM

Ho

rz B

-C T

ilt B

CMH0

CMH1

CMH2

CMH3

CMH4

CMH5

CMH6

CMH7

CMH8

CMH9

CMH10

CMH11

CMH12

CMH13

CMH14

CMH15

CMH16

CMH17

CMT0

CMT1

CMT2

CMT3

CMT4

CMT5

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

CMT7

CMT6

CMT10

CMT11

CMT8

CMT9

PD1

PD2

LED+

LED-

Coil+

Coil-

Coil+

Coil-

Con

trol

Mas

s H

ori

zo

nta

l A

Con

trol

Mas

s H

ori

zo

nta

l C

-D

Con

trol

Mas

s T

ilt B

Con

trol

Mas

s H

ori

zo

nta

l B

-C

Con

trol

Mas

s T

ilt D

CM

T6

CM

T7

CM

T8

CM

T9

CM

T1

0

CM

T1

1

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Male

DB

-25 p

in d

istr

ibuti

on

Figure C.18: Control mass connections circuit diagram.


11

22

33

44

DD

CC

BB

AA

55

Ele

ctro

sta

tic

Co

ntr

ol

Sig

na

ls

19

/06/2

00

9

Ele

ctro

static.S

ch

Doc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J12

Ele

ctro

static C

ontr

ol

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J14

Ele

ctro

static A

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J15

Ele

ctro

static C

EST0

EST1

EST2

EST3

EST4

EST5

GND

EST6

EST7

GND

EST8

EST9

EST10

EST11

EST12

EST13

EST14

EST15

SIG4

HV4

Vbias

+15V

OSC4

SIG1

HV1

GND

OSC1

-15V

SIG2

HV2

Vbias

+15V

OSC2

SIG3

HV3

GND

OSC3

-15V

AUX0

AUX1

AUX3

AUX2

AUX4

AUX5

AUX6

AUX8

AUX7

AUX9

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J13

Au

xil

iary

AUX10

AUX11

AUX12

AUX13

AUX14

AUX17

AUX19

AUX15

AUX16

AUX18

AU

X1

9

AU

X1

8

AU

X1

7

AU

X1

6

AU

X1

5

AU

X1

4

AU

X1

3

AU

X1

2

AU

X1

1

AU

X1

0

EST16

EST17

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Fem

ale

DB

-25 p

in d

istr

ibu

tion

Male

DB

-25 p

in d

istr

ibuti

on

Male

DB

-25 p

in d

istr

ibuti

on

Figure C.19: Electrostatic board connections circuit diagram.


11

22

33

44

DD

CC

BB

AA

19

Co

ntr

ol

Ele

ctro

nic

s B

ack

pla

ne

19

/06/2

00

9

Backpla

ne_2

00

8.S

chD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

PD

[0..7

]

CS

[0..

3]

Ho

rz[0

..2

3]

Ho

rizon

tal

Ho

rizon

talB

ck_

20

08

.SchD

oc

Vert

[0..7

]

CS

4

PD

[8..9

]

Vert

ical

Vert

ical

Bck_2

00

8.S

ch

Doc

RL

[0..7

]

PD

[10..

13

]

Rob

erts

Lin

kag

e

Rob

erts

Lin

kB

ck

_2

008

.SchD

oc

CS

[5..

7]

PD

[14..

19

]

CM

H[0

..17

]

Con

trolM

assH

or

Con

trolM

assH

orB

ck

_2

008

.SchD

oc

PD

[0..7

]

CS

[0..

3]

CS

4

PD

[8..9

]

PD

[10..

13

]

FL

T[5

..9]

CS

[5..

7]

PD

[14..

19

]

PD

[20..

23

]

CS

[8..

9]

DS

P1

00

Con

necto

rSu

bD

100

Bck_2

00

8.S

chD

oc

Ho

rz[0

..2

3]

Vert

[0..7

]

RL

[0..7

]

FL

T[0

..4]

CM

H[0

..17

]

CM

T[0

..11

]

DB

25

Con

necto

rsD

B25

Bck

_2

00

8.S

chD

oc

FL

T[0

..4]

FL

T[5

..9]

Fil

ter

Fil

terB

ck_2

00

8.S

chD

oc

CM

T[0

..11

]

PD

[20..

