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Download date: 18. Jul. 2018

Design, Fabrication and Testing of Silicon Resonators

for Opto-Acoustic Parametric Amplifiers

Author:Francis Achilles Torres,B.Sc., M.Sc.

Supervisors:Prof. David Blair

A/Prof. Ju LiDr Alexey Veryaskin

Dr Mariusz Martyniuk

This thesis is presented for the degree of Doctor of Philosophy

of The University of Western Australia.

School of Physics

2013

A father once put gold in the ground and years later, forgot where it lay. The man

asked his sons to dig and they worked tirelessly. They could not find it.

Instead they found a land that was now ripe and ready for grape harvest. It yielded

in abundance and they made a fortune.

—A lessons on ‘fixation of purpose’, by the philosopher Francis Bacon.

i

ABSTRACT

Design, Fabrication and Testing of Silicon Resonators for

Opto-Acoustic Parametric Amplifiers

by Francis A. Torres, B.Sc., M.Sc.

Supervisors: Prof. D.G. Blair, A/Prof. L. Ju,

Dr A. Veryaskin and Dr M. Martyniuk

The development of gravitational-wave detectors has involved pushing the frontiers of

technology in many areas. One of these efforts, km-long optical cavities with high optical

power, led to the realisation that 3-mode opto-acoustic interactions could occur in these

systems [1, 2]. In researching this problem, Zhao et al. [3] recognised that the concepts

could be utilised to make new ultra-sensitive opto-acoustic devices.

The thesis reports the development of acoustic resonators that would form the heart

of a 3-mode opto-acoustic parametric amplifier (OAPA), and presents the design of such

an amplifier and predictions of its performance.

Finite element modelling (FEM) is used to choose an appropriate design for an acoustic

resonator to be incorporated in a novel OAPA device. Acoustic resonator prototypes range

from single-paddle designs [3] to 3-paddle designs inspired by Davis et al. [4]. Predictions

of optical coating losses are presented as a function of size and location of coatings on the

3-paddle resonator design. Four models are identified as showing the most promise.

Various resonator fabrication techniques and experimental setups to test resonators

are discussed. Nodes and antinodes from the acoustic wave of a torsional mode (the mode

of interest) in the frame are modelled with FEM to determine the optimal suspension

locations of a 3-point contact setup.

High quality-factors of 8.6 105 and 7.5 105, close to the design target, are observed at

room temperature, for resonators made with laser machining and dry etching techniques.

The torsional mode is identified using a Michelson interferometer and optical lever method,

consistent with FEM predictions to within 5%. The relationship between suspension

contact locations and highest obtained quality-factors is discussed.

Thermal effects of silicon resonators from laser heating are investigated; these include

the frequency shift of the acoustic modes and the relaxation times from dissipation. A

thermal model is developed and compared to these results. This model is used to predict

effects expected in a tabletop OAPA, based on a near-self-imaging cavity design. This

ii

OAPA uses the millimetre-scale silicon resonator as an end mirror, which exhibits a tor-

sional vibration mode with a frequency in the 105−106 Hz range. The OAPA design gives

a tuning coefficient of 2.46 MHz mm−1, which allows tuning between amplification and

self-cooling regimes. Based on these demonstrated resonator parameters, the OAPA is

predicted to achieve parametric instability with 1.6 μW of input power, and mode cooling

by a factor of 1.9 104 with 30 mW of input power. An improved resonator design from

silicon-on-insulator technology is developed to address thermal and suspension issues in

an OAPA device.

This body of work contributes to the development of 3-mode OAPAs as novel high-

sensitivity signal transducers.

iii

ACKNOWLEDGMENTS

I would like to extend my sincere gratitude to Prof. David G. Blair, Dr Li Ju, Dr

Alexey Veryaskin, and Dr Mariusz Martyniuk. Without the assistance and support of my

supervisors, writing this thesis would not have been possible.

Special thanks to Prof. David Blair whose extensive knowledge and expertise in all

areas of physics were essential and invaluable in leading my research and directing my

publications.

I would like to address my thanks to Dr Li Ju for her support. Her daily assistance

and easy going attitude provided just the right environment to foster a sense of team.

Dr Alexey Veryaskin always made me feel confident and his enthusiasm for life and

science was contagious.

I want to thank Dr Mariusz Martyniuk, from the School of Electrical, Electronic and

Computer Engineering, for your support and kindness. It was always a pleasure and

privilege to chat in your office. You always had fresh perspective, both on papers and on

different experimental avenues to explore.

Dr Zhao’s experience with optical cavities and machines in the lab was a major factor

in getting the experiments working in this research. Though he is not listed as a supervisor,

he acted like one for the duration of this PhD.

A special thanks to Dr. Gras, who patiently assisted my learning of the ANSYS

software and his codes, which proved instrumental in my research.

I started my research, learning and experimenting with wet KOH etching in the West-

ern Australia Centre for Semiconductor Optoelectronics and Microsystems (WACSOM) at

the School of Electrical, Electronic and Computer Engineering of the University of West-

ern Australia, Here in Perth. I extend my thanks to all in the lab who have significantly

contributed to my knowledge, especially Nir Zvison and Thuyen Nguyen. Thanks for your

patience and beautiful attitude.

I would like to thank Dr. Kai-Yu Liu for all of the dry-etched silicon resonators made in

iv

the Australian National Fabrication Facility Ltd (ANFF-Q) in Queensland. Many batches

of samples were made, and your long efforts and persistence made a giant contribution to

my experiments.

I thank Isabelle Roch-Jeune for her incredible hard work, helping to coordinate the

efforts in the fabrication of resonators with optical coatings, as well as performing the

dry etching and final steps. These coated resonators were fabricated in a collaboration

of several facilities in France: fine polishing of the wafers in the Societe Europeenne de

Systemes Optiques (SESO), dry etching in the Institute of Electronic, Micro-electronic

and Nanotechnology (IEMN), and ion-beam sputtering (IBS) of Bragg mirrors in the

Laboratoires de Materiaux Avances (LMA).

My sincere gratitude to Dr. Chao and his students Howard Pan and Huong Wei Ja, in

Taipei, Taiwan. This research team from the Institute of Photonics Technologies (IPT) at

the National Tsing Hua University, Hsinchu, in Taiwan, coordinated laser micromachining

at LEGEND Laser Inc, in Taipei, Taiwan. The highest quality factors obtained in this

research was made by them. Thank you!

Also, my visit to Taiwan was both fruitful and heartwarming. Thank you for hosting

me and showing me the wonders of beautiful Taiwan (and the hair cut).

Special thanks to Dr. Hou Wei, a visiting professor from the Institute of Semiconduc-

tors, Chinese Academy of Sciences, in Beijing, China. Dr. Hou Wei developed a Michelson

interferometer which allowed the torsional mode of silicon resonators to be identified.

I would like to thank Liu Jian and Ma Yubo, who designed an optical cavity in which

my resonators are used as an end mirror, and form the heart of an opto-acoustic parametric

amplifier. Our combined effort which led to a manuscript submitted to Applied Optics,

presented in Chapter 6, was a major contribution to the work presented in this thesis.

It was my pleasure to supervisor a few summer students during my PhD, who have

significantly assisted the collection of data from my experiments. Thank you Melanie,

Phillip, Jason, and Simon. You were a pleasure to work with and I hope I treated you

well!

I would like to thank Dr Wayne McRae, Dr Bruce Hartley, Dr Ron Burman, and

Mr Howard Golden for their assistance and support throughout this PhD, as well as for

proof-reading chapters and sections of my thesis and discussing ideas for my research.

I would also like to thank Mr Golden for playing a central role in my decision to fly over

to Australia and take part in this research, as well as the many networking connections in

geophysics which greatly assisted my work. Howard, you are my academic uncle.

v

I would like to thank the workshop staff, especially John, Gary, Dave and Steve, for

their workmanship, suggestions and friendly disposition on any day. It was always a

pleasure to come visit with new requests for my experiments.

I salute Ian McArthur, by far the best head of school I’ve ever heard of. The same

goes for Jay Jay. Thank you for all the help and true friendship. I extend my thanks to

the administration crew on the 4th floor. You are all guardian angels in my book.

To my friends and colleagues who gave me much appreciated support and made my

time at UWA such a pleasure, I give my thanks. Thank you Andrew Sunderland, Shaun,

Jean-Charles, Lucienne, Carl, Fang Qi, Chichi, Jiayi, Sundae, Akhter, Fan, Pablo, Eric,

Andrew Wooley, Haixing, Zhongyang, Sunil, and many others.

To my friends outside of work, who made my experience in Australia a true adventure

and milestone in my life, words can’t express my gratitude. The list here would be too

long. You know who you are. Thank you so much.

Finally, I would like to express my sincere gratitude to my parents. Thanks for your

love and support and always believing in me.

I dedicate this book to my first nephew, Nicodemus Gabriel Torres, and goddaughter,

Teagan Lahaie. You are the future.

vi

Contents

Table of contents x

List of figures xiv

List of tables xv

List of abbreviations xv

Useful formulae xvii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 From Euclid to LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Parametric Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Thesis Aim and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Opto-Acoustic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Resonator Requirements for a 3-mode OAPA Device . . . . . . . . . . . . . 17

1.7 Silicon Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.8 Silicon Microresonator Manufacturing Techniques . . . . . . . . . . . . . . . 23

1.8.1 Optical Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.9 Acoustic Resonators Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.10 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Finite Element Modelling 33

2.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Prototype Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Finite Element Modelling (FEM) . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Single Paddle Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1 Square Paddle Design . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vii

2.4.2 Circular Paddle Design . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Three-Paddle Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.1 Detailed Analysis of Model D . . . . . . . . . . . . . . . . . . . . . . 44

2.5.2 Summary of Predictions for the 3-Paddle Designs . . . . . . . . . . . 46

2.6 Optical Coating Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.1 Calculations of Quality Factor Losses from Optical Coatings . . . . 47

2.6.2 Optical Coatings Impact . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Fabrication and Testing Methods 51

3.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Resonator Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Methods for Testing Silicon Resonators . . . . . . . . . . . . . . . . . . . . 54

3.3.1 Optical Lever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.2 Metal Clamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.3 Yacca Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.4 Electrostatic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.5 3 Wire Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.6 Three-Point Suspension Setup . . . . . . . . . . . . . . . . . . . . . 62

3.4 FEM Analysis of Nodal Positions . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Analysis of Results 73

4.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Fabricated Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Results from Prior Testing Methods . . . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Metal Clamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.2 Yacca Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.3 Electrostatic Actuation . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.4 3-Wire Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4 Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Interferometer Method . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4.2 Sample UQB3C-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.3 Sample UQB3D-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4.4 Sample TB2C-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

viii

4.5 Three-point Suspension: Initial Configuration . . . . . . . . . . . . . . . . . 103

4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.5.2 Resonator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.5.3 Experimental Setup and Methods . . . . . . . . . . . . . . . . . . . . 109

4.5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.5.6 Effect of the Quadrant Photodetector Orientation . . . . . . . . . . 116

4.5.7 Air Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.5.8 Effects of Sample Positions via Acoustic Translations . . . . . . . . . 119

4.5.9 Nodal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.6 Three-Point Suspension: Improved Configuration . . . . . . . . . . . . . . . 124

4.6.1 Reproducibility of Results . . . . . . . . . . . . . . . . . . . . . . . . 127

4.6.2 Quality Factor vs Loading Positions . . . . . . . . . . . . . . . . . . 128

4.6.3 Surface Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5 Thermal Effects 137

5.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.3 Heat Transfer Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.3.1 Thermal Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.3.2 Radiation Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.4 Finite Element Modelling of Frequency Shift with Temperature . . . . . . . 147

5.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.5.1 Frequency Drift and Thermal Relaxation Times . . . . . . . . . . . . 148

5.5.2 Optical Coating Effects . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.5.3 Equivalence of Heating and Cooling . . . . . . . . . . . . . . . . . . 155

5.6 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.6.1 Predictions for an OAPA device . . . . . . . . . . . . . . . . . . . . 159

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6 Optical Cavity Design 161

6.1 Three-mode OAPA Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

ix

6.1.3 Silicon Resonator with Optical Coatings . . . . . . . . . . . . . . . . 166

6.1.4 Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.1.5 Predicted performance of a Practical NSI Cavity Setup . . . . . . . 173

6.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2 Silicon-on-Insulator (SOI) Resonator Design . . . . . . . . . . . . . . . . . . 178

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7 Conclusions and Future Work 183

7.1 Review of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.1.1 The Story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.2 Summary of the Results and Conclusions . . . . . . . . . . . . . . . . . . . 188

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

A Appendix 0 205

B Appendix 1 207

C Appendix 2 209

C.1 Modal Predictions with ANSYS 14.0 . . . . . . . . . . . . . . . . . . . . . . 210

C.2 Coating Loss Calculation with MATLAB . . . . . . . . . . . . . . . . . . . 213

C.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

D Appendix 3 217

D.1 Resonator Fabrication Details . . . . . . . . . . . . . . . . . . . . . . . . . . 217

D.1.1 Wet Etching in WACSOM . . . . . . . . . . . . . . . . . . . . . . . . 217

D.1.2 Dry Etching in Queensland . . . . . . . . . . . . . . . . . . . . . . . 221

D.1.3 Dry Etching and Optical Coating in France . . . . . . . . . . . . . . 223

D.1.4 Laser machining in Taiwan . . . . . . . . . . . . . . . . . . . . . . . 225

x

List of Figures

1.1 Many important contributions to science from Albert Einstein... . . . . . . 2

1.2 Four km-long detector built in... . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 A laser interferometer consisting of... . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Coupled cavity consisting of a tuning and main cavity.. . . . . . . . . . . . . 8

1.5 Parametric interaction between photons and phonons... . . . . . . . . . . . 10

1.6 In a 2-mode interaction, the carrier mode... . . . . . . . . . . . . . . . . . . 12

1.7 Three-mode opto-acoustic interactions within a free-space... . . . . . . . . . 13

1.8 Electromagnetic field patterns of radiation... . . . . . . . . . . . . . . . . . . 14

1.9 Silicon spindle resonator with an acoustic torsional... . . . . . . . . . . . . . 15

1.10 Diagram of planes, orientations and directions... . . . . . . . . . . . . . . . . 19

1.11 Frame of reference axes on silicon wafer... . . . . . . . . . . . . . . . . . . . 21

1.12 Step by step process of wet etching... . . . . . . . . . . . . . . . . . . . . . . 25

1.13 Schematic illustration of a plasma dry-etching process... . . . . . . . . . . . 26

1.14 Ion beam sputtering of tantalum pentoxide and silica... . . . . . . . . . . . . 27

1.15 Set of resonators described in the literature... . . . . . . . . . . . . . . . . . 28

2.1 Single paddle resonator design... . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Square paddle prototype... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Circular paddle design... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Silicon nitride membrane resonators... . . . . . . . . . . . . . . . . . . . . . 39

2.5 Three-paddle design, providing isolation of the middle paddle from the frame... 40

2.6 Acoustic mode shapes in which mostly the 3 paddles... . . . . . . . . . . . . 41

2.7 Acoustic mode shapes in which the frame of the sample... . . . . . . . . . . 42

2.8 Acoustic mode spectrum of designs C and D... . . . . . . . . . . . . . . . . 43

2.9 Acoustic mode spectrum of design C, near the three modes... . . . . . . . . 43

2.10 Design shape of model D, with three paddles and a wafer thickness... . . . . 44

2.11 The acoustic torsional mode of the middle paddle (mode of interest)... . . . 45

xi

2.12 The mode of interest of model D is surrounded by other acoustic modes... . 45

2.13 ANSYS view of the optical coatings... . . . . . . . . . . . . . . . . . . . . . 47

2.14 Finite element modelling of the impact of optical coatings... . . . . . . . . . 48

3.1 Optical lever on a large-mass optical table. . . . . . . . . . . . . . . . . . . 55

3.2 Close-up of a piezoceramic transducer (PZT)... . . . . . . . . . . . . . . . . 55

3.3 Measuring Qs with the spectrum analyser... . . . . . . . . . . . . . . . . . . 56

3.4 Metal clamping system ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Yacca bonding system... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 Yacca bond metal rods... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.7 Side and front view of the result of Yacca bonding. . . . . . . . . . . . . . . 60

3.8 Electrostatic excitation setup... . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.9 The 3-wire sample holding system, expected to minimize the clamping losses... 61

3.10 Close-up of a DC-PZT, with an AC-PZT glued to the end surface... . . . . 62

3.11 3-point suspension in the tank... . . . . . . . . . . . . . . . . . . . . . . . . 63

3.12 Initial system of 3-point suspension... . . . . . . . . . . . . . . . . . . . . . . 63

3.13 Improved system of 3-point suspension... . . . . . . . . . . . . . . . . . . . . 63

3.14 Loading samples on the three-point suspension with... . . . . . . . . . . . . 64

3.15 Nine plastic jigs (positioning tools)... . . . . . . . . . . . . . . . . . . . . . . 65

3.16 The resonator surface directions parallel to the torsion rod long axis... . . . 65

3.17 Resonator and three suspension point locations, forming a rigid triangle... . 67

3.18 Three modes are investigated with the FEM nodal study method... . . . . . 67

3.19 Full view of the vibration amplitude map on the frame... . . . . . . . . . . . 68

3.20 Profiles of the vibration amplitudes of the frame along the paddle-axis... . . 69

4.1 These are samples diced from the edges of the round silicon... . . . . . . . . 75

4.2 Wet etching of 3-paddle designs in the Western Australia... . . . . . . . . . 76

4.3 Wet etching of 1-paddle designs... . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 UQB1- and UQB3-series fabricated at the Australian National... . . . . . . 77

4.5 Back and front sides of samples from the UQB2-series... . . . . . . . . . . . 77

4.6 Optical microscope view of the UQB2-series... . . . . . . . . . . . . . . . . . 78

4.7 Optical microscope measurements of the sample UQB3C-1... . . . . . . . . . 78

4.8 Samples from the TB1- and TB2-series were made from laser... . . . . . . . 80

4.9 Sample FD-1, with optical coatings. (a) Image from... . . . . . . . . . . . . 80

4.10 Metal clamping method. (a) Image showing the sample... . . . . . . . . . . 82

xii

4.11 Diagram of various metal clamping locations... . . . . . . . . . . . . . . . . 83

4.12 Spectrum analyser screenshot of the... . . . . . . . . . . . . . . . . . . . . . 85

4.13 Yacca bonding method. (a) Image of the resonator bonded... . . . . . . . . 85

4.14 Three-wire suspension system and finite element modelling... . . . . . . . . 88

4.15 Graphic analysis to estimate the suspension locations... . . . . . . . . . . . 89

4.16 Three-wire suspension system. (a) Ringdown... . . . . . . . . . . . . . . . . 90

4.17 Regions of the resonator relative to the edges... . . . . . . . . . . . . . . . . 92

4.18 Interferometer method, developed by Dr. Hou Wei... . . . . . . . . . . . . . 93

4.19 The lines scanned with the interferometer... . . . . . . . . . . . . . . . . . . 94

4.20 The modes of interest. (a) Common mode... . . . . . . . . . . . . . . . . . . 95

4.21 Finite element modelling predictions of mode shapes... . . . . . . . . . . . . 97

4.22 Amplitude ratio measurements made with the optical lever... . . . . . . . . 98

4.23 Amplitude ratio measurements made with the optical lever... . . . . . . . . 98

4.24 Interferometer measurements on sample UQB3C-1... . . . . . . . . . . . . . 99

4.25 Interferometer measurements done with sample UQB3D-1... . . . . . . . . . 101

4.26 Amplitude ratio measurements done with sample TB2C-1 . . . . . . . . . . 102

4.27 Micro-mechanical resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.28 Schematic drawing of the experimental setup. . . . . . . . . . . . . . . . . . 110

4.29 Comparison of modelling and data. . . . . . . . . . . . . . . . . . . . . . . . 112

4.30 Observed high quality factor. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.31 The quadrant photodiode (QPD)... . . . . . . . . . . . . . . . . . . . . . . . 117

4.32 Plot of the Q of sample UQB3C-1... . . . . . . . . . . . . . . . . . . . . . . 118

4.33 Repeated measurements of the Q... . . . . . . . . . . . . . . . . . . . . . . . 119

4.34 Method of moving the sample on the 3-point suspension... . . . . . . . . . . 120

4.35 Finite element modelling used to determine new... . . . . . . . . . . . . . . 122

4.36 Repeatability of Q measurements... . . . . . . . . . . . . . . . . . . . . . . . 127

4.37 Quality factors measured at different loading positions . . . . . . . . . . . . 130

5.1 Tabletop setup. Copper plate stage resting on... . . . . . . . . . . . . . . . . 138

5.2 The silicon resonator structure and an electrical circuit representation... . . 140

5.3 Schematic of radiation heat transfer mechanisms... . . . . . . . . . . . . . . 143

5.4 Relaxation time vs emissivity of silicon... . . . . . . . . . . . . . . . . . . . . 146

5.5 Finite element modelling (FEM) prediction of frequency change... . . . . . . 147

5.6 Diagram depicting the 650 nm laser power, and power meter... . . . . . . . 149

5.7 Measurement of the frequency drift of sample UQB3C-1 . . . . . . . . . . . 150

xiii

5.8 Observed frequency drift with laser exposure time... . . . . . . . . . . . . . 150

5.9 Measurement of the frequency drift of sample UQB3C-1... . . . . . . . . . . 151

5.10 Frequency drift measurements on sample UQB3C-1... . . . . . . . . . . . . . 152

5.11 Frequency drift of the coated resonator FD-1 from France... . . . . . . . . . 154

5.12 Diagram depicting the 650 nm laser power, and power meter... . . . . . . . 155

5.13 Experimental measurements of frequency drift with laser on (heating)... . . 156

6.1 Micromechanical silicon resonator with optical coating... . . . . . . . . . . . 167

6.2 Near-self-imaging optical cavity concept and key parameters... . . . . . . . . 169

6.3 Mode gap as a function of the position of the micromechanical... . . . . . . 172

6.4 Experimental setup of the near-self-imaging (NSI) cavity... . . . . . . . . . . 173

6.5 Expected parametric gain for given laser input power... . . . . . . . . . . . 175

6.6 Resonator design from a 2 mm thick silicon-on-insulator wafer... . . . . . . 178

6.7 Torsional mode shape (mode of interest) predicted by FEM... . . . . . . . . 179

6.8 Zoom-in to three regions of interest, showing... . . . . . . . . . . . . . . . . 180

6.9 A narrow strip of the resonator along the left support... . . . . . . . . . . . 181

D.1 Steps for wet etching processing: cleaving, cleaning, applying SiNx mask

layers, photolithography, RIE to expose silicon, KOH etching, and HF strip

to remove the masks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

D.2 Diamond tip scriber to cut the wafer into 2 cm by 2cm square samples. . . 218

D.3 Prior to KOH etching, the silicon samples are covered in SiNx masking

layers. (d) Microscope examination reveals scratches and bubbles formed

in the masking layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

D.4 Photolithography. (a) Hood for spin-coating of AZ2035 negative photore-

sist. (b) UV exposure machine. . . . . . . . . . . . . . . . . . . . . . . . . . 219

D.5 Glass plate with printed pattern for UV exposure during photolithography. 220

D.6 Steps for DRIE process in Queensland, using a layer of Omnicoat to strip

the SU8 resin after DRIE etching. . . . . . . . . . . . . . . . . . . . . . . . 222

D.7 Steps for DRIE etching and optical coating performed in France (SESO,

IEMN, LMA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

D.8 A diagram of the layers in optical coatings, for a configuration of L(HL)NH2L,

consisting of ‘N’ pairs, a bottom layer of silica, another layer of tantalum

after the N pairs, and finished with a thicker layer of silica. . . . . . . . . . 225

D.9 Steps for laser machining and optical coating of samples made in Taiwan. . 226

xiv

List of Tables

1.1 Stiffness C (in GPa) and compliance S (in 10−12 Pa)... . . . . . . . . . . . . 20

1.2 Young’s modulus (in GPa), Poisson’s ratio and... . . . . . . . . . . . . . . . 21

1.3 Thermal properties of silicon at different temperatures. . . . . . . . . . . . . 22

1.4 Reflectivity of titanium-doped optical coatings... . . . . . . . . . . . . . . . 27

2.1 Material properties of silicon... . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Young’s modulus E, shear modulus G and Poisson ratio ν... . . . . . . . . . 36

2.3 Three-paddle designs, models A, B, C, and D... . . . . . . . . . . . . . . . . 40

2.4 Finite element modelling predictions of the 3 types of torsion... . . . . . . . 46

3.1 Fabrication techniques used to produce sample series. . . . . . . . . . . . . 52

4.1 Fabricated resonator characteristics... . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Review of mechanical resonators... . . . . . . . . . . . . . . . . . . . . . . . 106

4.3 Comparison of the measured frequencies of mechanical... . . . . . . . . . . . 113

4.4 Table of the relative vibration amplitudes . . . . . . . . . . . . . . . . . . . 125

4.5 Highest Qs obtained from resonators... . . . . . . . . . . . . . . . . . . . . . 127

4.6 Table of calculated surface to volume ratios . . . . . . . . . . . . . . . . . . 132

5.1 Material properties of components in the vacuum tank... . . . . . . . . . . . 139

5.2 Summary of the frequency drift measurements at different pressures, per-

formed on sample UQB3C-1, on the three-point suspension system depicted

in Fig. 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.3 Estimates of the emissivity and absorptivity of silicon . . . . . . . . . . . . 158

6.1 Table of the relative vibration amplitudes (local divided by maximum... . . 182

D.1 SiNx deposition conditions, using the Oxford instrument Plasma 80+ ICPCVD.217

D.2 SiNx removal conditions with RIE, using the Oxford instrument Plasma

100+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

xv

xvi

Useful formulae

Quantity Formula Unit

Stokes mode ωS = ω0 − ωm frequencyAnti-Stokes mode ωA = ω0 + ωm frequency

Quality factor (spectrum) Q = fδf unitless

Quality factor (ringdown) Q = πfτ unitless

Quality factor formula Q = 2πEsEd

Loss from coating Q−1net = Q−1

i +Q−1c (ΔE/E)

Parametric Gain (3-mode) R = 8IinQ0Q1Qm

mω0ωmL2Λ

1+(Δω/γ1)2unitless

xvii

Chapter 1

Introduction

1.1 Background

Recent progress in the fields of gravitational wave physics and micro-electro-mechanical

systems (MEMS) technology provide an exciting opportunity to create novel sensing tools

for earth- and space-based exploration. Building on these advances, this thesis describes

the first stages in the creation of a new type of sensor technology: the 3-mode opto-acoustic

parametric amplifier (OAPA), focusing mainly on the development of the opto-acoustic

resonator that is required at the heart of the device.

Opto-acoustic parametric amplifiers could be useful in a number of applications. For

example, it could be used to create a sensitive opto-acoustic magnetometer with the ad-

dition of an inductive current loop. In a cryogenic implementation it could be used to

generate tripartite entanglement or to cool macroscopic objects to the quantum ground

state.

The work described in this thesis was undertaken in a research group focussed on, and

motivated by the detection of gravitational waves. The research itself is a direct spin-off

from gravitational-wave detector technology. For this reason I want to begin this thesis

by briefly reviewing the historical background of this exciting frontier, so that the reader

can appreciate the context of my project.

1.2 From Euclid to LIGO

The research presented would not have been possible without a long history of scientific

accomplishments. In 300 B.C, Euclid developed the foundation of flat-space geometry

used for the next two millennia: Euclidean Geometry [5, 6]. In 1687, Sir Isaac Newton

presented classical mechanics [7, 8], based firmly in the belief of Euclid’s flat-space.

1

2 1. Introduction

It was Gauss, in 1828, who challenged these beliefs and refused to take Euclidean

Geometry at face value, believing instead that one should go out and measure the shape

of space for himself. He devised a method to achieve this, described in his Theorema

Egregium (remarkable theorem) [9], which states that the curvature of space can be deter-

mined by measuring angles, distances and their rates, without reference to how this space

is embedded in Euclidean space. He allegedly attempted these measurements in what is

referred to as the 3 mountain top experiment.

In 1854, Gauss trained his PhD student Riemann to develop the mathematical de-

scription of curved-space [10], which was instrumental in producing Einstein’s theory of

general relativity in 1915 [11, 12].

Figure 1.1: Many important contributions to science from Albert Einstein: special rela-tivity, general relativity, Brownian motion, predictions such as curvature of spacetime andthe existence of gravitational waves. Picture taken from the public domain.

The theory of general relativity links the curvature of ‘spacetime’ with the mass, energy

and momentum within its fabric, as shown in Einstein’s field equations [12]:

Gμν =8πG

c4Tμν , (1.1)

where Gμν is the Einstein tensor, G is Newton’s gravitational constant, c is the speed

of light and Tμν is the stress-energy tensor. This equation bears similarities with the

equation of elasticity which has wave-like solutions. Einstein resolved the field equations

1. Introduction 3

in a similar way to the equation of elasticity, using a solution in the form of a wave [13],

and so predicted the existence of ‘gravitational waves’. These waves are perturbations in

the fabric of spacetime, traveling like ‘ripples on a cosmic sea’ [14].

According to the theory of general relativity, gravitational waves (GWs) travel at

the speed of light carrying vast amounts of energy, and originate from sources such as

inspiraling massive stellar bodies (i.e: binary neutron stars, black holes). As a binary

system orbits around a center of mass, GWs are emitted and this slow dissipation of energy

eventually causes the two objects to collide. Orbital periods are expected to decrease with

time.

An indirect observation of GWs was obtained when Hulse and Taylor observed the

orbital period decay of PSR1916-13 [15], the first binary pulsar system ever discovered.

Hulse and Taylor reported a strong agreement between general relativity predictions and

observations of the orbital period decay. Though to date there has been no direct obser-

vations of GWs reported, it is of great interest to physicists and astronomers to detect

GWs, as this would allow important observations, such as:

Black hole coalescence, the most energetic events since the Big Bang.

The birth of neutron stars in supernova explosions.

The transient quasi-normal mode vibrations of black holes in the moments after they

have formed [13].

Resonant-bar detectors were designed to measure acoustic signals induced in massive

bars from coupling to GWs [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Several problems

prevented their success: they had a narrow bandwidth and only had high sensitivity

to burst events. Modelling showed that the bursts from supernova explosions were much

weaker than first anticipated. Thermal noise was too high and the detectors were unable to

approach the standard quantum limit (SQL), which is a fundamental limit on measurement

accuracy at quantum scales. It refers to the minimum level of quantum noise, directly

related to the fundamental Heisenberg uncertainty principle. For optical transducers, this

limit arises from the combined contribution of shot noise and radiation pressure noise.

The shot noise limit is due to the statistical fluctuations in the number of detected

photons. The relative size of the fluctuation, and therefore the shot noise limit, decreases

with laser power. The radiation pressure noise comes from the same statistical fluctuations

that cause shot noise. They exert varying radiation pressure forces on mirrors. They

dominate the noise at low frequency and high laser power.

4 1. Introduction

All quantum noise can be understood in terms of vacuum fluctuations entering the

dark port of an interferometer. The SQL is the trade-off point when the shot noise limit

and radiation pressure limit are equal. This limit can be circumvented with advanced

state preparation (squeezed light [26, 27]) and measurement schemes.

In 1962, Gertsenshtein and Pustovit explicitly suggested using optical interferometers

for GW detection, which was later supported by a feasibility study and detailed design by

Weiss [28]. In the 1970s, Forward et al. constructed the first working prototype of a laser-

interferometer GW detector [29], and Drever developed a Michelson interferometer with

Fabry-Perot arms to multiply the intracavity power [30]. This Michelson interferometer

design [31] uses powerful laser beams and high quality optics in kilometre-long interfer-

ometer arms constructed as two optical cavities in an L-shaped design. In the following

years, modern day detectors, such as GEO 600 [32], CLIO [33], VIRGO [34, 35], and LIGO

[36, 37, 38] (see Fig. 1.2) were developed. An Australian Consortium for Interferometric

Gravitational Astronomy (ACIGA) research facility with 80-m arms is located in Western

Australia [39].

Figure 1.2: Four km-long detector built in USA, in Hanford, Washington. The Laser-Interferometer Gravitational-Wave Observatory (LIGO). Image courtesy of LIGO Labo-ratory.

To explain the concept of the laser-interferometer GW detector, see Fig. 1.3. A laser

1. Introduction 5

Figure 1.3: A laser interferometer consisting of two perpendicular km-long arms withFabry-Perot cavities is designed to detect small variations in cavity length changes, dueto the passing of gravitational waves. Image courtesy of Dr. McRae.

beam is divided by a beam splitter and sent in two orthogonal directions into Fabry-Perot

cavities [40]. The light which is multiply reflected between the Fabry-Perot mirrors and

built up by resonance, returns from the cavities to the beam splitter. In order to minimize

noise [13], the system is tuned to be at antiphase, so that without any perturbations, the

interferometer would produce a dark fringe.

Waves passing through this detector cause an elongation of one arm, while simultane-

ously causing a compression in the other arm for half a cycle. In the second half of the

cycle the effect is reversed. This creates an optical path difference between the two arms,

resulting in a change of the interference pattern. The signal is then filtered from the noise

using sophisticated data processing algorithms [41].

In order for GWs to be detected, advanced detectors [42] require high optical power to

reduce the shot noise. This high optical power allows for higher sensitivity of a detector,

but also increases the radiation pressure forces. Braginsky et al. [1, 2] predicted that

in long optical cavities, a nonlinear interaction mediated by the radiation pressure force

could create ‘parametric instability’.

This instability is an opto-acoustic coupling between 2 optical modes in a cavity and a

mechanical (acoustic) mode of the cavity mirror, and hence is termed a 3-mode interaction.

This can lead to a ring up of the cavity mirror vibrations and cause the cavity to lose lock,

and disrupt the operation of a GW detector.

This phenomenon that was identified as a potential problem for GW detectors provides

the basis for the opto-acoustic parametric amplifier (OAPA) discussed in this thesis. In

the next section, I will review the physics of parametric instability.

6 1. Introduction

1.3 Parametric Instability

Laser-interferometer GW detectors in which optical and acoustic fields can couple may

be described as parametric oscillators, in which the system is driven by the variation of

one or many parameters. A simple conceptual case of a parametric oscillator is a child

on a pendulum swing, alternating between squatting and standing up. The change in

center of mass, timed carefully with the oscillation of the pendulum swing, can increase

the maximum height achieved by the swing. Instability occurs if the driving introduces

energy at a rate faster than the natural damping.

As mentioned above, parametric instability (PI) was first predicted by Braginsky et

al. [1, 2], in the context of advanced laser interferometer GW detectors. The basis of this

effect is the excitation of the cavity mirrors via radiation pressure from the beating of two

optical modes of the cavity, as shown later in Fig. 1.7. This can occur when the optical

energy stored in a cavity main mode is high enough and a frequency matching condition

is met. Details of opto-acoustic interactions will be discussed further in Section 1.5.

The Gravity research group at the University of Western Australia (UWA) has done

extensive work modelling PI and developing approaches to control it. In particular, Zhao

et al. [43] extended the modeling of PI in laser interferometers to 3-D using recent detector

parameters, showing that PI is more likely to occur in current configurations. The esti-

mation of the strength of the opto-acoustic coupling (parametric gain R, see Section 1.5)

was extended to include multiple-mode interactions by Gras et al. [44], and comparisons

of different test mass materials [45]. Such simulations were carried out for over thousands

of acoustic modes of test masses, considering the diffraction losses for each mode [46].

The opto-acoustic coupling in long optical cavities was confirmed when Zhao et al.

reported first observation of 3-mode opto-acoustic interactions (see Section 1.5) in an

80-m long cavity [39].

Susmithan et al. demonstrated a thermal tuning technique, consisting of directly

heating a cavity mirror using a CO2 laser [47], capable of tuning and controlling PI. The

laser power is varied, changing the radius of curvature of a mirror, which changes the

optical mode gap. The acoustic mode frequency that satisfies the resonance condition

is swept over a range of ultrasonic mirror acoustic modes. The 3-mode interaction is

observed over a range of acoustic modes and the system effectively displays the spectrum

of the test mass ultrasonic modes. This work demonstrates the use of 3-mode interactions

to monitor test mass modes, as well as a method to control PI.

Using a similar method, Blair et al. reported successful driving of an acoustic mode of

1. Introduction 7

a test mass at ∼ 180 kHz, using 0.4 W of the TEM01 mode and 1 kW of the main carrier

mode [48]. They observed an acoustic amplitude ∼ 10−13 m, an overlap factor of 2.7±0.4,

and a parametric gain ∼ 4 10−3 (see Section 1.5).

Other methods to control PI in laser interferometers were investigated by Ju et al. [49].

A heating ring at the front or back of a test mass was used to compensate for thermal

lensing and reduce the parametric gain by tuning cavity mode frequencies [50]. Another

method consisted of using ring dampers to lower the acoustic quality factors of the test

masses [51, 52], or resonant dampers (small lossy spring-mass resonators) to specifically

target dangerous acoustic modes.

Another approach was examined which uses an optical feedback to suppress one of the

optical modes in the 3-mode interaction (see Section 1.5) by injecting a beam with the

appropriate amplitude, frequency and phase to create destructive interference [53]. By

detecting the optical signal at the interferometer dark port and suppressing the higher-

order mode, PI can be controlled.

While PI is a concern for the field of GW detection, the physics of opto-acoustic inter-

actions (see Section 1.5) offers opportunities in the fields of high-sensitivity transducers

and quantum measurements of macroscopic objects. As will be discussed later, the para-

metric gain can be higher when the mass of the acoustic resonator and the length of the

cavity is small. Despite inherent difficulties to miniaturising a 3-mode opto-acoustic setup,

such as finding an optical mode gap small enough to match an achievable acoustic mode

frequency, smaller instruments are often easier to set up and more practical for sensor

applications. The Gravity research group at the UWA has done work in this direction.

In a tabletop design, Miao et al. [54] demonstrated that adding a tuning cavity to a

main Fabry-Perot cavity can improve the stability of the cavity as well as allow tuning

between the instability (PI) and self-cooling regimes (when the parametric gain R <0).

This is illustrated in Fig. 1.4. The tuning between the two regimes is achievable with mm

displacements of a mirror and a lens.

Using this tuning cavity design, Zhao et al. introduced the ‘3-mode opto-acoustic

parametric amplifier’ (OAPA) as a tool for quantum experiments [3]. They demonstrated

that PI can be achieved in a design using a mg-scale resonator of MHz acoustic frequency

and quality factor ∼ 105, and 1 mW of laser input power.

A method for reducing acoustic losses from optical coatings is discussed in Section

Proposed experiment of [3]. Multilayer SiO2/Ta2O5 coatings are inherently lossy. For this

reason, the coating mass should be small compared to the resonator mass to reduce the

8 1. Introduction

Figure 1.4: Coupled cavity consisting of a tuning and main cavity [54]. Proposed ex-perimental setup to observe 3-mode opto-acoustic interactions in a tabletop setup with atorsional microresonator (M2) [3]. Image courtesy of Dr. Miao [3].

impact of the coatings on the acoustic quality factor (Q). The authors modelled a silicon

spindle torsional resonator in which the strain amplitude is located in the spindle arms

and away from a coating area, thereby reducing the coupling to the lossy coatings.

Recently, Chen et al. demonstrated the first observation of PI in a tabletop opti-

cal cavity, using a thin silicon nitride (SiNx) membrane between two mirrors [55]. This

achievement paves the way for other compact setups with small resonators to produce

strong opto-acoustic coupling, such as the design presented by Zhao [3] where an end

mirror serves as the acoustic resonator, instead of a membrane in the middle of a cavity.

1.4 Thesis Aim and Scope

The work of Zhao et al. [3] constitutes the starting point of this thesis. The design of Fig.

1.4 and the silicon torsional microresonator design are the first steps in this research. The

aim is to fabricate high Q resonators in the millimetre and milligram scale to meet the

requirements of a 3-mode OAPA device.

The scope of this project is limited to the design, fabrication and testing of microme-

chanical silicon resonators, assessing their performance in terms of Qs, acoustic mode

shapes and spectra, thermal effects, as well as preliminary design of a practical OAPA

cavity.

As will be demonstrated in subsequent chapters, the resonators developed and tested

for this thesis match and even outperform similar prototypes found in the literature,

demonstrating their potential for future applications such as new sensor devices.

In the rest of this chapter, I will present a brief overview of (1) opto-acoustic inter-

actions and the achievements in this field, (2) the requirements for an OAPA device, (3)

1. Introduction 9

material properties of silicon, which is the choice of substrate for the resonators studied

here, (4) manufacturing techniques for making micromechanical resonators, and (5) previ-

ous studies with resonators, discussing designs, fabrication techniques, performance, and

intended applications.

10 1. Introduction

1.5 Opto-Acoustic Interactions

It is instructive and appropriate to consider opto-acoustic interactions from a quantum

mechanical viewpoint, particularly because the OAPA devices are intended to operate

at the quantum level. Such devices are designed to have high opto-acoustic coupling to

facilitate interactions such as those illustrated in Fig. 1.5.

In one case, a photon of frequency ω0 is scattered from a phonon, creating a lower

frequency photon of frequency ω1 = ω0 − ωm and a phonon of frequency ωm. This is the

Stokes process which increases the occupation number of the acoustic mode. In the other

case, a photon is scattered from a phonon, creating a higher frequency photon of frequency

ω1 = ω0 + ωm. This is the anti-Stokes process, and requires that the acoustic mode loses

a phonon, thus reducing the occupation number of the acoustic mode. These two effects

happen simultaneously and can cancel out.

Figure 1.5: Parametric interaction between photons and phonons, causing either ampli-fication or damping (cooling) of the mechanical vibrations of an acoustic resonator. a)Stokes process where mechanical vibrations are amplified. In this situation, a phononof frequency ωm is created. b) Anti-Stokes process where the acoustic mode is damped(cooled), i.e: a phonon is destroyed (absorbed). ω0 is the carrier frequency and ω1 is thescattered optical frequency. Image courtesy of Dr. Ju [44].

In optical cavities it is possible to vary the relative strength of the Stokes and anti-

Stokes processes. From a classical viewpoint, the input laser will be scattered by a vi-

brating mirror surface, creating two sidebands. If both sidebands have equal amplitude,

the Stokes and anti-Stokes processes are balanced and there is no net effect on the mirror

acoustic mode.

On the other hand, if one sideband is enhanced more than the other by an imbalance in

the cavity, one process will dominate. The beating of this sideband and the input carrier

will couple to the mirror motion through the radiation pressure force acting on the mirror.

This imbalance can be created in two ways, as illustrated in Fig. 1.6 (a) and (b). In

1. Introduction 11

one case, an acoustic mode scatters an input beam to create a pair of sidebands within

the cavity bandwidth of the same cavity mode. The input beam is slightly detuned from

the cavity fundamental TEM00 mode to create an unbalance between the sidebands, and

one process will dominate. One optical mode (supporting three separate frequencies) and

one acoustic mode of a mirror are resonant, and this is termed a 2-mode interaction. An

example of the anti-Stokes mode in 2-mode interactions is illustrated in Fig. 1.6 (a).

In another case, an input beam and scattered beam are in different cavity transverse

modes. In this interaction, two optical modes and one acoustic mode are involved, and this

is termed a 3-mode interaction. The input carrier mode is tuned to the cavity fundamental

TEM00 mode and only one of the sidebands coincides with a high-order TEMmn mode.

This is made possible by the asymmetry of the mode structure in a cavity, as shown in

Fig. 1.6 (b). The mode structure is repeated at every free spectral range (FSR), given

by FSR = c/2L (see Fig. 1.6 c), where c is the speed of light and L is the length of the

cavity.

When the upper sideband (ω1 = ω0 + ωm, anti-Stokes mode) is enhanced, this results

in the cooling of the acoustic mode, referred to as ‘self-cooling’ in the literature. When the

lower sideband (ω1 = ω0−ωm, Stokes mode) is enhanced, this results in the amplification

of the acoustic mode (phonon lasers).

Optical cavities can be free-space, in which the laser light propagates in the empty

space between two cavity mirrors (see Fig. 1.6 (c)) or medium-filled, where the light

propagates inside a medium (e.g., glass sphere illustrated in Fig. 1.6 (d)), such as a

whispering gallery cavity [56, 57, 58].

In a medium-filled cavity, the refractive index variation induced by an acoustic wave

acts as a grating in the medium through which light propagates. This causes scattering

within the medium. If this process occurs in an optical cavity, the scattered light can

stimulate acoustic wave excitation in the medium [59, 60]. Such scattering with positive

feedback is called stimulated Brillouin scattering (SBS) [61].

Another mechanism for scattering in a medium-filled cavity is from an electromagnetic

field intensity gradient inducing a force on the medium [62]. In both SBS and gradient

force cases, photons are scattered by bulk phonons in the frequency range of ∼ THz. Such

systems have been shown to produce 2- and 3-mode opto-acoustic interactions [63, 58, 57]

in which a mechanical resonator mode was cooled close to the ground state.

In a free-space cavity, 2- and 3-mode interactions occur when an acoustic resonator

(mirror) creates frequency modulation by scattering photons from surface phonons. Nu-

12 1. Introduction

Figure 1.6: (a) In a 2-mode interaction, the carrier mode of frequency ω0 is offset fromthe center of an optical cavity transverse mode. The scattering from a mirror creates twosidebands, which fall within the cavity bandwidth. In this example, the right sidebandis enhanced more than the other, resulting in the anti-Stokes mode. (b) In a 3-modeinteraction, the sideband ω1 and the carrier mode ω0 are in different cavity transversemodes TEMmn, creating a triple resonant system. The laser carrier mode frequency ω0 istuned to the fundamental mode of the cavity TEM00 and ω1 is scattered in the TEM01

mode (see Fig. 1.8). Diagram of the cavity optical mode structure, showing asymmetry ofthe spectrum. The modes are repeated at frequency gaps equal to the free spectral range(FSR). (c) Free-space cavity where the length L of the cavity is the distance between thetwo mirrors. The cavity mode structure is given by the FSR = c/2L. (d) Medium-filledcavity where the light is trapped inside a transparent dielectric material (e.g., glass sphere,from the public domain) from total internal reflection [56]. Standing waves in the mediumconcentrated near its surface are called whispering gallery modes. On the right is anexample of a spherical glass cavity of diameter 0.3 mm.

1. Introduction 13

merous experiments have reported 2-mode interactions in free-space cavities, and they

have been widely used for mode cooling [64, 65, 66, 67, 68], optical spring experiments

[69, 70], and in creating linear motion transducers [71].

In one particular case, the normal mode of a 1.5 tonne Nb bar was cooled to 5 mK

by using a microwave parametric transducer [20]. In another case, a silicon-on-insulator

nanomechanical resonator in the shape of a beam was cooled from 20 K to the quantum

ground state [72]. Other interesting applications of 2-mode interactions are discussed

in literature, such as teleportation of quantum states [73] and observation of stationary

quantum entanglement between photons and phonons [74, 75].

The work in this thesis consists of developing a suitable resonator to enable 3-mode

opto-acoustic interactions in free-space cavities, in which the scattering process is charac-

terized by a simultaneous triple resonance (product of three Qs, see Eq. 1.2) of a carrier

optical mode, a high-order TEM mode, and an acoustic mode as illustrated in Fig. 1.7.

Figure 1.7: Three-mode opto-acoustic interactions within a free-space optical cavity. Thecarrier mode is pumped into a cavity, where it experiences scattering on the mirror surfaceprofiled by an acoustic mode of frequency ωm. If the mirror surface motion spatiallyoverlaps the electric field distribution of a transverse optical mode, the carrier mode canbe scattered into this cavity mode. The carrier and scattered light beat together and drivethe acoustic mode via the radiation pressure force. The asymmetry of the modal structureof optical cavities ensures that only one sideband participates in the interaction (see Fig.1.6 b). Image courtesy of Dr. Ju [44].

As mentioned in Section 1.3, using an 80 m cavity, Zhao et al. [39] have shown

experimentally that the interaction of photons with test mass surface vibrations does

occur. This was the first evidence that cavities may sustain 3-mode interactions. More

recently, 3-mode parametric interactions were used to drive ultrasonic acoustic modes [48].

The 3-mode interactions have advantages over the 2-mode interactions, such as reduced

susceptibility to laser phase and amplitude noise [76], and reduced laser power require-

ment. The strength of 3-mode opto-acoustic interactions is defined by the dimensionless

14 1. Introduction

Figure 1.8: Electromagnetic field patterns of radiation measured in a plane perpendicularto the propagation of a beam. The transverse modes supported by a cavity are quan-tized, due to the boundary conditions. In most lasers, transverse modes with rectangularsymmetry (TEMmn) are formed, defined by the parameters m and n, representing thehorizontal and vertical orders of the pattern. For example, the TEM00 mode (see Fig.1.6) is the fundamental transverse mode of a laser resonator, corresponding to a Gaussianbeam. Picture from the public domain.

parameter — the parametric gain R [2] — as follows:

R =8IinQ0Q1Qm

mω0ωmL2

Λ

1 + (Δω/γ1)2, (1.2)

where Iin is the input laser power, Q0, Q1 and Qm are the quality factors of the two

optical modes ω0 and ω1, and the acoustic mode of the mechanical resonator, m is the

(effective) mass of the mechanical resonator, ω0 is the carrier frequency of the laser, ωm

is the acoustic mode frequency, L is the length of the optical cavity, Δω is the frequency

detuning equal to |ω0 − ω1| − ωm (zero in the resonant tuned case) and γ1 is the half

linewidth of the cavity transverse mode, equal to ω1/(2Q1).

The overlap factor Λ, describing the acoustic and optical mode-matching, is important

as it forces a special requirement to achieve parametric gain: the one acoustic and two

optical modes must share significant spatial overlap. To visualize a mode-shape matching,

first consider the optical transverse cavity modes given by the Hermite-Gaussian TEMmn

modes illustrated in Fig. 1.8. Acoustic mode shapes must be found in which the spatial

distribution pattern is similar to an optical mode pattern shown in Fig. 1.8. An example

of a good match was presented by Zhao et al. [3], considering the torsional vibration of

1. Introduction 15

a silicon spindle resonator and the transverse optical TEM01 mode. This is illustrated in

Fig. 1.9. Other combinations of optical and acoustic modes can provide a suitable mode

match. For example, the fundamental optical TEM00 mode shares a good spatial overlap

with an acoustic drum mode, which can lead to 2-mode interactions. For the purposes of

this thesis, as will be described later, the acoustic torsional mode and TEM01 mode will

be used in a 3-mode OAPA design.

Mathematically, three modes — consisting of the input laser carrier tuned to the cavity

TEM00 mode, the high-order mode TEMmn, and the acoustic mode ωm — must have a

spatial match defined by the integral

∫ψ0(�r⊥)ψ1(�r⊥)uzd�r⊥, (1.3)

where ψ0 and ψ1 are the optical field distributions over the mirror surface for the TEM00

and TEMmn modes, respectively. The spatial displacement vector for the acoustic mode

�u has a component uz normal to the mirror surface. Acoustic modes vary in the fraction

of the total mass which vibrates. Such a fractional mass, called the effective mass, refers

to the excitation susceptibility of an acoustic mode. If the effective mass is large, then

the energy required to induce vibration is also high. A normalised overlapping parameter

between the ith high order mode and jth acoustic mode can be written as:

Λ =V∫ |�u|2dV

(∫ψ0(�r⊥)ψ1(�r⊥)uzd�r⊥)2∫ |ψ0|2d�r⊥

∫ |ψ1|2d�r⊥ , (1.4)

where V/∫ |�u|2dV is the ratio of the test mass to the effective mass of the acoustic mode.

When R > 0, the opto-acoustic interaction leads to amplification of the mirror motion

(increased acoustic mode amplitude) via the Stokes process. If the strength of the inter-

action is large enough (R >1), then the acoustic mode amplitude can ring-up and grow

exponentially. This is the ‘parametric instability’ (PI), which was discussed in Section 1.3.

Figure 1.9: (a) Silicon spindle resonator with an acoustic torsional vibration. (b) Acoustictorsional mode shape. (c) The optical TEM01 mode (see Fig. 1.8), showing a suitablespatial overlap with the acoustic torsion mode shown in (b). Image courtesy of Dr. Zhao[3].

16 1. Introduction

On the other hand, if the cavity is tuned such that the anti-Stokes mode is excited (R<0),

then the mirror mode can in principle be cooled to the quantum ground state.

As mentioned in Section 1.3, Chen et al. demonstrated the first observation of 3-mode

PI in a tabletop free-space cavity, using a thin silicon nitride (SiNx) membrane between

two mirrors. In order for a compact 3-mode opto-acoustic setup to be useful for novel

sensing applications, the resonator is required to support a sensor. This is not practical

with a SiNx membrane-in-the-middle cavity design. For this reason, the tabletop 3-mode

OAPA design proposed by Zhao et al. [3] is more suited to the creation of a novel sensing

device. As mentioned previously, the work of Zhao et al. [3] will serve as the starting

point of this research.

1. Introduction 17

1.6 Resonator Requirements for a 3-mode OAPA Device

In order to make a compact 3-mode OAPA device feasible, certain criteria must be met:

The resonator must demonstrate a high acoustic Q.

The cavity must have high optical Qs for the relevant modes.

The resonator must have optical coatings to act as an end mirror in a cavity.

The acoustic mode frequency of the resonator must match the mode gap between

the cavity TEM00 mode and a certain cavity TEMmn mode.

The acoustic and optical mode shapes must spatially overlap.

A compact optical cavity (small cavity length L and small effective mass m) is

required.

A cavity design with tunable transverse modes is required.

High Qs are important in order to increase the strength of the 3-mode opto-acoustic

interaction, as given by Eq. 1.2. This is directly related to the performance of an OAPA

device. The acoustic Q can be improved through careful resonator design and substrate

choice. For example, materials such as aluminium, niobium and sapphire have different

acoustic Q values.

While aluminium and niobium are more readily available in large masses, their Qs of

7 107 and 2.3 108 [20] respectively are lower than those of silicon and sapphire which

are 2 109 [77] and 3 109 [13] respectively. Furthermore, silicon is much easier to pattern

with micromachining.

Many factors can influence the Q of a resonator, such as: the surface finish, impurities

and defects, mounting and bonding stresses, temperature, gas damping, interfering nearby

modes and the driving level.

The acoustic resonator must be designed to scatter a TEM00 mode into a specific

TEMmn mode of the cavity. This is achieved by (1) ensuring the desired acoustic mode

frequency matches the frequency gap between the two cavity transverse modes, and (2)

choosing an acoustic mode shape that shares significant spatial overlap with the field

distribution of the optical modes. Experience with a membrane cavity 3-mode setup [55]

has demonstrated the importance of the alignment and relative sizes of the optical and

acoustic modes in order for the overlap to be large.

18 1. Introduction

Optical coatings are required to use the acoustic resonator as an end mirror in a cavity

designed as the OAPA device. The reflectivity of these coatings should be high to increase

the cavity finesse, in order to have high optical Qs. The coatings must be large enough

to allow a match between the acoustic mode shape and the optical TEM00 and TEMmn

mode beam spot sizes. The coating must be larger than the spot sizes to reduce the loss

from diffraction.

Thin film optical coatings require many layers to achieve high reflectivity, and due

to the higher loss of the coatings compared with that of the resonator material, this can

potentially reduce the acoustic Q. This can be addressed by careful design of the resonator

and proper choice of the coating location, to minimize the elastic strain on the coatings.

To allow a compact configuration of the OAPA device, a small cavity length L implies

a small-scale resonator (small size and mass). When a cavity is small, the free spectral

range is higher and the frequency gap between transverse optical modes becomes large.

This forces a requirement that the acoustic mode frequency be high, within the MHz

range. Also, in order for a sensor to be attached to the back of a resonator, a lower limit

is imposed on the resonator size, based on the size of small sensors.

Finally, for an OAPA device to be used as a small signal transducer, high-sensitivity

sensor, and for quantum measurements, the cavity design should allow tuning of the

transverse modes. The design proposed by Miao et al. [54] provides practical solutions.

One possible application of a 3-mode OAPA which uses high Q resonators is an opto-

acoustic high-sensitivity magnetometer. The performance of such a device was evaluated

by Dr. Miao [78].

The proposed device comprises an inductive loop attached to a resonator to measure

the Ampere’s force �F = I(t) �B �l from coupling between an ambient magnetic field �B and

a modulated electric current I(t) in a loop �l. The motion of the resonator is monitored by

an optical readout, from which the magnetic field can be extrapolated.

The performance was calculated assuming a milligram resonator with acoustic Q of

106, a frequency of 1 kHz, a current of 10 mA, an integration time of 1 s, and a wire (loop)

length of 1 cm.

A quantum limit to the sensitivity is predicted as ∼ 10−12 Tesla.

With the calculations of Dr. Miao and the previously mentioned work by Zhao et al.

[3], which predict that an acoustic resonator with a Q of ∼ 105 can be used to achieve

parametric instability, a target value of 106 is chosen for resonator Qs. Such a target would

enable a strong opto-acoustic coupling. As will be discussed later, this target is higher

1. Introduction 19

than Q values for many silicon microresonators available in the literature.

Based on these requirements, a silicon torsional microresonator seems best suited for

the aim of this thesis. The acoustic torsional mode shares strong overlap with a TEM01

mode and restricts the elastic strain to torsion rods holding a rigid-body central paddle,

where coating is applied. This design will be discussed in further details in the body of

this thesis.

1.7 Silicon Material Properties

Silicon has properties which make it a good choice of substrate for an acoustic resonator in a

3-mode OAPA device. It has low intrinsic acoustic loss, it is widely used and understood,

and many techniques are readily available to manufacture various shapes and designs.

Some of these techniques are discussed in the next section.

An excellent review of silicon material properties is given by Masolin et al. [79], sum-

marizing mechanical and thermal properties, useful for analysis with finite element mod-

elling. These include elasticity, conductivity, diffusivity, specific heat, thermal expansion

and emissivity, amongst others.

At room temperature, silicon has a density of 2329 kg m−3. It has a diamond cubic

crystal structure, depicted in Fig. 1.10 (a). As an anisotropic material, its properties

depend on the relative orientation within the crystal lattice. Silicon is also described as

an orthotropic material, as it has at least two orthogonal planes of symmetry.

Figure 1.10: Diagram of planes, orientations and directions in standard silicon (100)wafers. (a) Diamond cubic crystal structure of silicon. Picture from the public domain.(b) Standard p-doped (100) wafer. The primary flat indicates the (110) plane. (c) 3-Dview of the wafer and perpendicular (110) plane. The normal vector to the (100) waferplane is the [100] direction.

20 1. Introduction

Elasticity is of primary concern in predicting acoustic resonances in silicon resonators,

and is described by Hooke’s law between stress and strain. This equation uses fourth rank

tensors for the stiffness C and compliance S, and second rank tensors for the stress σ and

strain ε in the expected linear range:

σij = Cijklεkl, (1.5)

and

εij = Sijklσkl, (1.6)

These equations are greatly simplified since the fourth rank tensors C and S can be

specified with only three independent components, due to the cubic symmetry of silicon

and the equivalence of the shear conditions. These components are accurately measured

by Hall et al. [80] as well as Masolin et al. [79], and are shown in Table 1.1.

Table 1.1: Stiffness C (in GPa) and compliance S (in 10−12 Pa) independent components,measured at 298 K [80, 81].C11 (GPa) C12 (GPa) C44 (GPa) S11 (10−12 Pa) S12 (10−12 Pa) S44 (10−12 Pa)

165.64 63.94 79.51 7.69 -2.14 12.6

The bulk modulus B can be obtained from the values of stiffness as follows:

B =C11 + 2C12

3, (1.7)

which gives a value of B = 97.8 GPa at room temperature. This value agrees remarkably

well with experimental measurements of B = 99.5± 0.5 GPa [82].

For orthotropic materials such as silicon, the elastic properties can be given in terms

of fundamental elastic quantities, such as Young’s modulus E, Poisson’s ratio ν and the

shear modulus G. These quantities can be derived for a given orientation (see Fig. 1.10

b and c). For a standard (100) silicon wafer, with frame of reference in the [110], [110]

and [001] directions (see Fig. 1.10 b), the elastic values are given in Table 1.2, with wafer

directions depicted in Fig. 1.11.

As most silicon wafers are not pure silicon and contain amounts of chemical impuri-

ties, the effects of doping (impurities) must be taken into account to determine material

properties. As it turns out, elasticity will change with heavy doping by less than 3%, and

can therefore be ignored for the purpose of this study [79].

Young’s modulus of silicon changes with temperature. The individual components

of the elasticity tensor have different temperature coefficients [79]; however, the elastic-

1. Introduction 21

Table 1.2: Young’s modulus (in GPa), Poisson’s ratio and shear modulus (in GPa) fordifferent wafer orientations and directions ‘x’ and ‘+’ shown in Fig. 1.11 a and b.

Wafer (100)x (100)+ (111)+

Ex 130 169 174

Ey 130 169 174

Ez 130 130 188

Gxy 79.6 79.6 60.5

Gyz 79.6 79.6 60.5

Gzx 79.6 50.9 70.0

νxy 0.278 0.064 0.241

νyz 0.278 0.362 0.166

νzx 0.278 0.362 0.166

Figure 1.11: Frame of reference axes on silicon wafer. (a) Frame of reference ‘x’ given bydirections [100], [010] and [001]. (b) Frame of reference ‘+’ given by directions [110], [110]and [001].

22 1. Introduction

ity variations are almost uniform in all directions, roughly -60 ppm/K [83, 84] at room

temperature. This value changes to -75ppm/K at ∼ 100 K.

As resonators made from silicon substrate will be part of opto-acoustic experiments

and applications, it is important to know their thermal properties. In particular, the

values of conductivity κ, diffusivity D, specific heat Cp and thermal expansion αT are

listed in Table 1.3 at different temperatures.

Table 1.3: Thermal properties of silicon at different temperatures.T (K) κ (W cm−1 K−1) D (cm2s−1) Cp αT 10−6 (W cm−1K−1)

200 2.66 0.557

300 1.56 0.86 0.713 2.626

400 1.05 0.52 0.785 3.253

500 0.80 0.37 0.832 3.614

Emissivity is difficult to estimate as it depends on many factors, such as thickness

of the material, doping, which is often difficult to know, and surface defects. At room

temperature, the emissivity of silicon is generally near zero and it rises smoothly to an

upper limit of 0.7 at temperatures above 1000 K [79].

Ravindra et al. [85] and Sato et al. [86] report emissivity values between 0.076 and 0.7

for p-doped (boron-doped) silicon (the doping type most relevant to the research in this

thesis) of thickness 700 μm, depending on the amount of impurities and the wavelength.

With silicon products, in particular processed and fabricated resonators, it is best to

measure the emissivity experimentally. However, this requires an elaborate setup beyond

the scope of this project [87].

Some studies where silicon wafers were bulk-micromachined to form suspended can-

tilevers, report on properties measured in different conditions. Young’s modulus of silicon

nanoplates was investigated at different temperatures and plate thickness, by Wang et al.

[88]. These studies indicate that while Young’s modulus varies greatly with thickness for

thin nanoplates between 1 and 10 nm, varying as much as ∼ 40 GPa, it then reaches a

plateau value of elasticity.

In another similar study of silicon cantilevers by Sadeghian et al. [89], Young’s modulus

was measured and modelled depending on temperature, thickness of the cantilever and

on surface defects expected from manufacturing techniques [90]. Similar to the study

with silicon nanoplates, a plateau is observed in the value of Young’s modulus when the

thickness of the cantilever reaches ∼ 600 nm.

In the case of nanoplates, the sample was a simple plate from a silicon (001) wafer,

with measurements and calculations of Young’s modulus in the [100] direction. In the

1. Introduction 23

case of silicon cantilevers, they were made from silicon-on-insulator (100) wafers, and are

thicker than the nanoplates. The main conclusion from these studies is that for resonators

made from silicon wafers of thickness greater than ∼ 100 μm, a value of Young’s modulus

should be fairly independent of thickness, but nevertheless dependent on temperature and

surface defects. This thesis uses silicon wafers that are more than 100 μm in thickness.

Nawrodt et al. studied the intrinsic acoustic loss in silicon bulk samples of thickness

of 24 mm in a cryostat environment. They report Qs as high as 1.2 108 at 36 K and

low pressures [91], while at 200 K and 300 K (room temperature) the Qs are 1.5 107

and 1.5 106, respectively. They also found that while Qs at room temperature were

higher for a quartz sample of the same size (24 mm thick), silicon Qs were higher at lower

temperatures. Below 18 K, the Qs of silicon decrease slightly down to temperatures of 5

K, though they remain higher than the values for quartz.

The highest Q ever measured for boron-doped bulk silicon (100) crystal is 2 109 at 3.5

K [77], and it is expected that pure crystal silicon without impurities and surface defects

could have significantly higher Qs. The highest Qs reported in other pure materials are

4.2 109 for quartz [92] and 5 109 for sapphire [77]. However, as silicon is much easier

to use in manufacturing resonator shapes, it is the substrate selected for the resonators

studied in this thesis.

1.8 Silicon Microresonator Manufacturing Techniques

Silicon has become the most intensely studied and developed material in microelectronics.

It dominates as the material of choice for manufacturing computer microchips and is

commonly used in micro-electro-mechanical systems (MEMS), to form cantilevered beams

for multiple applications, e.g. biomedical applications [93, 94].

Silicon wafers, as discussed above, have high Qs that make them a substrate of choice

for microresonators. Different micromachining techniques are available, including wet

etching, dry etching, vapor etching and laser machining, that seem best suited for the

requirements in this project [95].

An excellent review on fabrication technologies compares wet and dry etching, as well

as vapor etching and laser machining on the basis of process complexity, etch rates, safety,

availability, resulting silicon surface roughness, isotropy and anisotropy of etching and cost

[96].

As discussed in Section 1.6, I have focused on a torsional acoustic resonator for this

research. Due to the sharp angles and straight edges required for this type of design, only

24 1. Introduction

anisotropic etching methods have been seriously considered for fabrication. Vapour etching

technologies are mainly isotropic and represent a poor choice of fabrication technique.

Both liquid (wet etching) and plasma (dry etching) etchants offer suitable anisotropic

etching and comparable silicon etch rates. Laser machining can produce straight edges,

but potentially suffers some irregularities from laser damage observable under an optical

microscope [97].

Alkali-OH wet etching and dry plasma etching seem to be the best candidates. The

former is substantially cheaper and common, therefore readily available in most commer-

cial cleanroom environments, while the latter is expensive, rare and raises many hazardous

concerns.

The dry etching techniques allow etching through the full thickness of a silicon wafer,

as well as offer nearly perpendicular walls [98]. Wet etching on the other hand, is slower

and the wall angles depend on the crystalline orientation of the flat surface of the wafer.

For instance, KOH etching of a (100) orientated silicon wafer (see Fig. 1.10) gives rise to

pyramidal holes with 54.74 faces, while using the (110) orientated silicon wafers gives rise

to trenches almost perfectly perpendicular to the wafer plane [95].

In wet etching, etch masks are made using silicon nitride layers with one side patterned

with a positive photoresist layer printed with photolithographic processes. Expected etch

rates are around ∼1 μm/min for exposed silicon and ∼1 nm/min for silicon nitride mask

layers [95, 96]. Possible difficulties include undercutting and survival of the silicon nitride

layer during etching. This process is summarized in Fig. 1.12.

Dry etching is a similar technique. The difference is the use of a plasma instead of

a wet solution. The sample is placed in a plasma chamber, and undergoes a series of

plasma etchings of the exposed silicon, and coatings to protect the exposed vertical walls.

This is repeated until the entire sample thickness is etched through (see Fig. 1.13). The

advantage with dry etching is that near vertical walls can easily be achieved.

Laser micromachining is an ablation process using pulsed or continuous wave lasers

focused on a substrate. Photons of a laser are absorbed by the electrons of the substrate

at power densities high enough to melt and vaporize the material. The type of laser must

be chosen according to the material and its spectral absorptivity. In order for the molten

material to be removed, the process can be assisted by gas applied coaxially to the laser

beam. This increases the ablation rate as the incident beam is not absorbed further by

the molten material.

In laser micromachining of silicon, the low temperature threshold of surface damage,

1. Introduction 25

(a) Deposition of a silicon nitride (SiNx) mask. (b) Spin-coating of a photoresist layer.

(c) Align the glass plate with a printed pattern. (d) Expose the photoresist to UV light.

(e) Remove mask and develop the exposed pho-toresist layer.

(f) Etch the exposed SiNx mask.

(g) Remove PR by dissolving it in Acetone.(h) Wet etching of the substrate, resulting in ta-pered walls.

Figure 1.12: Step by step process of wet etching, including masking, photolithography andplasma and wet etching, resulting in tapered walls. The process of dry etching is similar.A different photoresist is used, and instead of the wet etching in (g), the substrate issubjected to alternating intervals of ion etching and passivation layer depositions, resultingin straighter walls.

26 1. Introduction

Figure 1.13: Schematic illustration of a plasma dry-etching process. (Left) The processalternates from etching the exposed silicon with SF6 ions, and deposition of a passivationlayer to protect the side walls from plasma etching. This process is repeated ∼ 100 - 1000times, until etching is performed through the thickness of the wafer. (Right) Example ofaspect ratios and shapes obtained from dry etching. Picture from the public domain.

the sensitivity to high laser intensity and pulse duration must be considered. Approaches

to reduce laser damage include optimizing process parameters to enhance the machined

surface quality by using shorter wavelengths and pulse durations, multi-wavelength exci-

tation, ultrasonic-aided laser machining, and liquid-assisted laser ablation.

Excellent results can be obtained from recent advances in laser micromachining of

silicon [99].

1.8.1 Optical Coatings

As discussed in Section 1.6, optical coatings are required to provide a high reflectivity

of the resonator, which is an important requirement in order to use the resonator as a

cavity mirror in an OAPA device. Ion beam sputtering (IBS) is used to apply alternating

titanium-doped thin film layers of tantalum pentoxide (Ta2O5) and silica (SiO2) to a

silicon substrate [100], as shown in Fig. 1.14. The reflectivity as a function of optical

coating thickness is given in Table 1.4.

As Table. 1.4 demonstrates, thicker coatings give a higher reflectivity, but at the cost of

larger mass added to the silicon resonator, which leads to increased losses associated with

coatings. Also, there is a limit to the thickness that can be applied using lift-off techniques,

to have localised coating areas. These techniques were used in the fabrication of optically

1. Introduction 27

Figure 1.14: Ion beam sputtering of tantalum pentoxide and silica (target materials) ontoa silicon substrate. An ion beam of Ar+ is focused onto a target material to releaseatoms or molecules of the target material and sputter onto a silicon substrate. The targetmaterial is then firmly bonded to the substrate. Alternating layers are created by rotatingthe target stage, alternating between silica and tantalum pentoxide. Image courtesy ofthe Laboratoires des Materiaux Avance’s (LMA), in France.

Table 1.4: Reflectivity of titanium-doped optical coatings of Ta2O5/SiO2 thin films.Number of layers Reflectivity (%) Thickness (μm)

10 99.84 3.3

14 99.997 5.1

20 99.9999 6.3

30 99.999999 9.3

28 1. Introduction

coated resonators, and more details are presented in Chapter 3 and in Appendix D.

1.9 Acoustic Resonators Review

This section presents an overview from the literature of available acoustic resonators,

varying in size, shape, material and performance, as well as intended applications. Various

resonators are compared, focusing mainly on resonator mass, size, material, mode shape,

frequency and acoustic Qs, as these are the relevant properties to this project (see Fig.

1.15).

Figure 1.15: Set of resonators described in the literature [4, 101, 102]. (a) Davis et al.silicon nitride membrane 3-paddle design. On the right, three typical torsion modes:common, differential, and torsion of the middle paddle. (b) Arcizetet al. silicon-on-insulator beam resonator. (c) Serra et al. silicon-on-insulator micro-oscillator for quantumopto-mechanics. All pictures reproduced with authors permissions.

Nanomechanical resonators were studied by Davis et al. [4] (see Fig. 1.15 (a)), and a

Q of 2 103 was reported for a torsional mode with effective mass of 0.1 pg at frequency

21 MHz. Measurements were performed at room temperature and pressure ∼ 10−7 Pa.

A pattern of 3 ‘paddles’ (rectangular elements) along a rod of width 100 nm was used

to isolate the vibration of the central paddle from vibrations of the surrounding silicon

nitride membrane. This relatively low Q could be attributed to acoustic coating losses,

as the entire sample was coated on one side with 10 nm of permalloy, for use as a torque

magnetometer.

Similar nanomechanical resonators were fabricated and studied by Chabot et al. [103].

They achieved mechanical Qs of 1.2 104 for torsional modes at 120 kHz, with effective

mass of 0.7 ng, measured at room temperature and pressure of 13 Pa. These resonators

1. Introduction 29

were made by etching silicon (100) boron-doped wafers, resulting in multiple double-torsion

paddle designs. A small magnetic film dot of 3 μm diameter was coated onto the upper

paddle to produce magnetometers and force sensors in nuclear magnetic resonance force

microscopy.

A French team led by Arcizet et al. fabricated and studied mm-scale silicon-on-

insulator chip resonators [101] (see Fig. 1.15 (b)), made using double-sided photolithog-

raphy and deep reactive-ion etching (DRIE). They reported Qs of 1.5 104, for a (0,4)

transverse mode of 2.8 MHz, with effective mass of 210 μg, measured at room tempera-

ture and pressure of 0.1 Pa. These 1 mm by 1 mm beam resonators were optically coated

for high-sensitivity optical monitoring of moving micromirrors.

Recently, Serra et al. reported [102] Qs as high as 1.5 105 for torsional modes of a

mm-scale central paddle micromechanical silicon resonator (see Fig. 1.15 (c)), resonating

at 85.5 kHz at room temperature and pressure of 10−3 Pa. By varying the thickness

of different parts of the sample and using vibration isolation paddles, they were able

to shield the main resonating paddle mode from wafer modes. The aim of their design

was to generate non-classical states of light by opto-mechanical coupling, and to produce

devices suitable for the production of ponderomotive squeezing and entanglement between

macroscopic objects and light.

Kuhn et al. [104] fabricated and studied 1 mm long, 240 μm wide, triangular nanopil-

lars made of crystalline quartz (known for low loss). They obtained a Q of 1.8 106 at 3.66

MHz for a longitudinal mode of the pillar measured at 10−1 Pa and room temperature.

Unfortunately, though quartz does have a lower intrinsic loss than silicon, it is not as

easily manufactured into the 3-paddle design intended for this research, which is why this

substrate was not considered.

Buser et al. used wet etching techniques on a monocrystalline silicon wafer and pro-

duced a torsional resonant structure which shows high Qs of 6 105. Over two decades

ago, these were the highest Qs ever reported for micromachined silicon resonators. This is

still high compared to most resonators presented here. They report results of the Q and

resonant frequency (around 24 kHz) as a function of pressure and temperature. Quality

factors of about 1.5 104 are observed at atmospheric pressure.

The above mentioned mm-scale resonators can be significantly outperformed by larger

bulk silicon resonators. Results were obtained for boron-doped bulk silicon cylindrical

resonators with length between 6 and 75 mm, and common diameter of 76.2 mm. Nawrodt

et al. [91] report a Q as high as ∼ 3.5 106. This corresponds to the fundamental drum

30 1. Introduction

mode of a cylindrical sample of length 12 mm, for a mode frequency ∼ 14 kHz at room

temperature and pressure of 10−3 Pa. The highest Q obtained for the same sample,

measured at 5.6 K, was 4.5 108.

As will be discussed in Chapter 4, the acoustic resonators developed for this thesis not

only meet but outperform these silicon resonators, with the exception of the bulk samples

[91] which do not meet the resonator size requirements (see Section 1.6) for a 3-mode

OAPA device.

1.10 Thesis Outline

This thesis represents the first steps in developing acoustic resonators designed to form

the heart of a 3-mode opto-acoustic parametric amplifier (OAPA) device. It entails (1)

the design, manufacture and testing of a silicon micromechanical resonator, a crucial

component of this device, and (2) an assessment of the feasibility of this device. The

actual development and manufacture of this device is outside the scope of this research,

and is left as a continuation of this project for future PhD students.

The patience required to accept the small part in the long process of creating something

new reminds me of an inspirational saying by the Chinese philosopher Lao-Tzu (604 BC

— 531 BC), from The Way of Lao-Tzu:

A journey of a thousand miles begins with a single step.

In the following chapters, I present the modelling and testing of our micromechanical

resonators, an experimental study of the thermal properties of these resonators, designs

of an OAPA device and its predicted performance, and improved resonator prototypes for

future work.

In Chapter 2, I will present finite element modelling (FEM) of the resonator prototypes

used to explore different designs and choose a suitable model to meet the requirements

presented in this introductory chapter. Acoustic loss from optical coatings of different

sizes is predicted using FEM. The methods used to fabricate and test resonators will be

treated in Chapter 3, using various suspension and actuation methods. A FEM method

is described to analyse Qs based on the support locations of a three-point suspension.

The results from the experiments described in Chapter 3, as well as a comparison of the

acoustic mode spectrum with FEM predictions will form Chapter 4. Some of the results

in this chapter were published in the Journal of Applied Physics [105] which reports the

1. Introduction 31

highest Qs obtained at the time of submitting that paper. Later experiments provided

even higher Qs with different resonators.

Thermal effects from laser heating of fabricated acoustic resonators are presented in

Chapter 5. These results are used to predict expected frequency variations in an OAPA

device using similar resonators.

Chapter 6 presents a design for a compact 3-mode OAPA, using a silicon micromechan-

ical resonator with optical coatings. An improved resonator design is presented, based on

silicon-on-insulator (SOI) technology, similar to resonators studied by Serra et al. [102].

I summarize the findings of this research in the Conclusion and present suggestions

for future work that could be undertaken in order to improve the fabrication and testing

methods, and resonator designs.

32 1. Introduction

Chapter 2

Finite Element Modelling of

Acoustic Resonators

Calculatus Eliminatus always helps an awful lot.

The way to find a missing something is to find out where it’s not.

— The Cat in the Hat

The above quote illustrates the finite element modelling (FEM) approach to choosing

a suitable resonator design for an opto-acoustic parametric amplifier (OAPA) device by

considering a variety of models and discarding the ones that are unsuitable.

2.1 Preface

In this chapter, FEM is used to assist in choosing an appropriate design for an acous-

tic resonator to be incorporated in an OAPA device. As was discussed in the previous

chapter, in order for an OAPA device to operate as intended, the strength of the opto-

acoustic interaction between two optical modes and one resonator acoustic mode must be

sufficiently high — the gain R must be large (see introductory chapter, Section 1.5). This

requires high quality factor (Q) resonators that satisfy certain conditions relating to the

acoustic frequency, mode shape matching, size and reflectivity.

Acoustic resonator prototypes are presented, ranging from a variety of single-paddle

designs [3] to 3-paddle designs inspired by Davis et al. [4]. I will also discuss design

geometries, material properties and FEM using the software ANSYS Classic 14.0 [106]

to produce modal predictions of resonant frequencies and mode shapes. Predictions of

optical coating losses are also presented as a function of size and location of coatings on

the 3-paddle resonator design.

33

34 2. Finite Element Modelling

2.2 Prototype Requirements

As explained in the introductory chapter, in Section 1.6, in order to make 3-mode opto-

acoustic parametric interactions possible and develop a novel OAPA sensor, a small

milligram-scale resonator with an appropriate acoustic resonance is required. This res-

onator should exhibit an acoustic mode in which the frequency is high to match an achiev-

able mode gap between two optical transverse modes in a small cavity.

The resonator should be kept small to increase the interaction strength as the para-

metric gain R is inversely proportional to the effective mass (see Chapter 1, Section 1.5),

while still large enough to sustain a sensor attached to the back of a resonating paddle

and to support optical coatings on the front side of the resonator. These coatings should

provide a high reflectivity and be much larger than the Gaussian profile of the laser beam,

to minimize scattering losses.

The resonator design should produce an acoustic mode which has (1) a frequency

between 0.1 to 1 MHz, (2) a rigid-body movement of an optically coated paddle with a

strain energy distribution showing minimum energy stored in the vibrating paddle, and

(3) a minimized energy coupling between the vibrations in the frame of the resonator and

of the coated paddle.

The stress energy stored in the vibrating paddle as it deforms must be minimized to re-

duce the acoustic coupling to the optical coatings, which would otherwise cause significant

reduction of the Q.

In general, the relationship between the Q and the coating losses is given as follows

[44]:

Q−1net = Q−1

i +Q−1c (ΔE/E) (2.1)

where Qi and Qc are the intrinsic and coating quality factors, respectively. The ratio of

the strain energy stored in the coatings ΔE, and the total strain energy E of the acoustic

mode are obtained by FEM. This is described in Section 2.6.

Typical coating loss values are Q−1c ≈ 4 10−4 [100], while silicon wafer intrinsic Qs

can be Qi = 107, though there is uncertainty in this value due to the contribution of

surface losses which are hard to predict.

The resonators will be clamped or suspended on the frame designed for this purpose.

The energy coupling between the frame and paddle must therefore be low in order to

minimize the coupling to suspension losses.

2. Finite Element Modelling 35

Here I model different resonator prototypes to assess which ones meet the requirements

mentioned above. As demonstrated by Zhao et al. [3], an acoustic torsional mode meets

the requirements of a rigid body movement to reduce coupling to coating losses as well as

the energy coupling to the frame. The vibration energy in the coating and frame can be

minimised by concentrating the energy in the torsion rods.

For the purpose of this thesis, I focused on simple designs and materials that would

be readily available to our collaborators to facilitate fabrication and ensure that testable

samples would be produced in a timely manner. In Chapter 6, I will present a more

elaborate design from silicon-on-insulator wafer products.

2.3 Finite Element Modelling (FEM)

The technique of FEM used to determine the mode shapes and frequencies of the acoustic

modes of resonators is a set of mathematical operations performed on specific model

geometries. These 3-D model geometries are represented by a finite set of nodes which are

grouped into elements of different types, depending on the variety of materials considered.

An in-depth review of these calculations is presented in the PhD Thesis of Slawomir

Gras [107] from which the majority of the codes used in this project originate. The codes

used here are available in Appendix C. Meshing of the models and modal studies was

performed using the ANSYS Classic 14.0 software [106]. The Bloc-Lanczos method [108],

a standard ANSYS approach, was used to solve the structural system.

The resonators were meshed using on average 74000 nodes, with a finer meshing of

size 50 μm in the area of interest (the paddles), and 300 μm in the frame of the resonator.

The models were meshed with Solid45 and Solid46 3-D elements, for the silicon substrate

and optical coatings, respectively.

For the prototypes studied in this chapter, the material properties of silicon [86, 81]

and of the optical coatings [100] were included in the FEM. The optical coatings consist

of multiple thin alternating layers of silica (SiO2) and tantalum pentoxide (Ta2O5). The

silicon substrate considered is a standard p-doped (100) monocrystal silicon wafer. The

properties of these materials, which are relevant to the modelling and predictions, are

shown in Table 2.1.

As silicon is an anisotropic material, the value of 169 GPa for Young’s modulus is only

an approximation [81]. In order for the modelling to be more accurate, material property

values given for different orientations were used [109]. The values for Young’s modulus,

shear modulus and Poisson ratio specific to the crystal orientation of a standard (100)

36 2. Finite Element Modelling

Table 2.1: Material properties of silicon substrate and silica-tantalum pentoxide opticalcoating thin films. Only Young’s modulus, Poisson ratio and density were relevant tomodal solutions.Material Young’s modulus (GPa) Poisson ratio Density (kg/m3)

Silicon (substrate) 169 0.22 2330

Silica (coating) 70 0.17 2200

Tantalum pentoxide (coating) 140 0.23 8200

silicon wafer were used in the FEM presented in this chapter. These values are listed in

Table 2.2.

Table 2.2: Young’s modulus E, shear modulus G and Poisson ratio ν of the anisotropicsilicon substrate [109], used for the FEM of acoustic mode shapes and frequencies.

Ex 169 GPa

Ey 169 GPa

Ez 130 GPa

Gxy 50.9 GPa

Gyz 79.6 GPa

Gzx 79.6 GPa

νxy 0.064

νyz 0.36

νzx 0.28

The most important material property (with the largest impact on predictions) is

Young’s modulus. As will be discussed in Chapter 5, small changes to Young’s modulus,

from laser heating for example, will cause significant changes to the predicted frequencies

of acoustic modes.

2.4 Single Paddle Prototypes

Resonator designs were first inspired by the work of Zhao et al. [3], who introduced the 3-

mode OAPA for macroscopic quantum mechanics experiments. Their design can be part

of a tunable centimeter-scale OAPA device, capable of cooling milligram-scale acoustic

resonators to the quantum ground state with practically achievable parameters [3].