23

]

CS

[8..

9]

Con

trolM

assT

ilt

Con

trolM

assT

iltB

ck_2

008

.SchD

oc

Figure C.20: Backplane board block diagram.


11

22

33

44

DD

CC

BB

AA

29

Ho

rizo

nta

l S

ign

als

Dis

trib

uti

on

19

/06/2

00

9

Ho

rizon

talB

ck_

20

08

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J1 Ho

rizon

tal

1 C

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J2 Ho

rizon

tal

1 A

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J3 Ho

rizon

tal

2 B

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J4 Ho

rizon

tal

2 D

-15

VC

C

15

VC

C15

VC

C

-15

VC

C-1

5V

CC

15

VC

C

-15

VC

C

15

VC

C

PD

[0..7

]

PD0

PD1

PD2

PD3

PD4

PD5

PD6

PD7

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

CS

0C

S1

CS

2C

S3

CS

[0..

3]

Ho

rz[0

..2

3]

PD

1 I

N

Coil

-

LE

D -

LE

D +

Coil

+

PD

2 I

N

Horz0

Horz1

Horz3

Horz5

Horz4

Horz2

Horz6

Horz8

Horz10

PD

1 I

N

Coil

-

LE

D -

Horz7

Horz9

Horz11

PD

2 I

N

Coil

+

LE

D +

Horz12

Horz14

Horz16

PD

1 I

N

Coil

-

LE

D -

Horz13

Horz15

Horz17

PD

2 I

N

Coil

+

LE

D +

Horz18

Horz20

Horz22

PD

1 I

N

Coil

-

LE

D -

Horz19

Horz21

Horz23

PD

2 I

N

Coil

+

LE

D +

Ho

rz[0

..2

3]

PD

[0..7

]

CS

[0..

3]

F

or

East

arm

IT

M:

H

ori

zon

tal

1 A

= N

ort

h

H

ori

zon

tal

2 B

= E

ast

H

ori

zon

tal

1 C

= S

ou

th

H

ori

zon

tal

2 D

= W

est

Figure C.21: Backplane horizontal axis signal distribution circuit diagram.


11

22

33

44

DD

CC

BB

AA

39

Ver

tica

l S

ign

als

Dis

trib

uti

on

19

/06/2

00

9

Vert

ical

Bck_2

00

8.S

ch

Doc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J5 Vert

ical

1 C

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J6 Vert

ical

1 A

-15

VC

C

15

VC

C15

VC

C

-15

VC

C

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 I

N

Coil

-

LE

D -

LE

D +

Coil

+

PD

2 I

NP

D1

IN

Coil

-

LE

D -

PD

2 I

N

Coil

+

LE

D +

Vert

[0..7

]

PD

[8..9

]

Vert2

Vert3

Vert4

Vert5

Vert6

Vert7

Vert1

Vert0

PD8

PD9

CS

4

CS

4

PD

[8..9

]

Vert

[0..7

]

F

or

East

arm

IT

M:

V

ert

ical

1 C

= S

ou

th

V

ert

ical

1 A

= N

ort

h

CS

4

Figure C.22: Backplane vertical axis signal distribution circuit diagram.


11

22

33

44

DD

CC

BB

AA

49

Ro

bert

s L

ink

age

Sig

nal

s D

istr

ibu

tio

n

19

/06/2

00

9

Rob

erts

Lin

kB

ck

_2

008

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J7 Rob

erts

Lin

kage 1

D

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J8 Rob

erts

Lin

kage 1

A

-15

VC

C

15

VC

C15

VC

C

-15

VC

C

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 I

N

Coil

-

LE

D -

LE

D +

Coil

+

PD

2 I

NP

D1

IN

Coil

-

LE

D -

PD

2 I

N

Coil

+

LE

D +

RL

[0..7

]

RL0

RL1

RL2

RL3

RL4

RL5

RL6

RL7

RL

[0..7

]

PD10

PD11

PD12

PD13

PD

[10..

13

]P

D[1

0..

13

]

F

or

East

arm

IT

M:

Robert

s L

ink

age

1 A

= N

ort

h

Robert

s L

ink

age

1 D

= W

est

Th

e o

rien

tati

on i

s th

e a

ctu

al p

osi

tio

n o

f th

e

s

hado

w s

en

sor.

No

rth m

easu

res E

ast

-West

dis

pla

cem

ent,

whil

e W

est

measu

res N

ort

h-S

ou

th

dis

pla

cem

ent.

Figure C.23: Backplane Roberts linkage signal distribution circuit diagram.


11

22

33

44

DD

CC

BB

AA

59

Fil

ters

Sig

na

ls D

istr

ibu

tion

19

/06/2

00

9

Fil

terB

ck_2

00

8.S

chD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J9 Fil

ter

0

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J10

Fil

ter

1

-15

VC

C

15

VC

C15

VC

C

-15

VC

C

FL

T[0

..4]

FL

T[5

..9]

FLT1

FLT2

FLT3

FLT4

FLT0

FL

T[0

..4]

FLT5

FLT6

FLT7

FLT8

FLT9

FL

T[5

..9]

La C

ost

eR

L D

RL

B

RL

A

RL

CR

L B

RL

DL

a C

ost

e

RL

A

RL

C

O

upu

t to

Hig

h C

urr

ent

pow

er s

up

pli

es

(DB

-15

)

F

LT

0 -

La C

ost

e

F

LT

1 -

RL

D

F

LT

2 -

RL

C

F

LT

3 -

RL

B

F

LT

4 -

RL

A

F

rom

DS

P S

UB

D-1

00

F

LT

5 -

RL

A -

Pin

87

F

LT

6 -

RL

B

- P

in 8

8

F

LT

7 -

RL

C

- P

in 8

9

F

LT

8 -

RL

D -

Pin

90

FL

T9 -

La C

ost

e -

Pin

92

Figure C.24: Backplane filter signal distribution circuit diagram.


11

22

33

44

DD

CC

BB

AA

69

Co

ntr

ol

Mas

s H

ori

zon

tal

Sig

na

ls

19

/06/2

00

9

Con

trolM

assH

orB

ck

_2

008

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J11

Con

trol

Mas

s 1

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J12

Con

trol

Mas

s 1

A

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J13

Con

trol

Mas

s 2

B-C

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J14

Con

trol

Mas

s 2

C-D

-15

VC

C

15

VC

C15

VC

C

-15

VC

C-1

5V

CC

15

VC

C

-15

VC

C

15

VC

C

PD

[14..

19

]

PD14

PD15

PD16

PD17

PD18

PD19

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

CS

5C

S6

CS

7

CS

[5..

7]

CM

H[0

..17

]

PD

1 I

N

Coil

-

LE

D -

LE

D +

Coil

+

PD

2 I

N

CMH0

CMH1

CMH3

CMH5

CMH4

CMH2

CMH6

CMH8

CMH10

PD

1 I

N

Coil

-

LE

D -

CMH7

CMH9

CMH11

PD

2 I

N

Coil

+

LE

D +

CMH12

CMH14

CMH16

PD

1 I

N

Coil

-

LE

D -

CMH13

CMH15

CMH17

PD

2 I

N

Coil

+

LE

D +

PD

1 I

N

Coil

-

LE

D -

PD

2 I

N

Coil

+

LE

D +

CM

H[0

..17

]

PD

[14..

19

]

CS

[5..

7]

F

or

East

arm

IT

M:

Contr

ol

Mass

1 A

= N

ort

h

Contr

ol

Mass

2 B

-C =

Sou

th-E

ast

Contr

ol

Mass

2 C

-D =

Sou

th-W

est

Figure C.25: Backplane control mass horizontal signal distribution circuit diagram.