Their design is shown in Fig. 2.1 (a), as a silicon resonator in the milligram scale,

consisting of a single square paddle with spindle arms (labelled ‘torsion rods’ from now

on) and optical coating on the paddle. The acoustic torsional mode (Fig. 2.1 (b)) can be

coupled to the optical transverse mode TEM01, shown in Fig. 2.1 (c), and have an overlap

factor value ∼1.

As demonstrated by Zhao et al., a torsional mode couples well to the TEM01 optical

cavity transverse mode [3]. For this reason designs in which the paddle vibrates with a

2. Finite Element Modelling 37

Figure 2.1: Single paddle resonator design. (a) Silicon acoustic resonator of dimensions1 mm by 0.8 mm by 0.5 mm, ∼ milligram mass with torsional resonant frequency inthe MHz range. The strain energy is mainly kept in the spindle arms (torsion rods) andminimized in the vibrating paddle. (b) Acoustic mode shape of the resonator, showinga simple torsional vibration. (c) The optical TEM01 mode (see Chapter 1, Section 1.5)is chosen as it has a suitable spatial overlap with the acoustic torsional mode. Imagecourtesy of Dr. Zhao [3].

torsion motion were considered.

2.4.1 Square Paddle Design

Single paddle designs were considered, using two torsion rods and a surrounding frame

of 20 mm by 20 mm, and thickness of 500 μm. These were investigated to obtain a

high frequency of the acoustic torsional mode and to minimise the elastic strain energy

on the middle paddle. Square paddle shapes and torsion rods shaped as long rectangles,

connecting the paddle to the frame, are illustrated in Fig. 2.2.

Figure 2.2: Square paddle prototype. (a) Dimensions: Wafer thickness of 500 μm, squarepaddle of dimension 1 mm by 1 mm, and torsion rods of 0.5 mm by 0.3 mm. (b) Thisdesign gives rise to an acoustic torsion mode at 334 kHz. (c)-(d) Strain energy distribution,zoomed-in and full frame view. (e)-(f) Amplitude map plots of the torsion mode.

This model, with a thickness of 500 μm, paddle of size 1 mm by 1 mm, and torsion rods

38 2. Finite Element Modelling

of 0.5 mm by 0.3 mm (length and width), achieves an acoustic torsional mode frequency of

334 kHz, which is suitable to meet requirements. The total strain distribution is suitable,

with minimal energy on the center of the paddle, which is the area intended for optical

coating.

The ratio of the vibration amplitude of the wafer to the amplitude on the torsion

paddle, referred to from now on as the ‘energy coupling’, is a bit high, reaching between

8 to 15% (see Fig. 2.2 (f)).

A low energy coupling is necessary to produce higher acoustic Qs of torsional modes, by

minimizing the impact of suspension losses. Since the energy coupling is not suitable with

the square paddle model, more modelling was required in order to find a better isolation

between the paddle and the frame.

2.4.2 Circular Paddle Design

A design with a circular paddle (sample thickness still 500 μm), shown in Fig. 2.3 was

also considered. The circular paddle has a diameter of 1 mm, and the torsion rods are the

same as for the square paddle design (0.5 mm by 0.3 mm). This design has a smaller mass

of vibrating paddle, which gives rise to a higher torsion frequency and a higher parametric

gain R (see Chapter 1, Section 1.5). The frequency of torsion from FEM predictions is

398.5 kHz, which is within the suitable range.

Figure 2.3: Circular paddle design. (a) Dimensions: Wafer thickness of 500 μm, circularpaddle of diameter 1 mm, and torsion rods of 0.5 mm by 0.3 mm. (b) This design gives riseto an acoustic torsional mode at 398.5 kHz. (c)-(d) Strain energy distribution, zoomed-inand full frame view. (e)-(f) Vibration amplitude map plots of the torsion mode.

The total strain distribution is qualitatively similar to the square paddle design, and

2. Finite Element Modelling 39

the energy coupling is between 6 to 14% (see Fig. 2.3 (f)). Though this is an improvement,

the level of energy coupling is still not suitable to meet requirements.

2.5 Three-Paddle Designs

Inspired by Davis et al. [4] who presented acoustic resonators with 3-paddle designs

(see Fig. 2.4), models with isolation paddles on either side of the torsion paddle were

considered. The 3-paddle design illustrated in Fig. 2.5 reduces the coupling between the

central vibrating paddle and the frame, which could lead to higher Qs.

Figure 2.4: Silicon nitride membrane resonators [4]. (a) Microscope image. (b) Finiteelement model of the common torsional mode (all three paddles moving in phase). (c)Differential torsional mode (out of phase isolation paddles moving only). (d) Middle paddletorsional mode. The color scale indicates the relative amplitudes. Image courtesy of Daviset al. [4].

More complicated designs are available in the literature [102, 103, 110]. These were

deemed too complex for the choice of materials and manufacturing techniques considered

here. Four prototypes of 3-paddle designs were developed for this thesis. From now on

they will be referred to as models A, B, C, and D, as shown in Table 2.3. Each model has

the same frame of dimension 20 mm by 20 mm, with a standard p-doped (100) monocrystal

silicon wafer thickness of 500 μm.

Models A, B, C, and D all have the same 3 paddle dimensions. The middle paddle

is 1 mm by 1 mm, and both side paddles are 1.8 mm vertically, and 1 mm horizontally

(see Fig. 2.5). The difference between these models is in the dimensions of the torsion

rods (4 in total) between the paddles, and between the side paddles and the frame. These

dimensions are given in Table 2.3.

40 2. Finite Element Modelling

Figure 2.5: Three-paddle design, providing isolation of the middle paddle from the frame.The torsion rods connect the paddles to each other, and to the frame. These are blocks oflength L0 and L1 and width W0 and W1, for the inner and outer rods, respectively. Theexample shown here is the model C.

Table 2.3: Three-paddle designs, models A, B, C, and D. Dimension variations in thetorsion rod lengths L0 and L1, and widths W0 and W1, as defined in Fig. 2.5. All modelshave the same paddle dimensions and a wafer thickness of 500 μm, except for samplesmade in France with optical coating (thickness of 300 μm).

Model L0 (mm) L1 (mm) W0 (mm) W1 (mm)

A 1.0 0.5 0.3 0.3

B 1.0 0.5 0.5 0.5

C 0.5 0.5 0.3 0.3

D 0.5 0.5 0.5 0.3

2. Finite Element Modelling 41

The four models are predicted to produce similar deformation and strain profiles, and

the amount of energy coupling differs only marginally. The main difference in these designs

is the value of the predicted frequency of the middle paddle torsional mode. Finite element

modelling plots for only one model are shown in Subsection 2.5.1 to avoid repetition. The

results for all models are summarized in a table in Subsection 2.5.2.

Acoustic modes of the resonators are categorized as ‘paddle modes’ (see Fig. 2.6)

and ‘frame modes’ (see Fig. 2.7). Because of the large number of modes in the acoustic

spectrum, only a subset of modes that serve to illustrate the main features of the resonators

will be discussed.

Figure 2.6: Acoustic mode shapes in which mostly the 3 paddles and 4 torsion rods areinvolved in the vibration, labelled ‘paddle modes’. These modes are expected to havesmaller coupling to the suspension system (presented in Chapter 3) and therefore providehigher Qs. Descriptive names were chosen for the mode shapes (a-e).

The paddle modes have less displacement of the frame and are expected to have lower

suspension loss, compared to frame modes, which have large displacements of the frame.

Figure. 2.6(a-c) shows three torsional modes: common, differential and middle paddle

torsion.

42 2. Finite Element Modelling

The middle paddle torsional mode is the ‘mode of interest’, for an OAPA device. This

mode in particular is chosen, rather than the common or differential torsional modes, as

it offers (1) a higher frequency and (2) a lower energy coupling between the middle paddle

and the frame.

The common mode has a lower strain energy distribution on the middle paddle, as all

three paddles and inner torsion rods are kept as a rigid body. The disadvantage is the

higher energy coupling which would cause more suspension losses and a lower acoustic

frequency (near ∼ 100 kHz).

Figure 2.7: Acoustic mode shapes in which the frame of the sample is involved in thevibration, labelled ‘frame modes’. These modes are expected to have greater coupling tothe suspension losses.

Frame modes are present all over the acoustic spectrum, with the fundamental drum

mode occurring within the first 10 or 20 kHz of the spectrum. These modes typically have

high acoustic loss due to a strong coupling to the suspension.

A problem arises when a frame mode is (1) too close to the mode of interest, and (2)

has a bandwidth large enough to overlap with this mode. This introduces losses in the

Q of the mode of interest, which reduces the strength of 3-mode interactions. For this

2. Finite Element Modelling 43

reason, designs are required which offer a good spectral separation between frame modes

and the mode of interest.

An example of acoustic spectra predicted for the designs C and D is shown in Fig. 2.8.

Spectra for designs A and B differ only quantitatively.

Figure 2.8: Acoustic mode spectrum of designs C and D, wafer thickness of 500 μm.Spacings between the modes vary greatly, ranging from a few 100 Hz, to several kHz. Redspectral lines indicate the common torsion, the differential torsion and the middle paddletorsion.

The important modes to consider are closest to the mode of interest, as they have

a stronger impact on the Q of this mode. The design dimensions can be modified until

nearby modes are far enough apart to reduce this impact. The mode spacing for design C

near the modes of interest is shown in Fig. 2.9.

Figure 2.9: Acoustic mode spectrum of design C, near the three modes of torsion. Redspectral lines indicate the common torsion at 105.8 kHz, the differential torsion at 159.2kHz and the middle paddle torsion at 378.3 kHz.

According to these predictions, the mode of torsion at 378.3 kHz is at least 5.8 kHz away

from the nearest mode, which is far enough to limit the impact of nearby modes. These

values will change depending on the dimensions, and once the resonators are fabricated,

44 2. Finite Element Modelling

FEM from measured dimensions will give a more accurate estimation of the mode gap.

2.5.1 Detailed Analysis of Model D

Model D has inner torsion rods with length of 0.5 mm and width of 0.5 mm, and outer

torsion rods with length of 0.5 mm and width of 0.3 mm (see Table 2.3). Figure 2.10 shows

the dimensions of the model, the acoustic torsion mode shape and strain distribution.

Figure 2.10: Design shape of model D, with three paddles and a wafer thickness of 500μm. (a) Dimensions of the paddles. (b) Deformation plot of the acoustic torsional modeof the middle paddle. The side paddles exert a small motion, an order of magnitude lowerthan the middle paddle. (c) and (d) are strain distribution plots (zoomed in and full viewof the resonator), showing the maximum strain energy located on the inner torsion rods,and the middle kept as a rigid body. (e)-(f) Vibration amplitude map plots of the torsionmode.

Model D offers the highest acoustic torsion mode frequency of all the 4 models consid-

ered, at 569.6 kHz. The strain energy distribution is not significantly different from the 3

other models (plots not shown), nor from models with 300 μm thickness, indicating that

the thickness of the wafer and the widths W0 do not impact the ability of the central pad-

dle to move like a rigid body. However, these dimensional changes do affect the torsional

mode frequency and the energy coupling (see Fig. 2.10 (f)).

Figures 2.10 (e)-(f) illustrate the acoustic torsional mode shape in a 3-D graph, showing

an energy coupling of 2%, compared to obtained values of 0.5%, 3% and 0.8% for models

A, B and C respectively (not shown). The wider torsion rods (larger values of W0), as

expected, cause greater energy coupling than models A and C. The higher frequency of

models B and D therefore come at the cost of reducing the isolation between the frame

and the paddles and potentially lowering the Q.

2. Finite Element Modelling 45

A distinction must be made between torsional modes of the middle paddle in which

only the paddles vibrate (pure), and in which non-negligible amounts of vibrations on the

frame are present (duplicates). The mode of interest is the pure torsion mode, whereas

the others are duplicates of the torsional mode. These modes will behave in a similar

rigid-body motion, with the addition of acoustic vibrations in the frame.

It is expected that these modes will be easier to excite through the frame (see Chapter

3) and will have lower Qs. Some duplicates of the mode of interest (middle paddle torsion)

are shown in Fig. 2.11. The modes nearest to the pure torsion of the middle paddle are

shown in Fig. 2.12.

Figure 2.11: The acoustic torsional mode of the middle paddle (mode of interest) occursseveral times at different frequencies, with different degrees of vibrations in the frame.

Figure 2.12: The mode of interest of model D is surrounded by other acoustic modes. Thenearest mode is at 567.6 kHz, at a spectral distance of 2 kHz. The second nearest modeis at 572.9 kHz, 3.3 kHz away.

In order to preserve the high Q of the pure torsion mode, other modes need to be

46 2. Finite Element Modelling

sufficiently far away and have linewidths (related to the Q) that ensure their mode spectral

profile does not overlap with the mode of interest. Since both modes closest to the torsional

modes are at least 2 kHz away, design D seems suitable.

2.5.2 Summary of Predictions for the 3-Paddle Designs

The following is a summary of the FEM predictions for the 3-paddle designs A, B, C, and

D. The three main torsional mode frequencies for each design are listed in Table 2.4, along

with the energy coupling between the frame and the middle paddle.

Table 2.4: Finite element modelling predictions of the 3 types of torsion modes for modelsA, B, C, and D, as well as energy coupling specifically for the mode of interest. All modelshave a thickness of 500 μm. Acoustic mode frequencies are given in kHz.Torsion Modes A B C D

Common, all 3 paddles in phase (kHz) 110.2 152.6 105.8 111.8

Differential, side paddles in anti-phase (kHz) 145.8 214.35 159.2 202.6

Torsion of middle paddle (kHz) 286.3 438.2 378.3 569.6

Energy coupling for the middle paddle torsional mode 0.6% 3% 0.5% 2.5%

The acoustic torsional mode of the middle paddle (mode of interest) is highest for

models B and D, due to the wider inner torsion rods (larger value of W0). However, the

drawback is the increased energy coupling from the frame vibrations.

Model C has the lowest energy coupling and higher Qs are expected from this design.

2.6 Optical Coating Losses

Another important aspect of the resonator design is the impact of optical coatings on the

torsion paddle. Having found suitable model designs, the requirement of high reflectivity

is then considered in order for a resonator to be part of an optical cavity (acting as an

end mirror) in an OAPA device.

One simple way to obtain mirrors of high reflectivity is to apply optical coatings on the

whole wafer sample (frame, paddles and torsion rods). This is by far the easiest approach,

in terms of manufacturing procedure. However, this would cause too much acoustic loss

due to energy coupling to the optical coatings, which have higher loss than silicon [111].

An alternative is to restrain the coating area to the middle paddle (see Fig. 2.13 (a))

and away from the torsion rods, where all the strain energy is located; an idea presented in

the work of Zhao et al. [3]. This requires careful planning of operations between superfine

polishing, dry etching [98] and applying optical coatings [112].

2. Finite Element Modelling 47

Figure 2.13: ANSYS view of the optical coatings. (a) applied only to the middle paddle,on a square area of 800 μm by 800 μm. (b) 3 mm by 3 mm square of optical coatings,reaching the inner torsion rods and half of each isolation side paddles.

2.6.1 Calculations of Quality Factor Losses from Optical Coatings

A complete description of the calculations of optical losses is found in the PhD Thesis

of Dr. Slawomir Gras [107]. The FEM codes from Dr. Gras were customised to fit our

3-paddle designs and used to perform the modelling of Q losses, as explained below. The

quality factor is a ratio of the total stored energy Es from an acoustic mode to the total

dissipation over one cycle Ed, given as follows:

Q = 2πEs

Ed. (2.2)

The total dissipation in a cycle can be written as Ed = 2π/ω ΔE, where ω is the

mode angular frequency and ΔE is the dissipation. The term ΔE can be written as

ωEs(φ/√φ2 + 1), where φ is the loss angle. From these, one integrates the energy stored

and the loss over the volume of the resonator, for a given mode [113]. This is given by the

following equation:

Qn =

∫En(�r)dv∫

En(�r)[φ(�r)√φ2(�r)+1

]dv, (2.3)

where Qn and En are the Q and the energy stored for a given mode n. This equation was

used to estimate the losses from coatings.

Finite element modelling is used to measure the amount of energy stored in every node

of the resonator volume for one acoustic mode. The ratio of energy stored in the substrate

and the energy stored in the optical coatings is calculated. Based on intrinsic loss values for

the substrate and the optical coatings, typically ∼ 10−7 and ∼ 4 10−4 respectively, the

48 2. Finite Element Modelling

net Qs are calculated for different optical coating sizes, keeping the coating to a common

thickness of 5.0 μm, chosen for a reflectivity of 99.997%, which meets requirements.

The intrinsic losses of the coating materials (silica and tantalum-pentoxide) are fre-

quency dependent [114]. Additionally, some acoustic mode shapes have stronger coupling

to the coated area, based on the mode-specific strain energy distribution. For these rea-

sons, the impact of optical coatings on the acoustic loss will be stronger for some modes,

and kept low for the mode of interest, by design.

2.6.2 Optical Coatings Impact

The impact of various optical coating sizes on the resonators was investigated, ranging from

small coatings fitting within the 1 mm by 1 mm area of the middle paddle to coatings

of the whole sample (20 mm by 20 mm), shown in Fig. 2.14. This figure shows that the

coating losses become important when the coating area overlaps with the torsion rods.

This is expected as the strain distribution has higher concentration in these rods, and

coatings in these regions will create a strong coupling to coating losses.

Figure 2.14: Finite element modelling of the impact of optical coatings on resonatorperformance, assuming different values of silicon intrinsic quality factor (Qi). Acousticlosses from optical coating area, estimated from Eq. 2.3. Beyond the point of 0.8 squaremm of optical coating area, the quality factors experience a steep decrease.

These calculations are performed for three different values of the intrinsic Q of the

silicon substrate: Qi = 106, 107 and 108. This is because there is some uncertainty in the

Q of the substrate. Figure 2.14 illustrate that when the coating is larger than the 1 mm

2. Finite Element Modelling 49

square paddle and overlaps the torsion rods, the losses from coating dominate and the net

Q of all three curves converge. Note that other losses are ignored. For coating areas small

enough, other losses from surface defects and suspension would dominate.

From the requirements mentioned above, I have determined that a coating area of 800

μm by 800 μm is optimal, as it allows for a reasonable beam size to minimize scattering

losses and does not incur large acoustic losses from excess mass or coupling to the strain

energy. The effect on the mode frequency is negligible, with a maximum decrease ∼ 4 kHz

when the resonator surface is completely coated.

2.7 Discussion

The use of FEM assisted the choice of prototypes and coating sizes for fabrication of

resonators to be tested in this project. Initial one-paddle prototypes developed with FEM,

with square and circular paddle shapes, show a high frequency of torsion and good mode

spacing, and a suitable strain energy distribution. However the energy coupling between

the frame and the paddles is too high for the mode of interest.

Designs with isolation paddles to address this issue were developed. Four models of

3-paddle designs were examined, labelled A, B, C, and D. These models vary only in the

dimensions of the torsion rods connecting the three paddles together, which affects the

value of the torsion frequency. These designs are predicted to perform according to the

requirements: provide an acoustic torsional mode at a frequency in the ∼ 100 kHz, have

a high acoustic Q, a resonance in which the middle paddle moves like a rigid body, and

good isolation from frame motion, which reduces coupling to suspension losses.

A coating area of 800 μm by 800 μm is chosen as it will prevent diffraction losses, and

will not introduce excessive acoustic loss.

As mentioned previously, more complicated designs could have been considered, such as

those described in the literature [102, 103]. Simple 3-paddle designs were selected because

of their ease of fabrication, and because they serve as suitable models for proof-of-principle

of a torsional resonator for an OAPA device.

Future work could make use of more complicated materials and manufacturing tech-

nologies to achieve higher Qs by reduced coupling to the frame and to the coatings.

50 2. Finite Element Modelling

Chapter 3

Methods of Fabrication and

Testing of Acoustic Resonators

3.1 Preface

Subsequent to the careful assessment of various microresonator designs using finite element

modelling (FEM), as described in Chapter 2, I have identified four models that show the

most promise to produce high quality-factors for an opto-acoustic parametric amplifier

(OAPA) device.

In this chapter, I present: (1) a brief overview of the fabrication techniques used to

make the silicon microresonators studied in this research; (2) the methods and experiments

used to test these resonators, varying in suspension design and excitation techniques; and

(3) FEM of the vibrational mode shapes to help determine the optimal locations for contact

with the suspension. This is important to obtain higher quality factors (Qs) by minimizing

the coupling between the acoustic mode of the resonator and the suspension.

The Qs obtained from various experiments performed on the fabricated resonators will

be discussed in Chapter 4, and analysed with the FEM nodal study method presented

here.

51

52 3. Fabrication and Testing Methods

3.2 Resonator Fabrication Techniques

Finite element modelling was used to identify four simple 3-paddle designs that showed

the highest potential to meet the requirements listed in Chapter 1, Section 1.6. These

designs have been identified as A, B, C, and D, using different dimensions of the torsion

rods connecting the paddles together (see Chapter 2).

Samples were fabricated using silicon (Si) wafers with patterns defined by wet etching

[96, 110], dry etching [98] and laser micromachining techniques [99]. To serve as a control

group to evaluate the impact of optical coatings, some samples were coated with an opti-

cally reflective set of thin films (total thickness ∼ 5.0 μm), using an ion beam sputtering

(IBS) technique [100].

The samples are grouped into various series according to their fabrication methods and

are summarized in Table 3.1.

Table 3.1: Fabrication techniques used to produce sample series. More details on the tech-niques are in Appendix D. Unless specified otherwise, all series are double-side polished.

Sample Location Technique Coating Thickness HighestSeries Q-factor

Obtained

UW1 WACSOM Wet KOH etching No 325 μm 1.4 104

UW2 WACSOM Dry etching No 325 μm -+ ANFF-Q

UQB1 ANFF-Q Dry etching No 530 μm 4.8 105

Single-side polished

UQB2 ANFF-Q Dry etching No 530 μm -

UQB3 ANFF-Q Dry etching No 515 μm 7.5 105

+ OmnicoatTM

F SESO, IEMN Dry etching Yes 365 μm 1.4 105

and LMA + SPR-220

TB1 LEGEND Lasers Laser Yes 520 μm 3.0 105

Inc + IPT micromachining

TB2 LEGEND Lasers Laser No 670 μm 8.66 105

Inc micromachiningSingle-side polished

Wet etching of the UW1-series was performed in the Western Australia Centre for

Semiconductor Optoelectronics and Microsystems (WACSOM) at the School of Electrical,

Electronic and Computer Engineering of the University of Western Australia, in Perth.

Photolithography and reactive-ion etching (RIE) techniques where used to define the wet

etching SiNx masking pattern on the Si wafer. Subsequently, the resonator’s shape was

obtained using KOH wet etching. This is known as Si bulk micromachining [96].

Dry etching of the UW2-, UQB1-, UQB2-, and UQB3-series was performed at the

3. Fabrication and Testing Methods 53

Australian National Fabrication Facility Ltd (ANFF-Q) in Queensland. A layer of SU8

2025 was spin-coated and patterned as a masking material for the deep reactive-ion etching

(DRIE) Bosch process [98], using a Plasma Therm DRIE machine. For the UQB3-series,

in order to facilitate the post-process removal of the SU8 photoresist mask, a layer of

OmnicoatTM was applied before the photoresist, and subsequently removed after the dry

etching was performed, using a lift-off technique [115] consisting of immersing the etched

samples in a solution of PG remover (NMP). Both the SU8 2025 and the OmnicoatTM

were obtained from MicroChem Corp [116].

A similar method was used to make the F-series in the Institute of Electronic, Micro-

electronic and Nanotechnology (IEMN) in France. These French samples were first pol-

ished in the ‘Societe Europeenne de Systemes Optiques’ (SESO). The SPR-220 thick pho-

toresist [117] was used as a masking material for dry etching, and a lift-off technique [115]

similar to that of ANFF-Q was used with a PMGI SF19 resin. Square holes of ∼ 0.8 mm

were patterned in the resin so that the optical coatings could be localised on the wafer.

The lift-off was performed by immersing the samples in a SVC14 solution until the PMGI

SF19 resin was completely dissolved.

Laser micromachining of the TB1- and TB2-series was performed at LEGEND Laser

Inc, in Taipei, Taiwan. A UV laser of 355 nm, 2W power and a beam spot of size 15 μm

were used to cut through the wafer.

All of the F- and some of the TB1-series were coated with an ion-beam sputtered

optical Bragg mirror consisting of alternating thin films of Ta2O5 and SiO2 of quarter

wavelength thickness [112, 100], in the ‘Laboratoires de Materiaux Avances (LMA) in

France and in the Institute of Photonics Technologies (IPT) at the National Tsing Hua

University, Hsinchu, in Taiwan.

Wet etching, dry etching, laser micromachining and optical coating process details are

in Appendix D.

In subsequent chapters, results are presented for individual samples from various series,

according to the design of the sample (A, B, C, and D). For example, a sample of the

UQB3-series, fabricated from dry etching in the ANFF-Q with a design C will be labelled

UQB3C-#, where UQB3 refers to the sample series described in Table 3.1, C refers to

the 3-paddle design, and # refers to the sample number from a selection within the same

series and design.

Similarly, samples made in France and Taiwan are labelled FD-1 and TB2C-2 (as

arbitrary examples). Note that FD-1 has no indication of the batch number, as only one

54 3. Fabrication and Testing Methods

batch was made in France. A subset of samples were tested and the results are discussed

in the next chapter.

It is important to note that while wet etching was not particularly successful in this

project, this technique has shown good results in the past (1990). Resonators similar to

the ones fabricated here were made with wet etching, producing Qs as high as 6 105 at

room temperature [110], surpassing any results from micromachined silicon resonators at

that time.

3.3 Methods for Testing Silicon Resonators

Different sample suspension/actuation techniques were used to characterize the fabricated

silicon resonators. These techniques included: metal clamping, Yacca bonding, electro-

static actuation and a vertical system of three-wires clamping the resonators on the edges

of the frame.

Characterisation of samples via these suspension/actuation techniques did not result

in measurements of high Qs. A subsequent method of resting the sample horizontally on

3 supports, labelled the 3-point suspension method, was developed. This method proved

useful in obtaining the highest Qs in this thesis. Results for all testing methods are

reported in Chapter 4.

3.3.1 Optical Lever

The optical lever method is a common component to the configuration of my experiments,

and is illustrated in Fig. 3.1. It consists of a laser beam reflected from the surface of the

resonator, which is subsequently focused by a focusing lens (FL) onto a quadrant photo-

diode (QPD) for detection. The QPD sends a voltage signal to the spectrum analyser.

This signal corresponds to the acoustic vibrations of the resonator under study.

The laser beam source is a 650 nm laser diode with a 26.5 mW output. The resonator

is suspended in a stainless steel cylindrical vacuum tank of length of 20 cm and a radius

of 20 cm. The spectrum analyser is used to process the QPD signal coming from the

reflected beam.

An oscilloscope was used to verify good alignment between the quadrant photodetector

and the reflected beam. The spectrum analyzer is also used to send a source signal to a

piezoelectric transducer (PZT), in contact with the resonator for actuation. This PZT is

shown in Fig. 3.2, where one side, not covered by the green rubber, is used for exciting

the sample (actuation surface).

3. Fabrication and Testing Methods 55

Figure 3.1: Optical lever on a large-mass optical table. (a) Experiment on the optical table.(b) Schematic representation of the setup, where the laser is incident on the resonator,and the reflected beam is focused with a focusing lens (FL) onto a quadrant photodetector(QPD). The signal received by the QPD is transformed into a voltage signal and sent tothe spectrum analyser (SA).

Figure 3.2: Close-up of a piezoceramic transducer (PZT), model AE0203D04F from Thor-Labs Inc. Two opposite sides are used for actuation (actuation surfaces). The rest of thePZT is covered in a green rubber.

56 3. Fabrication and Testing Methods

As illustrated in Fig. 3.3, the spectrum analyser can be used to measure the acoustic

spectrum (frequency domain) and the ringdown of a specific acoustic mode (time domain,

or time-trace). The Qs were measured in frequency domain and in time domain, depending

on the magnitude of the Q.

In frequency domain, the acoustic mode resonant frequency is observed by sweeping

the source with a chirp signal. The peak response frequency is measured and divided by

its bandwidth. The quality factor (Q) is given by the relationship Q = f/Δf , where f is

the acoustic frequency and Δf is the bandwidth.

Figure 3.3: Measuring Qs with the spectrum analyser. (a) Frequency domain, using a chirpsource, and measuring the center frequency (f) and bandwidth (Δf) of the peak. Thequality factor is given by Q = f/Δf . (b) Time domain, using a fixed sine source, excitingthe resonance and then switching off the exciting. A ringdown is observed, and fittedwith an exponential decay function Y (t) = A0e

−(t−t0)/τ + C0, where A0 is the resonantamplitude before the ringdown, t0 is the time at which the excitation is switched off, andC0 is the noise level.

In time domain, the resonance is excited by a fixed sinusoidal source, and the ringdown

is measured after the source is switched off. In order to do this, the acoustic mode

frequency is first determined in the frequency domain, as illustrated in Fig. 3.3 (a). Then

the source sent to a PZT is set to a fixed sine at the acoustic mode frequency. This drives

only that mode. The signal is then switched off and the ringdown of the acoustic mode is

observed, as illustrated in Fig. 3.3 (b).

The ringdown observed in time domain is fitted with an exponential decay:

Y (t) = A0e−(t−t0)/τ + C0, (3.1)

where A0 is the amplitude of resonance, or at time t = 0, t0 is the starting time of the

ringdown (when the source is switched off), and C0 is the noise level observed in the signal,

3. Fabrication and Testing Methods 57

without excitation or long after the ringdown. The exponential decay is characterised by

the relaxation time τ , and the quality factor is obtained as Q = πfτ .

The time domain (time-trace) method is more suitable for Qs higher than ∼ 104, as

the bandwidth becomes smaller than is practical to measure in frequency domain. These

Q measuring techniques were used for all of the experiments described below.

3.3.2 Metal Clamping

The first sample holding method consisted of using a metal clamp to squeeze a PZT and

resonator within two metal blocks, as shown in Fig. 3.4. The clamp was made from

aluminium, holding the PZT and sample firmly together. The surface area clamped in

this system was allowed to vary between measurements, in efforts to minimise suspension

losses. The location of the clamping was also varied, from clamping the corner to clamping

the center of the sample edge, in order to explore optimal clamping locations where higher

Qs could be obtained. Clamping areas that were too small introduced a risk of cracking

the sample frame.

Figure 3.4: Metal clamping system to hold our resonators vertically, in the plane of thelaser propagation. A piezoelectric transducer is held with the sample to induce acousticexcitation.

3.3.3 Yacca Bonding

A method of suspending the samples by bonding with Yacca gum was explored. This

consisted of bonding a sample to the extremity of a thin steel rod, and then clamping the

other end of the rod in the metal clamp described above. This is illustrated in Fig. 3.5.

58 3. Fabrication and Testing Methods

Yacca gum is a known material of low intrinsic loss, previously demonstrated not to

degrade the Q significantly [118, 119]. It was expected that this would limit suspension

loss.

Figure 3.5 illustrates the procedure used to bond the samples to the rod. Yacca gum

was grated into a fine red powder. A sample corner was put in contact with the Yacca

powder (see Fig. 3.5 b) and the bar was heated to 85 with a hot plate to melt the powder.

The temperature of the metal rod and sample was monitored with a thermistor (see Fig.

3.5 c). The hot plate was turned off once the powder turned black and liquid. The bond

was left to cool for an hour to solidify, and then the bar was clamped, as illustrated in

Fig. 3.5 (d).

As illustrated in Fig. 3.6 (a), one end of the rod was machined into a small protruding

area of 1 mm by 1 mm for bonding to the resonator frame. This reduced the contact area

with the sample, in the hope of reducing suspension losses. A metal sheet with a round

punch-hole is used to consistently distribute the same small amount of Yacca powder to

the protruding edge of a metal rod, as shown in Fig.3.6 (b).

Figure 3.7 shows the front and side view of the sample and thin rod bonded by Yacca

gum, using the improved rods with a small protruding area. Different bonding locations

were explored to attempt reducing the coupling losses and produce higher Qs.

The Yacca bond was brittle and great care was required to load the rod and sample

onto the metal clamp without breaking the bond. The bond survived acoustic waves from

the PZT excitation. However, small vibrations from the vacuum tank pumps or other

seismic sources sometimes caused the bond to break, dropping the sample.

3.3.4 Electrostatic Excitation

The electrostatic excitation method was a variation in the method for actuating acoustic

vibrations in the sample. Rather than using a PZT actuator, electrostatic forces from an

electrode were used a short distance behind the resonator, as shown in Fig. 3.8.

The idea is to selectively excite the torsional mode of the middle paddle by aligning

the electrodes to the middle paddle, with a slight offset. The electrode position was varied

with three translation stages (3-D).

The sample was first held in the metal clamp, shown in Fig. 3.4. Later, in attempts

to improve the experiment, a three-point suspension method was developed to reduce

clamping losses. The sample rested on three vertical supports and the electrode was

positioned underneath the sample.

3. Fabrication and Testing Methods 59

Figure 3.5: Yacca bonding system. (a) The Yacca gum powder is placed on the edge of ametal rod. (b) Two rods hold the sample in place, the rod with Yacca powder is placedunderneath the location chosen for bonding (bottom left corner in example). (c) A blockis pressing the sample and Yacca powder together while a hot plate heats the rods, sampleand Yacca powder. A thermistor is used to measure the temperature of the sample. Atemperature of 85 is required to melt the powder. (d) The bond is solidified after slowlycooling for an hour. The rod and sample are ready for loading carefully in the metal clampexperiment.

Figure 3.6: Yacca bond system. (a) Improved metal rods with a small 1 mm by 1 mmsquare protuberance are made to decrease the surface of contact. (b) Metal punch-hole toreproduce similar amounts of Yacca powder in every bonding attempt.

60 3. Fabrication and Testing Methods

Figure 3.7: Yacca bonding system. Side (a) and front (b) view of the result of Yaccabonding.

Figure 3.8: Electrostatic excitation. (a) A system of three translation stages is usedto move an electrode behind the vertical sample, held by a metal clamp. A PZT isheld with the sample to switch from PZT acoustic direct-contact excitation, to a non-contact electrostatic force excitation. (b) A close-up view of the positioning system forthe electrode.

3. Fabrication and Testing Methods 61

This 3-point suspension was later developed into the most important experiment in

this thesis, where the highest Qs were observed. This is discussed in Subsection 3.3.6.

3.3.5 3 Wire Systems

A holding system was designed and fabricated for vertical suspension using three wires, as

shown in Fig. 3.9. This was equivalent to a wall with three small arms (2 on one side, 1 on

the other) holding the sample outwards, 1 cm away from the wall. The main appeal of this

method is that it is suitable for holding a resonator firmly in an opto-acoustic parametric

amplifier (OAPA) device.

Figure 3.9: The 3-wire sample holding system, expected to minimize the clamping losses.(a) Different shapes of wires for reducing the transfer of vibrations from the apparatusand the samples. (b) View of a sample held by 3 wires. (c) Needle tip for excitation bycontact, using an AC- and DC-PZT.

The PZT excitation was designed as a movable needle, and various locations of exci-

tation could be explored. This method was improved by using a DC-PZT for positioning

and retracting of the PZT needle (see Fig. 3.10). This was done in an attempt to obtain

higher Qs by letting the resonator acoustic mode ringdown without a point of contact with

the PZT, which was thought to introduce excessive damping.

62 3. Fabrication and Testing Methods

Figure 3.10: Close-up of a DC-PZT, with an AC-PZT glued to the end surface, and a metalprodding needled glued to the end of the AC-PZT. This is used in the 3-wire systems.

3.3.6 Three-Point Suspension Setup

A suspension method was developed, consisting of supporting the resonators in a horizontal

position on three pins. The idea for this method came from efforts with the electrostatic

excitation method, as discussed above.

The laser beam was redirected with a 45 mirror, fixed inside the vacuum tank, onto

the middle paddle of the resonator. The beam was then redirected back to the QPD, as

shown in Fig. 3.11.

The initial 3-point system is shown in Fig. 3.12. Two brass columns were spaced 15

mm apart and the third point was a small steel pyramid glued on the top actuation surface

of a PZT (see Fig. 3.2). This third point was 12 mm below the two brass points. By

gluing the third support to the PZT, the excitation became part of the suspension.

An improved setup was developed based on finite element modelling (FEM) which

identified better locations for two of the three supports. This is presented in Section 3.4

below. The two brass column supports were replaced with steel needles of radius of 2.5

mm and height of 1 cm, with a fine point of ∼ 100 μm cross-sectional area as the surface

of contact with the sample. This setup is shown in Fig. 3.13.

3. Fabrication and Testing Methods 63

Figure 3.11: (a) View of the system installed in the vacuum tank. (b) Schematic diagramof directing the laser beam onto a sample supported by the 3-point system. The DC-PZTis used for retracting the actuation needle, to reduce damping of the ringdown.

Figure 3.12: Initial system of 3-point suspension, with two brass columns of 1 cm heightand 0.5 mm radius with a flat top surface of contact.

Figure 3.13: Improved system of 3-point suspension, with two steel needles of 1 cm heightand 2.5 mm radius. The point-like tips decrease the surface of contact with the sample,minimizing the suspension losses.

64 3. Fabrication and Testing Methods

Additionally, jigs (positioning tools) were fabricated to allow precise sample positioning

relative to the three-point supports in order to test the optimal suspension locations. This

also provided a means for repeatable results, by allowing samples to be loaded and re-

loaded in the same positions (see Fig. 3.14), within the precision of the manufactured jigs

and uncertainties related to manipulation.

Figure 3.14: Loading samples on the (1) three-point suspension with the plastic jigs. (2)Place the jig, push it against the back of the two support points, made possible by thesquare angle of the jig. (3) Load the sample onto the 3-points and lightly push until thesample is flush with both perpendicular jig walls. (4) Carefully push the jig tool backwardsby means of the handling extension, and remove the jig.

These tools were made in nine different shapes (see Fig. 3.15), allowing for a distance

between the two needle tips and the closest edge of the wafer along the paddle axis (see

Fig. 3.16) to vary between 5.75 mm and 9.75 mm, spaced 0.5 mm apart. The last position

(9.75 mm) is precariously close to the tipping point (10 mm) of the 20 mm side length

samples. This tipping point is referred to as the ‘equator line’ from now on (see Fig. 3.16).

As discussed in the next chapter, this 3-point suspension method proved useful in

obtaining high Qs from resonators studied here. For this reason, more emphasis was put

on this method, and the next section describes a finite element modelling (FEM) study of

the optimal loading positions for this method.

3. Fabrication and Testing Methods 65

Figure 3.15: Nine plastic jigs (positioning tools). (a) Jig with 7.0 mm spacing. Red linesshow the outlines of the resonator edge and paddles of the sample when it is loaded againstthe positioning tool walls. (b) Four of the nine separate jigs made, used to explore theacoustic standing waves in the frame of the resonator, for different acoustic modes.

Figure 3.16: The resonator surface directions parallel to the torsion rod long axis is referredto as the ‘Rod Axis’, shown on the bottom edge of the wafer. The directions parallel tothe length of the paddles is the ‘Paddle Axis’, shown on the left edge of the wafer. The‘Equator Line’ (dashed line) passes through all three paddles, along the rod axis. The‘Middle Paddle Line’ (dashed line) passes through the middle paddle along the paddleaxis.

66 3. Fabrication and Testing Methods

3.4 FEM Analysis of Nodal Positions

The Q measured from resonators tested with the 3-point suspension setup is expected to

vary according to the position of the 3 supporting pins. The torsional mode is predicted to

have most of the deformation in the middle paddle. However, some vibrations are present

in the frame, thereby increasing the coupling of the acoustic mode to the suspension.

A strong coupling to the supporting pins may degrade the measured values of the Qs.

The amount of coupling depends on the vibration amplitudes at the points of contact

between the resonator and the suspension points. By identifying the nodes and antinodes

of the acoustic vibrations in the frame, predictions can be made on the optimal suspension

locations where higher Qs could be measured.

To understand how the Q varies with the suspension locations and predict optimal

locations to maximize the Qs, this FEM nodal study will provide a comparison between

the experimentally determined optimal suspension positions, where the highest Qs were

measured (described in the next chapter), and the optimal suspension locations predicted

by FEM. The study described in this section is therefore a tool for analysing the Q results

reported in the next chapter as a function of the suspension locations.

The suspension system presented in Subsection 3.3.6 has three support locations which

are fixed, represented as a triangle in Fig. 3.17 (a). The jigs provide precise positioning

along the paddle axis, corresponding to translations of the support triangle along that

axis, as shown in Fig. 3.17 (b). While the position of the triangle of pins is allowed to

vary along the paddle axis, the position along the rod axis is fixed. The ‘left and right

support’ pins are 2.3 mm away from the edges of the sample (see Fig. 3.17 (b)), and the

third point is aligned with the centre of the equator line. The optimal distance of 2.3 mm

for the left and right support was determined by FEM of the torsional mode amplitude

distribution, as discussed below.

The vibration amplitudes on the frame of the resonator are calculated with FEM at

each supporting pin position, for the nine loading positions, which span three lines of 4

mm in length parallel to the paddle axis (dashed line in Fig. 3.17 (b)). The vibration

amplitudes of the resonator will vary according to the acoustic mode shape.

For the purpose of describing the modelling in this section, the torsional mode will be

analysed (see Fig. 3.18 a).

The snake and piston modes are also relevant as their acoustic frequencies are near

the torsional mode frequency. These modes give a signal similar to that of the torsional

mode detected from the optical lever method, described in Section 3.3, and are thus easily

3. Fabrication and Testing Methods 67

Figure 3.17: Resonator and three suspension point locations, forming a rigid triangle. (a)The triangle is upside down with a vertical height of 8.3 mm and a horizontal length of 15.4mm. For simplicity, the three pins are labelled: ‘left support’, ‘right support’ and ‘centresupport’, from now on. The centre support is the PZT actuator with a steel needled gluedon top of it. (b) As the sample is shifted downwards (white arrow) from loading positionsprovided by the jigs, the three points shift upward on the sample along the paddle axis.

Figure 3.18: Three modes are investigated with the FEM nodal study method. (a) Tor-sional mode of the middle paddle, the mode of interest for 3-mode OAPAs. (b) Snake-likemode, in which paddles flex along the rod axis. (c) Piston-like mode, where the paddlesmove in and out of the frame plane.

68 3. Fabrication and Testing Methods

confused with the mode of interest.

Figure 3.19: (a) Full view of the vibration amplitude map on the frame for the middlepaddle torsional mode. The maximum amplitude is normalised to 1 on the middle paddle.Dashed-line boxes indicating regions labelled A, B, C surrounding each support pin (left,right and centre supports). (b) Specific regions A, B and C of the vibration amplitudemap with thin strips parallel to the paddle axis, aligned with the fixed position of the pinson the rod axis. Amplitude profiles are calculated along these thin strips.

The vibration amplitude of the resonator is mapped across the resonator surface for

the torsional mode and the regions corresponding to the three support locations, labelled

A, B and C in Fig. 3.19, are examined. Thin strips on the three regions of interest (visible

as long thin rectangles in Fig. 3.19 (b)) are examined in more detail to plot the amplitude

profiles along the paddle axis lines which are aligned with each of the three support pins.

The profile of vibration amplitudes for the positions of the left and centre supports

are shown in Fig. 3.20. The profile of the left support line spans 20 mm, which is the

side length of the resonator. From the nodes and antinodes visible along this profile,

an acoustic wavelength of ∼ 4.4 mm is observed on the frame for this mode. Optimal

locations will therefore be spaced at ∼ 2.2 mm apart along that line.

The jigs will allow a suspension position range of 4 mm to be tested, starting near

the equator line. This 4 mm range is shown as red and blue insets in Fig. 3.20 (a). The

amplitude profiles along these ranges are replotted in Fig. 3.20 (b), where nodes and

support pin positions provided by the jigs (labelled A to I ) are indicated.

The amplitudes are null at the acoustic wave nodes, and these locations are optimal

to minimize the coupling between the acoustic mode and the pins. The loading positions

that will coincide best with these nodes are expected to produce the highest Qs.

Recall that the centre support consists of a PZT actuator with a steel needle on the top

surface. This third support is also used for actuation of the acoustic modes in the resonator.

3. Fabrication and Testing Methods 69

Figure 3.20: (a) Profiles of the vibration amplitudes of the frame along the paddle-axisthin strips from Fig. 3.19 (b), for the left and center supports. The left support profilespans 20 mm and the center support profile spans 8 mm, both parallel to the paddleaxis. (b) Close-up of the inset blue and red boxes, which correspond to the range of pinpositions provided by the nine jigs, for the left support and centre support, respectively.Red circles indicate the nodes (zero amplitude) of the acoustic waves in the frame for thetorsional mode. The thin blue rectangles indicate the nine possible positions of the pins,provided by the jigs. These jigs are labelled from A to I from now on.

70 3. Fabrication and Testing Methods

Therefore, in order for the torsional mode to be excited by this pin, the amplitude at the

location of this pin should not be zero.

The optimal situation would be to have zero amplitudes on the left and right support

pins and an antinode at the centre pin, to allow excitation of this mode with minimal losses

from the two fixed supports. This would create a problem in ringdown measurements,

however, once the actuation was switched off. A trade-off must be found. This complicates

the optimisation of this method.

The FEM nodal study method will be used in the next chapter to examine the relation-

ship between the measured Qs and the offset between the nodal points. As an example,

from the amplitude profile along the left support for the torsional mode shown in Fig.

3.20 (b), the optimal position is expected from using the loading tool D, with second best

positions from tools A and I.

From the profile along the centre support, the optimal position for actuation is from

loading tool C, with the second best positions provided by tools B and D. According to

these results, a sample of model C with dimensions close enough to those specified in the

design should have higher Qs measured when the sample is loaded with the position tools

B, C and D.

A similar reasoning is used to determine optimal locations for the other modes in Fig.

3.18 (b-c). As fabricated samples will differ in dimensions, thickness, and in acoustic mode

frequency, choosing the optimal 3-point locations is challenging.

Comparison between FEM predicted optimal locations and Q measurements reported

in Chapter 4 will give an indication of the validity of this modelling.

3.5 Summary

The silicon resonator fabrication techniques were briefly described and more details on

separate processes are available in Appendix D. The experimental setup used to measure

Qs of resonators, with different suspension and actuation techniques, were also described.

Here, the 3-point suspension method was described in more detail as this method al-

lowed measurements of high Qs, which are reported in Chapter 4. This 3-point suspension

method allows a means to incorporate the actuation as part of the suspension system by

gluing a support pin to a PZT. The minimal surface of contact between the sample and

the support pins reduces the coupling to suspension losses.

In the next chapter, samples are examined under microscope and carefully measured

dimensions are used in FEM predictions of acoustic modes. The nodal study in this chapter

3. Fabrication and Testing Methods 71

will be used to predict accurately the optimal suspension locations required to produce

higher Qs. The modelling presented here will be used as a basis of comparison between

the FEM prediction of optimal suspension locations and the experimentally measured

locations where higher Qs were obtained with the positions provided by the nine jigs.

72 3. Fabrication and Testing Methods

Chapter 4

Suspension Losses and Mode

Shapes

4.1 Preface

This chapter demonstrates that the resonators fabricated in this thesis can produce high

quality factors, close to the target of 106, as required for a 3-mode opto-acoustic parametric

amplifier (OAPA) device.

The long efforts of developing and testing suspension methods led to one method which

outperformed all others in this thesis. As presented in Chapter 3, the methods evolved

from metal clamping, Yacca bonding, electrostatic actuation, 3-wire suspension, to the

final 3-point suspension, described in Subsection 3.3.6.

The 3-point suspension method is used to examine various contact positions between

the resonator frame and the suspension. The optimal support pin locations, which mini-

mize coupling to the suspension, are explored by examining the relationship between the

quality factor (Q) and the contact locations. Using this method, the highest Qs in this

thesis were obtained.

Equally important for an OAPA device is the identification of the acoustic torsional

mode shape, as described in Chapter 2. This is required to couple two optical transverse

modes and reduce losses from coatings. Estimates of the acoustic mode shapes were

obtained using the optical lever method described in Chapter 3, as well as with a Michelson

interferometer described in this chapter. Finite element modelling (FEM) was used to

assist the investigation of the mode shapes measured at different frequencies.

Another goal of this chapter was to evaluate suitable methods of fabricating resonators

for an OAPA device. This is attempted by comparing the performance of samples fabri-

73

74 4. Analysis of Results

cated in different facilities, using manufacturing techniques described briefly in Chapter 3

with process details in Appendix D.

This chapter is structured as follows: First, I present a summary of the resonators

fabricated by wet etching, dry etching and laser machining is presented, including samples

with optical coatings. This is followed by a summary of results from prior testing methods

which provided lower Qs. Next, the mode shape measurements of different samples are

discussed. These were measured using the optical lever method and the interferometer.

Finally, Qs obtained from a 3-point suspension are presented.

The high Q results obtained with the 3-point suspension method were published in

the Journal of Applied Physics [105], and this publication forms the bulk of Section 4.5.

Further results, not included in the publication, are also presented in this section.

Based on the findings of this experiment, an improved configuration of the 3-point

suspension was developed and used to evaluate a wider range of samples that yielded even

higher Qs. These results are analysed in Section 4.6.

4. Analysis of Results 75

4.2 Fabricated Resonators

A set of resonators were fabricated using different techniques such as wet etching [95, 96],

dry etching [98], laser machining [99], and optical coating [100]. However, only a handful

were suitable for testing. Samples were rejected if they contained excessive surface damage,

rough pattern profiles or if they were made from the rounded edges of the silicon wafer,

resulting in non-square samples (see Fig. 4.1). These corner samples were rejected as

they were incompatible with the 3-point suspension method that I chose for the majority

of the experiments.

Most attempts at making samples from wet etching (UW1-series, see Chapter 3) in

the Western Australia Centre for Semiconductor Optoelectronic and Microsystems (WAC-

SOM) resulted in unusable samples, due to severe surface damage. The middle paddle

disappeared as a result of this etching (see Fig. 4.2), leaving only side paddles in small

diamond shapes, instead of rectangular paddle shapes as intended. This is due to the

higher etch rate of the (111) plane of the silicon wafer [96] which is aligned at 45 from

the sides of the paddles.

Single paddle designs also rarely survived the wet etching. Two samples made with

1-paddle designs of a square paddle of size 2 mm by 2 mm are shown in Fig. 4.3. The

poor results from this method are due to my lack of experience and time to dedicate to

explore this method, which was not my main objective. The best results from wet etching

are shown in Figs. 4.2 and 4.3.

Samples made from dry etching provided the majority of the samples suitable for

testing. UQB1- and UQB3-series provided good quality samples of 3-paddle designs (see

Fig. 4.4).

Figure 4.1: These are samples diced from the edges of the round silicon wafers, resultingin one or more rounded corners. They were rejected in order to focus strictly on squareframe samples, better suited for comparisons between samples.

76 4. Analysis of Results

Figure 4.2: Wet etching of 3-paddle designs in the Western Australia Centre for Semicon-ductor Optoelectronic and Microsystems (WACSOM) facilities at the University of West-ern Australia (UWA), Perth. (a) Masking pattern after photolithography and reactive-ionetching (RIE) of the selected parts of the masking layer. (b) Result after wet etching. Thesample is then dipped in a hydrofluoric acid (HF) solution to remove the remaining SiNx

masks.

Figure 4.3: Wet etching of 1-paddle designs. Pinholes and undercutting create manyunwanted scratches and damage to the surface of the resonators.

4. Analysis of Results 77

Figure 4.4: UQB1- and UQB3-series fabricated at the Australian National Fabrication Fa-cility Ltd (ANFF-Q) in Queensland. (a) UQB1-series were made from single-side polishedsilicon (100) wafers with SU8 resin residue, giving a diffuse reflection on the surface. (b)UQB3-series were made from double-side polished silicon (100) wafers and the SU8 resinwas properly removed using a lift-off technique [115] involving OmnicoatTM [116], givinga higher surface reflection than the UQB1-series.

The first batch (UQB1-series) with single-side polished silicon (100) wafers was made

according to the designs presented in Chapter 2. The resonators from this batch had

suitable straight edges and sharp angles, however there appeared to be a diffuse reflection

on the polished side of the samples. This diffuse reflection is probably from post-process

resins and chemicals left on the sample.

The second batch of resonators were made from double-side polished wafers (UQB2-

series) and had a better reflectivity, possibly due to steps taken to limit the residue of

photoresist. However, the dry etching was not successful in etching correctly all the way

through the wafer, and as a result the back side had a distorted unfinished pattern (see

Fig. 4.5 and Fig. 4.6), which rendered these samples unsuitable for testing.

Figure 4.5: Back and front sides of samples from the UQB2-series. The dry etching didnot cut through the full thickness uniformly, resulting in shapes unsuitable for testing.

78 4. Analysis of Results

Figure 4.6: Optical microscope view of the UQB2-series, showing a rough pattern on theback of the sample.

The third batch (UQB3-series), using the OmnicoatTM lift-off technique [115, 116],

produced the best samples from the ANFF-Q in Queensland. The reflectivity of the

samples and shape of the pattern were adequate for testing. Sample dimensions, obtained

from optical microscope measurements, are shown in Fig. 4.7.

Figure 4.7: Optical microscope measurements of the sample UQB3C-1, providing precisedimensions used for the updated FEM predictions of acoustic frequencies. L0 and W0

are the torsion rod length and width, respectively. LMP and WMP are the middle paddlelength and width respectively. W1 is the width of the torsion rod between the right paddleand the frame, which is designed to be the same as W0 for this design (C).

4. Analysis of Results 79

Two batches of laser micromachined [99] samples were made at LEGEND Laser Inc, in

Taipei, Taiwan (see Fig. 4.8). The first batch (TB1-series) contained three samples, one of

which was made without a protective film. The other two samples used a protective film

and were optically coated (Fig. 4.8 a) in the Institute of Photonics Technologies (IPT)

at the National Tsing Hua University, Hsinchu, Taiwan. These coatings were ion-beam

sputtered in a manner similar to the coatings made in the Laboratoires de Materiaux

Avances (LMA) in France.

As illustrated in Fig. 4.8 (b), the coatings were misaligned with the centre of the

middle paddle, resulting in coatings on the torsion rod. The mode of interest is the

torsional mode of the middle paddle, in which the strain energy is concentrated in the

torsion rods supporting the middle paddle. As discussed in Chapter 2, optical coatings on

torsion rods are expected to reduce the Q due to increased strain energy located in the

coating area.

The second batch (TB2-series) contained 4 samples, two of which were used for prelim-

inary testing of the laser micromachining technique (Fig. 4.8 (c), (d)). These test samples

were made from single-side polished silicon wafers of thickness ∼ 675 μm of unspecified

doping and wafer orientation. Laser damage, probably from overheating, is visible along

the edges of the pattern in Fig. 4.8 (c).

Sample TB2C-2 in Fig. 4.8 (d) shows a chip and a thin crack from the edge of the

sample. This may introduce surface losses. More importantly, sample TB1C-2 in Fig. 4.8

(b) shows rough undulating pattern walls, which could significantly affect the Q.

The remaining 2 samples were made from double-side polished (100) monocrystal sili-

con boron-doped wafers of thickness ∼ 520 μm. They contained optical coatings and were

corner pieces, and therefore unsuitable for testing with our methods.

Of these two batches, 4 samples were suitable for testing (2 of each), and as demon-

strated later in this chapter, one of these samples did produce the highest Qs obtained in

this project.

Dry etched and optically coated samples were made in the Institute of Electronic,

Micro-electronic and Nanotechnology (IEMN) in France (F-series). These French samples

were polished in the Societe Europeenne de Systemes Optiques (SESO). They are kept safe

from exposure to oxygen and dust and are locked in a cleanroom for future optomechanical

experiments, which will follow from this project.

Two samples were selected for examination and testing, to compare with other res-

onators. Microscope examination of optical coatings is illustrated in Fig. 4.9. The French

80 4. Analysis of Results

Figure 4.8: Samples from the TB1- and TB2-series were made from laser machining tech-niques at LEGEND Lasers Inc, in Taipei, Taiwan. (a) Sample TB1C-2, with opticalcoatings on the frame and on the middle paddle and torsion rod. (b) Close-up of coatingon sample TB2C-1, showing that the coatings intended to be centered on the middle pad-dle were instead applied partly on the torsion rod and middle paddle, due to misalignment.(c) Sample TB2C-1, which produced the highest Qs in this research. (d) Sample TB2C-2with major crack damage on the edge of the frame.

Figure 4.9: Sample FD-1, with optical coatings. (a) Image from scanning electron mi-croscope at IEMN, in France. (b) Measurement of the coating area dimension from anoptical microscope.

4. Analysis of Results 81

resonator coatings have a measured size of 830 μm by 830 μm for both samples, and are

assumed to be the same for all 13 samples produced. The resonator pattern appears to

be slightly curved. This is not observed on samples made in the ANNF-Q or by laser

micromachining.

The measured dimensions of resonators fabricated by different methods are summa-

rized in Table. 4.1. The UQB1-series have dimensions which are closest to the design

specifications, with variations under 3 μm in the torsion rod widths (the most relevant

parameter). By comparisons, the UQB3-series have dimensions ∼ 50 μm lower than the

specified widths, and the TB1- and TB2-series differ the most from intended dimensions,

with as much as 100 μm discrepancy. The F-series are within 40 μm of the torsion rod

dimensions.

It is not clear why the UQB1-series are so much more accurate than subsequent series.

Laser machined samples, however could probably be easily improved with more careful

procedure, and smaller focal length of the laser beam used for ablation. These measured

dimensions were used in FEM predictions of the acoustic spectra, discussed in Section 4.4.

Table 4.1: Fabricated resonator characteristics from a subset of samples made from wetetching using a potassium hydroxide (KOH) solution, deep reactive-ion etching (DRIE)which is a dry etching technique, laser micromachining (L.M.), and optical coating usingan ion-beam sputtering (IBS) technique. The labels W0, W1, L0, and L1 refer to thewidth and length dimensions of the inner (subscript 0) and outer (subscript 1) torsionrods, as described in Chapter 2.

Sample Name Fabrication Th. (μm)Dimensions (μm)

W0 W1 L0 L1

UW1 KOH 325 160 545 - -

UQB1A-1 DRIE 530 299 301 1052 542UQB1B-1 DRIE 530 501 497 1049 537UQB1C-1 DRIE 530 302 302 497 497UQB1D-1 DRIE 530 498 302 494 492

UQB3C-1 DRIE 515 254 252 546 537UQB3D-1 DRIE 515 452 249 545 537

FC-1 DRIE + IBS 365 264 264 532 534FD-1 DRIE + IBS 365 474 272 528 530

TB1C-1 L.M. 520 220 221 511 510TB1C-2 L.M. + IBS 520 202 207 510 518TB2C-1 L.M. 670 224 222 510 511TB2C-2 L.M. 675 218 220 510 508

82 4. Analysis of Results

4.3 Results from Prior Testing Methods

As mentioned in Chapter 3, several experiments were used in testing silicon resonators

in this thesis. All experiments included the optical lever component and Q measuring

techniques (time and frequency domain) described in Chapter 3, Subsection 3.3.1.

Many of these methods were found unsuitable for measuring high Qs, as the suspension

losses they registered were too high. The results of these methods are summarized below,

while the rest of the chapter will focus mainly on results from the three-point suspension

system, which produced the highest Qs in this thesis.

4.3.1 Metal Clamping

Experiments began with an aluminium block metal clamp to hold the resonator. The

sample was squeezed forcefully to a piezoelectric transducer (PZT), as shown in Fig. 4.10

(a). I tested the first available resonators from the UQB1-series, made from dry etching.

Samples from wet KOH etching were unsuitable as mentioned before.

The Qs obtained at first were no higher than ∼7 103 at pressures of 10−5 mbar. These

were measured in the frequency domain (see Chapter 3, Subsection 3.3.1), where the Qs

were calculated by dividing the centre frequency (f) of the acoustic mode by the bandwidth

(Δf) at half-maximum, according to the relationship Q = f/Δf . This is illustrated in

Fig. 4.10 (b).

Figure 4.10 (c) shows Qs measured at different pressures to investigate the air damping

effects. The Q measurements were performed as follows: First, the vacuum tank was

pumped to ∼10−5 mbar. The Q was measured at this low pressure. Then, by progressively

leaking air into the tank, the pressure was stabilised at different values, where more Q

measurements took place. This followed until atmospheric pressure was reached.

Figure 4.10: Metal clamping method. (a) Image showing the sample and piezoactuator(PZT) clamped together. (b) Example of an individual quality factor measurement infrequency domain: Q = f/Δf , where f is the acoustic peak frequency and Δf is thewidth at half-maximum. (c) Plot of the Q as a function of pressure in the vacuum tank.

4. Analysis of Results 83

Figure 4.11: Diagram of various metal clamping locations and area sizes. The minimumarea was 1 mm by 2 mm, below which the risk of cracking the sample was too high,indicated as ‘Optimal’, in the lower right of the image.

As seen in Fig. 4.10 (c), the Qs were not limited by air damping for pressures lower

than 1 mbar. At pressures lower than 1 mbar, the Qs were limited instead by suspension

losses. At pressures above 1 mbar, air damping was dominating the results.

By exploring different locations of clamping and also reducing the area of clamping,

higher Qs were obtained. Different clamping locations and clamping sizes are illustrated

in Fig. 4.11. This demonstrated that the coupling between the acoustic modes of the

resonator and the suspension was important. Quality factors of ∼4 104 were obtained.

This was exciting at the time, considering there was no guarantee that any of the resonators

fabricated in this thesis would produce high Qs.

The optimal clamping location was found at the corner of the sample, for a clamping

area ∼ 2 mm2 (see Fig. 4.11). Smaller clampings introduced too much risk of damaging

and breaking the sample.

As the Qs reached even higher levels, a method more suitable for measuring them

was utilised. Instead of measuring the frequency and bandwidth, the amplitude versus

time signal (time domain) was analysed after switching off the excitation. The frequency

and time domain measurements are illustrated in Fig. 4.12. The exponential fit and

calculation of the Q from a ringdown (see Fig. 4.12 d) is described in more detail in

84 4. Analysis of Results

Chapter 3, Subsection 3.3.1.

Figure 4.12 (d) shows the highest Q measured with this method, which was 1.18 105

at 10−5 mbar, still higher than many results from resonators found in the literature [4,

103, 101].

A similar silicon wafer resonator in the literature, suspension method unspecified, had

been reported to produce Qs as high as 6 105 [110]. For this reason, the Qs measured

with the metal clamping were deemed still limited by the clamping losses.

Further investigations with other resonators and different clamping locations did not

yield higher Qs and so I undertook a reanalysis of the problem and emerged with innovative

solutions that were applied experimentally to reduce clamping (or suspension) loss.

4.3.2 Yacca Bonding

Another method was introduced to lower the losses from clamping by instead bonding the

resonator with Yacca gum. The bonding area was smaller than the contact with metal

clamping. A metal punch-hole was used to confine the bonding area to 1 mm by 1 mm.

Details of this method and how the bond was made are in Chapter 3, Section 3.3.3.

Figure 4.13 illustrates the resonator bonded to a thin rod, using Yacca gum. The metal

clamp system described above is now used to hold the thin rod, together with a PZT. The

excitation is transfered from the PZT to the rod, and into the sample via the Yacca bond.

Similar to the metal clamping method, the location of the bonding was allowed to

vary. Different bonding locations are illustrated in Fig. 4.13 (b). The advantage of this

method was that less pressure was exerted on the sample from this suspension method.

The Yacca gum was easily removed using heat applied on the sample after experimental

runs.

A similar method was used by Schediwy et al. [118], comparing losses from bonding

a niobium resonator to mirrors using Yacca gum, cyanoacrylates and epoxy resin. In this

experiment, a small Q degradation was observed when using Yacca resin, from a Q of

3.31 106 without bonding, to 2.69 106 with Yacca bonding. This was compared to 1.69

106 and 1.00 106 for bonding with cyanoacrylates and epoxy resin respectively. Qin et

al. also used Yacca gum to bond a silicon nitride (SiNx) membrane in an optomechanical

experiment [55, 119], obtaining Qs ∼ 106.

Despite these previous high Q results with this resin, I did not find an improvement

in my experiment, with Qs produced no higher than ∼6 104 at pressures ∼10−4 mbar,

as shown in Fig. 4.13 (c). I also observed that in this setup, the acoustic mode excitation

4. Analysis of Results 85

Figure 4.12: (a) Spectrum analyser screenshot of the spectrum. (b) Screenshot of theringdown time-trace signal. (c) Recorded time-trace when the excitation frequency isslightly offset from the acoustic mode frequency, creating a beating signal. (d) Ringdownwhen the source and acoustic mode frequencies are equal. The ringdown starts when thesource is switched off. Highest Q obtained with the metal clamping method, near 1.2 105

at a pressure ∼10−5 mbar. The ringdown was measured and fit with an exponentialfunction (described in SubsectionOpticalLever) to determine the relaxation time τ . TheQ was calculated from the relationship: Q = πfτ , where f is the acoustic mode frequency.

Figure 4.13: Yacca bonding method. (a) Image of the resonator bonded to a thin rod withYacca gum. The rod is then held in the metal clamp discussed in the previous subsection.(b) The bonding location was varied to explore optimal suspension location. The bondingarea was fixed to 1 mm by 1 mm. (c) The highest Q obtained with this method was∼6 104.

86 4. Analysis of Results

required higher driving, due to the PZT no longer being in direct contact with the res-

onator. Vibrations from the PZT had to transfer through the entire 10 mm height of the

metal rod, and then transfer through the small Yacca bond, onto the sample.

Discrepancies with other high Q results in [55, 118, 119], where the resonators were

actuated with capacitive plates, could be attributed to the coupling to suspension still

being too high. As the results for metal clamping and Yacca bonding were still dominated

by suspension losses, an improved setup was required to obtain Qs closer to the target of

106.

4.3.3 Electrostatic Actuation

Electrostatic excitation avoids contact between the actuation and the resonator, and con-

centrates the excitation directly on the middle paddle. The main motivation to try this

method was that all supports could be made of low-loss materials, such as brass, instead

of a contact with the PZT, which has higher intrinsic loss.

I used electrodes with small cross-sectional areas aligned to the middle paddle with a

slight offset to selectively drive the middle paddle of the resonator, in the hope that this

would excite only the pure torsion mode. As excitation through the frame was already

demonstrated to be difficult using metal clamping, I expected that this localised excitation

would be required.

Contrary to expectations, the results were not an improvement, with a maximum Q

∼2 104 measured at 10−5 mbar. This excitation method only worked for a limited time

(a few days), despite long efforts with various different setups and electrode designs.

In principle, this method should provide less loss to the Q. Electrostatic excitation has

been widely used for exciting acoustic modes in silicon cantilevers [120], fused silica fibers

[121] and bulk silicon samples [91].

The resonators in this research are different to the ones in the literature, as the res-

onating paddle is much smaller and stiffer than long cantilevers. Even so, it is still unclear

why the method only worked for a limited time and what caused such difficulties getting

this method to work.

One positive outcome of the efforts with this method was the development of a hor-

izontal 3-point suspension, where the sample is simply resting on 3 pins. This method

later proved instrumental in obtaining the highest Qs measured in this thesis, and this is

discussed later in Section 4.5.

4. Analysis of Results 87

4.3.4 3-Wire Suspension

The 3-wire suspension system was introduced, consisting of clamping the edges of the

resonator frame, furthest away from the paddles, and holding the resonator vertically.

The motivation of trying this method was that it allows holding the sample in a manner

suitable for an opto-acoustic parametric amplifier (OAPA) device.

Figure 4.14 illustrates a FEM technique used to determine the optimal positions of

suspension on the left and right edges of the resonator frame, as well as a view of the

resonator, held by the 3 wires. These wires were bent into hooks and soldered to a plastic

board (see Fig. 4.14 b). Using FEM, the vibration amplitude of the torsional mode was

scanned on the left and right edges of the resonator frame (see Fig. 4.14 a). At the nodal

points, where the vibration amplitude is zero, the suspension contact is expected to couple

least to the acoustic mode (optimal). The right side of the frame was in contact with one

wire, located in the centre of the frame edge (see Fig. 4.14 b, the right edge and wire),

where FEM predicts a nodal point at the sample edge centre, as illustrated in Fig. 4.14

(d). On the left side, the 2 wires were placed 10 mm apart from each other, and 5 mm

away from the corners of the sample, where 2 other nodal points were predicted. This is

illustrated in Fig. 4.14 (c). The left and right side of the resonator are equivalent. I just

chose one side as an example here.

A method to estimate the points of contact between the 3 wires and the sample was

developed, consisting of analysing pictures taken after loading the samples at different

positions. The sample holders and the camera were fixed to repeat this measurement for

different runs, with different wire positions. The known side length of the square frame

(L = 20 mm) was used to determine the 3 wire positions. This is illustrated in Fig. 4.15.

This was only a rough estimate as the precise positioning of the wires was difficult.

The actuation in this method was done by a needle that was glued to a PZT, located

behind the resonator. This method was described in Chapter 3, Section 3.3.5.

Quality-factors varied with the sample position relative to the three wires. They also

varied with the location of the actuation needle, though to a smaller degree. I proceeded

to develop a method by which the resonator could be excited and then allowed to ringdown

without still being in contact with the excitation needle. This consisted of connecting an

AC-PZT with needle to a DC-PZT stack, to move the excitation in and out of contact

with the resonator. This was devised to assess whether the limit on Qs was from the

suspension or from the contact with the needle.

At first, this DC-retraction proved complicated. When a DC source was switched

88 4. Analysis of Results

Figure 4.14: Three-wire suspension system and finite element modelling (FEM) to deter-mine optimal suspension positions. (a) Scan lines to find optimal suspension locations onthe edges of the resonator. (b) Image of a 3-wire sample holder, with a resonator held bythe left and right edges of the frame. (c) FEM vibration amplitude profile along the leftedge of the resonator frame. (d) FEM vibration amplitude profile along the right edge ofthe resonator frame.

4. Analysis of Results 89

Figure 4.15: Graphic analysis to estimate the suspension locations after loading the res-onators on the 3-wire system. A picture is taken after loading, the frame side length Lprovides a known scale of 20 mm for reference. Lengths L1, L2, L3, and L4 are measuredgraphically and provide estimates of the 3 wire locations along the resonator frame. Thewire on the right side is meant to be in the centre (L3 = L4 = L/2).

off, instead of removing the contact monotonically, the DC-PZT seemed to oscillate and

induce large oscillations in the initial segment of the ringdown, as illustrated in Fig.4.16

(a). The Q could still be calculated, however the retraction method was then decreasing

the value instead of improving it (see Fig. 4.16 b).

A method was developed consisting of pulling out the small coaxial cable feeding

the DC-PZT, followed by switching off the power supply, and carefully re-connecting the

small coaxial cable. This proved successful in minimizing the oscillations induced from the

retraction method. At this point, the DC-retraction method improved the Qs measured

from AC-PZT switch method, as illustrated in Fig.4.16 )c).

The highest Q measured with the 3-wire suspension, obtained with the DC-retraction,

was 1.3 105 measured at 5.4 10−3 mbar, as illustrated in Fig.4.16 (d). A ∼ 20% increase

in Q was the maximum improvement from using the DC-retraction method, and never

beyond the highest Q mentioned above. For this reason, I concluded that the system was

not dominated by damping from the actuation needle, but still by suspension losses.

Continuing to introduce incrementally better suspension systems, a method was de-

veloped, using three vertical supports, on which the resonator rested horizontally like a

table. This was inspired by previous efforts with electrostatic actuation. The modification

here was that a PZT was now simultaneously acting as the actuator and as a support, by

mounting a steel needle on top of the PZT.

The sample was not held or clamped, but only supported on the three points. This

90 4. Analysis of Results

Figure 4.16: Three-wire suspension system. (a) Ringdown using the DC-retraction methodwith improved grounding, with a Q of 1.26 105, measured at 6.3 10−3 mbar. (b) Ring-down with AC-PZT with a Q of 1.27 105, measured at 6.3 10−3 mbar. (c) Measurementsdemonstrating that the DC-retraction provides an increase of 20% of the Q value com-pared to the AC-PZT switch method, once the technique was improved. (d) The highestQ obtained with the 3-wire system was 1.3 105 measured at 5.4 10−3 mbar, using theDC-retraction technique.

4. Analysis of Results 91

method allowed easy removal and reloading, and a possibility of exploring the optimal

location of the sample on the three points (see Chapter 3, Subsection 3.3.6 for a detailed

description).

A complex evolution of the suspension incorporating innovation, serendipity, and learn-

ing from experience led to the 3-point suspension that enabled results to be obtained that

approached the target Q. The results from this method are the main focus of the rest of

this chapter.

4.4 Mode Shapes

There are many modes that would give a strong signal when measured on the middle

paddle, and possibly give high Qs. Since the torsional mode is most relevant to this

project, it is important to first identify the mode shapes, in order to match the Q results

to specific modes of the resonator.

It should be pointed out that some discrepancies are expected between the FEM and

experimentally observed modes. These discrepancies arise from the dry etched walls not

being perfectly perpendicular, the undulating pattern lines from laser micromachining,

and different dimensions observed in the front and back sides of the samples. Also, the

mode shape measurements performed with the interferometer and the optical lever are

not using the same suspension and actuation methods. Some modes are being excited by

one method while not by the other.

For simplicity, when presenting mode shapes I will refer to the regions on the resonator

as identified by Fig. 4.17. Mode shapes at different frequencies and for different resonators

studied in this chapter were obtained by two means: (1) using the optical lever method

to measure the angular amplitudes of modes and compare their amplitudes on a set of

locations on the resonator surface, and (2) using an interferometer, which allows the phase

and displacement amplitudes to be measured.

The optical lever method is described in Chapter 3, Subsection 3.3.1, and is the basic

optical configuration used for all the experiments in this research. The ∼ 1 mm diameter

beam incident on several locations of the resonator was redirected to a quadrant photodi-

ode (QPD), and amplitudes of the vibration signal were recorded. The amplitudes were

compared over a set of locations, and this is referred to as the ‘amplitude ratio’ measure-

ment. This does not allow the phase of the signal to be measured and only gives a first

approximation of the mode shapes.

A better method was developed to allow the measurement of the response amplitude

92 4. Analysis of Results

Figure 4.17: Regions of the resonator relative to the edges of the wafer and to the equatorand middle paddle (MP) axis lines. The equator line passes through the middle of eachpaddle, parallel to the ‘rod axis’. The MP axis line passes through the middle paddle(widthwise), parallel to the ‘paddle axis’.

and phase, using interferometric principles. The interferometer is the work of Dr. HouWei,

a visiting professor from the Institute of Semiconductors, Chinese Academy of Sciences,

in Beijing, China. I have participated in the measurement of the mode shapes of some

samples, while Dr. Hou Wei developed the method and the technique, and performed

most of the measurements presented here. The interferometer is described in the next

subsection.

4.4.1 Interferometer Method

The Michelson interferometer uses the optical path difference to measure the resonator

paddle vibration amplitude. The interference pattern is detected by the photodiode (PD),

which creates a corresponding voltage signal sent to the spectrum analyzer. This signal

is proportional to the displacement amplitude of the resonator at the location of the laser

beam spot.

The interferometer is tuned to the point between maxima and minima in the inter-

ference pattern, such that the derivative of the pattern is largest. This provides larger

signals from perturbations (tapping on the optical table, as a test) and from the resonator

4. Analysis of Results 93

Figure 4.18: Interferometer method, developed by Dr. Hou Wei. The Helium-Neon lasersource beam is focused to a spot size of ∼ 80 μm, for several measurements along thepaddle lengths. The control loop, consisting of a low-pass filter and high-voltage ampli-fier, continuously corrects the reference mirror position for optimal signal output. Bycomparing the source output signal (Driving signal) to the signal measured by the pho-todiode (PD), the spectrum analyser measures the phase of the acoustic mode relativeto the source signal. This is repeated at several beam spot locations. Mode shapes aremapped across the surface of the resonator, using the phase and amplitude information.

vibrations (the signal).

A control loop, consisting of a low-pass filter and high-voltage amplifier (see Fig. 4.18),

is used to lock the interferometer output at the middle point between a bright and dark

fringe (half intensity), as mentioned above. This is done by applying a feedback voltage

to the PZT2 (see Fig. 4.18), to correct the arm length difference which slowly drifts at

low frequencies, due to temperature variations and seismic noise.

The acoustic modes of the sample are resonantly excited by a source signal sent to

the PZT1, clamped with the sample, and the spectrum analyzer displays the acoustic

spectrum and time-trace.

By simultaneously measuring the signal coming from the PD and the source used for

excitation of the sample, the spectrum analyser reads the phase of the vibration relative

to the source. This way, the phase is measured at different locations on the resonator to

identify parts that are in phase or antiphase, providing more information on the mode

shape.

Combining the phase and the amplitude measurements from scans along the paddles

of the resonator (Fig. 4.19), the mode shape can be mapped more accurately than with

the optical lever method.

A first assessment of the modes is performed by scanning on the paddles only, to

94 4. Analysis of Results

Figure 4.19: The lines scanned with the interferometer (Fig. 4.18). Following paddle axislines across the three paddles, marked L (left paddle), R (right paddle) and C (centre ormiddle paddle). Some measurements were performed beyond the paddle length limits, andonto the wafer to measure the amplitudes on the frame. The PZT actuator is clamped inthe bottom left corner of the sample.

4. Analysis of Results 95

identify the modes of interest. These are the torsion modes, where (1) all paddles are

moving in phase with each other (Fig. 4.20 (a)), (2) only the side paddles are moving,

and they are in antiphase (Fig. 4.20 (b)), and (3) mostly the centre paddle moves, with

smaller vibrations of the side paddles, which are both in antiphase with the centre paddle

(Fig. 4.20 (c)). Those modes are described in more detail in Chapter 2.

Figure 4.20: The modes of interest. (a) Common mode, (b) Differential mode, (c) CentrePaddle Torsion mode.

This first step is to qualitatively observe how the amplitude changes when scanning

along the paddle axis. The modes of interest are the ones where the amplitude is a

minimum on the equator line (see Fig. 4.17), and a maximum near the top and bottom

edges of the paddles, corresponding to the expected behavior from the modes of interest.

The identified candidates are examined in further detail, measuring the amplitude and

phase at 100 μm intervals along the paddle axis from the top to the bottom edge of the

frame (see Fig. 4.17). This is done to compare vibrations on the paddles to vibrations on

the frame.

As discussed in Chapter 2, the torsional mode of the centre paddle is the mode of

interest. There are a few modes which have a mode shape similar to the one shown in Fig.

4.20 (c). They differ from the pure torsion mode only in the amount of vibrations present

in the frame of the resonator. It is important to distinguish these ‘duplicate’ modes from

the mode of interest. Only the pure torsion mode will have significantly low coupling to

the suspension.

The following subsections present the mode shape results of a subset of samples from

dry etching and laser machining. These were obtained from the optical lever method and

the interferometer, and compared to FEM performed with measured dimensions of the

samples.

96 4. Analysis of Results

4.4.2 Sample UQB3C-1

The sample UQB3C-1 was made using a dry etching technique[98], in a process described

in Chapter 3 and with more detail given in Appendix D. The sample was observed under

an optical microscope and based on careful dimension measurements, FEM predicted: a

common torsional mode at ∼ 100 kHz, a differential torsional mode at ∼ 146 kHz, and a

pure torsion mode at ∼ 349.6 kHz. This was done using parameters for a standard (100)

p-doped silicon wafer.

These predictions are different from the results in Chapter 2, where perfect design

dimensions were assumed. The predicted mode shapes from measured dimensions are

shown in Fig. 4.21. As mentioned above, some discrepancies are expected between the

FEM and experimentally observed modes. Only a short subset is shown here.

The optical lever method was used to measure experimentally the modes predicted

with FEM. The Figs. 4.22 and 4.23 show that the modes at 294 kHz and 401.5 kHz

are both possible candidates of the pure torsion mode. The amplitude is a maximum

on the centre paddle, and a minimum on the frame, as expected for this mode shape.

The FEM predicts the torsion frequency at 349.6 kHz, which is between these values, but

discrepancies are expected with FEM, as discussed above.

The FEM predicted modes that were not observed experimentally with this method.

It is expected that some modes will couple more to the suspension and actuation method,

and therefore, some modes will be harder to excite. As a result, only a subset of modes

was identified with this method.

Amplitude ratio measurements gave strong evidence of torsion of the middle paddle at

294 kHz and 401 kHz (see Figs. 4.22 and 4.23). It should be noted that this optical lever

method is not able to distinguish precisely between the torsional mode and other paddle

modes, and is therefore only an approximation.

Both of these modes produced high Qs, particularly the one at 401 kHz. This made a

convincing case that one of these was the mode of interest, as it was expected that this

mode would couple least to the suspension losses.

As a consequence of this lower coupling, these modes should be harder to excite through

the frame, where the piezo transducer is located. These modes were indeed harder to

excite, further making the case that both modes at 294 kHz and 401 kHz could be the

torsional mode. One mode could be the pure torsion and the other, a duplicate of torsion

with more vibrations in the frame.

The interferometer method was operational only after high Q results were published

4. Analysis of Results 97

(a) Common Torsion mode at 100 kHz (b) Differential Torsion mode at 146 kHz

(c) Pure Torsion 349 kHz (d) Wafer Torsion 298 kHz

(e) Snake-like mode at 302 kHz (f) Piston-like mode at 410 kHz

Figure 4.21: Finite element modelling predictions of mode shapes for the sample UQB3C-1, based on measured dimensions. (a) Common torsional mode at 100 kHz. (b) Differentialtorsional mode at 146 kHz. (c) Pure torsion mode at 349 kHz. (d) Torsional mode withwafer vibrations at 298 kHz. (e) Snake-like mode at 302 kHz. (f) Piston-like mode at 410kHz.

98 4. Analysis of Results

Figure 4.22: Amplitude ratio measurements made with the optical lever method for themode at 294 kHz, using the x-output and y-output channels of the quadrant photodiode.The pure torsion mode is expected to have a maximum amplitude measured with they-output channel on the edge of the middle paddle.

Figure 4.23: Amplitude ratio measurements made with the optical lever method for themode 401.5 kHz, showing results similar to Fig. 4.22.

4. Analysis of Results 99

[105]. These are presented in the next section.

The mode shapes of the sample UQB3C-1 were measured again, this time using the

interferometer method (Fig. 4.18), to confirm the mode shapes expected from previous

amplitude ratio measurements. It was not possible to excite the modes at 294 kHz and

401 kHz with the interferometer method, for reasons mentioned above.

A mode at 332.5 kHz was excited and demonstrated remarkable evidence of the pure

torsion mode. As shown in Fig. 4.24 (a), the scans along the lines of the paddles, measuring

the amplitudes and the phase, demonstrate the expected behavior of the centre paddle.

This demonstration is stronger evidence than that obtained from the optical lever due to

the phase information.

The torsional mode was confirmed as the pure torsion in Fig. 4.24 (b), in which the

amplitude is measured on the resonator frame, showing a noticeable decrease of amplitude.

This is consistent with FEM of the pure torsion mode shape. Moreover, the frequency of

the torsional mode predicted by FEM is much closer to the 332.45 kHz observed with the

interferometer than to the candidates (294 kHz and 401 kHz) determined by the optical

lever measurements.

Figure 4.24: Interferometer measurements on sample UQB3C-1, for the mode 332.5 kHz,(a) showing indications of a centre paddle mode of torsion, where the centre paddle hasstronger vibrations than the side paddles. The side paddles are in phase, and the centrepaddle is in antiphase with the side paddles. (b) Scan lines extending to the frame region,showing smaller amplitudes on the wafer. This is consistent with the FEM predicted modeshape for the pure torsion mode.

4.4.3 Sample UQB3D-1

The sample UQB3D-1 is made from the same series as the sample UQB3C-1, presented

above. The design ‘D’ has wider torsion rods supporting the middle paddle (500 μm

100 4. Analysis of Results

instead of 300 μm, see Chapter 2), resulting in a higher torsional mode frequency.

Finite element modelling, based on careful dimension measurements, predicted a pure

common torsion at ∼ 99.9 kHz, a pure differential torsion at ∼ 191.8 kHz, and a pure

torsion mode of the middle paddle at∼ 576.9 kHz, with duplicate mode shapes surrounding

each of these modes. In particular, there is a torsion mode at 596.6 kHz with larger

vibrations in the frame of the resonator. The plots of FEM predictions are not shown as

they are qualitatively similar to those shown in Fig. 4.21.