11

22

33

44

DD

CC

BB

AA

79

Co

ntr

ol

Mas

s P

itch

Sig

nal

s

19

/06/2

00

9

Con

trolM

assT

iltB

ck_2

008

.SchD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J15

Con

trol

Mas

s 3

B

1B

2A 2B

1A

2C

4A 4B

4C

6A 6B

6C

8A 8B

8C

10

A

10

B

10

C

12

A

12

B

12

C

14

A

14

B

14

C

16

A

1C

3A

3B

3C

5A

5B

5C

7A

7B

7C

9A

9B

9C

11

A

11

B

11

C

13

A

13

B

13

C

15

A

15

B

15

C

17

A

17

B

17

C

19

A

19

B

19

C

21

A

21

B

21

C

23

A

23

B

23

C

25

A

25

B

25

C

27

A

27

B

27

C

29

A

29

B

29

C

31

A

31

B

31

C

16

B

16

C

18

A

18

B

18

C

20

A

20

B

20

C

22

A

22

B

22

C

24

A

24

B

24

C

26

A

26

B

26

C

28

A

28

B

28

C

30

A

30

B

30

C

32

A

32

B

32

C

J16

Con

trol

Mas

s 3

D

-15

VC

C

15

VC

C15

VC

C

-15

VC

C

PD

1 O

ut

PD

2 O

ut

PD

1 O

ut

PD

2 O

ut

PD

1 I

N

Coil

-

LE

D -

LE

D +

Coil

+

PD

2 I

NP

D1

IN

Coil

-

LE

D -

PD

2 I

N

Coil

+

LE

D +

CM

T[0

..11

]

PD

[20..

23

]

CMT0

CMT1

CMT2

CMT3

CMT4

CMT5

CMT8

CMT9

CMT7

CMT6

PD20

PD21

PD22

PD23

CS

[8..

9]

CS

8

PD

[20..

23

]

CM

T[0

..11

]

CS

[8..

9]

CS

9 CMT10

CMT11

Figure C.26: Backplane control mass tilt signal distribution circuit diagram.


11

22

33

44

DD

CC

BB

AA

89

Ba

ckp

lan

e D

B-2

5 S

ign

als

19

/06/2

00

9

Con

necto

rsD

B25

Bck

_2

00

8.S

chD

oc

Tit

le

Siz

e:

Nu

mb

er:

Date

:

Fil

e:

Revis

ion

:

Sheet

of

A4

Dra

wn

by:

Pablo

Barr

iga

1

20

2

21

3

22

4

23

5

24

6

25

7

8

9

10

11

12

13

14

15

16

17

18

19

27

26

J17

Ho

rizon

tal

Ho

rz[0

..2

3]

Horz0

Horz1

Horz2

Horz3

Horz5

Horz6

Horz7

Horz8

Horz10

Horz11

Horz12

Horz13

Horz15

Horz16

Horz17

Horz18

Horz4

Horz9

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ical

& R

L

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[0..7

]

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Vert2

Vert3

Vert4

Vert5

Vert6

Vert7

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[0..7

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trol

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CMH12

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Figure C.27: Backplane DB–25 signal distribution diagram.


11

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Figure C.28: Backplane Sub–D 100 pin signal distribution diagram.

Appendix D

Publications List

D.1 Principal author

D.2 References

[1] P. Barriga, J.-C. Dumas, C. Zhao, L. Ju, D. G. Blair, “Compact vibration iso-

lation and suspension system for AIGO: Performance in a 72 m Fabry-Perot

cavity,” Rev. Sci. Instrum., 2009, submitted.

[2] P. Barriga, M. A. Arain, G. Mueller, C. Zhao, D. G. Blair, “Optical design of

the proposed Australian International Gravitational Observatory,” Opt. Express

17 (2009) 2149–2165.

[3] P. Barriga, B. Bhawal, L. Ju, D. G. Blair, “Numerical calculations of diffraction

losses in advanced interferometric gravitational wave detectors,” J. Opt. Soc.

Am. A 24 (2007) 1731–1741.

[4] P. Barriga and R. DeSalvo, “Study of the possible reduction of parametric insta-

bility gain using apodizing coatings in test masses,” Technical Report, T060273-

00-Z, LIGO, (2006).