Interferometric measurements of the mode shapes of the sample UQB3D-1 are pre-

sented in Fig. 4.25. The modes at 90 kHz, 191 kHz and 576 kHz were observed and

match well to the FEM predictions of the common, differential and middle paddle tor-

sional modes. The measurements show a mode at 594 kHz observed on UQB3D-1 with a

mode shape of torsion of the middle paddle with larger vibrations on the frame than on

the paddles (Fig. 4.4.3), consistent with FEM.

4.4.4 Sample TB2C-1

The sample TB2C-1 was fabricated from a wafer of thickness ∼ 670 μm polished only on

one side, patterned using laser micromachining [99]. The precise specifications are not

known, as this sample was originally meant as a test for the laser machining technique. It

later turned out to be the best performing sample.

The FEM predictions of the modes of interest are 109.4 kHz for the pure common

torsion mode, 163 kHz for the pure differential torsion mode, and 384 kHz for the pure

torsion mode of the middle paddle.

Amplitude ratio measurements in Fig. 4.26 seem to indicate a torsional mode of the

middle paddle at 394 kHz, and the common torsional mode at 87.9 or 100.8 kHz. The

differential mode was not identified in the amplitude ratio measurements. These observed

mode shapes are in agreement to the FEM predictions, to within ∼ 10 kHz.

Unfortunately, the sample TB2C-1 broke due to accidents in the manipulations before

it could be measured with the interferometer method.

Acoustic modes at 239, 304, 314, 394, and 445 kHz all produced high Qs, despite the

fact that some of them (in particular the mode at 445 kHz) has most of its mode vibrations

in the frame, as illustrated in Fig. 4.26 (d). This is contrary to the expectation that higher

Qs should be obtained from modes where mostly the paddles are involved. These are less

coupled to the frame where the suspension supports are located. Modes with vibrations

on the frame should couple more to suspension losses and have lower Q.

4. Analysis of Results 101

(a) Mode 90 kHz (b) Mode 191 kHz

(c) Mode 576 kHz (d) Mode 596 kHz

Figure 4.25: Interferometer measurements done with sample UQB3D-1, for modes 90, 191,576, and 596 kHz. (a) Common torsional mode at 90 kHz, where all three paddles movein a torsional motion, in phase with each other. (b) Differential torsional mode at 191kHz, in which the side paddles move in a torsional motion, in antiphase with each other.The central paddle is not involved in this vibration. (c) Pure torsion mode at 576 kHz,where the centre paddle has stronger vibrations than the side paddles, and the frame hasminimal vibrations. The side paddles are in phase, and the centre paddle is in antiphasewith the side paddles. This is consistent with the FEM predicted mode shape for thepure torsion mode. (d) Centre paddle torsional mode at 597 kHz, with larger vibrationamplitudes on the frame.

102 4. Analysis of Results

(a) Modes 88 and 98 kHz (b) Modes 100 and 239 kHz

(c) Modes 304 and 314 kHz (d) Modes 394 and 445 kHz

Figure 4.26: Amplitude ratio measurements done with sample TB2C-1, for modes 88, 98,100, 239, 304, 314, 394, and 445 kHz.

4. Analysis of Results 103

From the amplitude ratio measurements, the modes at 239 kHz and 314 kHz have

mode shapes of torsion of the middle paddle. However, only the mode at 394 kHz shows

vibrations almost strictly on the middle paddle, and is therefore much more likely to be

the pure torsion mode. Finite element modelling predictions indicate that a mode near

305 kHz is a torsional mode of the middle paddle with vibrations mainly on the wafer,

which could correspond to the experimentally measured 314 kHz mode, mentioned above.

4.5 Three-point Suspension: Initial Configuration

The mode shapes of the sample UQB3C-1 were predicted by FEM and measured using

an optical lever method and a Michelson interferometer method. This sample was tested

with the horizontal 3-point suspension, consisting of two brass columns as the left and

right supports and one steel needle mounted on a piezoelectric transducer (PZT) as the

bottom center support. Measurements were carried out by Phillip Meng and myself.

The results from this experiment were published in the ‘Journal of Applied Physics ’

[105]. My contribution to this paper amounts to 80% of the experimental work, all of

the FEM, and 80% of the manuscript preparation. This paper forms the bulk of this

section, with additional subsections on material which did not figure in the scope of this

publication, but which are still useful to this chapter.

It should be noted that at the time of testing this sample with a three-point suspension

method, the interferometer method was not yet operational, as mentioned in the previous

section. For this reason, the results in this publication are from the mode at 401.5 kHz,

which was believed to be the pure torsion mode of the middle paddle. This was supported

by amplitude ratio measurements.

Later measurements with the interferometer method revealed that the torsional mode

of the middle paddle was at 332.45 kHz instead. The only change to this publication is

that the high Qs for the mode of 401.5 kHz do not correspond to a pure torsion mode of

the middle paddle, but rather possibly to a piston-like mode predicted at 410 kHz and

shown in Fig. 4.21 (f).

Similar experiments were performed with an improved 3-point suspension setup on

a set of resonators from different fabrication techniques. Quality factors above the ones

presented in the publication were obtained, and they are reported in a later section.

104 4. Analysis of Results

JOURNAL OF APPLIED PHYSICS 114, 014506 (2013)

High Quality Factor mg-scale Silicon Mechanical

Resonators for 3-mode Optoacoustic Parametric

Amplifiers

By F. A. Torres1, P. Meng1, L. Ju1, C. Zhao1, D. G. Blair1, K.-Y. Liu2, S. Chao3, M.Martyniuk4,I. Roch-Jeune5, R. Flaminio6, and C. Michel6

1School of Physics, University of Western Australia, 35 Stirling Highway, Crawley, WesternAustralia 6009,Australia.2Australian National Fabrication Facility (Queensland Node), Australian Institute forBioengineering and Nanotechnology, The University of Queensland, Brisbane QLD 4072,Australia3Institute of Photonics Technologies and E.E. Department, National Tsing Hua University, 101Kuangfu Rd. Sec. 2, Hsinchu 300, Taiwan4School of Electrical and Electronic Engineering, University of Western Australia, 35 StirlingHighway, Crawley, Western Australia 6009, Australia5Plate-Forme RENATECH IEMN, Ave Poincare-BP 60069, 50652 Villeneuve dAscq cedex,France6Laboratoire des Materiaux Avancees (LMA), IN2P3/CNRS, Universite de Lyon, F-69622Villeurbanne, Lyon, France

(Received 28 March 2013; accepted 17 June 2013; published online 3 July 2013)

ABSTRACT

Milligram-scale resonators have been shown to be suitable for the cre-ation of 3-mode optoacoustic parametric amplifiers, based on a phenom-ena first predicted for advanced gravitational-wave detectors. To achievepractical optoacoustic parametric devices, high quality-factor resonatorsare required. We present millimetre-scale silicon resonators designed toexhibit a torsional vibration mode with a frequency in the 105 − 106 Hzrange, for observation of 3-mode optoacoustic interactions in a compacttable-top system. Our design incorporates an isolation stage and mini-mizes the acoustic loss from optical coating. We observe a quality factorof 7.5 105 for a mode frequency of 401.5 kHz, at room temperatureand pressure of 10−3 Pa. We confirmed the mode shape by mappingthe amplitude response across the resonator and comparing to finite el-ement modelling. This study contributes to the development of 3-modeoptoacoustic parametric amplifiers for use in novel high-sensitivity signaltransducers and quantum measurement experiments.

2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4812731]

4. Analysis of Results 105

4.5.1 Introduction

Three-mode optoacoustic parametric interactions were first introduced by Braginsky et al.

[1, 2] who presented the phenomena and evaluated the potential risk of 3-mode parametric

instability in long optical cavities for gravitational-wave detectors [36].

The physics of the 3-mode optoacoustic parametric amplifier is similar to that of an

optical parametric amplifier, in which the interaction occurs through the Kerr effect. The

3-mode optoacoustic interaction occurs when 2 optical modes inside an optical cavity are

resonantly coupled to a mechanical mode of the cavity mirror. The coupling is mediated

by the radiation pressure forces due to the beating of the two optical modes. Three-

mode interactions have been observed and studied extensively in long optical cavities

[39, 122, 123].

As triply resonant devices, 3-mode optoacoustic parametric amplifiers can in principle

have high sensitivity to torsional motion, achieve strong optoacoustic coupling and require

less laser power than a 2-mode interaction [36, 66], where only 2 modes are resonant.

For this reason, the 3-mode interaction reduces susceptibility to laser phase noise and

amplitude noise [76].

The creation of 3-mode parametric amplifiers requires a cavity design in which the

mechanical motion of the mirror couples the main laser carrier mode to a transverse mode.

For example, an optimal optoacoustic coupling can be achieved by using a torsionally

resonant mirror, which has optimal spatial overlap to the TEM01 optical cavity mode. The

mechanical mode of frequency near the MHz range can be matched to the gap between

the TEM00 and TEM01 modes in a specially designed optical cavity, as discussed by Miao

et al. [124]. This paper describes mechanical resonators designed for table-top 3-mode

parametric amplifiers [3].

In a small optical cavity, 3-mode optoacoustic parametric instability was first observed

by Chen et al. with a silicon nitride membrane of thickness 50 nm, within a Fabry-

Perot cavity [55]. Miao et al. [124] have proposed a tunable compact table-top system,

designed to observe 3-mode optoacoustic parametric interactions by adding a tuning cavity

to a main optical cavity. This would allow continuous tuning between the positive gain

regime (characterized by the Stokes mode, or amplification of the acoustic mode) and the

negative regime (Anti-Stokes mode, referred to as ‘self-cooling’ in the literature [125]), by

small adjustments of a lens and mirror.

In order to produce 3-mode optoacoustic parametric amplifiers in a compact table-top

setup, a mechanical resonator is required to be in the mm- and mg-scale ranges, have a

106 4. Analysis of Results

high mechanical quality factor, and a mechanical oscillation near the 0.1 to 1 MHz range.

Table 4.2: Review of mechanical resonators reported in the literature with high qualityfactors and frequencies in the range of interest.

Authors Mass Mode Shape Frequency Q(300K)

Davis et al.[4] 0.1 pg Torsion 21 MHz 2000

Chabot et al.[103] 0.7 ng Torsion 120 kHz 12000

Arcizet et al.[101] 210 μg (0,4) mode 2.8 MHz 15000

Serra et al.[102] 3 mg Torsion 85.5 kHz 145000

Kuhn et al.[104] 1 pg Longitudinal 3.66 MHz 1.8 106

Mechanical resonators have been reported in the literature and have characteristics

which are comparable to these requirements. Nanomechanical resonators were studied

by Davis et al. [4], and a quality factor of 2000 was reported for a torsional mode with

effective mass of 0.1 pg at frequency 21 MHz. Measurements were performed at room

temperature and pressure of ∼ 10−7 Pa. A pattern of 3 ‘paddles’ (rectangular elements)

along a rod of width 100 nm was used to isolate the vibration of the central paddle from

vibrations of the surrounding silicon nitride membrane. This relatively low quality factor

could be attributed to acoustic coating losses, as the entire sample was coated on one side

with 10 nm of permalloy, for use as a torque magnetometer.

Similar nanomechanical resonators were fabricated and studied by Chabot et al. [103].

They achieved mechanical quality factors of 12 000 for torsional modes at 120 kHz, with

effective mass of 0.7 ng, measured at room temperature and pressure of 13 Pa. These

resonators were made by etching silicon (100) boron-doped wafers, resulting in multiple

double-torsion paddle designs. A small magnetic film dot of 3 μm diameter was coated

onto the upper paddle, in view of making magnetometers and force sensors in nuclear

magnetic resonance force microscopy.

A French team led by Arcizet et al. fabricated and studied mm-scale silicon-on-

insulator chip resonators [101], made using double-sided photolithography and deep reactive-

ion etching (DRIE). They reported quality factors of 15 000, for a (0,4) transverse mode

of 2.8 MHz, with effective mass of 210 μg, measured at room temperature and pressure of

0.1 Pa. These 1 mm by 1 mm beam resonators were optically coated for high-sensitivity

optical monitoring of moving micromirrors.

Recently, Serra et al. reported [102] quality factors as high as 1.5 105 for torsional

modes of a mm-scale central paddle micromechanical silicon resonator, resonating at 85.5

kHz at room temperature and pressure of 10−3 Pa. Vibration isolation paddles and varying

the thickness of different parts of the sample were used to shield the main resonating paddle

mode from wafer modes. The aim of their design was to generate non-classical states of

4. Analysis of Results 107

light by opto-mechanical coupling, and to produce devices suitable for the production of

pondero-motive squeezing, and entanglement between macroscopic objects and light.

Kuhn et al. [104] made and studied 1 mm long, 240 μm wide, triangular nanopil-

lars made of crystalline quartz (known for low loss). They obtained a quality factor of

1.8 106 at 3.66 MHz, for a longitudinal mode of the pillar, measured at 10−1 Pa at

room temperature. Table 4.2 summarises the results obtained by the above mentioned

researchers.

For various reasons, the resonators presented above are not directly useful to use as

3-mode optoacoustic parametric amplifiers. However, they have helped to inspire our

design, presented in the next section.

The above mentioned mm-scale resonators can be significantly outperformed by larger

bulk silicon resonators. Results were obtained for boron-doped bulk silicon cylindrical

resonators with length between 6 to 75 mm, and common diameter of 76.2 mm. Nawrodt et

al. [91] report a quality factor as high as ∼ 3.5 106. This corresponds to the fundamental

drum mode of a cylindrical sample of length 12 mm, for a mode frequency ∼ 14 kHz at

room temperature and pressure of 10−3 Pa. The highest quality factor obtained for the

same sample, measured at 5.6K, was 4.5 108.

Most resonators have quality factors far below what is obtained for bulk silicon res-

onators. However, this paper presents resonators that have quality factors that do come

close.

Our motivation, comparable to the goals of Serra et al. [102], was to test 3-mode

optoacoustic parametric amplifiers in table-top setups, that are capable of being tuned

between positive gain and self-cooling [66].

In this paper, we report our design and finite element modelling (FEM) of millimetre-

scale silicon mechanical resonators, and various methods of fabrication. We describe the

experimental setup and methods used for measurements. We report our results, which are

similar to the low end of quality factors for bulk resonators [91], and a substantial quality

factor improvement from mm-scale resonators reported above.

4.5.2 Resonator Design

The primary purpose of our design, inspired by previous work with a silicon nitride mem-

brane [4], was to achieve a torsional mode that has adequate vibration isolation, a fre-

quency ∼ 400 kHz, and an adequate spatial overlap with a supported optical transverse

mode: a requirement for 3-mode optoacoustic parametric interactions [3, 126]. For reasons

108 4. Analysis of Results

of simplicity and low cost of fabrication, we avoided designs that required step changes

in thickness. Silicon was chosen for its low cost, availability, ease of fabrication and low

acoustic loss [91, 101, 4]. It is important to note that while a good vibration isolation

is required to observe mechanical modes of interest with high quality factor, too much

isolation is also a problem, if we want to excite the mode of interest via a different part

of the resonator which is not directly on the torsional paddle itself. A balance must be

reached to have low enough isolation to allow excitation of the central paddle mode, via

piezo excitation of the wafer, and high enough to observe a high quality factor of that

mode.

Our design consists of a 20 mm by 20 mm wafer with a central 1 mm by 1 mm paddle

and two side paddles (1 mm by 1.8 mm each). These paddles are on a narrow rod of

length 5 mm and width of 0.3 mm (see Fig. 6.1). The entire sample, made of silicon (100)

boron-doped monocrystal, has a uniform thickness of 500 μm.

Figure 4.27: Micro-mechanical resonator. (a) Wafer design: a 20 mm by 20 mm wafer ofthickness 500 μm, with a pattern of three paddles on a 5 mm long torsion rod. (b) Opticalmicroscope image of the central element of the resonator: a square mm paddle connectedby torsion rods of length 0.5 mm and width 0.3 mm. (c) Finite element modelling viewof the 3 paddles, each with a width of 1 mm and thickness 500 μm. Side paddles have alength of 1.8 mm. This design was chosen to obtain a torsional mode with high frequencyand a central paddle to act as a rigid body.

The central paddle is designed to undergo minimal elastic deformation and act as a

rigid body. The elastic stress is thus confined to the torsional rods, and an area of minimal

elastic stress of 0.8 mm by 0.8 mm, centered on the central paddle, is reserved for optical

coating [3]. This is important to reduce the loss due to stress on the coating material,

leading to unwanted contributions to the acoustic loss.

Simulations by FEM, using ANSYS 14.0 and a mesh size of 200 μm for the wafer, and

50 μm for the central paddle, showed that the fundamental wafer drum mode appeared

4. Analysis of Results 109

around 10.64 kHz, while higher frequency modes were mostly paddle vibration modes.

Modelling indicated that the 3-paddle system should exhibit 3 torsion modes, of which the

mode with highest quality factor (mode of interest) would have most vibrations localised

in the central paddle. In particular, the torsional mode of interest was predicted at 401.9

kHz. The other two modes occurred at frequencies around 100 kHz (all 3 paddles vibrating

in phase) and 160 kHz (side paddles vibrating in antiphase and central paddle not moving).

Our modelling also showed that this torsion mode of interest was separated from any low

quality factor wafer modes by at least 5 kHz, which ensured low contamination to the

mechanical loss of the torsion mode.

We experimented with three methods of fabricating our resonators: wet etching using

a potassium hydroxide (KOH) solution, laser cutting, and DRIE with a mask of SU8

resin. DRIE was chosen as it provided sharper features, compared to tapered walls and

other complications (pinholes and undercutting) associated with our efforts with KOH

wet etching. Laser cutting was also used and provided a significant improvement over wet

etching. However, we observed lower quality factors than for resonators fabricated using

DRIE.

The pattern was formed onto a 500 μm thick silicon boron-doped (100) wafer, DRIE

etching through the entire thickness of the wafer. A layer of Omnicoat was applied under-

neath the SU8 2025 resin that was used for photolithography patterning. The Omnicoat

layer allowed easy removal of the SU8 2025 resin after the DRIE was done.

The resonator reported here was without optical coating, however we have defined a

process by which coating could be put on the sample without covering the torsional rods.

This is for future experiments in a cavity designed for 3-mode optoacoustic parametric

amplifiers.

The design presented in this section could lead to excess coupling of the torsional mode

to wafer modes. Great care was required to design a suspension system to overcome this

problem. This is discussed in the next section.

4.5.3 Experimental Setup and Methods

The experimental setup used to characterise the fabricated resonator is sketched in Fig.

5.1. The light source was a class III laser diode operating at 26 mW and λ = 650 nm

(red). The laser beam is reflected off a 45 ◦ mirror and directed onto the central paddle of

the resonator, with a waist size of ∼ 300 μm. The reflected beam is redirected through a

focusing lens and onto a quadrant photodetector (QPD).

110 4. Analysis of Results

Figure 4.28: Schematic drawing of the experimental setup. Brass columns (BC); Piezo-electric transducer (PZT) with a glued-on steel pyramidal tip; Focusing lens (FL); Quad-rant photodetector (QPD); Spectrum analyser (SA). Red lines indicate the laser beampath. Black lines indicate the electronic connections. The sample is kept inside a vacuumtank, at room temperature.

Clamping the sample at one corner (1 mm by 3 mm area) caused excess losses on the

quality factor of the mode of interest. A maximum quality factor value of 9 104 was

obtained by clamping the wafer at various locations along the edges and corners. Bonding

our sample with Yacca gum, a natural resin with low intrinsic loss [118], also produced

quality factors too low for our requirements (6 104), as we found the isolation provided

by the paddles and torsion rods were insufficient for our required level of quality factors.

We adopted a 3-point support system consisting of two static brass columns of diameter

500 μm and height 10 mm, and one pyramidal steel tip glued to a piezo-electric transducer

(PZT). We chose the 3-point suspension because it allowed a convenient way to excite the

resonator by incorporating a PZT into a suspension point.

As discussed in the previous section, despite the isolation design, there would still be

some energy coupling between the torsional mode and some wafer modes. For this reason,

the wafer was required to be carefully positioned so that all 3 suspension points were close

to a node in the acoustic standing waves on the wafer surface. Finding the correct wafer

position could be challenging. However, for the correct position, high quality factors have

been observed and are reported in the next section.

The sample was mounted in a vacuum tank as shown in Fig. 5.1, in which pressure

4. Analysis of Results 111

could be reduced to 10−3 Pa. The experimental setup was kept at room temperature.

A spectrum analyser (SA) generated a driving signal sent to the PZT for excitation of

the mechanical modes of the resonator. The QPD measured angular variations in the

reflected beam and a corresponding voltage signal was sent to the SA. The driving signals

were generated as a fixed sinusoidal or a periodic chirp, both at values between 1 to 40

mV, kept low enough to avoid overdriving, which could cause nonlinear response from

the resonator and displacements of the sample. A periodic chirp was used to scan and

observe a spectrum of resonant frequencies, while the fixed sinusoidal signal was used to

specifically excite a target mode. From the latter, we measured a ringdown curve when

the source was switched off. The decay of a ringdown was then fitted with the following

formula:

A(t) = A0e−πftQ + C0, (4.1)

where A(t) is the amplitude (in Vrms) at any time t, A0 is the amplitude at the time when

we switch the source off, f is the mechanical mode frequency (in Hz), Q is the quality

factor and the parameter of interest, and C0 is the noise floor (in Vrms), at which the

amplitude eventually settles after sufficient time. We repeated the measurement to obtain

a value for Q with sufficient accuracy.

4.5.4 Results and Discussion

In order to identify the torsional mode in which only the central paddle is vibrating, we

scanned the wafer surface to measure the angular variations distribution (see Fig. 4.29

(b) and (d)) as a result of PZT excitation. Low frequency modes showed a distribution

in which all regions of the wafer are vibrating, while for some high frequency modes, the

vibrations are localised on the 3 paddles.

We measured the frequencies of the mechanical modes of the resonator between 10

to 500 kHz. We compared these mode frequencies to FEM predictions. Careful mea-

surements were made to obtain correct dimensions of the sample to enable calibration of

FEM predictions. The torsion mode of interest and the fundamental drum mode of the

wafer were identified experimentally at 401.5 kHz and 10.7 kHz, respectively. This was in

good agreement to FEM estimates of 401.9 kHz and 10.64 kHz (Fig. 4.29 (c) and (a)),

respectively.

We studied pressure dependence of quality factors and found that acoustic losses from

coupling to residual gas became negligible below pressures of 10 Pa. This ensured our

112 4. Analysis of Results

Figure 4.29: Comparison of modelling and measured data for a low and high frequencymode. (a) Fundamental drum mode of the wafer expected at 10.64 kHz. (b) Observed am-plitude distribution of angular vibration across the wafer at 10.7 kHz clearly demonstratesthe acoustic energy is mostly on the wafer. (c) Torsional mode expected at 401.9 kHz.(d) Observed amplitude distribution of angular vibration at 401.5 kHz, where acousticenergy is concentrated on the central paddle. Measured frequencies agree with modellingpredictions to within 400 Hz.

4. Analysis of Results 113

measurements at lower pressures were not dominated by gas damping.

Figure 4.30: Observed quality factor of (7.5± 0.2) 105 at a pressure of 10−3 Pa and roomtemperature. A ringdown curve is observed (red data points) when we switch off the fixedsinusoidal signal source at 401.5 kHz. The quality factor is calculated from the exponentialfit (blue dashed line) of the ringdown, with error bar obtained from the standard deviationof multiple measurements.

Table 4.3: Comparison of the measured frequencies of mechanical modes at room temper-ature and pressure of 10−3 Pa to modelling predictions. Both resonator modes (401.5 and296.2 kHz) have high quality factors (∼ 105), while both wafer modes (35.5 and 10.7 kHz)have low quality factors (∼ 103). † see Fig. 4.29 (a) and (c), for mode shapes.Mode Exp. Freq. (Hz) FEM Freq. (Hz) Q

Torsion 401 550† 401 900 7.45 105

Snake 296 220 299 800 6.56 105

Twist 35 500 33 450 2.16 103

Drum 10 700† 10 645 0.98 103

We obtained a quality factor of (7.5 ± 0.2) 105 for a torsional mode at 401.5 kHz

measured at room temperature and pressure of 10−3 Pa. Ringdown measurements, as

shown in Fig. 4.30, were repeated to obtain an estimate of the statistical error on the

quality factor. This value is higher than quality factors obtained from other small-scale

silicon resonators [103, 101, 102] presented in Table 4.2, and is close to quality factor values

obtained from silicon bulk samples [91].

Using similar methodology, we studied low and high frequency modes and we summa-

rize results for four modes in Table 4.3. Both high frequency modes show quality factors

significantly higher than for both low frequency modes. When vibrations overlap the lo-

114 4. Analysis of Results

cations of the 3-point suspension system, this can create stronger mechanical losses and

hence lower quality factor values. Given that both low frequency modes are predicted by

FEM to have vibrations on most of the sample area, this could explain why they have qual-

ity factors much lower than those of both the high frequency modes, which have vibrations

localised on the 3 paddles.

The ‘twist’ mode presented in Table 4.3 is a low frequency mode predicted at 35.5

kHz by FEM and observed experimentally at 33.45 kHz. The mode shape is described by

considering that any two adjacent corners of the wafer are moving in opposite directions.

Both low frequency modes in Table 4.3 have quality factors in the 103 range, low in

comparison to higher frequency modes. The second high quality factor mode shown in

Table 4.3 is termed ‘snake’ mode, in which the chain of 3 paddles undergo an S-shaped

motion, as predicted by FEM, on a X-Z plane along the Y= 0 line (see axes on Fig. 4.29

(b) or (d)). This snake mode was predicted at 299.8 kHz and found experimentally at

296.22 kHz. The snake mode also has vibrations localised on the 3 paddles, which explains

the relatively high quality factor.

The 3-point suspension system had the disadvantage that it made our results sensitive

to small displacements of the sample relative to the 3 points. The quality factors varied

greatly as a function of position, reaching a minimum of 40 000 for the torsion mode of

interest, for example. Using low frequency and large excitation, we were able to shift the

position of the wafer inside the closed tank, and obtain the optimal quality factor.

4.5.5 Conclusion

This study has shown that micro-mechanical torsional resonators, designed to optimize

spatial overlap with the TEM01 mode in an optical cavity for 3-mode interactions, can

have high mechanical quality factors. We have reported a quality factor of (7.5 ± 0.2)

105 for a mg-scale resonator at 401.5 kHz and at room temperature. The resonator

performance matches the requirements for a proposed 3-mode optoacoustic parametric

amplifier. The resonator is designed to ensure low acoustic stress in the optical coating,

so as to minimize optical coating losses. We’ve shown that the resonator can be excited

through the substrate without incurring unacceptable suspension losses. FEM predictions

were compared to amplitude response across the resonator and the wafer, matching the

predicted and measured frequencies and mode shapes.

The resonator performance achieved is close to the performance requirement presented

by Zhao et al. [3]. At room temperature, it should enable the creation of novel sensors

4. Analysis of Results 115

such as electromagnetic sensors and thermal noise sensors. Based on previous cryogenic

acoustic loss measurements in bulk silicon [91], the resonators presented here, if placed

in a cryogenic environment, could be expected to achieve quality factors in the range of

107 − 108. Such values would allow a range of experiments in quantum measurements,

tripartite entanglement between photons and phonons [124] and groundstate cooling. We

are currently working with a prototype optoacoustic parametric amplifier, using an opti-

cally coated resonator in a 12 cm-long optical cavity. Results will be reported in a future

paper.

ACKNOWLEDGMENT

This work was supported by the Australian Research Council, the Australian National

Fabrication Facility, the French RENATECH network, and a SIRF scholarship from the

University of Western Australia. We would like to thank Slawomir Gras and Stefan Danil-

ishin for collaboration on the finite element modelling.

116 4. Analysis of Results

As mentioned in the preface of this section, some of the results obtained with the

three-point suspension setup with brass columns were not included in the paper presented

above, as they did not fit within the scope of the manuscript. Those results are still

important to compare with the results from other resonators in the next section, and they

are presented here.

4.5.6 Effect of the Quadrant Photodetector Orientation

In the experiment using the optical lever, the vibration of the resonator causes a jitter in

the reflected beam. Ideally, a pure torsional vibration will cause the beam jitter to trace a

line on the quadrant photodiode (QPD) which is perfectly aligned to the y-axis. In reality,

a small offset angle between the jitter and the QPD y-axis is expected. As shown in Fig.

4.31 (b), the plane traced by the jitter cuts the QPD at an angle θ to the y-axis of the

QPD.

In order to optimize the signal-to-noise ratio while measuring results from the torsional

mode for this sample, I mounted the quadrant photodiode (QPD) on a rotation stage (see

Fig. 4.31 a). I then tested different angles, attempting to align the beam jitter from the

torsional mode to the y-axis of the QPD.

It is important to note that these measurements were made assuming that the modes

at 294 kHz and 401 kHz were torsional modes. These results should still be valid for

the torsional mode found with the interferometer and identified at 332.5 kHz. Any slight

misalignment would only reduce the signal received at the QPD.

The plot of signal amplitude versus QPD angle in Fig. 4.31 (c) shows alternating

maximum signals for the x-output readout and y-output readout. The torsional mode

signal is a maximum along one axis, and as the QPD is rotated by 90 , the signal becomes

a maximum in the other axis.

4.5.7 Air Damping

Having optimised the signal-to-noise ratio with QPD alignment, Qs were measured at

different vacuum tank pressures, to investigate the gas damping limit [127]. This required

measuring the Q factor at different pressures, which itself required to stabilize the pressure

at different values. This was achieved by first pumping that vacuum tank to a pressure of

10−5 mbar, closing a valve between the tank and the pump, and incrementally leaking air

into the system. This allowed measurements at different stable pressure values.

Figure 4.32 shows the Q of several resonant modes of sample UQB3C-1, measured with

4. Analysis of Results 117

Figure 4.31: (a) The quadrant photodiode (QPD) has 4 regions on the detector: A, B,C, D. The x-output reads the signal as (A+C) - (B+D). The y-output reads the signal as(A+B) - (C+D). The QPD is mounted on a rotation stage. (b) The beam jitter from thetorsional mode is expected to be at some offset angle θ from the vertical axis. (c) Plot ofthe QPD signal vs rotation angle of the QPD provided by the rotation stage.

the initial configuration of the 3-point suspension (described in the manuscript above), as

a function of pressure.

As seen in Fig. 4.32, the Qs were relatively stable for pressures below 1 mbar, and

not limited by air damping. The Qs begin to diminish as the pressure is increased to

atmospheric pressure (∼ 103 mbar), as expected.

I then repeated the measurements of Q as a function of pressure for the same sample,

investigating more acoustic modes this time. As previously, the measurements were per-

formed in the order of low pressures first, and slowly leaking to reach subsequent higher

pressures, until the atmospheric pressure.

Figure 4.33 shows measurements of the Q vs pressure for additional modes of the same

sample (total of 11 modes considered), in a shorter range of pressures. The range is shorter

to demonstrate an irregularity found in the ‘last’ measurement (higher pressure), discussed

below.

It was observed that when an acoustic mode with high response was excited, the system

was susceptible to sample displacements because of the strong coupling of the mode to

the suspension. This is termed ‘acoustic translations’ in this chapter and is discussed in

the next subsection.

It can be seen in Fig. 4.33 that the measurements at pressure of 102 mbar, performed

last, show an increase in Q from the previous measurement at pressure 40 mbar, for the

mode of 401.5 kHz. The Q should have decreased from air damping as this effect is greater

at higher pressures. However, the measurements made at this pressure induced resonator

118 4. Analysis of Results

Figure 4.32: Plot of the Q of sample UQB3C-1 for varying pressures, measured with thebrass columns 3-point suspension. At 1 mbar, the gas damping no longer dominates theresults.

4. Analysis of Results 119

vibrations sufficient to displace the sample (acoustic translation). This new position,

though random and hard to predict, produced a higher Q of the 401.5 kHz mode than

expected from air damping.

A change in the position of the sample, when large enough, is observed from the shifted

position of the laser beam with respect to the centre of the middle paddle. Sometimes

this shift is large enough that the beam is no longer incident on the middle paddle, and

recalibration of the system is required.

Figure 4.33: Repeated measurements of the Q of UQB3C-1 as a function of pressure forthe brass columns 3-point suspension. Due to overexcitation of a low frequency mode, thesample undergoes translation from acoustic vibrations, causing an increase of Q for somemodes (example shown with connected data points), where a decrease of Q was expectedfrom higher-pressure air damping.

4.5.8 Effects of Sample Positions via Acoustic Translations

During the experiment, I found that some modes were easily excited due to strong coupling

to the setup, as mentioned above. When overexcited, this resulted in displacements of the

whole resonator relative to the support pins. While this effect was undesirable during Q

measurements, I used this effect to explore the influence of the suspension position change

on the Q. I labelled this sample position shifting method acoustic translations, and it

allowed to observe one of the highest Qs in this thesis.

120 4. Analysis of Results

The disadvantage of acoustic translation is that it is difficult to predict the direction in

which a sample will move from these acoustic translations. The advantage is that multiple

sample positions can be explored when the vacuum tank is still closed and the experiment

is running. The alternative is time consuming. One must shut down the vacuum pumps,

turn off the equipment, open the tank, reload the sample at a different position, and pump

back to low pressures for every change in sample position.

Acoustic translations were repeated to explore the Q measurements at random po-

sitions in a much shorter time than the alternative. Another method would be to use

remotely controlled PZT sample positioners, however those options were not considered

as the tank setup was too confined. This could work well in a larger setup.

Figure 4.34 shows the Q results for 3 modes which were previously observed to produce

high Qs, including both the modes at 294 and 401 kHz.

Figure 4.34: Method of moving the sample on the 3-point suspension without openingthe tank. Measurements performed on the first 3-point system, with the sample UQ3C-1.Three frequencies are measured: 294 kHz, 296 kHz and 401 kHz. The X-axis representsdifferent iterations of acoustic translations, not in sequence.

These measurements show that the Qs are, as expected, sensitive to the suspension

locations. Quality-factors ranged from 4 104 to 7.5 105 for the 401 kHz mode, while

the mode at 294 kHz varied less, between 1 105 and 2 105. This demonstrates a key

4. Analysis of Results 121

difference between these modes. One mode, at 401 kHz, shows strong coupling to the

suspension. It is therefore likely that this mode involves significant vibrations in the

frame. The opposite is true for the mode at 294 kHz, and this is marginally supported by

the optical lever mode shape mapping of these modes, illustrated earlier in Fig. 4.22 and

Fig. 4.23.

Figure 4.34 demonstrates the importance of finding an optimal position of the sample

relative to the three contact supports. It also shows that some modes are more sensitive

to the position of the support pins. The torsional mode is expected to be less sensitive to

support positions, due to the reduced coupling to the frame.

From the results shown here, the mode at 294 kHz is more likely than the mode

at 401 kHz to be the torsional mode, judging from the stability of the Q. Though as

interferometer measurements have previously confirmed that the pure torsion mode is in

fact at 332.5 kHz, the mode at 294 kHz could then be a duplicate of torsion with a small

amount of vibrations in the frame (as opposed to large amplitudes which would increase

the sensitivity to displacements).

4.5.9 Nodal Points

Based on the results observed above, the torsional mode Q may be higher for specific

locations of the support pins in the suspension. The acoustic translation method, while

providing a means to explore different resonator positions, is inherently random, and a

more rigorous method was developed.

In order to properly identify the optimal location of the supports in the 3-point sus-

pension method, two changes were made to the experiment: (1) loading tools were made

to precisely position the samples and repeat measurements for the same positions, and (2)

the area of contact between the supports and sample was reduced to diminish the coupling

to the suspension. The loading tools and improved setup were described in more detail in

Chapter 3.

Finite element modelling was used to produce resonator amplitude maps of the tor-

sional mode, according to the measured dimensions of the sample UQB3C-1. This was

used as a reference to identify better locations for the 3 support points of the suspension

(Chapter 3, Section 3.4).

The amplitude maps in Fig. 4.35 show that there is low vibration amplitude on the

equator line, as shown in Line 2 of Fig. 4.35 (c). In perpendicular lines, 2.3 mm away

from the left and right edges of the sample (Line 1 on left side), there are also narrow

122 4. Analysis of Results

Figure 4.35: Finite element modelling used to determine new and improved locationsfor the 3 supports of the 3-point suspension. (a) The mode shape found with FEM,showing the torsional vibration of the middle paddle for sample UQB3C-1. (b) Vibrationamplitude plot of this mode, normalised to the maximum amplitude on the edges of themiddle paddle. (c) Section of the left side of the resonator, mapped for more details ofthe relative amplitudes. (d) Line 1 which is along the paddle axis (see Fig. 4.17), showinga low level of vibrations near the torsion rod axis. (e) Line 2, showing the vibrationamplitudes from the left side of the resonator to the left boundary of the three paddles,along the torsion rod axis (equator line). The vibrations along Line 2 are below 0.2% ofthe maximum values calculated on the middle paddle, indicating a good location to applythe suspension contacts. More details of this FEM are in Chapter 3, Section 3.4.

4. Analysis of Results 123

corridors of low amplitudes. Good locations for the left and right supports, now labelled

top supports, are at the intersections of Line 1 (found on each side), and Line 2.

Two optimal locations for the top supports are therefore at 20 - (2 2.3) = 15.4 mm

away from each other and 2.3 mm from the left and right edges of the frame. The optimal

location for the third support point varies too much from the FEM mapping of one sample

to the next for any reliable optimal choice to be made. Therefore, a practical choice was

made to place it 8.3 mm below the other two points, in order to allow a range of sample

positions to be explored.

A distance greater than 8 or 9 mm makes it impossible to probe sample locations

where the two top points are close to the equator, as the bottom point becomes close to

the bottom edge of the sample (at or beyond 10 mm below the equator).

124 4. Analysis of Results

4.6 Three-Point Suspension: Improved Configuration

The suspension setup was modified accordingly by replacing the two brass columns with

two steel needles of diameter 2.5 mm and fine needle tip for a lower surface area of contact

(∼ 100 μm measured with an optical microscope) between the supports and the sample

substrate. The position of the steel needles was chosen from FEM of optimal suspension

locations, as discussed above.

This method allowed the highest Qs to be obtained, as well as precise sample loading

positions. For these reasons, I chose this setup to test samples from different fabrication

techniques and compare their Qs.

Given the nine loading positions from loading tools presented in Chapter 3 and the

measurements of the sample dimensions, FEM was used to measure the expected ampli-

tudes on the frame for different acoustic modes. Mapping the amplitudes on the wafer

gives an estimate of the acoustic waves in the frame for given modes.

This FEM technique was presented in Chapter 3, Section 3.4, showing a variation

of acoustic wavelengths in the frame according to mode shape, frequency and sample

thickness. This information is important in order to assess the impact of suspension

losses when the sample is suspended on strategic nodal points, or offset from these points.

Examples of this FEM nodal investigation based on the samples UQB3C-1, UQB3D-1 and

TB2C-1 are summarized in Table 4.4.

The values in Table 4.4, for the pure torsion mode at frequencies 349 kHz, 576 kHz

and 384 kHz, show reduced energy coupling, compared to other modes, as expected. A

few exceptions include lower energy coupling values at bottom support positions for the

mode at frequency 379 kHz, for the sample TB2C-1.

The predicted optimal support positions for the top (left and right) supports generally

coincide, as expected from the symmetry of the torsional mode shape. The loading tools

to provide these positions do not coincide, however, with the optimal positions predicted

for the bottom support.

As an example, consider the energy coupling values for sample UQB3D-1 at 576 kHz

(see Table 4.4). The optimal top support positions are obtained with the loading tools A,

E and F, while the optimal positions of the bottom support are obtained with tools C, G

and H. A similar mismatch is found for samples UQB3C-1 and TB2C-1.

This mismatch between the optimal positions for all three support pins, and corre-

sponding loading tools, makes identifying the optimal suspension a serious challenge.

Sample TB2C-1 is a special case, as this sample broke before all positions could be

4. Analysis of Results 125

Table 4.4: Table of the relative vibration amplitudes (local divided by maximum) atdifferent suspension positions, with respect the specific acoustic modes of samples UQB3C-1, UQB3D-1 and TB2C-1. Labels A to I refer to the 9 possible loading positions providedby the loading tools of 4.5 mm to 8.5 mm described in Chapter 3. The table includes ameasure of the acoustic wavelength in the frame for each mode: λx and λy along the rodaxis and paddle axis, respectively (see Fig. 4.35).

Samples

UQB3C-1 UQB3D-1 TB2C-1

Frequency (kHz)

347 349 357 570 576 596 379 384 391

Left Support

A 0.18 0.0011 0.11 0.15 0.0029 0.058 0.011 0.00032 0.20

B 0.48 0.0023 0.31 0.39 0.0075 0.14 0.026 0.00094 0.17

C 0.58 0.0017 0.47 0.45 0.0087 0.22 0.053 0.0016 0.11

D 0.46 0.00066 0.56 0.30 0.0063 0.26 0.086 0.0021 0.031

E 0.17 0.0039 0.57 0.026 0.0013 0.24 0.12 0.0026 0.083

F 0.18 0.0065 0.48 0.26 0.0041 0.14 0.15 0.0030 0.21

G 0.44 0.0070 0.31 0.43 0.0077 0.024 0.17 0.0033 0.33

H 0.51 0.0047 0.092 0.41 0.0080 0.20 0.15 0.0033 0.43

I 0.37 0.000090 0.15 0.20 0.0048 0.030 0.11 0.0031 0.48

Right Support

A 0.18 0.00065 0.11 0.15 0.0028 0.045 0.20 0.00033 0.0071

B 0.48 0.0016 0.31 0.39 0.0074 0.14 0.17 0.00097 0.026

C 0.58 0.00095 0.47 0.44 0.0087 0.22 0.11 0.0016 0.053

D 0.46 0.0011 0.56 0.30 0.0064 0.26 0.028 0.0021 0.089

E 0.17 0.0038 0.57 0.026 0.0014 0.24 0.080 0.0026 0.13

F 0.18 0.0059 0.48 0.26 0.0040 0.14 0.20 0.0029 0.16

G 0.43 0.0061 0.31 0.43 0.0076 0.016 0.33 0.0031 0.17

H 0.50 0.0038 0.091 0.41 0.0080 0.20 0.43 0.0031 0.16

I 0.36 0.00061 0.15 0.20 0.0048 0.33 0.47 0.0030 0.11

Bottom Support

A 0.023 0.0035 0.072 0.52 0.010 0.23 0.00072 0.0037 0.095

B 0.12 0.0051 0.30 0.39 0.0078 0.42 0.00074 0.0029 0.28

C 0.21 0.0053 0.47 0.091 0.0022 0.50 0.00038 0.0015 0.38

D 0.23 0.0042 0.56 0.21 0.0032 0.41 0.0066 0.00028 0.38

E 0.18 0.0026 0.58 0.38 0.0062 0.18 0.0014 0.0020 0.29

F 0.047 0.00083 0.51 0.36 0.0058 0.14 0.0043 0.0034 0.0043

G 0.12 0.00053 0.38 0.18 0.0030 0.45 0.00093 0.0043 0.018

H 0.25 0.0013 0.21 0.062 0.00051 0.65 0.0043 0.0046 0.067

I 0.30 0.0016 0.0014 0.26 0.0031 0.65 0.0014 0.0046 0.075

Acoustic Wavelength (mm)

λx 7.3 5.0 4.5 5.0 4.6 3.9 6.0 5.0 5.0

λy 4.6 4.6 7.9 4.5 4.6 5.8 8.5 14.6 8.0

126 4. Analysis of Results

tested. As a result, Qs are presented only for the positions provided by the loading tools

A, B, C, and D. At these four positions, the results of Table 4.4 show that the optimal

support positions for the top supports are the complete opposite of the bottom support.

The optimal loading tool for the top supports coincide with the worst for the bottom

support, and vice-versa, for Sample TB2C-1. This provides a unique opportunity to

determine which of the supports introduce more suspension losses. This is achieved by

comparing the behaviour of the Q as a function of loading positions, and matching to the

opposite predictions of the top and bottom supports.

The acoustic wavelengths along the rod axis (λx) and paddle axis (λy) give an indica-

tion of the sensitivity to sample displacement relative to the optimal suspension location.

Acoustic modes with longer wavelengths should be less sensitive to loading position errors.

According to Table 4.4, sample TB2C-1 is expected to be less sensitive to offsets along

the paddle axis.

Measurements were made at pressures below 10−3 mbar to ensure that gas damping

losses are negligible (see Fig. 4.32). An assumption was made that the only other losses

are the intrinsic loss and the suspension loss. Thermoelastic loss was considered and this

is discussed in a later section.

The samples made from wet etching [96] were damaged to a point where it was not

practical to compare them with other samples. With the highest Qs observed of 1.4 104,

at pressures of 10−5 mbar, these samples were not tested further.

A set of samples from dry etching [98], laser micromachining [99] and in some cases,

with optical coatings [100], have been tested. The highest Q obtained from each sample

is shown in Table 4.5.

It is important to note that, while the sample UQB3C-1 initially produced Qs of

7.5 105, this sample since then ceased to provide such high Qs. The same sample later

produced Qs of limited to ∼ 105 for the mode at 401.5 kHz, and 1.6 105 at 294 kHz. Due

to the extent of testing of this sample, it is likely that damage and contamination of the

sample have occurred. This could account for the change in performance.

The high Q of 7.5 105 of sample UQB3C-1 was repeatedly observed in the previ-

ous configuration of the 3-point suspension, but not reproduced during the study in this

section. The latest two highest Qs from this table are then from samples TB2C-1 and

UQB3D-1, with Qs of 8.6 105 at 394 kHz, and 6.3 105 at 594 kHz, respectively. Results

from these samples are discussed further in Subsection 4.6.2.

4. Analysis of Results 127

Table 4.5: Highest Qs obtained from resonators made in Taiwan, Queensland, Perth andFrance. Modes where the middle paddle behaves like in the pure torsion mode and theframe contains higher levels of deformation than the ‘pure torsion’ are labelled ‘torsionduplicates’, as mentioned previously. * This sample achieved its highest Qs in the 2-brasscolumn 3-point suspension, and later stopped producing high Qs, possibly due to damage.

Sample F (kHz) Th (μm) Qm Mode Shape

TB2C-1 394.3 670 8.6 105 Pure torsion

TB2C-2 446.6 675 3.0 105 Piston-like

TB1C-1 379.7 530 3.1 105 Snake-like

TB1C-2 437.1 530 2.7 105 unknown

UQB1C-2 370.2 535 4.1 105 unknown

UQB1D-1 593.9 535 4.6 105 Torsion duplicate

UQB3C-1 * 401.5 515 7.5 105 * Piston-like

UQB3C-1 * 332.45 515 7.1 105 * Pure torsion

UQB3C-1 294.5 515 1.7 105 Torsion duplicate

UQB3C-5 344.2 515 1.3 105 Pure torsion

UQB3D-1 576.6 515 4.7 105 Pure torsion

UQB3D-1 594.5 515 6.3 105 Torsion duplicate

FC-1 366.0 365 2.0 105 unknown

UW1 403.1 325 1.4 104 unknown

4.6.1 Reproducibility of Results

In order to assess the confidence level of the Q measurements, I selected one sample for

repeated measurements, namely the sample TB2C-1 from laser micromachining. This

sample was positioned using the loading tools C and D (described in Chapter 3). The

highest Qs in this thesis were obtained from sample TB2C-1 at these two positions.

Figure 4.36: Repeatability of Q measurements from the laser micromachined sampleTB2C-1, for different modes measured with positions provided by the loading tools Cand D. These measurements were made at pressures below 103 mbar.

The repeatability measurements of sample TB2C-1 are shown in Fig. 4.36, where the

Q of multiple modes are repeatedly measured at positions provided by two loading tools.

128 4. Analysis of Results

Average Q values of (7.7±0.7) 105 and (8.1±0.8) 105 are obtained for the pure torsion

mode at 394 kHz, with positions provided by the loading tools C and D, respectively. The

error is the standard deviation of the four measurements, and indicates the reliability of

the Q measurements, with an uncertainty of about 10%.

The error in measurement is quite large, and limits what can be interpreted from the

results in this section. While the Q for the torsion mode is greater when using the loading

tool D, the difference between this value and the Q obtained with tool C is within the

error in measurement. I am unable to confirm, therefore, that a sample position provided

by the loading tool D is more suitable for a high Q of the torsion mode. The results in

Fig. 4.36 at least set lower limits of the achievable Qs from these resonators, using this

suspension method.

4.6.2 Quality Factor vs Loading Positions

The samples TB2C-1 and UQB3D-1 were tested at multiple locations to determine the

optimal loading positions at which the highest Qs can be obtained. The results were com-

pared to FEM predictions based on the acoustic wavelengths on the frame, and vibration

amplitudes at the 3 support locations (two at the top and one at the bottom), listed in

Table 4.4.

The sample TB2C-1 has an acoustic wavelength of 14.6 mm for the torsional mode at

frequency 394 kHz, along the paddle axis, aligned with the two top (left and right) support

points. Therefor, the optimal top support positions, where the vibration amplitudes are

close to zero (a node), are predicted at the centre (i.e., loading tool A), and also ∼ 7.3

mm above and below the equator line.

Sample TB2C-1 was loaded at the positions where the top supports were spaced 0.25

mm, 0.75mm, 1.25 mm and 1.75 mm above the equator line (i.e., tools A, B, C, and D).

Further positions (tools E, F, G, H, and I) were not tested as the sample broke before I

had a chance to try other positions.

Figure 4.37 (a) shows Qs for acoustic modes of the sample TB2C-1, measured at

different positions. The highest Q for the torsional mode at 394 kHz was measured using

loading tool D. The second highest Q was measured with loading tool C, though within

the measurement error of the highest Q. This value decreased with tool B, and reached a

minimum of 2.7 105 at the position provided by the tool A, well beyond the measurement

error.

This result is inconsistent with FEM predictions in Table 4.4, where the optimal top

4. Analysis of Results 129

support positions are obtained from tools A, and the worst position for tool D, as discussed

previously. Recall, however, that the predictions of the optimal positions are opposite for

the top and bottom supports.

The results for the torsional mode at 394 kHz are consistent with the predictions for

the bottom support, which consists of a steel needle mounted on a PZT. This indicates

that suspension losses may be more influenced by the coupling to the bottom pin. This

can be explained by the PZT incorporated in this support point. Compared to the steel

needles of the top two supports, the PZT is expected to have higher intrinsic loss.

Figure 4.37 (b) shows the Qs for multiple modes of the second highest performing

sample from Queensland, using dry etching (sample UQB3D-1), which has wider torsion

rods (W0 = 500 μm instead of 300 μm) supporting the central paddle.

The acoustic wavelength in sample UQB3D-1 is ∼4.5 mm along the paddle axis, which

means that a span of at least 2.25 mm is required to properly investigate this wavelength.

With loading positions between 4.5 mm (loading tool I) and 8.5 mm (loading tool A), this

was easily achieved.

The highest Q measured for this sample was 6.3 105 for the acoustic mode at frequency

of 596 kHz. This corresponds to the torsional mode of the middle paddle with significant

vibrations present in the frame (also known as a duplicate of torsion). The sample pro-

duced Qs near 3.8 105 for the pure torsion mode, at the time of these measurements (see

Fig. 4.37 (b), ‘First Tests’).

It should be noted that the above results were obtained before the sample was loaded

in the interferometer, for mode shape measurements. After the interferometer operations,

I could no longer excite the mode of frequency 596 kHz, and the highest Q for any mode

was below 105 at any suspension position.

Attributing this discrepancy to surface contamination of the sample, and hoping there

was no structural damage, I proceeded to clean the sample with acetone. Immediate

improvements were observed.

The Qs climbed back to high values for most modes. In the case of pure torsion mode

at 576 kHz, the Q exceeded previous results and now reached 4.7 105, with little variation

between loading positions.

The mode at 596 kHz, however, now produced Qs below 105, despite numerous mea-

surements at all positions. The cause for this drop in Q is unknown.

The relationship between the support positions and the observed Qs of the pure torsion

mode is unclear, for sample UQB3D-1. The variations in Q from using different loading

130 4. Analysis of Results

Figure 4.37: Quality factors measured at different loading positions. (a) The highest Qobtained for sample TB2C-1 was 8.6 105, for the pure torsion mode at 394 kHz, usingloading tool D. (b) The sample UQB3D-1 was tested at all nine positions. Initial mea-surements at three positions (B, C and D) produced the highest Q from this sample, at8.6 105 for a duplicate of the torsional mode (with vibrations in the frame) at 594.7 kHz.The repeated measurements showed an increase of the Q of the pure torsional mode (575.8kHz) and a reduction in the 594.7 kHz mode.

4. Analysis of Results 131

tools is within the uncertainty level determined from the repeatability tests, illustrated in

Fig. 4.36.

It is also not clear why the torsional mode Q of this sample appears relatively immune

to the loading position, while the Q of sample TB2C-1 varies significantly with positions

(between 2.6 105 to 8.6 105).

One possible explanation for the discrepancy in the results from sample UQB3D-1

is the sample position shift induced by vacuum pump vibrations. After the first tests

were completed, I discovered that the flexible tube connecting the vacuum pump and the

vacuum tank had a resonance at ∼500 Hz, and this was excited while the vacuum pump

was ramping up its rotation speed from 0 to 1500 Hz. I then devised a method to damp

these vibrations by supporting the flexible tube at the half point, and covering it with

heavy cloth during pumping.

This improvement may have led to the relative stability of the Q of the pure torsion

mode of sample UQB3D-1 (see Fig. 4.37 b). It is hard to conclude anything with the

limited amount of results obtained at this point.

4.6.3 Surface Losses

The possibility that the experiment was not dominated by suspension losses must be

addressed. Surface losses could explain why the laser machined samples of ∼ 500 μm

thickness produced much lower Qs than the 670 μm thick samples.

As stated by Gretarsson et al., the Q depends on the bulk and surface layer properties

of a resonator [121]. These losses are combined in a frequency-independent loss term φc

as follows:

φc = φbulk(1 + μdsV/S

), (4.2)

where φbulk is the loss in the material, typically of order 10−8, μ is a geometrical factor

which depends on the shape of the resonator and on the acoustic mode shape, ds is the

dissipation depth which is the integral of the product of the losses and Young’s modulus

within a thin surface layer, V is the volume of the resonator and S is the surface area.

The parameters that change between the TB1- and TB2-series of laser machined sam-

ples are the thickness of the samples and the polishing (on one or both sides of the sample).

The samples from the TB1-series are double-side polished with a thickness of ∼520

μm, while the TB2-series are single-side polished with a thickness of ∼670 μm. While the

latter probably has more surface defects due to single-side polishing instead of double-side,

132 4. Analysis of Results

Table 4.6: Table of calculated surface to volume ratios and related prediction of the Q (seeEq. 4.2) for a subset of samples from paddle design C. The last column is the expectedQs from thermoelastic loss calculations [128].

SampleThickness S/V

Qm 105Relative Q Q from

(μm) 104m−1 expected TE loss 105

TB2C-1 670 1.36 8.6 highest 5.9

UQB3C-1 515 1.45 7.5 medium 6.6

UQB3D-1 515 1.45 6.6 medium 25.0

TB1C-1 520 1.45 3.1 medium -

FC-1 365 1.61 2.0 lowest -

the influence of the thickness might overcome the surface roughness.

A tentative comparison of resonators is shown in Table 4.6, where a correlation does

appear between thickness and Qs.

It is also possible that the thinner samples have more surface defects from poor wall

patterning (see Fig. 4.8 b) which makes the surface losses more important and causes

significantly lower Qs.

Despite claims that thermoelastic (TE) loss can be ignored for pure torsion modes, due

to negligible heat generation from the nearly isochoric deformation [129], finite element

modelling (FEM) was performed to calculate the expected thermoelastic dissipation loss

of the resonator geometries considered here. This FEM was carried out by Dr. Heinert

[128], a post-doctoral student of Dr. R. Nawrodt, who recently reported on the mechanical

losses in silicon cantilevers [130].

The TE loss was calculated for samples UQB3C-1, UQB3D-1 and TB2C-1, using the

software COMSOL. The results from these calculations are in the last column of Table

4.6.

While analytical relationships between bending modes and TE loss are fairly well

understood [130, 129], the relationship between the geometry of a torsional resonator and

TE loss is not well understood. Therefore discrepancies were expected.

The FEM results are inconsistent with the results presented in this chapter for those

three samples. The TE loss is greater for the thicker TB2C-1 sample, while the exper-

imental results indicate the opposite. This is not well understood, though one possible

explanation is that surface losses are expected to diminish with increased thickness, and

the observed results could be limited by surface loss. On the other hand, at room tem-

perature, the surface losses are expected to be negligible compared to TE losses [130]. It

is not possible to determine the dominating loss for this experiment at this stage.

4. Analysis of Results 133

4.7 Discussion

Silicon microresonators were fabricated and tested with various suspension and actuation

methods. The main conclusions that can be extracted from the results in this chapter are

as follows:

1. Quality-factors close to the target of 106 can be achieved at room temperature from

micromechanical resonators made from dry etching and laser micromachining.

2. The highest Q achieved here was 8.6 105 for a laser micromachined resonator (sam-

ple TB2C-1) of thickness 670 μm.

3. The next higher Qs were 7.5 105 and 6.3 105 from samples UQB3C-1 and UQB3D-

1, respectively. These were from 515 μm thick wafers and patterned using dry etching

procedures.

4. When optical coatings are applied, Qs near 3 105 can be achieved. This is only a

lower limit.

5. The Michelson interferometer allowed mapping and identifying the torsional mode

of the middle paddle (mode of interest). This interferometer was developed by Dr.

Hou.

6. Resonator designs C and D (see Chapter 2) are suitable to produce a torsional mode

of high frequency and high Q. Models A and B were not tested thoroughly in the

time frame of this project.

7. The 3-point suspension, where a sample is resting on 3 supports, is shown to allow

measuring superior Qs from resonators, compared to other methods tested in this

research.

8. The Q does vary with the locations of the 3 support points, and careful design of

the suspension is required to minimise coupling losses.

After a long series of experiments, a final configuration of suspension and actuation

was found which allowed Qs close to the target of 106. The measured Qs, which were

initially in the scale of 103, finally reached a value of 8.6 105. This was measured with

the sample TB2C-1, made from laser micromachining.

That the best results were obtained from a laser micromachined sample is somewhat

of a contradiction. I expected this fabrication technique to produce resonators with lower

134 4. Analysis of Results

Qs due to uneven edges along the rim of the patterns, inherent in this fabrication method

(see Fig. 4.8).

The chemical processes involved in dry etching and possible left-over chemicals on the

samples (visible to the naked eye as stains of different colours, shapes and sizes) could have

caused losses from excess mass on the resonators, potentially reducing the net acoustic

Q. Despite careful cleaning [131] of the samples, some process resin is difficult to remove

(e.g., SU8 photoresist), and this could account for the Qs of dry etched samples being

lower than the laser micromachined sample TB2C-1.

Another possible explanation is the difference in thickness. Sample TB2C-1 is thicker

(670 μm) than other samples from Taiwan and Queensland, which have a thickness between

500 μm and 540 μm. Finite element modelling (FEM) does suggest that thicker samples

will be less sensitive to the suspension.

The samples from laser micromachining with wafers of thickness 520 μm (TB1-series)

are a better choice to compare with dry etched samples. Two laser micromachined samples

(TB1C-1 and TB1C-2), with thickness comparable to dry etched samples in Queensland,

have produced Qs no higher than 3.1 105. This is less than half the Q value obtained

for the dry etched sample UQB3C-1 of similar thickness. This sample produced a Q

of 7.5 105, while a similar dry etched sample, of same thickness and different design

(UQB3D-1), produced a Q of 6.3 105.

Results indicate that the sample loading position influences the value of the Q, however,

a relationship with the expected optimal support locations (predicted by FEM) was not

confirmed by the results. The only correlation that was found is for the sample TB2C-1,

where the Qs seemed to follow the trend predicted by the optimal positions of the bottom

support, comprising of a steel needle mounted on a PZT. Further investigation is required

to confirm this correlation.

I would like to point out that this study does not rule out the possibility of using

electrostatic excitation as a means to excite the torsional mode. This method of excitation

is used extensively [91, 120, 121] and it was surprising to find difficulties with this method.

Perhaps the choice of electrodes could be revised. For example, using a larger surface area,

while targeting the rest of the frame might yield better results.

Also, while the samples from wet etching were of poor quality, other work with sim-

ilar shaped resonators made from wet etching produced high Qs, as high as 6.0 105 at

pressures of 10−3 mbar [110]. This fabrication method is therefore not abandoned.

4. Analysis of Results 135

This study shows that, from the set of testing methods considered, a 3-point suspension

was better suited for producing high Qs. In comparison to other suspension methods used

in this project, the 3-point suspension was superior in both practicality and in performance

of the samples. This method, however, is not practical for holding a resonator in an opto-

acoustic parametric amplifier (OAPA) device. The 3-wire suspension would be better

suited if difficulties could be overcome and high Qs obtained with this method.

136 4. Analysis of Results

Chapter 5

Thermal Effects on Silicon

Microresonator Acoustic Modes

5.1 Preface

Frequency stability in instrumentation for geophysical exploration is required for practical

operation of detection tools, such as EM sensors and magnetometers (using crystal ovens

and other techniques to control the temperature of the system). Most devices, therefore,

have temperature controls or compensation systems.

In order to assess thermal and temperature control requirements in a 3-mode opto-

acoustic parametric amplifier (OAPA) device, silicon microresonators are investigated in

laser heating experiments. This is done using the optical lever method and the 3-point

suspension method described in Chapter 3.

The results from laser heating are compared to a model developed in this chapter.

Predictions are made on the expected acoustic frequency shift and thermal loading of the

silicon resonators used in a 3-mode opto-acoustic parametric amplifier (OAPA) device,

based on the results and the thermal model presented here.

Silicon resonators fabricated with and without optical coatings are investigated, and

results from both will be compared, to extrapolate results to future coatings of different

designs and specifications.

137

138 5. Thermal Effects

5.2 Experiment

The experiment for investigating thermal effects uses the optical lever, which was used

previously for measuring the acoustic modes and their quality factors. The laser power is

partially absorbed by the silicon resonator, and dissipation depends on the experimental

configuration, which is summarized below and in Fig. 5.1.

The sample is in a stainless steel cylindrical tank, and rests on a 3-point system (2

brass columns and a piezo-ceramic actuator with a steel tip glued on top) attached to a

copper plate base. This copper base is attached to a larger aluminium breadboard. There

is also a round tank window and a 45 mirror above the sample.

Figure 5.1: Tabletop setup. Copper plate stage resting on an aluminium breadboard. TheSilicon resonator is on a 3-point suspension, attached to the copper plate, which is fixedto the breadboard. A 45 mirror is used to redirect a laser beam onto the sample, and thereflected beam is redirected to a quadrant photodiode. The vacuum tank is made fromstainless steel. Image courtesy of Dr. McRae.

The laser source is a 650 nm laser diode, with a measured 26.5 mW output. The

pressure in the tank can be reduced to 10−5 mbar.

The copper base is 2.5 mm thick, with an area of 60 mm by 65 mm. The stainless

cylindrical vacuum tank has a length of 20 cm, diameter of 20 cm and a rectangular

5. Thermal Effects 139

aluminium breadboard fits inside with a thickness of 10 mm, area of size 10 cm by 16 cm.

The 45 degree mirror is 30 mm by 30 mm in area and ∼ 50 mm above the resonator (as

measured from the center of the mirror to the flat surface of the resonator).

Table 5.1: Material properties of components in the vacuum tank: Material type, density,specific heat, thermal conductivity, emissivity and thermal expansion coefficient.Material ρ (g cm−3) cp (J kg−1 K−1) K (W m−1K−1) ε β (10−6 K−1)

Copper 8 386 401 0.64 16.5

Stainless Steel 7.9 502 16.2 0.28 17.3

Aluminium 2.7 900 237 0.03 - 0.16 23.1

45 Mirror 2.2 710 156 0.01 2.6

Silicon (100) 2.3 700 148 0.064 - 0.7 2.6

Table 5.1 is a list of the components and relevant material properties needed to anal-

yse the thermal effects measured in the experiments, described in the next section. As

the mathematical analysis will show, only the properties of silicon will be important for

predicting thermal effects.

5.3 Heat Transfer Mechanisms

In an OAPA device that uses a laser to sense the motion of the resonator, the silicon

resonator will absorb laser heat, then dissipate energy by (1) convection to the gas in the

tank, (2) conduction through the rest of sample volume and onto the three support points,

and (3) blackbody radiation.

At atmospheric pressures, the dominating heat dissipation is by convection with the

gas in the tank. This effect is minimized once the tank is closed and the pumps bring the

pressure inside the tank to ∼ 10−5 mbar. At this stage, blackbody radiation is the most

important dissipation, as heat transfers through the supports by conduction are assumed

small, due to the poor thermal contact of this suspension.

Since the objective is to build a model to predict what would happen with a practical

OAPA device, which would operate at low pressures, I will focus mainly on the scenario

at low pressures.

The next subsections will show that thermal conductivity time scales are orders of

magnitude smaller than those of blackbody radiation, and the resulting larger time scales

are consistent with the results presented in a later section. Due to the poor thermal

contact between the resonator and the enclosure, thermal conductivity will play a minor

role compared to blackbody radiation - the dominating dissipation at low pressures.

140 5. Thermal Effects

5.3.1 Thermal Conduction

In this subsection, a mathematical derivation of the relaxation time by conduction heat

transfer is presented. As the three points of contact with the resonator are needles with

small cross section areas, it is assumed that the conduction of heat from the sample to

the tank via these three needles is negligible. The resonator is considered as ‘floating’ in

the vacuum tank. Therefore, the heat transfer described here is from the heated centre

paddle to the side paddles, and to the frame of the resonator, via the torsional rods.

The scenario modelled here is the aftermath of laser heating, when the laser is switched

off after having heated the middle paddle of the resonator to a temperature Ts = T0+ΔT ,

where T0 is the room temperature and the temperature of the resonator frame (initially),

and ΔT is a temperature rise in the middle paddle for an arbitrary amount of laser heating.

The thermal transfer of heat within the volume of a three-paddle silicon micromechan-

ical resonator can be modelled with an analogue electrical circuit of capacitors (middle

plate, side plates and frame of the resonator), and thermal resistors (torsion rods of length

0.5 mm, width 0.3 mm) between the plates and frame. This is shown in Fig. 5.2.

Figure 5.2: The silicon resonator structure and an electrical circuit representation. Themiddle and side paddles, as well as the frame of the resonator are considered thermalcapacitors, storing heat. The torsion rods in between paddles are considered thermalresistors, impeding the heat transfer between paddles and the frame.

The thermal resistance of the circuit is given by Eq. 5.1:

RT =dΔT

dQ/dt=

d

KA, (5.1)

where d is the thickness of the thermal resistor (torsion rods, labelled R1 and R2 in Fig.

5.2), ΔT is the temperature difference between the heated paddle and the rest of the

resonator (temperature gradient), and dQ/dt is the heat rate between this paddle and the

rest of the resonator.

5. Thermal Effects 141

when considering the specific case of thermal conduction, dQ/dt is = KA, where K

is the thermal conductivity of silicon and A is the cross-sectional area of the torsion rod

face in contact with the thermal capacitor (paddle).

The thermal capacitance of the circuit is given by Eq. 5.2:

CT = ρV cp = mcp, (5.2)

where ρ is the density of silicon, V is the volume of the paddle, m is the mass of the paddle

and cp is the specific heat of silicon (see Table 5.1).

Considering that our sample is 500 μm thick, the area of the middle paddle is 1 mm2,

and the volume is 0.5 mm3 (or 5 10−10 m3), the thermal capacitance is calculated from

Eq. 5.2, giving 8.15 10−4 J/K.

The first torsion rod to the right is a block of volume 300 μm by 500 μm by 500 μm,

where the faces touching the middle and side paddles have an area of 500 μm by 300 μm.

This gives a thermal resistor value of 4.44 10−2 K W−1, according to Eq. 5.1.

Considering the system of the middle and right paddle, the thermal relaxation time τ

is given

τ = CTRT , (5.3)

and this gives a relaxation time of 18 ms. Thus heat is quickly transferred from the middle

paddle to the right and left paddles.

A similar calculation for the conduction exchange between the side paddles and the

frame of the resonator yields a relaxation time of 33 ms. These results indicate that our

system is relatively isotropic, if we consider time steps bigger than 50 ms. This allows

considerable simplifications when modelling the blackbody radiation dissipation. It also

means that blackbody radiation is the process responsible for observed relaxation times

of the order of ∼ 100 s (see results in later sections). This time scale shows the required

duration to reach a stable frequency, i.e: when the laser heat absorbed is balanced by the

dissipation from radiation. This is described in the next subsection.

5.3.2 Radiation Heat Transfer

As mentioned earlier, radiation is the dominant mechanism of heat dissipation at low

pressures, due to the absence of (or reduced) convection with gas in the tank.

The time scale of radiation heat transfers is given by the same analogy to an electrical

circuit (Eq. 5.3). The difference with thermal conductivity is in the term of thermal

142 5. Thermal Effects

resistivity given by the Eq. 5.1, and the starting temperature of the sample Ts, which

now corresponds to the temperature reached after a longer period (∼s) of laser heating.

A thermal relaxation time from radiation is expressed as follows:

τBB =mCΔT

(dQ/dt)rad, (5.4)

where m is the mass of the silicon, C is the specific heat, and ΔT is the difference in

temperature between the resonator surface and the enclosure surface. An expression for

(dQ/dt)rad from radiation is derived later in this subsection.

The starting temperature of the silicon resonator (Ts) is at equilibrium, i.e: the heat

rate absorbed from the laser (a fixed amount) is balanced by radiation dissipation, which

depend on the temperature difference between the sample and the enclosure (∼ T 4s −T 4

0 ).

The scenario modelled here is for a fixed temperature gradient between the resonator and

the enclosure at time t = 0, when the laser is switched off.

Consider the radiation heat transfer between the heated microresonator and the en-

closure at room temperature T0, which is the tank and suspension system. The different

mechanisms involved in the heat exchange between the sample at temperature Ts and the

tank components at T0 are shown in Fig. 5.3. The total irradiance G is equal to the

reflected, transmitted and absorbed power [132], such that:

G(W/m2) = ρG+ τG+ αG, (5.5)

where ρ, τ and α are reflectivity, transmissivity and absorptivity of silicon respectively,

satisfying 1 = ρ+ τ + α.

Because the irradiance is coming from the blackbody radiation of other components

in the tank, all of which are at a temperature similar to that of the sample (within a few

K, at most), then, by the laws of Kirchhoff, the absorptivity is equal to the emissivity of

the sample, α(T ) = εs(T ), where εs(T ) is the emissivity of silicon at temperature T .

The silicon resonator does not allow any 650 nm laser light to go through the substrate,

which indicates that τ is 0, corresponding to an opaque substrate. From Eq. 5.5, this

implies that 1 = ρ+ α, and ρ = 1− α = 1− ε.

Radiosity J (W m−2) is the power emitted at a surface which can be expressed for the

inside of the tank and for the resonator. In the case of the silicon resonator, it is the sum

of the reflected and re-emitted power: (1 − ε)G + εσT 4s , where Ts is the temperature of

the silicon resonator at equilibrium, as defined earlier.

5. Thermal Effects 143

Figure 5.3: Schematic of radiation heat transfer mechanisms. Irradiance G is the powerincident on the resonator, coming from the enclosure. This is partially reflected and ab-sorbed, and partially transmitted. The silicon resonator is re-emitting power as blackbodyradiation. The sum of this emission and the reflected power is called the radiosity J .

The rate of heat leaving the resonator surface can be expressed as:

dQs/dt = As(Js −Gs), (5.6)

where As is the total area of the silicon microresonator, Js is the radiosity coming from

the resonator surface, and Gs is the radiation incident on the resonator (see Fig. 5.3).

Substituting Gs from the definition of radiosity (above), this becomes

dQs/dt = As(Js − Js − εsσT4s

1− εs) =

Asεs1− εs

(σT 4s − Js). (5.7)

A surface resistance to radiation can be defined as:

Rs =1− εsAsεs

, (5.8)

such that

dQs/dt =σT 4

s − JsRs

. (5.9)

A similar approach is used to define the rate of heat leaving the inside surface of the

enclosure, dQt/dt.

144 5. Thermal Effects

The next step is to define the heat transfer taking place between the silicon resonator

surface and the surface inside the tank, which completely surrounds the resonator. The

total heat leaving the resonator must be received by the enclosure (vacuum tank).

The net radiation heat transfer from the silicon surface to the surface inside the tank,

At, is given as follows:

dQst/dt = AsFstJs −AtFtsJt, (5.10)

where Jt is the radiosity of the enclosure, and the view factor Fij is a geometrical parameter

which defines the fraction of the total heat coming from surface i which is directly incident

on surface j. For example, the outside surface of a sphere enclosed in a larger sphere will

have a view factor equal to 1, as all of the radiative heat leaving the smaller sphere is

directly incident on the inside of the larger sphere.

In the case described here, Fst and Fts are the view factors from the silicon resonator to

the tank surface, and vice-versa. There is a rule of reciprocity which states that AsFst =

AtFts, which simplifies Eq. 5.10 to:

dQst/dt = AsFst(Js − Jt). (5.11)

Similar to the surface resistance in Eq. 5.9, a ‘space resistance’ to radiation can be

defined as follows:

Rst =1

AsFst, (5.12)

which simplifies Eq. 5.11 to:

dQst/dt =Js − JtRst

. (5.13)

The value of dQs/dt can be obtained by solving the equations dQs/dt = dQst/dt and

dQt/dt = dQts/dt, for Js and Jt. It can then be shown [87] that the heat loss by the

silicon can be expressed as:

dQs/dt =σ(T 4

s − T 40 )

1−εsεsAs

+ 1AsFst

+ 1−εtεtAt

, (5.14)

where εt is the emissivity of the tank enclosure. This can be an average of all material

properties of the enclosure, or extra surfaces for every material can be used, adding terms

in Eq. 5.14. This makes the system a lot harder to solve, and as shown below, such

5. Thermal Effects 145

complications are unnecessary as, from further derivations, the tank material properties

become irrelevant.

The Eq. 5.14 can be further simplified in the case of a convex closed body within an

enclosure, to the expression:

dQs/dt =Asσ(T

4s − T 4

0 )1εs

+ AsAt

( 1εt− 1)

. (5.15)

Given that the area of the silicon As is much smaller than the enclosed area of the

container (As � At), this equation simplifies to an expression where the emissivity and

area of the container become irrelevant:

dQs/dt = Asεsσ(T4s − T 4

0 ). (5.16)

Because the silicon resonator is only expected to increase by a few degrees K, the

temperature terms can be expanded to the form:

(T 4s − T 4

0 ) ≈ 4T 30ΔT, (5.17)

where Ts = T0 +ΔT .

Substituting Eq. 5.16 and Eq. 5.17 into Eq. 5.4, the following blackbody relaxation

time expression is obtained:

τBB =mcΔT

4AsεsσT 30ΔT

=mc

4AsεsσT 30

. (5.18)

Equation. 5.18 shows that for small resonators in large enclosures, the blackbody

relaxation time does not depend on tank material properties or the temperature gradient,

but rather on the room temperature, the area and material properties of the resonator.

The end result is a model of the expected relaxation time for different values of silicon

emissivity, shown in Fig. 5.4.

According to literature [79, 85], values of silicon emissivity for a standard (100) silicon

wafer at room temperature range from 0.07 to 0.6, which gives relaxation times between

100 and 1000 seconds. Only when these values are compared to experimental results will

a better estimate of emissivity be found.

Measuring the emissivity of silicon is not trivial. The modelling presented here is

later compared to experimental results of the relaxation times to get an estimate of the

emissivity of the resonators tested.

146 5. Thermal Effects

Figure 5.4: Relaxation time vs emissivity of silicon, calculated from Eq. 5.18. Theserelaxation times will be matched to experimental results presented in a later section.

At different pressures the amount of gas in the tank available for thermal exchange

with the silicon varies greatly. As a result, there is an extra term of heat dissipation to

consider in the relaxation time Eq. 5.18.

Heat dissipation from convection depends on the pressure, and this term should con-

tribute to the total dissipation. A term for the heat dissipation from conduction between

the sample and the enclosure is added to the general total heat rate Eq. 5.19. The heat

transfer through the poor thermal contacts is expected to be negligible, and this term will

be assumed nil for the present calculations.

(dQ/dt)tot = (dQ/dt)rad + (dQ/dt)conv + (dQ/dt)cond, (5.19)

where (dQ/dt)rad is the radiation dissipation discussed in the previous subsection, (dQ/dt)conv

is the heat dissipation from convection, which is a function of the pressure in the enclosure,

and (dQ/dt)cond is the heat dissipation from thermal conduction between the sample and

the enclosure, assumed negligible.

A full description and modelling of the thermal relaxation time at high pressures is

beyond the scope and purpose of this chapter.

5. Thermal Effects 147

5.4 Finite Element Modelling of Frequency Shift with Tem-

perature

As stated in the literature, changes to Young’s modulus from temperature variations is

the dominant parameter in determining acoustic mode frequency shift, and other material

property variations can be ignored [120, 133].

It is reported that Young’s modulus of standard (100) silicon wafers is 169 GPa at room

temperature [81], and this value changes with temperature by -60 ppm/K [83]. Different

experimental studies show a reasonable consensus of these values [84, 134].

Note that silicon is an anisotropic material with elastic constants that vary with crystal

orientations, and the value of 169 GPa is an acceptable approximation for the thermal mod-

elling study presented here. A set of matrix values, however, were used in the modelling

of acoustic modes, presented in Chapter 2 and Chapter 4 to obtain accurate predictions.

Figure 5.5: Finite element modelling (FEM) prediction of frequency change with tem-perature, modelled with variations to the Young’s modulus according to the literatureconsensus of -60ppm K−1 [81].

Changes to Young’s modulus are calculated for temperature variations of 1, 5, 10, and

20 K. These values are inserted in FEM simulations and the torsion mode frequency is

obtained in a manner described in Chapter 2.

The results are shown in Fig. 5.5. This figure indicates that a temperature rise of 1

K is enough to cause a decrease in frequency of 10 Hz, while a rise of 20 K will lower the

resonant frequency by 200 Hz.

These values will be compared to experimental frequency drift measurements in the

148 5. Thermal Effects

next section in order to estimate the temperature rise, which can not otherwise be mea-

sured in the experiment used here.

5.5 Experimental Results

5.5.1 Frequency Drift and Thermal Relaxation Times

Measurements of the acoustic mode frequency as a function of laser exposure time are

repeated at different laser power levels, dividing the power with a beam splitter and

comparing results to measurements with a full power beam. Measurements are also taken

at vacuum pressures ∼ 103 to 10−6 mbar, using samples with and without optical coatings.

The amount of frequency shift and the thermal relaxation times are compared for different

scenarios.

While the laser is shining on the substrate, the resonator temperature rises and causes

a decrease in the acoustic frequency. An opposite effect is observed when the laser is

switched off, and the resonator is allowed to cool back to room temperature.

The physics of the heating and cooling process modelled earlier in the chapter was

assumed equivalent in terms of relaxation times and temperature variations. Measure-

ments of the heating and cooling process are compared to validate calculations based on

the model for cooling, presented earlier.

In order to estimate the laser energy being absorbed by a silicon resonator, the laser

power was measured before and after reflection on the silicon resonator. For a sample

held outside the enclosure (not shown in Fig. 5.6), the incident power on the sample is

measured as 26.5 mW, and after reflection the power is ∼ 8.8 mW. Not all of the incident

beam is reflected directly to the power meter, however these measurements at least put

an upper limit on the fraction of laser power absorbed.

Results are shown in Fig. 5.6 from measurements when the resonator is placed inside

the tank. An incident power of 21 mW is focused on the middle paddle of the resonator.

However, the beam size is comparable to the paddle size, and some of the beam will not

be contained within the surface of the middle paddle, incident instead on the copper base

below the resonator. The power meter reads 6.5 mW after the beam is reflected from the

mirror, and 5.5 mW after passing through the glass window.

About 1/3 of the incident laser power (∼ 8.8 mW and 6.5 mW) is reflected from

the silicon. This indicates that the silicon resonator absorbs a maximum of ∼2/3 of the

incident power, assuming there is no transmission through the 500 μm silicon substrate

5. Thermal Effects 149

Figure 5.6: Diagram depicting the 650 nm laser power, and power meter measurementsat different stages of the experiment. The output of the laser is ∼ 26.5 mW, and about5.5 mW reaches the quadrant photodetector (QPD).

and no losses due to scattering or diffuse reflections.

From observations that no laser light seems to pass through the resonator, it is a safe

assumption that transmission is negligible. However, the silicon surface is not perfectly

flat or smooth, and the reflection must be partially diffusive.

The frequency drift measurements are done inside the tank, with the setup described

in Fig. 5.1. Results of different experimental runs with variations to the parameters - such

as pressure, laser intensity and sample surface (coated, non-coated) - are presented below.

Figure 5.7 Shows an example of a frequency shift at low pressures for a single mode.

The frequency shift in a 10−5 mbar environment is observed to vary between 240 and 450

Hz and with relaxation times in the range of 250 to 450 s. These values depend on the

pressure and room temperature variations. For example between 102 and 103 mbar, the

frequency decrease is around 17 Hz, and the relaxation time, in both cases, is 40 seconds.

The major difference in relaxation time between room pressure and low pressures is due

to the thermal exchange with the surrounding gas in the tank, which is greater at high

pressures.

This frequency shift varies with initial frequency, but as the normalized plot of Fig.

5.8 shows, the relative shift is similar for all three modes. The temperature of the room

is usually stable within 1 K, however, the air conditioning system has failed a few times

without the experimenter immediately noticing, and larger than usual temperature varia-

tions could cause significant changes to the same experiment. This effect can be especially

important if the room temperature drops from excessive air conditioning.

150 5. Thermal Effects

Figure 5.7: Measurement of the frequency drift of sample UQB3C-1 with laser exposuretime, at pressure 1.4 10−5 mbar and frequency of 294 kHz, showing a thermal relaxationtime of ∼ 274 seconds.

Figure 5.8: Observed frequency drift with laser exposure time, for 5 different modes, allat 2.2 10−5 mbar. These frequencies are 99.7, 294.1, 296.4, 334.5, and 401.5 kHz. Therelaxation times measured here are the same for every mode, near 450 s. As seen in anormalized plot, the relative shift for different modes is similar, near 0.05%.

5. Thermal Effects 151

Figure 5.9: Measurement of the frequency drift of sample UQB3C-1, using the full laserpower, and half laser power divided by a beam splitter. A change in frequency drift isobserved, from 136 Hz to 60 Hz when the beam is divided by the beam splitter. Therelaxation time is the same for both measurements, at 250 s.

Another variable to consider is the laser beam intensity. By using a beam splitter to

divide the beam in half, a comparison is made of the frequency drift at different laser

intensities: at full power (26.6 mW) and half-power (13.3 mW). The result plotted in Fig.

5.9 demonstrates that the relaxation time does not depend on the laser power, and hence

does not depend on the temperature rise. The frequency shift however still depends on

the power used. This is consistent with the models described in the first section of this

chapter.

Comparing frequency drift experiments undertaken at different tank pressures (results

shown in Fig. 5.10) demonstrates two methods of heat transfer. At low pressures, black-

body radiation is the dominant mechanism and the thermal relaxation time, as well as

frequency change, are much larger than the values at higher pressures, where a small

frequency shift and short relaxation time are observed.

The entire set of measurements performed with the sample UQB3C-1 at different

pressures are quite similar to the results presented already. The results of the frequency

shift and fitted relaxation times are summarized in Table 5.2, for comparison at different

pressures.

152 5. Thermal Effects

Figure 5.10: Frequency drift measurements on sample ‘UQB3C-1’ at different pressures.The measurements at 100 mbar and 1000 mbar both have a drift of 17.5 Hz, while themeasurements at 1.2 and 1.6 mbar have a drift of 22 Hz. All four of these curves have acommon relaxation time of 40 s. At lower pressures, 2.5 - 4.0 10−5 mbar, the frequencydrift is ∼ 160 Hz, with a relaxation time of 300 s. This plot depicts two regimes of heattransfers, dominating at either range of pressures. At high pressure, heat transfers aredominated by convection with the gas. At low pressures, there is negligible gas in thetank, and convection can be ignored. Blackbody radiation is therefore dominant.