[5] P. Barriga, C. Zhao, L. Ju, D. G. Blair, “Self-Compensation of Astigmatism in

Mode-Cleaners for Advanced Interferometers,” J. Phys. Conf. Ser. 32 (2006)

457–463.

[6] P. Barriga, C. Zhao, D. G. Blair, “Optical design of a high power mode-cleaner

for AIGO,” Gen. Relat. Gravit. 37 (2005) 1609–1619.

313

314 APPENDIX D. PUBLICATIONS LIST

[7] P. J. Barriga, C. Zhao, D. G. Blair, “Astigmatism compensation in mode-cleaner

cavities for the next generation of gravitational wave interferometric detectors,”

Phys. Lett. A 340 (2005) 1–6.

[8] P. J. Barriga, M. Barton, D. G. Blair, A. Brooks, R. Burman, R. Burston, E.-J.

Chin, J. Chow, D. Coward, B. Cusack, G. de Vine, J. Degallaix, J.-C. Dumas,

M. Feat, S. Gras, M. B. Gray, M. W. Hamilton, D. Hosken, E. Howell, J. S.

Jacob, L. Ju, T.-L. Kelly, B. H. Lee, C. Y. Lee, K. T. Lee, A. Lun, D. E.

McClelland, K. McKenzie, C. Mow-Lowry, A. Moylan, D. Mudge, J. Munch, D.

Rabeling, D. H. Reitze, A. Romann, S. Schediwy, S. M. Scott, A. S. Searle, B.

S. Sheard, B. J. Slagmolen, P. J. Veitch, J. Winterflood, A. Woolley, Z. Yan, C.

Zhao, “Status of ACIGA High Power Test Facility for advanced interferometry,”

SPIE Proceedings: Gravitational Wave and Particle Astrophysics Detectors 5500

(James Hough and Gary H. Sanders) (2005) 70–80.

[9] P. Barriga, M. Barton, D. G. Blair, A. Brooks, R. Burman, R. Burston, E.-J.

Chin, J. Chow, D. Coward, B. Cusack, G. de Vine, J. Degallaix, J.-C. Dumas,

M. Feat, S. Gras, M. Gray, M. Hamilton, D. Hosken, E. Howell, J. S. Jacob, L.

Ju, T.-L. Kelly, B. H. Lee, C. Y. Lee, K. T. Lee, A. Lun, D. E. McClelland, K.

McKenzie, C. Mow-Lowry, A. Moylan, D. Mudge, J. Munch, D. Rabeling, D.

Reitze, A. Romann, S. Schediwy, S. M. Scott, A. Searle, B. S. Sheard, B. J. J.

Slagmolen, P. Veitch, J. Winterflood, A. Woolley, Z. Yan, C. Zhao, “Technology

developments for ACIGA high power test facility for advanced interferometry,”

Class. Quantum Grav. 22 (2005) S199–S208.

[10] P. Barriga, A. Woolley, C. Zhao, D. G. Blair, “Application of new pre-isolation

techniques to mode-cleaner design,” Class. Quantum Grav. 21 (2004) S951–S958.

D.3. ACIGA COLLABORATION 315

D.3 ACIGA collaboration

D.4 References

[1] J.-C. Dumas, P. Barriga, C. Zhao, L. Ju, and D. G. Blair, ”Compact suspension

systems for AIGO: Local control system,” Rev. Sci. Instrum. 2009, submitted.

[2] L. Ju, D. G. Blair, C. Zhao, S. Gras, Z. Zhang, P. Barriga, H. Miao, Y. Fan

and L. Merrill, “Strategies for the control of parametric instability in advanced

gravitational wave detectors,” Class. Quantum Grav. 26 (2009) 015002 (15pp).

[3] C. Zhao, L. Ju, Y. Fan, S. Gras. B. J. J. Slagmolen, H. Miao, P. Barriga, D.G.

Blair, D. J. Hosken, A. F. Brooks, P. J. Veitch, D. Mudge, J. Munch, “Obser-

vation of Three Mode Parametric Interactions in Long Optical Cavities,” Phys.

Rev. A 78 (2008) 023807 (6pp).

[4] H. Miao, C. Zhao, L. Ju, S. Gras, P. Barriga, Z. Zhang and D. G. Blair, “Three-

mode optoacoustic parametric interactions with a coupled cavity,” Phys. Rev. A

78 (2008) 063809 (6pp).

[5] D. G. Blair, P. Barriga, A. F. Brooks, P. Charlton, D. Coward, J-C. Dumas, Y.

Fan, D. Galloway, S. Gras, D. J. Hosken, E. Howell, S. Hughes, L. Ju, D. E.

McClelland, A. Melatos, H. Miao, J. Munch, S. M. Scott, B. J. J. Slagmolen,

P. J. Veitch, L. Wen, J. K. Webb, A. Wolley, Z. Yan, C. Zhao, “The Science

benefits and Preliminary Design of the Southern hemisphere Gravitational Wave

Detector AIGO,” J. Phys. Conf. Ser. 122 (2008) 012001 (6pp).

[6] C. Zhao, D.G. Blair, P. Barriga, J. Degallaix, J.-C. Dumas, Y. Fan, S. Gras, L.

Ju, B. Lee, S. Schediwy, Z. Yan, D. E. McClelland, S. M. Scott, M. B. Gray, A. C.

Searle, S. Gossler, B. J. J. Slagmolen, J. Dickson, K. McKenzie, C. Mow-Lowry,

A. Moylan, D. Rabeling, J. Cumpston, K. Wette, J. Munch, P. J. Veitch, D.

Mudge, A. Brooks and D. Hosken, “Gingin High Optical Power Test Facility,”

J. Phys. Conf. Ser. 32 (2006) 368–373.

[7] D. E. McClelland, S. M. Scott, M. B. Gray, A. C. Searle, S. Goßler, B. J. J.


Slagmolen, J. Dickson, J. H. Chow, G. de Vine, K. McKenzie, C. M. Mow-

Lowry, A. Moylan, D. S. Rabeling, B. S. Sheard, J. Cumpston, K. Wette, D. G.

Blair, L. Ju, R. Burman, D. Coward, C. Zhao, P. Barriga, E. Chin, J. Degallaix,

Y. Fan, S. Gras, E. Howell, B. Lee, S. Schediwy, Z. Yan, J. Munch, P. J. Veitch,

D. Mudge, A. Brooks, and D. Hosken, “Status of the Australian Consortium

for Interferometric Gravitational Astronomy,” Class. Quantum Grav. 23 (2006)

S41–S50.

[8] Z. Yan, C. Zhao, L. Ju, S. Gras, P. Barriga and D. G. Blair, “Automatic Rayleigh

scattering mapping system for optical quality evaluation of test masses for gravity

wave detectors” Rev. Sci. Instrum. 76 (2005) 015104 (5pp).

[9] L. Ju, M. Aoun, P. Barriga, D. G. Blair, A. Brooks, R. Burman, R. Burston, X.

T. Chin, E. J. Chin, C. Y. Lee, D. Coward, B. Cusack, G. de Vine, J. Degallaix,

J. C. Dumas, F. Garoi, S. Gras, M. Gray, D. Hoskens, E. Howell, J. S. Jacob,

T. L. Kelly, K. T. Lee, T. Lun, D. McClelland, C. Mow-Lowry, D. Mudge, J.

Munch, S. Schediwy, S. Scott, A. Searle, B. Sheard, B. Slagmolen, P. Veitch, J.

Winterflood, A. Woolley, Z. Yan, C. Zhao, “ACIGA’s High Optical Power Test

Facility,” Class. Quantum Grav. 21 (2004) S887–S893.