Table 5.2: Summary of the frequency drift measurements at different pressures, performedon sample UQB3C-1, on the three-point suspension system depicted in Fig. 5.1.

Pressure (mbar) Δf (Hz) τ (s)

1000 17.5 40

100 17.9 40

1.2 22.5 40

1.6 21.8 40

0.63 27 50

1.9E-4 81 190

5.4E-5 166 320

4.0E-5 160 300

2.5E-5 160 300

2.5E-5 133 290

2.5E-5 158 312

2.2E-5 192 450

1.9E-5 136 250

6.3E-6 310 *400-700

5. Thermal Effects 153

There are discrepancies in the measurements as the frequency shift varies from 133 Hz

to 192 Hz at 10−5 mbar, and reaches up to 310 Hz at 10−6 mbar. The relaxation time

varies substantially in pressures ∼ 10−5 mbar, ranging from 250 to 450 s. Presumably,

room temperature variations can affect these measurements.

With a predicted 10 Hz K−1 decrease, the variation in frequency shift by 59 Hz would

imply a room or enclosure temperature rise by 5.9 K. This is roughly double the expected

range of room temperature variations, however the temperature is not measured directly

inside the tank, and there could be a difference in temperature between the tank and the

room. This is unlikely, as the tank is not isolated from the environment.

The values measured at 2.2 10−5 and 6.3 10−6 mbar set a lower limit to the frequency

shift and relaxation time of strictly blackbody radiation heat transfers. Lower frequency

shifts and shorter relaxation times would occur if heat dissipation from convection and

conduction were introduced, as predicted by Eq. 5.19.

5.5.2 Optical Coating Effects

A resonator with optical coating, sample FD-1, was placed in the experiment to observe

thermal effects. Results were obtained with the laser beam incident on the coated middle

paddle. These were compared to results from laser beams incident on the frame of the

same sample, as shown in Fig. 5.11.

The change in frequency and in thermal relaxation time for the coated resonator was

smaller than for non-coated resonators.

A mirror coating is expected to restrict the laser heat transmitted to the resonator

substrate, and as a result, reduce the frequency shift. However, the results obtained with

this sample show an increase in frequency shift when the beam was incident on the coating,

as opposed to the frame. This was expected, as discussed below.

The coatings were designed to be part of an OAPA device, which operates with a laser

source of 1064 nm. The experiment used here, however, used a laser diode of 650 nm. The

poor reflectivity of the coatings is consistent with the expected transmissivity of optical

coatings of this design [100], which is low at the target spectrum between 1000 to 1200

nm, and high in the region of 650 nm.

From Fig. 5.12, it can be seen that less laser power (2.8 mW) reaches the quadrant

photodiode when the beam is focused on the optical coating. Indeed, for the same sample,

when the laser is incident on the silicon wafer, on the frame or on the side paddles, the

laser beam, reflected back to the quadrant photodiode, has a higher power reading of 5.5

154 5. Thermal Effects

Figure 5.11: Frequency drift of the coated resonator FD-1 from France, model design ‘D’(see Chapter 2). Measurements of the mode at frequency 405 kHz performed at 7.4 10−5

mbar, performed first with the laser beam incident on the coated paddle, then repeatedwith the laser incident on the frame. The decrease in frequency when the beam is incidenton the paddle is 37 Hz, while for the laser incident on the frame, it is reduced to 28.8Hz. The thermal relaxation time is the same for both measurements, as τ = 80 seconds,indicating that it does not depend on where the laser is incident.

5. Thermal Effects 155

mW. This is double the amount of power measured from the laser incident on the coatings,

and the same power as obtained from samples without coatings.

As a consequence, more laser heat is absorbed by the resonator through (1) laser light

transmitted through the coatings and (2) laser heat absorbed by the coatings and then

transmitted to the substrate by thermal conduction.

Figure 5.12: Diagram depicting the 650 nm laser power, and power meter measurementsat different stages of the optical lever experiment, with optically coated FD-1 sample. Theoutput of the laser is ∼ 26.4 mW, and about 2.8 mW reaches the quadrant photodetector(QPD).

This is confirmed in Fig. 5.11, where the frequency shift from a laser beam incident

on the coatings is greater than for a beam incident on the substrate of the same sample.

The thermal relaxation time, however, does not depend on the location of the beam.

It appears lower than results from experiments on non-coated resonators, shown in Fig.

5.7, regardless of where the laser beam is located on the coated sample.

It is expected that the coatings increase the average emissivity of the resonators. The

experimental results in Fig. 5.11 seem to support this as the thermal relaxation times are

shorter. This is consistent with Eq. 5.18, where the relaxation time is shorter for higher

resonator (sample) emissivity.

5.5.3 Equivalence of Heating and Cooling

The mathematical modelling presented in this chapter was derived from a conceptual

scenario where the resonator is at a higher temperature than the enclosure (the tank),

the laser is switched off and the resonator is slowly cooled by dissipation with a thermal

156 5. Thermal Effects

relaxation time which can be calculated from Eq. 5.18.

The cooling process is much simpler to model mathematically than the heating process,

due to the absence of a laser heating term (as discussed earlier). It is important to

verify that this modelling is valid when the resonator and the enclosure start at room

temperature, and laser heating is introduced. Experimental measurements discussed below

will show that the thermal relaxation time for both the heating process and the cooling

process are similar, indicating that the model is valid for both processes.

Figure 5.13: Experimental measurements of frequency drift with laser on (heating) andwith laser off (cooling). Comparison of thermal relaxation times for both processes. Mea-surements performed at a pressure of 1.6 10−5 mbar, at room temperature of 295 K.The frequency drift in both cases is ∼140 Hz and the relaxation times are 256.9 s for thecooling process and 223 s for the heating process.

Measurements of the change in frequency during laser heating (laser on) are presented,

as well as during cooling. The cooling process is measured as soon as the laser is switched

off and the resonator temperature is allowed to cool back to room temperature.

The measurements of the cooling process are not trivial, and the result obtained is only

an approximation of the real cooling time. The capture of data requires short temporary

periods with the laser switched on in order to record the frequency. The short duration

with the laser on is enough to cause (1) a pause in the cooling process, and (2) a small

amount of heating, thereby retarding the cooling process.

5. Thermal Effects 157

In order to minimize this unwanted heating and minimize the delay in cooling, the

measurements are made as fast as possible, within 4 seconds for the spectrum analyser to

integrate the signal, and the measurements are few: 5 or 6 points, compared to 20 to 30

points for the heating process.

As Fig. 5.13 shows, there is a discrepancy in the relaxation times of about 30 sec-

onds, which is consistent with the explanation above. The laser heating time during the

measurements must be deducted from the measured relaxation time of the cooling curves.

Another correction from this relaxation time would take account of extra heating

during measurements. For this reason, this discrepancy of 30 seconds can be explained

by a 4 second delay from every data point (4 times 5 data points), and the remaining 10

seconds can be attributed to the small temperature rises during the measurements.

These results are sufficient to justify modelling the cooling process to compare it with

the measured heating process (see calculations of heat transfer in Section 5.3).

5.6 Analysis of Results

The value of the emissivity of silicon varies significantly at room temperature and for low-

level doping [85]; between 0.076 to 0.68 for wavelengths between 1 to 10 μm. Measurements

of the emissivity require a sample to be in a vacuum chamber, completely isolated from any

thermally radiating bodies. These measurements are beyond the scope of this research.

For this reason, models based on emissivity values are compared to experimental results as

an attempt to obtain accurate estimates of the emissivity specific to the samples studied

here. Emissivity as well as the absorptivity of silicon are required in order to make a

prediction model for the expected thermal effects in a practical OAPA device.

The relaxation times range from 250 to 450 seconds, and the frequency shifts are

between 130 to 310 Hz, for pressures near and below 10−5 mbar. The FEM prediction of

the frequency change with temperature in Fig. 5.5 suggests that a rise of 10 K decreases

the frequency by 100 Hz. It is therefore reasonable to assume that the temperature rise is

between 13 and 31 K, based on the observed frequency shifts at low pressures.

In order to get an estimate of the emissivity and absorptivity of silicon, the equation

of balance between the heat absorbed from laser heating and dissipation from blackbody

radiation (Eq. 5.16) can be used.

The laser intensity is fixed at 26.5 mW out of the laser, and 21 mW directly incident

on the resonator. Since the measured reflected beam is about 0.33 of the output from

the laser, a maximum value of 0.67 is allowed for the absorptivity of silicon. The possible

158 5. Thermal Effects

values of emissivity are found by observing the parameter values at which Eq. 5.20 is

balanced, for reasonable values of the absorptivity.

The balanced equation is as follows:

αPin = Aεσ(T 4 − T 40 ), (5.20)

where α is the absorptivity of silicon at 650 nm, Pin is the laser power incident on the

resonator, measured as 21 mW inside the tank, A and ε are the surface area and emissivity

of silicon, and T0 is the room temperature. The temperature of silicon at equilibrium is

given by T = T0 +ΔT , and ΔT is the estimated rise in temperature.

Solving for α in Eq. 5.20 for different emissivity values and for temperature rise values

between 13 K to 31K, results are obtained and summarized in Table 5.3.

Table 5.3: Estimates of the emissivity and absorptivity of silicon in the experiment at lowpressures, based on the estimate of a temperature rise between 13 to 31 K. Values of therelaxation times are obtained from the modelling section, using Eq. 5.18. Absorptivity ofsilicon is listed at 13 K and 31 K.

Emissivity α(Δ T=13 K) α(Δ T=31 K) τ(s)

0.01 0.032 0.084 6660

0.06 0.19 0.50 2220

0.1 0.32 0.84 666

0.13 0.42 1.09 512

0.15 0.48 1.26 444

0.2 0.64 1.68 333

0.22 0.71 1.85 303

0.3 0.97 2.53 222

A contradiction occurs for an emissivity of 0.22 as the absorptivity is 0.71 at 13 K,

which is above the maximum value 0.67 from measurements of the reflected beam power.

This limit is reached at smaller emissivities if a 31 K increase in temperature is considered.

In this case, the emissivity value has an upper limit of 0.1, before the absorptivity reaches

the upper limit.

The temperature rise of 31 K is perhaps an overestimate, as room temperature vari-

ations could increase the sample temperature during the experiment, leading to larger

frequency shifts than those predicted in this chapter.

The results in Table 5.3 indicate that emissivity of the resonator studied here (mainly

sample UQB3C-1) may be between 0.06 and 0.2. This is consistent with the literature [79,

85] which state that polished metals usually have emissivities near 0.1, and in particular,

silicon can have an emissivity near nil at room temperature.

However, the lower values of emissivity (0.01 and 0.06) give a relaxation time of 6660

5. Thermal Effects 159

s and 2220 s, respectively, as shown in Table 5.3. These times are much longer than the

longest relaxation times measured here.

It could be argued that the experiment was not performed at pressures low enough to

measure the intrinsic time scale of blackbody radiation. A value of emissivity of 0.1 gives

a relaxation time of ∼ 670 s, which is a much better match to the results presented in

this chapter. Therefore, in this study, the value of the emissivity is estimated between 0.1

to 0.2, and the average value of 0.15 will be chosen for the predictions of operation in an

OAPA device, as presented below.

5.6.1 Predictions for an OAPA device

Recall that for these calculations, assumptions were made about the thermodynamic sys-

tem. These include the assumption that at low pressures (∼ 10−5 mbar), only blackbody

radiation was responsible for thermal dissipation.

The analysis suggests that even at these low pressures, conduction through the narrow

3-point support contacts could be a non-negligible contribution to the heat dissipation.

Further investigations at lower pressures should be undertaken to examine the true black-

body radiation relaxation time.

From the above analysis, the implications of the thermal effects on an OAPA device

can now be considered, assuming an emissivity of 0.15, and an absorptivity of 0.5. Optical

coatings will be used to match a laser source at 1064 nm.

The design for an OAPA device is presented in Chapter 6, along with an improved

resonator design which will address some of the thermal issues discussed here.

From the results of this research, and the expected upper limit of circulating power

of ∼1 kW, predictions are made of the expected temperature rise and frequency shift, for

different coating reflectivities. Coatings of alternating layers of tantalum and silica can

achieve transmissivities as low as 30 ppm to 1 ppm for 14 and 20 pairs of layers [100],

respectively. Lower transmissivities can be obtained with thicker coatings, at the price of

extra mass added to the resonators.

At a transmissivity of 1 ppm, 1.2 mW is absorbed from the silicon, from the 1.2 kW

circulating power in the OAPA device (see next chapter). For an emissivity of 0.15 and

absorptivity of 0.5, estimated from the results presented in this chapter, the predicted rise

of resonator temperature is ∼ 3.2 K, which would decrease the resonant frequency by ∼32 Hz, assuming a similar torsional frequency predicted between 300 to 400 kHz.

The temperature rise increases to ∼30 K and ∼70 K, for optical transmissivities of 10

160 5. Thermal Effects

ppm and 30 ppm, respectively giving a frequency shift of 300 to 700 Hz, which could still

allow normal functioning of a 3-mode OAPA device.

These temperature rises will become a problem when the OAPA device is used to cool

the macroscopic resonator to the quantum ground state. Optical coatings with trans-

missivities near 0.1 ppm may be required, which would allow a temperature rise ∼ 0.3

K. This amount of thermal loading is manageable by dilution refrigerator cooling power

limits, which are near 100 μW at low temperatures [135].

5.7 Discussion

In this chapter, the thermal effects of laser heating a micromechanical resonator were

described, modelled, measured, and analysed.

From a comparison with models, FEM predictions and experimental measurements,

an estimate of the emissivity of the standard (100) silicon wafer resonator is obtained

between 0.1 to 0.2, and a value of 0.5 for the absorptivity of silicon at 650 nm and room

temperature.

For a practical OAPA device, using similar resonators with optical coatings of higher

reflectivity, up to 99.9999% reflectivity with 20 alternating thin film layers (discussed in

Chapter 6), the laser power circulating in an optical cavity can be between a few μW to

an order of 1 kW.

At this higher power limit and reflectivity, the absorbed power from the incident laser

in the cavity can be near 1.2 mW, causing a drift in frequency of the resonators of -32 Hz

for the acoustic modes near 400 kHz. This amount of drift is not an issue for the operation

of an OAPA device.

Chapter 6

Three-mode OAPA and Improved

Resonator Design.

6.1 Three-mode OAPA Design

6.1.1 Preface

The paper which forms the first and largest portion of this chapter presents a compact

setup for a 3-mode opto-acoustic parametric amplifier (OAPA), using a silicon microme-

chanical resonator with optical coatings, to serve as an end mirror. Such resonators were

fabricated in a collaboration of several facilities in France: fine polishing of the wafers

in the Societe Europeenne de Systemes Optiques (SESO), dry etching in the Institute of

Electronic, Micro-electronic and Nanotechnology (IEMN), and ion-beam sputtering (IBS)

of Bragg mirrors in the Laboratoires de Materiaux Avances (LMA). These fabrication

techniques are briefly described in Chapter 3, with process details in Appendix D.

An optical cavity design is provided where self-cooling, amplification and instability are

achievable with reasonable laser input power and realistic experimental parameters. The

predicted performance of this device is based on demonstrated quality-factors described

in Chapter 4, and on thermal modelling from Chapter 5.

The manuscript, of which I am second author, has been submitted to ‘Applied Optics’.

Most of the cavity design work was performed by Liu Jian and Ma Yubo, summer intern-

ship students from the Beijing Normal University, in Beijing, China. My contribution to

this paper amounts to 10 % of the cavity design, all of the finite element modelling and

thermal analysis, all of the silicon resonator design and experimental work, and 80 % of

the manuscript preparation.

The second part of this chapter presents a silicon-on-insulator (SOI) micromechanical

161

162 6. Optical Cavity Design

resonator prototype, and FEM predictions of acoustic wave node locations for optimized

suspension. The use of SOI wafer technology allows a prototype with non-uniform thick-

ness, with a thicker frame and thin paddles. This could improve suspension losses and

isolation from frame modes, as well as reduce the impact of thermal effects, crucial for a

future OAPA device used in quantum measurement experiments.

6. Optical Cavity Design 163

APPLIED OPTICS

Near-self-imaging cavity for 3-mode optoacoustic

parametric amplifiers using silicon

microresonators

By J. Liu1,2, F. A. Torres2, Y. Ma1,2, C. Zhao2, L. Ju2, D, G. Blair2, Z.-H. Zhu1, S.Chao3, I. Roch-Jeune4, R. Flaminio5, C. Michel5, and K.-Y. Liu6

1Department of Astronomy, Beijing Normal University, Beijing 100875, China2School of Physics, University of Western Australia, 35 Stirling Highway, Crawley, WesternAustralia 6009, Australia3Institute of Photonics Technologies and E.E. Dept., National Tsing Hua University, 101Kuangfu Rd. Sec. 2, Hsinchu, Taiwan 3004IEMN UMR CNRS 8520, av Poincare, BP 60069, 50652 Villeneuve d’Ascq Cedex, France5Laboratoire des Materiaux Avances (LMA), IN2P3/CNRS, Universite de Lyon, F-69622Villeurbanne, Lyon, France6Australian National Fabricatoin Facility (Queensland Node), Australian Institute forBioengineering and Nanotechnology, The University of Queensland, Brisbane QLD 4072,Australia

(Submitted on October 23rd, 2013, accepted on December 11th, 2013, published onlineon February 5th, 2014)

ABSTRACT

Three-mode optoacoustic parametric amplifiers (OAPAs) in which a pairof photon modes are strongly coupled to an acoustic mode, provide ageneral platform for investigating self-cooling, parametric instability andvery sensitive transducers. Their realization requires an optical cavitywith tunable transverse modes and a high quality factor mirror res-onator. This paper presents the design of a table-top OAPA based on anear-self-imaging cavity design, using a silicon torsional microresonator.The design achieves a tuning coefficient of 2.46 MHz/mm, which allowstuning of the optical mode spacing to alternate between amplificationand self-cooling regimes of the OAPA device. Based on demonstratedresonator parameters (frequencies ∼ 400 kHz and quality-factors ∼ 7.5105) we predict that the OAPA can achieve parametric instability with

1.6 μW of input power, and mode cooling by a factor of 1.9 104 with30 mW of input power.

164 6. Optical Cavity Design

6.1.2 Introduction

Optoacoustic systems are attracting considerable interest for quantum optomechanics and

signal transduction. In such devices, modes of acoustic (or ‘mechanical’) resonators are

coupled to optical modes through a non-linearity provided by the radiation pressure force.

This coupling provides a tool to study quantum effects at the macroscopic scale and to

achieve quantum limited sensitivity in detecting small displacements and weak forces.

Laser-cooling of acoustic resonators to their quantum ground state has been achieved

in experiments with sufficient optoacoustic coupling [68, 58, 65, 57], and extremely high

sensitivity transducers have been demonstrated [39, 122, 123, 136, 137, 138].

Optoacoustic interactions represent a macroscopic form of Brillouin scattering [61] in

which electrostriction couples optical fields and acoustic waves. A travelling electromag-

netic wave scatters from a moving sound grating in a bulk medium (phonon frequencies

up to THz), producing a second electromagnetic wave. In the case of optoacoustic inter-

actions, an optical field scatters on the surface of a vibrating mirror (phonons in the MHz

range).

When considering optoacoustic interactions in an optical cavity, it is important to

distinguish between 2- and 3-mode interactions. In a 2-mode interaction, an acoustic

mode scatters an input beam to create a pair of sidebands within the bandwidth of the

same cavity mode. Depending on side-band asymmetry, this can lead to suppression of

the acoustic mode (self-cooling [125, 66]), or amplification of the acoustic mode. Four

frequencies are present, but supported by two modes - one acoustic and one optical.

In a 3-mode interaction, an input beam and scattered beam are in different cavity

transverse modes. Two optical modes and one acoustic mode are involved. This interaction

is analogous to a photonic molecule [139] and is a tuned system that allows the creation of

quantum-optical acoustic amplifiers capable of free oscillation (phonon lasers [140]), mode

cooling and amplification of small signals.

Three-mode interactions are qualitatively and quantitatively different from 2-mode

interactions, since power flows at three frequencies. In appropriately designed systems,

the optoacoustic coupling can be much greater as it depends on the product of three

high quality-factors (2 optical and 1 acoustic). We will show here that a practical 3-mode

optoacoustic parametric amplifier (OAPA) can achieve very high self-cooling factors ∼ 104,

compared with cooling factors ∼ 30 achieved with 2-mode self-cooling [64, 65, 57].

Braginsky et al. first predicted 3-mode optoacoustic parametric interactions [1, 2] in

the context of long-baseline laser-interferometer gravitational-wave detectors [42]. This

6. Optical Cavity Design 165

analysis was extended to three dimensions [43], including detailed modelling of acoustic

and optical mode structures by Zhao et al., while Ju et al. showed that multiple mode

interactions could contribute to parametric gain [44].

The theory was confirmed by Evans et al. [141], while Zhao et al. [39] and co-workers

[48] have confirmed the detailed physics of 3-mode interactions in long Fabry-Perot cavities.

Zhao et al. observed these interactions in an 80-m cavity [39], and recently reported the

first demonstration of ultrasonic acoustic modes driven by radiation pressure using 3-mode

parametric interactions [48].

Three-mode interactions have been observed in solid state resonator structures [142]

and recently, the first tunable free space 3-mode interaction system was demonstrated

using a 50 nm thick membrane between two mirrors [55].

A crucial requirement of 3-mode interactions is a high quality-factor acoustic resonator

that scatters light into a cavity transverse mode, with an acoustic mode frequency that

matches the frequency gap between both cavity modes. The high quality-factor is required

to reduce thermal noise and to increase the parametric gain R, a dimensionless quantity

which describes the strength of the optoacoustic interaction relative to the losses of the

acoustic resonator.

Assuming that the 3-mode system is tuned such that ωm = ω0 − ω1, where ω0, ω1

and ωm are the optical carrier mode, high order mode and acoustic mode frequencies

respectively, the parametric gain R is given by the following formula [1, 3]:

R =8IinQ0Q1QmΛ

mL2ω0ω2m

, (6.1)

where Iin is the input power, Q0 and Q1 are quality factors of the cavity modes, Qm is

the quality factor of the acoustic mode, m is the effective mass of the resonator, and L

is the cavity length. The overlap factor Λ [1, 44] between the TEM00, TEM01 and the

acoustic mode shapes determines the strength of 3-mode interactions. It is an integral of

the product of the three mode shapes. The magnitude of Λ depends on the alignment and

the relative sizes of the optical and acoustic modes. Thus, to ensure a suitable overlap

factor, it is important to optimise the beam spot sizes and alignment, in relation to the

resonator structure.

This paper presents the design and predicted performance of a practical OAPA device

developed from the concept first presented in reference [3]. We present the design and

validation studies of a compact tunable OAPA system based on a Fabry-Perot [40] cavity

with a high quality-factor mg-scale silicon microresonator [105] as an end mirror. This

166 6. Optical Cavity Design

could also provide a platform for studying parametric instability, an important issue in

advanced gravitational-wave detectors [42].

The torsional acoustic vibration of a flat mirror is an effective means for obtaining a

large spatial overlap between TEM00, TEM01 optical modes, and the acoustic mode. Our

design matches the acoustic resonator dimensions to the spot sizes for a TEM00 pump

mode and a TEM01 high order mode. The torsion mode frequency near 400 kHz dictates

the requirement for the optical mode gap for a reasonable cavity size.

Optoacoustic parametric amplifiers can be operated in two regimes: a) cooling, and

b) amplification and self-sustained oscillation. The design presented here is based on a

three-element near-self-imaging (NSI) cavity, which allows beam spot size optimisation

and tuning of the optical mode gap by a few MHz, sufficient to span the two operating

regimes.

This paper is organized as follows: In Section II, we present the design and properties

of a suitable high quality-factor resonator with optical coatings. In Section III, we present

the NSI cavity design that allows cooling, amplification and parametric instability to be

achieved. In Section IV, we predict the performance of the OAPA, which is limited by

radiation damage and thermal dissipation if operated at cryogenic temperatures.

6.1.3 Silicon Resonator with Optical Coatings

The long standing problem of the acoustic loss of multilayer optical coatings [100, 112]

restricts the options for the design of high quality-factor optical microresonators. To

minimise acoustic losses the resonator is designed to reduce the elastic strain in the optical

coatings by applying them only on a central element, designed to move as a rigid body in

a torsion motion. In the future this design could also allow an inductive sensor loop to be

attached without incurring large acoustic losses.

The resonator design is illustrated in Fig. 6.1. Resonators consist of three paddles,

inspired by previous work [4], connected by torsion rods, fabricated on 20 mm by 20 mm

standard silicon wafers. The paddle design consists of a 1 mm by 1 mm central paddle

supported by two isolation paddles (1 mm by 1.8 mm). The torsion rods have length of

0.5 mm and width of 0.3 mm. Wafer thickness of 365 μm, ∼ 500 μm and 670 μm were

used. For the design presented here, we will focus on a resonator with observed frequency

401.5 kHz and thickness 515 μm [105].

The finite element modelling (FEM) software ANSYS 14.0 was used to develop the

torsion resonator, with approximately 140 000 nodes to represent the resonator, a mesh

6. Optical Cavity Design 167

Figure 6.1: Micromechanical silicon resonator with optical coating. (a) Dimensions: 20mm by 20 mm frame, of thickness 500 μm, and 3 paddles (1 mm wide). The centralpaddle contains a 0.8 mm square 14-layer tantala/silica optical coating (total thickness ∼5.0 μm). (b) Photograph showing paddles and central coating. (c) FEM projection of thetorsion mode of interest in which the central paddle has maximum amplitude, shown bythe contour plot of deformation.

size of 300 μm on the frame and a finer mesh size of 50 μm on the paddles and torsion

rods. The resonator was modelled to determine the amplitude distribution of acoustic

modes and to ensure that torsion modes were separated in frequency from resonances of

the frame. The modes of interest (see Fig. 6.1, (c)) should have strain energy localised

on the torsion rods, and low energy in the frame. Such modes were expected to have

a good isolation from suspension losses. Localisation of the strain energy in the torsion

rod segments holding the middle paddle, and minimal strain energy on the paddle itself,

confirmed the near rigid body behaviour of the central paddle.

Resonators for this OAPA system have been fabricated at the Institute of Photonics

Technologies of the National Tsing Hua University in Taiwan, at the Institute of Electronic,

Micro-electronic and Nanotechnology (IEMN), the Societe Europeenne de Systemes Op-

tiques (SESO) and the Laboratoires de Materiaux Avances (LMA), in France, and at the

Australian National Fabrication Facility Ltd in Queensland.

Following careful metrology of resonators fabricated by dry etching [98] or laser mi-

cromachining [99], we used FEM modelling to help identify the torsion modes of interest.

The 401.5 kHz mode was consistent with the predicted torsion mode frequency for the

resonators fabricated from monocrystal (100) boron-doped silicon with thickness of 500 ±25 μm.

Thin film optical coatings were applied on 365 μm thickness resonators as a test of

the coating technique, within a square area of 800 μm by 800 μm, centred on the middle

168 6. Optical Cavity Design

paddle. The coatings consisted of 14 alternating layers of SiO2 and Ta2O5 with total

thickness 5.0 μm. The specified reflectivity was 99.997% [100, 112], at 1064 nm. Future

resonators will use higher reflectivity coatings and thicker wafers. This coated paddle is

intended as the end mirror in the near-self-imaging cavity design presented in the next

section.

FEM modelling was used to estimate the losses from coatings by calculating the ratio

of the strain energy stored in the coatings ΔE, and the total strain energy E of the acoustic

mode. In the absence of suspension losses, the acoustic quality-factor can be approximated

by [44] Q−1m = Q−1

i +Q−1c (ΔE/E), where Qi and Qc are the intrinsic and coating quality-

factors. For coatings on the 401.5 kHz resonator, the calculated energy ratio is 1.7 10−3.

Assuming a typical coating loss[100] of Q−1c = 2.5 10−4 , we expect to achieve Qm ∼

1.9 106 if Qi = 107, and Qm ∼ 7.0 105 if Qi = 106. There is some uncertainty on the

intrinsic quality-factor of thin wafers, due to the contribution of surface losses [130].

The acoustic loss contribution from the coatings is very sensitive to the wafer thickness.

This loss contribution could be considerably reduced using small changes to the coating

area and substrate thickness.

While the intrinsic quality-factor of bulk silicon and thin silicon flexures has been ob-

served to be very high [91, 130], the quality factor of wafers is uncertain because it is

difficult to distinguish between suspension losses and intrinsic losses. As stated by Chan-

dorkar et al. [129], the heat generation in the resonator due to deformation is negligible

for a pure torsion mode. For this reason, thermoelastic loss can be ignored.

We measured the mode frequencies and quality factors of resonators without coating

using an optical lever and piezo excitation system [105]. Quality factors in the range 5.0

105 - 8.6 105 were observed, reasonably close to our design goal.

We also observed that the quality factor depends greatly on the position of the wafer

holder, indicating that suspension losses are not negligible. A comparison of resonators

will be published elsewhere.

6.1.4 Cavity Design

An OAPA requires a cavity design that is well matched to the acoustic resonator. Here

we present a 3-element near-self-imaging (NSI) optical cavity design that meets the re-

quirements to achieve appropriate optical mode frequencies and spot size to enable strong

3-mode optoacoustic parametric interactions in a compact tabletop system.

In a single optical cavity, for a given cavity length L, the resonant frequency of the

6. Optical Cavity Design 169

TEMmn mode is given by

ν =c

2L[2p+

α

π(m+ n+ 1)], (6.2)

where c is the speed of light, p is the longitudinal mode index, m and n are transversal

indexes, and α is the Gouy phase shift of the TEM00 mode over one cavity length. If

α = 0, the cavity is fully degenerate, and all TEMmn modes and their linear combinations

resonate simultaneously. This is a self-imaging cavity [143]. Similar cavities have been

studied and used in multi-mode squeezing experiments [144].

Figure 6.2: Near-self-imaging optical cavity concept and key parameters. A 400kHz tor-sional microresonator (M1) interacts with the TEM00 mode and the TEM01 mode. We cancontinuously change the mode gap between the two optical modes by tuning the positionof the lens LT . W1 and W2 are the waist sizes before and after the lens LT , respectively.In a practical cavity the tuning lens LT is replaced with a tuning mirror, MT (shown laterin Fig.6.4).

The configuration of a self-imaging cavity is fixed after choosing the mirror radius of

curvature R0 and the lens focal length f . The spacing between components is given by

L1 + L2 = f + R0, (6.3)

and

L3 = f + f2/R0. (6.4)

A small adjustment to the position of any component in the self-imaging cavity breaks the

degeneracy, thereby creating the near-self-imaging (NSI) cavity, in which there is a small

tunable frequency gap between modes.

As stated in the introduction, the 400 kHz acoustic frequency of the torsional mi-

170 6. Optical Cavity Design

croresonator should match the frequency gap between the TEM00 and TEM01 modes in

our cavity design. The TEM01 mode is chosen for its suitable spatial overlap with the

torsional acoustic mode of the resonator.

Another requirement of the OAPA is that the optical mode size should match the size

of the acoustic mode. For a simple 2-mirror optical cavity, a mode gap near 400 kHz

is easy to achieve in a near-planar or a near-concentric cavity [54], but it is difficult to

achieve the correct mode size in a small cavity. This is easier in a 3-element NSI cavity,

using small adjustments of the position of LT (see Fig. 6.2) to adjust both the mode gap

and the mode size.

As shown in Eq. (6.2), the mode gap between the TEM00 mode and the TEM01 mode

is determined by the Gouy phase shift α, which is a function of the cavity length L, and

therefore of the radius of curvature R0 of mirror M0 and the focal length f of the lens LT .

Here we will derive the Gouy phase shift and the optimal positions of the components

in the cavity in order to produce 3-mode interactions. To simplify the analysis, the cavity

is divided into three parts: L1, L2 and L3. We use φ1 for the Gouy phase shift from M0

to the waist W1, φ2 for the phase shift from the waist W1 to the lens LT , and φ3 for the

phase shift from the lens LT to the resonator M1. The total Gouy phase shift in the cavity

is then:

α = φ1 + φ2 + φ3. (6.5)

For a Gaussian beam, the Gouy phase shift at any point is given by

φ = arctan(z/zR), (6.6)

where z is the displacement of the point relative to the waist, zR is the Rayleigh range

given by zR = πW 2i /λ, where λ is the laser wavelength, and Wi is either the waist before

(i = 1) or after (i = 2) the lens LT .

In the NSI cavity, the laser spot size on the resonator is waist W2, which must be

tailored to fit within the square mm surface of the torsion resonator. The distance L3

between the resonator M1 and the lens LT must satisfy:

φ3 = arctan(L3

zR), (6.7)

where zR = πW 22 /λ is the Rayleigh range after the lens.

The properties of the laser beam after passing through the lens are completely deter-

6. Optical Cavity Design 171

mined by the state of the beam before the lens and the transmission matrix of the lens.

The waist position L2 of the first waist, and the waist size W1 can be derived directly from

the ray transfer relation of a Gaussian beam, given by

q′ =fq

f − q, (6.8)

where f is the focal length of LT , q is the complex parameter of a Gaussian beam

defined by q=z+izR, and q′ is the complex parameter before the lens. From Eq. (6.8), we

obtain L2 and W1:

L2 = f +(L3 − f)f2

(L3 − f)2 + z2R, (6.9)

W1 =f2W 2

2

(L3 − f)2 + z2R. (6.10)

With the values of L2 and W1 determined, phase shift φ2 is given by

φ2 = arctan(L2

z′R), (6.11)

where z′R is the Rayleigh range before the lens, given by:

z′R =πW 2

1

λ. (6.12)

The radius of curvature R0 of the mirror M0, and the waist size W1 are related to L1

as follows:

R0 = L1(1 + (πW 2

1

λL1)2). (6.13)

From L1 we obtain φ1:

φ1 = arctan(L1

z′R). (6.14)

Equations 6.7, 6.11 and 6.14 show that the Gouy phase shift over the cavity length is

determined by four parameters: W2, R0, f and L3. We chose the size of W2 to balance two

requirements: (1) high overlap between the optical and acoustic modes; (2) low optical

diffraction loss to maintain a high cavity finesse. Considering the resonator size of 1 mm by

1 mm, a 0.2 mm beam waist was chosen to have an overlap factor of 0.1 and a diffraction

loss of 15 ppm. If we fix R0 and f by using specific components, the Gouy phase shift

172 6. Optical Cavity Design

will depend only on L3. Thus, by changing L3, we control the mode gap between optical

modes TEM00 and TEM01. Using Eq. (6.2), we obtain a relationship between the mode

gap Δν and L3 as follows:

Δν =c

2π(L1 + L2 + L3)(φ1 + φ2 + φ3 + nπ). (6.15)

Here n is an integer which indicates the difference in the longitudinal mode index of the

two optical modes (n = 1 in our case). We have considered different cavity configurations

(R0 and f values) to achieve a mode gap of Δν = ± 400 kHz. We found that a configuration

with R0 = 50 mm and f = 50 mm is suitable, as we only need to adjust the position of

the lens by 325 μm (the tuning gap) to adjust the cavity from amplification to cooling

regimes.

Figure 6.3: Mode gap as a function of the position of the micromechanical resonatorrelative to the lens, with fixed parameters of the radius of curvature and focal length,R0 = 50 mm and f = 50 mm. The tuning gap is 325 μm and the tuning coefficient is 2.46MHz/mm. The tuning is almost linear and symmetric around the self-imaging point.

For a self-imaging cavity, in which L3 = f + f2/R0 (Eq. (6.4)), using R0 = 50 mm

and f = 50 mm gives a value of the distance L3 = 100.0 mm. This corresponds to the

NSI average value of L3 for instability and cooling regimes, which is 100.16 mm and 99.84

mm, respectively. This indicates that the tuning of the lens LT in the cavity is symmetric

around the self-imaging point (Δν = 0).

Cavity tuning is demonstrated in Fig. 6.3, where the mode gap is plotted as a function

6. Optical Cavity Design 173

of the position of the lens relative to the resonator. A tuning coefficient of 2.46 MHz/mm

is achieved. When L3 is reduced below 100 mm, the OAPA enters the cooling regime.

When L3 is increased above 100 mm, it enters the positive gain regime.

6.1.5 Predicted performance of a Practical NSI Cavity Setup

Based on the design, given in the previous sections, we present here a practical experimen-

tal design and performance estimates for a 3-mode OAPA shown in Fig. 6.4. Rather than

using a lens as shown in Fig. 6.2, a practical OAPA uses a mirror to replace the tuning

lens. The pump beam requires a pre-mode cleaner (PMC) to insure that the injection is

a pure TEM00 mode, and must be mode matched and frequency locked using a standard

Pound-Drever-Hall (PDH) locking scheme [30].

Figure 6.4: Experimental setup of the near-self-imaging (NSI) cavity for three-mode inter-actions. The laser light passes through a pre-mode cleaner (PMC) to clean up high ordermodes, and is then phase modulated by an EOM for Pound-Drever-Hall (PDH) lockingof the laser frequency to the NSI cavity. A 400 kHz torsional microresonator acts as theend mirror of the NSI cavity, which interacts with the cavity TEM00 and TEM01 modes.The sum signal from the quadrant photodetector (QPD) is used for PDH locking whileits differential signal is used to monitor the resonator amplitude. The mode gap betweenthe two optical modes is adjusted by tuning the position of MT .

The NSI cavity needs to be operated in vacuum to achieve high acoustic resonator

quality-factors, and to be sufficiently rigid to maintain dimensional constraints. Piezo

mirror mounts for the input mirror and the tuning mirror can allow the required tuning

range. Experience with a membrane cavity three-mode interaction setup [55] has demon-

strated the importance of alignment and tuning to achieve maximal overlap between the

acoustic mode and TEM01 mode.

As shown in Fig. 6.4, a quadrant photodetector can be used in common mode for the

PDH locking, and in differential mode to monitor the OAPA signal, at 400 kHz. This signal

174 6. Optical Cavity Design

is a beat frequency between the TEM00 and TEM01 modes, which is proportional to the

amplitude of the torsion resonator. In the absence of external excitation, the QPD output

should record a thermally-driven acoustic mode amplitude of the torsional resonator. At

low input power this signal should represent the kT thermal energy of the resonator, but

as power increases the mode should be amplified or suppressed, as discussed further below.

To predict the performance of the OAPA described here, we need to determine the

maximum power the device will withstand. This depends on two factors: optical damage

and thermal heating. Using the continuous wave damage threshold for thin film optical

coatings, ∼ 1MW/cm2 for currently available technologies [100], and the designed waist

size at M1 of 0.2 mm, the circulating power limit is ∼ 1.2 kW.

The transmissivity and absorption of the resonator coatings will allow some of the

circulating power to be absorbed by the silicon substrate, which could alter the acoustic

torsion mode frequency and detune the NSI cavity.

The input mirror is designed for a transmissivity of 10−4. Combined with the 1ppm

loss of the coatings and 15 ppm diffraction loss of the resonator reflective coating, the

OAPA cavity can achieve a finesse of 6.3 104, calculated from the relationship: F =

2π/φ, where F is the finesse, and φ is the total loss. The circulating power limit sets the

input power limit to 30 mW.

Only a small fraction of the input power need be dissipated by the resonator. Since

the silicon substrate will strongly absorb 1064 nm light, the power absorbed by the res-

onator will be the sum of the coating transmissivity and absorption. Assuming 1 ppm

transmission, coating absorption of 0.25 ppm, and silicon absorption of 0.5, a maximum of

∼ 1 mW should be absorbed by the resonator. As discussed below, this could be reduced

in resonators designed for cryogenic operations.

We modelled the resonators using the thermal conductivity of 148 W/mK [86] to

predict the effects of laser heating. The small mass of the resonator structure and the large

thermal conductivity of silicon ensures that the resonator has a short thermal relaxation

time (∼ ms) to the wafer substrate. Thereafter it cools by blackbody radiation.

At room temperature, the 1 mW from laser heating is estimated to cause the wafer

temperature to rise by 3.2 K before equilibrium is reached with dissipation from blackbody

radiation. The blackbody radiation thermal relaxation time for the wafer is ∼ 10 minutes.

We used the temperature dependence of the Young’s modulus Y(T) of silicon to confirm

the above analysis, as changes to the Young’s modulus are the dominating cause of acoustic

mode frequency drift [120, 133]. The near-linear relation [86] ΔY(T)/ ΔT ≈ - 7 MPa/K

6. Optical Cavity Design 175

allows the mode frequency to be used as a temperature probe.

With the low power laser used in the optical lever for measuring acoustic modes, we

could apply known amounts of heat and measure the resonator temperature changes over

time. Results were consistent with the simple modelling, above, leading to an expectation

that, at maximum optical power, the mode frequency would drift by ∼ 32 Hz due to laser

heating. This is small compared with the optical mode linewidths, so should not vary the

OAPA gain.

Figure 6.5: Expected parametric gain for given laser input power values. Parametricinstability (gain R = 1) is achieved with 1.6 μW (left vertical dotted line). The red curveis the achievable acoustic mode amplification and instability (for R ≥ 1), and the bluecurve is mode cooling. Our system allows for 30 mW of input power before the damagethreshold of optical coatings is reached. With 30 mW of input power, a cooling factor ∼1.9 104 is reached (right vertical dotted line).

Two parameters can be used to characterise OAPA performance: a) power required to

achieve parametric instability (R = +1) and b) maximum possible cooling factor. Here

we will estimate these factors.

The effective acoustic mode temperature Teff reached by self-cooling (anti-Stokes

mode) is related to the parametric gain R (see Eq. (6.1)), as shown in Eq. (6.16) [3].

Teff =T0

1−R, (6.16)

where T0 is the thermodynamic temperature of the acoustic mode.

According to Eq. (6.16), if R = +1, Teff → ∞, corresponding to parametric instability.

Large negative values of R cause strong cooling.

Using Eq. (6.1) and the parameters reported above, the OAPA achieves R = 1 with an

176 6. Optical Cavity Design

incident power ∼ 1.6 μW (cavity finesse of 6.3 104, Q0 = Q1 = 1.3 1010, and Λ = 0.1,

resonator effective mass m = 1.17 mg, and experimentally observed [105] Qm = 7.5 105).