[10] J. S. Jacob, P. Barriga, D. G. Blair, A. Brooks, R. Burman, R. Burston, L. Chan,

X. T. Chin, E. J. Chin, J. Chow, D. Coward, B. Cusack, G. de Vine, J. Degallaix,

J. C. Dumas, A. Faulkner, F. Garoi, S. Gras, M. Gray, M. Hamilton, M. Herne,

C. Collit, D. Hoskens, E. Howell, L. Ju, T. L. Kelly, B. Lee, C. Y. Lee, K. T.

Lee, A. Lun, D. McClelland, K. McKenzie, C. Mow-Lowry, D. Mudge, J. Munch,

D. Paget, S. Schediwy, S. Scott, A. Searle, B. Sheard, B. Slagmolen, P. Veitch,

J. Winterflood, A. Woolley, Z. Yan, C. Zhao, “Australia’s Role in Gravitational

Wave Detection,” Publ. Astron. Soc. Aust. 20 (2003) 223–241.

D.5. LIGO SCIENTIFIC COLLABORATION 317

D.5 LIGO Scientific Collaboration

D.6 References

[1] B. Abbott,... P. Barriga, et al, “Search for gravitational waves from low mass

binary coalescences in the first year of LIGO’s S5 data,” Phys. Rev. D 79 (2009)

122001 (14pp).

[2] B. Abbott,... P. Barriga, et al, “Einstein@Home search for periodic gravitational

waves in LIGO S4 data,” Phys. Rev. D 79 (2009) 022001 (29pp).

[3] B. Abbott,... P. Barriga, et al, “Search of S3 LIGO data for gravitational wave

signals from spinning black hole and neutron star binary inspirals,” Phys. Rev.

D 78 (2008) 042002 (19pp).

[4] B. Abbott,... P. Barriga, et al, “Search for Gravitational-Wave Bursts from Soft

Gamma Repeaters,” Phys. Rev. Lett. 101 (2008) 211102 (6pp).

[5] B. Abbott,... P. Barriga, et al, “Beating the Spin-Down Limit on Gravitational

Wave Emission from the Crab Pulsar,” Astrophys. J. Lett. 683 (2008) L45–L49.

[6] B. Abbott,... P. Barriga, et al, “Implications for the Origin of GRB 070201 from

LIGO Observations,” Astrophys. J. 681 (2008) 1419–1430.

[7] B. Abbott,... P. Barriga, et al, “Search for gravitational waves associated with

39 gamma-ray bursts using data from the second, third, and fourth LIGO runs,”

Phys. Rev. D 77 (2008) 062004 (22pp).

[8] B. Abbott,... P. Barriga, et al, “Search for gravitational waves from binary in-

spirals in S3 and S4 LIGO data,” Phys. Rev. D 77 (2008) 062002 (13pp).

[9] B. Abbott,... P. Barriga, et al, “All-sky search for periodic gravitational waves

in LIGO S4 data,” Phys. Rev. D 77 (2008) 022001 (38pp).

[10] B. Abbott,... P. Barriga, et al, “Upper limit map of a background of gravitational

waves,” Phys. Rev. D 76 (2007) 082003 (11pp).


[11] B. Abbott,... P. Barriga, et al, “Searches for periodic gravitational waves from

unknown isolated sources and Scorpius X-1: Results from the second LIGO

science run,” Phys. Rev. D 76 (2007) 082001 (35pp).

[12] B. Abbott,... P. Barriga, et al, “Search for gravitational wave radiation associated

with the pulsating tail of the SGR 1806–20 hyperflare of 27 December 2004 using

LIGO,” Phys. Rev. D 76 (2007) 062003 (12pp).

[13] B. Abbott,... P. Barriga, et al, “Upper limits on gravitational wave emission from

78 radio pulsars,” Phys. Rev. D 76 (2007) 042001 (20pp).

[14] B. Abbott,... P. Barriga, et al, “First cross-correlation analysis of interferometric

and resonant-bar gravitational-wave data for stochastic backgrounds”, Phys. Rev.

D 76 (2007) 022001 (17pp).

[15] B. Abbott,... P. Barriga, et al, “Searching for a Stochastic Background of Gravi-

tational Waves with the Laser Interferometer Gravitational-Wave Observatory,”

Astrophys. J. 659 (2007) 918–930.

aspects, techniques and design of advanced interferometric ...€¦ · chapter five was undertaken...

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