Fig. 6.5 shows the predicted parametric gain as a function of input power.

Using the maximum input power of 30 mW, and the parameters given above, the max-

imum cooling gain is - 1.9 104. This gain would allow the resonator effective temperature

to be cooled from 300 K to ∼ 15 mK. If the resonator was cryogenically cooled to 76 mK,

it would allow self-cooling of the macroscopic resonator acoustic mode to the quantum

ground state, near 4 μK [3] (Eq. (6.16))).

6.1.6 Conclusion

A three-element NSI cavity has been shown to allow the creation of a high parametric gain

3-mode OAPA in a compact setup. We have presented a configuration that allows tuning

between positive and negative gain (from parametric instability to strong cooling) with

small relative position adjustments of the optical components. The device can be tuned

for different optical mode gaps and mechanical resonator frequencies.

We have predicted that the OAPA design presented here can achieve parametric in-

stability, corresponding to a parametric gain R = 1 with 1.6 μW of input power. The

maximum self-cooling factor, limited by the optical coating damage threshold, is 1.9 104

with 30 mW of input power. This cooling gain is sufficient to cool the resonator to the

quantum ground state from an initial temperature of 76 mK.

The above estimate does not account for the strong reduction of acoustic losses in

silicon at low temperatures. A resonator designed to be cooled to the ground state would

require reduced laser heating and reduced loss contributions from the optical coatings. If

a quality factor of 107 was achieved, combined with coating losses ∼ 0.1 ppm, the heat

load would fall below 100 μW, and the peak cooling factor would increase, making ground

state cooling achievable for this mg scale resonator [135].

An OAPA has many potential applications. Because the resonators have long relax-

ation times (∼ s), if cooled to the quantum ground state, they could be used as quantum

memory devices. Their mm scale makes them suitable to study macroscopic quantum

mechanics on scales larger than many other systems used in macroscopic quantum exper-

iments. They are sources of tripartite entanglement.

As room temperature devices, a simple current loop mounted on the central paddle

could enable an OAPA to be operated as a sensitive magnetometer or radio frequency

field detector. In the first instance we plan to test the predictions presented here, and

6. Optical Cavity Design 177

use the device for studying three-mode parametric instability and suppression methods

for advanced gravitational wave detectors [42].

We also propose to develop an improved resonator design using silicon-on-insulator

wafer technology, to allow a difference in thickness between the frame and the paddles [102].

A design with thin paddles and a thick frame (∼ mm) will allow simple thermal grounding

without incurring suspension losses, which would be necessary for efficient cooling in a very

low temperature environment.

ACKNOWLEDGEMENTS

This work was supported by the Australian Research Council, the Australian National

Fabrication Facility, the French RENATECH network, and Beijing Normal University, for

partial funding of the work.

178 6. Optical Cavity Design

6.2 Silicon-on-Insulator (SOI) Resonator Design

The resonators fabricated as described in this thesis were limited to simple designs and

fabrication methods, to facilitate the timely achievements of the first steps to create a

3-mode OAPA device. The research described in the previous chapters and in the above

manuscript [145], has demonstrated that (1) high quality-factor (Q) resonators can be

made using monocrystal silicon wafers with a simple 3-paddle design, and (2) a 3-mode

OAPA device is predicted to achieve parametric instability with reasonable parameters,

as well as cooling of a macroscopic object by a factor of ∼ 104, or to the quantum ground

state if pre-cooled.

The next step is to improve the performance of the OAPA device by creating higher Q

resonators, reducing the energy coupling between the frame and the paddles, and increas-

ing the thermal grounding to reduce the heat load in cooling applications. This section

will show that a resonator design fabricated from a silicon-on-insulator (SOI) wafer can

achieve all of the above.

The use of SOI wafer technology allows resonators to be made with a thicker frame

while keeping the same paddle pattern dimensions used in this thesis, as shown in Fig.6.6.

The design uses a 2 mm thick SOI wafer made of a ‘device layer’ of 0.5 mm for the resonator

paddles, and a supporting frame of 1.5 mm (handle layer), separated by a buried oxide

layer. Both sides are etched separately, using dry etching. Wet etching is used to remove

the exposed buried oxide layer underneath the resonator paddles. This avoids damping

the Q from the acoustic loss of the oxide layer.

Figure 6.6: Resonator design from a 2 mm thick silicon-on-insulator wafer. The resonatorpaddle pattern is on the top ∼ 500 μm wafer, similar to resonators presented in thisproject. The rest of the 1.5 mm thickness is used to make a thick frame surrounding thepattern.

6. Optical Cavity Design 179

Similar to the finite element modelling (FEM) of simple 3-paddle designs in Chapter 2,

the SOI resonator is modelled, and the torsional frequency of the middle paddle is predicted

to be 385 kHz. This mode is shown in Fig. 6.7, indicating that the implementation

of a thicker frame support does not change the resonant frequency significantly. This

conveniently allows the SOI resonator to replace the resonator of uniform thickness in the

OAPA device, without having to alter the cavity configuration.

Figure 6.7: Torsional mode shape (mode of interest) predicted by FEM at 385 kHz. (a)Displacement contour plot. (b) Normalised vibration in the torsional mode.

The map of vibration amplitudes on the SOI resonator, illustrated in Fig. 6.8, demon-

strates a major reduction in the energy coupling between the acoustic torsional mode and

the frame. The amplitude of the frame is ∼ 0.0004 of the maximum amplitude of the

middle paddle, corresponding to a coupling reduced by 1 order of magnitude, compared

to uniform thickness resonators (see Chapter 3 and 4).

Another advantage with this SOI design is the reduced sensitivity to the location of

the suspension points. The acoustic wavelength in the frame, for the torsional mode, is

above 17.5 mm along the paddle axis (see Fig. 6.9), compared to wavelengths of 4.6 mm

to 14.6 mm along the paddle axis for resonators made from silicon wafers of 515 μm and

670 μm thickness, respectively. This longer wavelength allows a larger surface of contact

between the suspension and the frame while still keeping the coupling between with the

paddle vibrations and the frame (for the mode of interest). This also allows for strong

clamping of the samples, which can be used for thermal grounding through the suspension.

Along specific thin strips of the frame surface, on either side of the 3-paddle region

(Fig. 6.9 b,c), the amplitude ratio between the frame and the middle paddle is no greater

180 6. Optical Cavity Design

Figure 6.8: Zoom-in to three regions of interest, showing small vibration amplitudes onthe frame, less than 0.0004 relative to the middle paddle.

6. Optical Cavity Design 181

than 0.00006, as shown in Fig. 6.9. This is 2 orders of magnitude lower than previous

designs. These locations are ideal for suspension contacts, and are expected to improve

the acoustic Qs.

Figure 6.9: A narrow strip of the resonator along the left support, parallel to the paddleaxis. The acoustic wavelength is measured between nodes, with a value of 17.5 mm.Along this strip, the maximum amplitude ration is 0.00006 of the middle paddle, showingextremely reduced energy coupling.

The fabrication process is not significantly more involved for SOI resonators, and more

improvements could be made to further isolate the paddles from the frame, and to reduce

the strain energy in the optical coatings. Nevertheless, these simple modifications already

improve the OAPA device performance. It is expected that these SOI resonators will have

a higher Q from reduced coupling to the suspension, and thermal grounding will reduce

the load on dilution refrigerators [135].

Table 6.1 compares the energy coupling between the frame and the acoustic mode by

showing amplitude ratios between the frame and the middle paddle at the specific left

support locations provided by positioning tools. Similar results are found for the right

and bottom supports (results not included here). These positioning (loading) tools are

described in Chapter 3.

Note the 2 orders of magnitude difference in the vibration amplitude ratios between

the SOI resonator design and the samples UQB3C-1 and TB2C-1. This will reduce the

coupling to the 3-point suspension.

The increase in acoustic wavelength along the paddle axis is significant (above 10

mm increase), between the SOI resonator and the 515 μm thick sample UQB3C-1. The

wavelength is only increased by ∼3 mm, from the value obtained for the 670 μm thick

sample TB2C-1. It is not clear from the modelling if further improvements to the acoustic

wavelengths can be obtained by further modifications to the thickness. Other design

variations could be explored, as this is still a simple design.

182 6. Optical Cavity Design

Table 6.1: Table of the relative vibration amplitudes (local divided by maximum) at theleft support positions (right and bottom support values not shown), with respect to theacoustic pure torsion mode of samples UQB3C-1, TB2C-1 and the SOI resonator design.Labels A to I refer to the 9 possible loading positions provided by the loading toolsdescribed in Chapter 3. The table includes a measure of the acoustic wavelength alongthe paddle axis and along the rod axis for each sample.

Sample UQB3C-1 TB2C-1 SOI model C

Thickness (μm) 515 670 2000

Frequency (kHz) 349 384 385

Left Support 10−3 10−3 10−5

A 1.09 0.317 0.265B 2.28 0.936 0.687C 1.69 1.55 1.40D 0.66 2.14 1.83E 3.90 2.65 2.52F 6.50 3.04 3.24G 6.97 3.26 3.69H 4.68 3.28 4.13I 0.09 3.12 4.32

Wavelength (mm)

λx 5.0 5.0 6.7λy 4.6 14.6 17.5

6.3 Discussion

Fabricated resonators have been shown suitable to be fitted in a near-self-imaging cavity

design which constitutes a 3-mode opto-acoustic parametric amplifier (OAPA) device.

This OAPA device, presented in the first part of this chapter, can cool a macroscopic

resonator to the quantum ground state if the resonator is previously cooled to an initial

temperature of 74 mK. This performance could be improved with a higher Q resonator

made from SOI wafers, with a thicker frame and thin paddles.

The second part of this chapter has demonstrated the potential for lowering suspension

losses by reducing the energy coupling between the thicker frame of an SOI resonator to

the vibrating paddles.

Furthermore, an SOI design provides the possibility of a stronger clamping contact

without incurring significant losses, which would allow thermal grounding of the resonator

and thus reduce the heat load.

A firm clamping, instead of the 3-point suspension, is a better suited method for

holding a resonator in an OAPA device. Reducing the heat load minimizes the material

property variations, such as Young’s modulus, to ensure an improved frequency stability

of the device.

Chapter 7

Conclusions and Future Work

In this final chapter, I present a summary of my work and findings, told through the story

of this research. I then emphasise conclusions drawn from this research. I complete this

chapter with a series of suggestions for future work to be undertaken as a continuation of

this exciting 3-mode opto-acoustic parametric amplifier (OAPA) project.

7.1 Review of the Research

The aims of the research in this thesis were as follows: (1) develop acoustic resonators

intended as the heart of a 3-mode opto-acoustic parametric amplifier (OAPA), (2) demon-

strate that these resonators meet the requirements for an OAPA, such as a high quality

factor (Q), torsional vibration in the ∼ MHz range, reduced strain energy on the optical

coatings, small resonator mass and size, and finally, (3) assess the feasibility of an OAPA

device, based on the demonstrated parameters of fabricated resonators.

The aims mentioned above were all met. High Q resonators in the mm- and mg-scale

were fabricated, and they exhibit an acoustic torsional mode in the ∼ MHz frequency

range, which is designed to reduce the strain on the coating area. An OAPA design

presented in Chapter 6 is predicted to achieve parametric instability and mode cooling

by a factor of ∼ 104 with practical cavity parameters. These theoretical predictions are

based on the demonstrated resonator properties measured in Chapter 4 and Chapter 5.

This device has tunable transverse modes, as required.

This is how it was done:

183

184 7. Conclusions and Future Work

7.1.1 The Story

The path to the achievements in this thesis is a long story of learning from mistakes and

dead-ends. Designs were examined with finite element modelling (FEM), with prototypes

ranging from single-paddle designs, inspired by Zhao et al. [3], to three-paddle designs,

inspired by Davis et al. [4]. The latter showed more promise and four such designs were

chosen for fabrication.

I spent a year learning and experimenting with wet KOH etching in the Western

Australia Centre for Semiconductor Optoelectronics and Microsystems (WACSOM) at the

School of Electrical, Electronic and Computer Engineering of the University of Western

Australia, in Perth. This fabrication method was successful in producing high Q silicon

resonators previously [110], however, I never managed to get good results.

The fabrication was then outsourced to commercial and academic institutions around

the world. These include the Australian National Fabrication Facility Ltd (ANFF-Q) in

Queensland, and the Institute of Electronic, Micro-electronic and Nanotechnology (IEMN)

in France, where dry etching of resonators took place. The resonators made in France were

first polished in the ‘Societe Europeenne de Systemes Optiques’ (SESO), in Grenoble,

France. Ion-beam sputtering of Bragg mirrors (optical coatings) onto the French samples

was done in the ‘Laboratoires de Materiaux Avances’ (LMA), also in France. A laser

micromachining fabrication technique was also used at LEGEND Laser Inc, in Taipei,

Taiwan, coordinated by a team who also performed optical coating in the Institute of

Photonics Technologies (IPT) at the National Tsing Hua University, Hsinchu, in Taiwan.

Producing optical coatings specifically located within the centre of a middle paddle

was not trivial and careful lift-off techniques were developed to achieve this requirement.

Once a few fabricated 3-paddle silicon resonators arrived back in Perth, I began a long

series of experiments, which would ultimately lead to a suitable method of suspending and

exciting these resonators, to enable them to reach high Qs.

The first method consisted of clamping the samples with an aluminium block. By

varying the area of clamping, and the clamping location, I determined the importance of

minimising suspension losses.

Initially, Qs measured were no higher than 7 103, and it first seemed as though the

designs were not promising to meet the requirements. Through minimising the clamping

area and exploring clamp locations, the Q reached 4 104, an exciting result at the time.

Later, samples of better quality arrived and testing revealed a new limit of Qs ∼ 105.

It was found that this system was intrinsically limited by suspension losses, and I moved

7. Conclusions and Future Work 185

on to new suspension methods.

Using a naturally occurring resin in Australia, called Yacca gum, the samples were

bonded, in the hopes of observing reduced coupling losses. The bonding area was reduced

and bonding locations explored, but Qs were limited to ∼6 104.

Electrostatic excitation was attempted, but I was unable to get this method to work

reliably. The positive outcome of these efforts was that an idea for suspending the resonator

on three vertical needles was developed, which later turned out to be instrumental in

successfully reaching the aims of this thesis.

Next, a suspension method was developed using three-wires to hold the sample verti-

cally from the outside edges of the resonator frame. This method, while only producing

Qs of 1.3 105, was at least a suitable method to use in an OAPA device, where samples

must be held firmly. A positive outcome of these efforts was an FEM technique devel-

oped to find optimal suspension locations for the 3 wires along the resonator’s edges. A

modification to this FEM technique was found useful for the final suspension method.

Finally, inspired by the efforts with electrostatic excitation and 3-wire suspension, a

3-point suspension method was developed. This consisted of resting the samples on three

vertical needles, with fine points of small cross-sectional areas to minimise suspension

losses. The positions of these needles were chosen according to FEM of nodal points of

acoustic waves in the frame, for a torsional mode.

The reduced area of contact, and the careful choice of locations was instrumental in

reducing the suspension losses. The highest Qs were obtained with this method, reaching

close to the target of 106, as discussed below.

It should be noted that locating 3 points which are all optimal for suspension is chal-

lenging. I certainly don’t presume to have found the best 3 points possible.

Loading tools were fabricated which allowed precise reloading of samples to the same

positions relative to the 3 support points. This allowed testing the repeatability of the

high Q at specific locations. It also allowed testing FEM predictions of the optimal loading

positions, and assess the validity of this modelling.

The highest Q obtained in this research was 8.6 105 for a laser micromachined res-

onator (sample TB2C-1) of wafer thickness of 670 μm. Other resonators were tested and

results were in the range of 2 105 to 7 105, for a variety of samples and designs, including

optically coated samples (which were limited to Qs of 2.7 105).

The loading tools provided nine different loading positions, spaced 0.5 mm apart along

one axis. Results from different positions demonstrated that the Q varied with loading

186 7. Conclusions and Future Work

position, though not clearly matching FEM predictions. Some reasons for the discrepancies

include possible contamination of samples after repeated experiments, and damage to

samples from manipulations.

Most likely, the reason for discrepancies was tank vibrations inducing small, yet impor-

tant, sample displacements. Even sub-mm displacements are predicted to have a strong

impact of the Q. In particular, when pumping to low pressures in a vacuum tank, the

ramping of the pump rotation speed from 0 Hz to 1500 Hz caused the flexible tube, used

between the pump and the tank, to resonantly vibrate at ∼500 Hz. This was addressed

by supporting the flexible tube and attempting to dampen the vibrations. It is not unrea-

sonable to think that Q variations and discrepancies are, in one form or another, related

to unwanted sample displacements of this nature.

This 3-point suspension method was shown suitable to produce high Q and repeatable

results. It should be noted however that this is a cumbersome method and is not suitable

for a 3-mode OAPA device, where the resonator is meant as an end mirror and should be

firmly held in place.

The fabrication methods of dry etching and laser micromachining have proved useful

to obtain high-Q resonators. The wet etching technique, however, should not be discarded.

The large amount of silicon that needed to be removed, combined with the inherent diffi-

culties related to an etching technique which follows crystal planes, made producing the

required dimensions difficult. As developing this specific method was not the aim, and

due to my lack of time for experimenting with this method, I pursued other fabrication

methods which worked well.

High Qs were only one of the requirements. A torsional mode was another important

requirement for a 3-mode OAPA. After many different attempts, a method for observing

the mode shape was developed. First attempts involved an optical lever where vibration

amplitudes were recorded over the resonator surface.

A better method was developed, which included phase information, consisting of a

Michelson interferometer. This was the work of Dr. Hou Wei, a visiting professor from

the Institute of Semiconductors, Chinese Academy of Sciences, in Beijing, China. The

interferometer allowed the torsional mode to be identified for two of the most important

samples (UQB3C-1 and UQB3D-1), to within 5% of the FEM modal predictions.

Using the optical lever and 3-point suspension, thermal effects were examined from

laser heating of the resonators. The thermal analysis consisted of modelling frequency

drift and thermal relaxation times, and comparing to experimental results. It was found

7. Conclusions and Future Work 187

that the acoustic torsional mode varied with temperature at about 10 Hz per K. This

indicates that the resonators could be used as sensitive temperature probes. Also, the

relationship could be exploited in an OAPA device for fine-tuning of the acoustic mode

frequency.

The thermal modelling proved useful in predicting thermal effects in an OAPA device,

showing that expected resonator temperature rises would not cause operational problems

when the system operating at room temperature. For quantum experiments performed

in cryogenic environments, however, the system might require higher reflectivity coatings

and firm clamping of the resonator to provide thermal grounding through the suspension.

Based on the Qs obtained with experiments and thermal analysis mentioned above,

and on a near-self-imaging cavity designed almost entirely by Liu Jian and Ma Yubo, an

OAPA device was predicted to achieve parametric instability with 1.6 μW of laser input

power, and mode cooling by a factor of ∼1.9 104 with 30 mW of input power. This

30 mW was determined as an upper limit to the laser power before the coatings of the

resonator would suffer laser damage.

This cooling factor is sufficient to bring a mg-scale resonator to its quantum ground

state from an initial temperature of 76 mK. At such low temperatures (∼mK) the heat

load from a 30 mW incident laser becomes close to the maximum cooling power of dilution

refrigerators [135]. This issue can be addressed with (1) higher reflectivity coatings which

reduce the transmitted power to the resonator substrate, and (2) by an improved resonator

design from silicon-on-insulator (SOI) technology.

An SOI resonator design was presented which incorporates a thicker frame structure to

support thin paddles of the designs fabricated here. The thicker frame reduces the energy

coupling between the acoustic torsional mode of the middle paddle and the frame. This

reduces the loss contribution from the suspension.

The SOI design is an improvement by two orders of magnitude in the reduction of

energy coupling between the frame and paddles in a torsional mode, compared to previous

uniform thickness resonators fabricated in this thesis. The SOI design has an acoustic

wavelength in the frame which is longer than in thinner resonators, which permits a larger

surface of contact area between the frame and the suspension.

By allowing a larger area of contact with the suspension, the SOI design effectively pro-

vides immunity to suspension losses, which in turn allows firm clamping of the resonators

without reducing the Q. This makes SOI resonators suitable for an OAPA device, and

allows thermal grounding through the suspension, which facilitates 3-mode self-cooling

188 7. Conclusions and Future Work

experiments to reach the macroscopic quantum ground state.

7.2 Summary of the Results and Conclusions

The findings of this reasearch are the following:

1. Quality-factors close to the target of 106 can be achieved at room temperature from

micromechanical resonators made from dry etching and laser micromachining.

2. The highest Q achieved here was 8.6 105 for a laser micromachined resonator (sam-

ple TB2C-1) of thickness 670 μm.

3. A three-point suspension method, consisting of resting a resonator horizontally on

three vertical supports, was shown suitable to measure high Qs.

4. The resonators fabricated here exhibit a torsional acoustic mode in the frequency

range of 0.3 to 0.6 MHz.

5. Torsional mode shapes were confirmed with a Michelson interferometer.

6. A 3-mode opto-acoustic parametric amplifier (OAPA) was designed. Based on

demonstrated resonator parameters, the OAPA is predicted to achieve parametric

instability and self-cooling by factors of ∼1.9 104.

7. A silicon-on-insulator (SOI) resonator design is shown to provide practicle solutions

to reduce the sensitivity of the Q to the suspension, and to provide a means of

thermal grounding of the resonator through the suspension.

From the findings in this thesis, the following conclusions can be drawn:

1. A workable design of an OAPA is demonstrated.

2. A suspension method developed here allows reproducible sample positions at which

Qs larger than 5 105 can be expected.

3. The method used, the three-point suspension, is somewhat cumbersome.

4. It would be best to develop resonator immune to the loading position (such as SOI

resonators).

5. Operating an OAPA at cryogenic temperatures to achieve self-cooling to the quan-

tum ground state might require a resonator to be firmly clamped.

7. Conclusions and Future Work 189

6. An SOI resonator design shown here is predicted to have limited energy coupling to

suspension, thereby allowing firm clamping. This makes an SOI resonator design a

good candidate for an improved OAPA design.

7.3 Future Work

In principle, the work here has demonstrated that an OAPA device using a coated silicon

resonator, similar to the ones fabricated in this project, could achieve strong opto-acoustic

coupling. Observing 3-mode interactions with the studied resonators is a serious goal and

far from trivial.

Developing an experimental near-self-imaging cavity will require careful tuning and

cavity locking. Observing 3-mode opto-acoustic parametric amplification is the natural

next step in this project. Following such observations, the next step would be to develop

a resonator with a sensor attached to the back of the middle paddle. For example, an

inductive loop applied by photolithographic procedures, similar to the ones used in wet

and dry etching of pattern masks, could serve as a prototype 3-mode OAPA magnetometer,

which could then be tested in the field.

Applying a sensor to the resonator and still maintaining a high Q may prove challenging

and require design alterations, such as thicker resonators, or larger resonating paddles, or

improved isolation systems.

With a silicon-on-insulator (SOI) resonator design, as presented in Chapter 6, some

improvements to the Q and thermal loading are expected. The technology for fabricating

SOI resonators is well established [102] and no great obstacles are anticipated for making

such resonators.

Further work pursuing the relationship between the suspension location along the

acoustic wave on the frame and the measured Qs may require improved suspension designs.

For example, the sample could be supported by fixed pins on nodal points (zero amplitude

on the frame) and the acoustic modes could be excited either by electrostatic actuation,

or by a retractable needle connected to a PZT, similar to the 3-wire suspension method

presented in Chapter 3.

An improvement could be made to the Michelson interferometer, in terms of the sample

holding method. Currently, the sample is clamped solidly on a corner, between a PZT

and a metal block. This induces losses in the Q of some modes, and as a result, some of

the acoustic modes observable with the 3-point suspension method are not observed with

the interferometer.

190 7. Conclusions and Future Work

By modifying the interferometer setup to hold the sample horizontally on 3 pins and

using a 45 mirror suspended above the sample, the setup might allow more modes to be

excited and their mode shapes identified.

Finally, to allow a more rigorous comparison between different samples and designs,

a common original wafer product should be used for all tested manufacturing techniques.

Various procedures from each technique should be explored.

7.4 Final Remarks

It is my hope that the findings of this research will serve in the creation of 3-mode OAPA

magnetometers and space-exploration tools. I am excited also by the prospect of my

resonators achieving self-cooling to the quantum ground state, and serving in various

quantum experiments. Subsequent PhD students should not hesitate to contact me for

discussions on research directions and collaborations.

Thank you for reading.

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Appendix A

Statement of my Contribution to

the Work in the Publications

The first publication included in my thesis is presented in Section 4.5, in Chapter 4, which

presents the main results of my thesis.

[1] Francis Achilles Torres, Phillip Meng, Li Ju, Chunnong Zhao, David Ger-

ald Blair, Kai-Yu Liu, Shiuh Chao, Mariusz Martyniuk, Isabelle Roch-Jeune, Raffaele

Flaminio, et al. High quality factor mg-scale silicon mechanical resonators for 3-mode

optoacoustic parametric amplifiers. Journal of Applied Physics, 114:014506, 2013.

My contribution to this paper amounts to 80% of the experimental work, all of the

finite element modelling of the resonator designs, and 80% of the manuscript preparation.

This paper forms the bulk of Section 4.5 in Chapter 4. The resonators used in these

experiments were made in the Australian National Fabrication Facility Ltd (ANFF-Q) in

Queensland, where dry etching took place.

205

206 A. Appendix 0

The second publication in my thesis is presented in Chapter 6, as the bulk of this

Chapter.

[2] J Liu, FA Torres, Y Ma, C Zhao, L Ju, DG Blair, S Chao, I Roch-Jeune,

R Flaminio, C Michel, and K-Y Liu. Near-self-imaging cavity for 3-mode optoacoustic

parametric amplifiers using silicon microresonators. Submitted to Applied Optics, 23

October 2013.

The manuscript, of which I am second author, has been published in ‘Applied Optics’.

The cavity design work was performed by Liu Jian and Ma Yubo, summer internship

students from the Beijing Normal University, in Beijing, China. My contribution to this

paper amounts to all of the finite element modelling and thermal analysis of silicon res-

onators (acting as a component of the cavity design), all of the silicon resonator design

and experimental work, and 80 % of the manuscript preparation.

The resonators that I tested produce quality factors from which the calculated per-

formance of the proposed device was determined. The resonators which were tested and

produced the highest quality factors (and most relevant to the discussion in the publi-

cation) were made by collaborators, such as those mentioned in the previous publication

(ANFF-Q), as well as the Institute of Photonics Technologies (IPT) at the National Tsing

Hua University, Hsinchu, in Taiwan, which coordinated laser micromachining at LEGEND

Laser Inc, in Taipei, Taiwan.

I tested resonators with optical coating in my experiments, as candidates of an end

mirror in the design cavity of the publication. These were fabricated in a collaboration

of several facilities in France: fine polishing of the wafers in the Societe Europeenne de

Systemes Optiques (SESO), dry etching in the Institute of Electronic, Micro-electronic

and Nanotechnology (IEMN), and ion-beam sputtering (IBS) of Bragg mirrors in the

Laboratoires de Materiaux Avances (LMA).

Appendix B

List of Publications Included or

Related to the Thesis

[1] Francis Achilles Torres, Phillip Meng, Li Ju, Chunnong Zhao, David Gerald

Blair, Kai-Yu Liu, Shiuh Chao, Mariusz Martyniuk, Isabelle Roch-Jeune, Raffaele

Flaminio, et al. High quality factor mg-scale silicon mechanical resonators for 3-mode

optoacoustic parametric amplifiers. Journal of Applied Physics, 114:014506, 2013.

[2] J Liu, FA Torres, Y Ma, C Zhao, L Ju, DG Blair, S Chao, I Roch-Jeune,

R Flaminio, C Michel, and K-Y Liu. Near-self-imaging cavity for 3-mode optoacoustic

parametric amplifiers using silicon microresonators. Submitted to Applied Optics, 23

October 2013.

[3] F.A. Torres, D.G. Blair, L. Ju, C. Zhao, and H. Miao. Three-mode

opto-acoustic interactions in optical cavities: introducing the three-mode opto-acoustic

parametric amplifier. In Proceedings of SPIE, volume 7579, page 75791A, 2010.

[4] F.A. Torres, J. De Lange, S. Shook, L. Ju, C Zhao, and DG Blair. Com-

parison of low-loss silicon micromechanical resonators from dry etching and laser micro-

machining. To be submitted.

207

208 B. Appendix 1

Appendix C

ANSYS 14.0 Codes for FEM

Here I present the ANSYS 14.0 finite element modelling codes utilised in obtaining predic-

tions of acoustic resonator frequencies and mode shapes, as well as optical coating losses.

This is a supplement to Chapter 2, where the simulated results are obtained from these

codes

209

210 C. Appendix 2

C.1 Modal Predictions with ANSYS 14.0

!! !! Generates 3D model of Si resonators !! Performs nodal analysis without loads !! !! By Francis Torres, !! earlier code by Dr. Slawomir Gras !! Latest updates in October 2013 !! ANSYS CLASSIC 14.0 !! !!--------------------------Step 0---------- !* /NOPR KEYW,PR_SET,1 KEYW,PR_STRUC,1 KEYW,PR_THERM,0 KEYW,PR_FLUID,0 KEYW,PR_ELMAG,0 KEYW,MAGNOD,0 KEYW,MAGEDG,0 KEYW,MAGHFE,0 KEYW,MAGELC,0 KEYW,PR_MULTI,0 KEYW,PR_CFD,0 /GO !* /COM, /COM,Preferences for GUI filtering

have been set to display: /COM, Structural !* /PLOPTS,INFO,3 /PLOPTS,LEG1,1 /PLOPTS,LEG2,1 /PLOPTS,LEG3,1 /PLOPTS,FRAME,1 /PLOPTS,TITLE,1 /PLOPTS,MINM,1 /PLOPTS,FILE,0 /PLOPTS,LOGO,1 /PLOPTS,WINS,1 /PLOPTS,WP,0 /PLOPTS,DATE,2 /TRIAD,LBOT /REPLOT !*

/ PREP7 !! !! !!------------------------STEP 1------- /units,SI MP,DENS,1,2329

MPTEMP,,,,,,,, MPTEMP,1,0 MPDE,NUXY,1 MPDE,NUYZ,1 MPDE,NUXZ,1 MPDE,PRXY,1 MPDE,PRYZ,1 MPDE,PRXZ,1 MPDATA,EX,1,,169E09 MPDATA,EY,1,,169E09 MPDATA,EZ,1,,130E09 MPDATA,PRXY,1,,0.064 MPDATA,PRYZ,1,,0.36 MPDATA,PRXZ,1,,0.28 MPDATA,GXY,1,,50.9E09 MPDATA,GYZ,1,,79.6E09 MPDATA,GXZ,1,,79.6E09 !!! Coating properties !! When running Coat loss codes !- Ta2O5 - !MP,EX,2,140E09 !MP,PRXY,2,0.23 !MP,DENS,2,8200 !--- SiO2 coating-------- !MP,EX,3,70E09 !MP,PRXY,3,0.17 !MP,DENS,3,2200 !!-------------------------STEP 2---- !!!!!!!element types !!element type 1 ET,1,MESH200 KEYOPT,1,1,6 !!!! defined 4 nodes KEYOPT,1,2,0 !!!!

212 C. Appendix 2

!!! Part 2 ! !! Step 1 /sol allsel,all ANTYPE,MODAL EQSLV,SPAR

MODOPT,LANB,5,345000,600000, ,ON !1st number is # modes; 2dn is start frequency; 3rd is stop frequency

MXPAND,,,,ON !no stress calculations if "OFF"

solve finish !! Step 2 /POST1 SET,LAST /VIEW,1,1,1,1 /ANG,1 /REP,FAST !! plot mode shape

SET,NEXT !* !!number of mode, numb /REPLOT,RESIZE !* /EFACET,1 PLNSOL, U,SUM, 0,1.0

C. Appendix 2 215

C.3 Final Remarks

I hope that a future student will not hesitate in contacting me for any questions regarding

the use and modification of these codes.

216 C. Appendix 2

Appendix D

Silicon Micromachining Details

Here I present more details on the fabrication methods used to make the resonators in

this project. This is a supplement to Chapter 3, where these methods were only briefly

described.

D.1 Resonator Fabrication Details

Subsections below describe the details of wet and dry etching, as well as laser microma-

chining and optical coatings applied by ion-beam sputtering.

D.1.1 Wet Etching in WACSOM

I used silicon wafers of 325± 25 μm thickness, p-doped (100) monocrystal, single- and

double-sided polish, with a flat edge indicating the wafer orientation. These products

were readily available in the lab, and suitable to test the wet etching technique. The wet

etching process is summarized in Fig. D.1.

The wafer was cut with diamond tip scriber into 2 cm by 2 cm square samples (see

Fig. D.2). These samples were cleaned using trichlorate, acetone and methanol, and using

an ultrasonic shaker.

Silicon nitride (SiNx) masking layers of ∼ 450 nm thickness were deposited on (see

Fig. D.3) both sides of the samples using Oxford instrument Plasma 80+ ICPCVD. The

deposition conditions are summarised in Table D.1.

Table D.1: SiNx deposition conditions, using the Oxford instrument Plasma 80+ ICPCVD.Sample series Gas chemistry Pressure RF Thickness (nm) Temperature

UW1SiH4 : 5 sccm

85 mTorr 150 W 450 200NH3 : 50 sccmN2 : 100 sccm

217

218 D. Appendix 3

Figure D.1: Steps for wet etching processing: cleaving, cleaning, applying SiNx masklayers, photolithography, RIE to expose silicon, KOH etching, and HF strip to remove themasks.

Figure D.2: Diamond tip scriber to cut the wafer into 2 cm by 2cm square samples.

D. Appendix 3 219

Figure D.3: Prior to KOH etching, the silicon samples are covered in SiNx masking layers.(d) Microscope examination reveals scratches and bubbles formed in the masking layers.

These layers protect selected portions of the silicon substrate from the wet etching

solution in the final step. The SiNx layer of 450 nm was suitable to survive wet etching

through 300 μm of silicon substrate, according to etch rates.

A layer of PR-AZ2035 negative photoresist is spin-coated (Fig. D.4 to the masked

samples, ready for photolithography with ultraviolet (UV) light.

Figure D.4: Photolithography. (a) Hood for spin-coating of AZ2035 negative photoresist.(b) UV exposure machine.

A pattern corresponding to the desired resonator pattern is printed on a glass plate

(see Fig. D.5), equivalent to the negative film of a camera.

The UV light is filtered through the patterned glass plate in order to shine UV light for

15 seconds on selected portions of the negative photoresist. A solution of AZ326 developer

is used to develop the UV-exposed photoresist.

The exposed SiNx mask is then removed by reactive ion etching (RIE) in a PLASMA

220 D. Appendix 3

Figure D.5: Glass plate with printed pattern for UV exposure during photolithography.

D. Appendix 3 221

100+ chamber. The conditions for RIE are summarised in Table D.2.

Table D.2: SiNx removal conditions with RIE, using the Oxford instrument Plasma 100+.Sample series Gas chemistry Pressure RF ICP Temperature

UW1O2 : 10 sccm

80 mTorr 75 W 100 W 20CF4 : 10 sccm

Selected areas of the silicon substrate are now exposed, ready for wet etching with

KOH.

A KOH solution is prepared by mixing 40 g of KOH pellets with 60 ml of DI water and

10 ml of isopropanol. The beaker with KOH is placed on a hot plate with a thermometer

to control the temperature of the solution. It is kept at 80 , and a small magnetic stirrer

(about 3 cm long) is spinning at a speed of 200 rpm. This spinning creates turbulence

in the solution, and ensures that small silicon particles are dislodged during wet etching.

Once the etching is done through the entire thickness of the samples, they are removed,

rinsed in DI water, dried and examined with a stylus surface profilometer Veeco Dektak

150. The remaining SiNx mask layers are removed by dipping the samples in a HF solution,

for 10 minutes. This removes the masks without damaging the silicon substrate.

This method has many drawbacks. The SiNx masks that I made were not suitable and

small pinholes and cracks caused exposure of the silicon substrate which are not intended

for etching. Furthermore, there is undercutting, where KOH will go under the mask and

etch the silicon that was meant to be protected.

Having found this method to be time consuming and not showing promise for ame-

lioration within a reasonable time frame for this project, other fabrication options were

considered.

D.1.2 Dry Etching in Queensland

The labs of Australian National Fabrication Facility Ltd (ANFF-Q) at the University

of Queensland were used to produce 3-paddle design resonators using dry etching: an

example of this technique is the Bosch process, or the deep reactive ion etching (DRIE)

[98]. Dry etching is a superior fabrication technique and offers the advantage of straighter

walls, compared to the tapered walls of wet KOH etching, with achievable aspect ratios

of 30.

Three batches of samples were made in Queensland: The UQB1-series from single-side

polished p-type silicon (100) 4 inch diameter wafers of thickness 500 μm; the UQB2-series

from double-side polished wafers and SU8-2025 photoresist resin; and the UQB3-series

222 D. Appendix 3

from double-side polished wafers and using a lift-off resin called OmnicoatTM [116] to

facilitate the removal of SU8-2025 resin, after DRIE.

While different results were obtained from each batch, the methods for fabrication

were similar and are summarised in Fig. D.6.

Figure D.6: Steps for DRIE process in Queensland, using a layer of Omnicoat to strip theSU8 resin after DRIE etching.

A layer of OmnicoatTM is spin-coated on the wafer. This layer allows easy removal

of the SU8-2025 photoresist resin. A total thickness of 40 nm of OmnicoatTM is required

to survive the DRIE through 500 μm of the silicon substrate. A 25μm layer of SU8-2025

is then spin-coated on the sample, and exposed to UV light filtered through a 3-paddle

pattern (see Fig. D.5) printed on a glass plate.

The exposed OmnicoatTM layer, from the developed SU8-2025 pattern, is removed

with reactive ion etching (RIE).

The DRIE process, or Bosch process, is a repetition (between 100 to 1000 iterations)

of two steps: the plasma etch of silicon, using SF6 ions; and deposition of a chemically

inert passivation layer of C4F8 to protect the vertical walls during plasma etching. This

is repeated until the whole wafer is etched through. The DRIE machine is the ‘PLASMA

THERM DRIE’ and was using an RF power of 2000W, giving etch rates of 3μm per

D. Appendix 3 223

minute.

The SU8 is removed by immersing the processed samples in a PG Remover (NMP)

solution, kept at 80 for a duration of 30 to 45 minutes. The OmnicoatTM layer is dissolved

which removes the SU8 layer from the substrate. Otherwise, SU8 resins are known to be

difficult to remove, and residues will remain on the silicon substrate. Excess material on

the resonators could contribute to surface losses and lower acoustic quality factors. This

would reduce the strength of opto-acoustic interactions, for which these resonators are

intended.

D.1.3 Dry Etching and Optical Coating in France

France collaborators participated in the fabrication of optically coated silicon resonators.

This involved superfine polishing of two silicon (100) 365 μm thick wafers in the ‘Societe

Europeenne de Systemes Optiques’ (SESO) in Grenoble, France. The wafers were polished

to a surface roughness of 2 A rms on the front side, and 3 to 10 A rms on the back side.

The polished wafers were sent to the Institute of Electronic, Micro-electronic and

Nanotechnology (IEMN) in France, which coordinated the rest of the process, including

arranging optical coatings in the ‘Laboratoires de Materiaux Avances (LMA), in France,

and DRIE processing in the IEMN labs. Only one wafer was processed for fabrication of

resonators. The other polished wafer is kept at IEMN, until the next run of resonator

fabrication is ready.

First, a PMGI SF19 lift-off resin is applied to the wafer, similar to the OmnicoatTM

layer used in Queensland, with square 0.8 mm by 0.8 mm holes designed to receive optical

coatings. The lift-off resin PMGI SF19 is sensitive to temperature and does not age well.

This forces IEMN and LMA to coordinate their operations in a way which ensures that

(1) once the lift-off resin is applied, it is promptly sent for coating at LMA, and (2) the

coating is performed at low temperatures, (3) the coated wafer is returned promptly to

IEMN to perform lift-off of the resin.

At this point, only the coatings on the small 0.8 mm squares are left on the silicon

wafer. This is sent back to LMA for post-baking of the coatings, which would have

otherwise compromised the lift-off resin. The wafer is returned to IEMN for the final

stages of fabrication.

A protective layer is applied to the whole wafer surface, in order to protect the optical

coatings from DRIE process and from cleaving at the end. Otherwise, projectiles could

land on the coatings and ruin their quality.

224 D. Appendix 3

Dry etching is performed to etch the resonator patterns, which are the same designs

A,B,C and D (presented in Chapter 2), with the addition of small square coatings on the

middle paddle.

This is followed by dicing of the wafer into square 2 cm by 2 cm samples. Finally, the

protective layer is removed, and the samples are carefully packaged and sent to the School

of Physics at University of Western Australia.

A summary of this fabrication method is shown in Fig.D.7.

Figure D.7: Steps for DRIE etching and optical coating performed in France (SESO,IEMN, LMA).

The LMA coatings are designed as mirrors for the wavelength of a YAG laser at

1064nm. These coatings are sputtered onto the wafer [112, 100] for a final optical coating

configuration of L(HL)14H2L, where L is a low index of refraction material, in this case

SiO2 (silica), and H is a high index of refraction material, in this case Ta2O5 (tantalum

pentoxide). The coatings were doped with titanium, which decreases the losses from

coatings [100].

The first layer on the silicon substrate is a quarter-wave thickness of silica (SiO2), a

thickness of 184.4 nm, given by Λ / 4 nL, where Λ is the wavelength 1064 nm, and nL

= 1.44, the index of silica. This is followed by 14 pairs of tantalum and silica layers, in

which the tantalum layers are of thickness 128.5nm, given by Λ / 4 nH , where nL = 2.07,

D. Appendix 3 225

the index of tantalum.

An extra layer of tantalum is added after the 14 pairs, and the final top layer of coating

is a double layer of silica (368.8nm). Such a configuration gives a total thickness of 5.08

μm, and a reflectivity 99.997% for a light source of 1064 nm.

Figure D.8: A diagram of the layers in optical coatings, for a configuration of L(HL)NH2L,consisting of ‘N’ pairs, a bottom layer of silica, another layer of tantalum after the N pairs,and finished with a thicker layer of silica.

The reflectivity depends on the thickness. For example, coatings with 10 pairs give a

thickness of 3.3 μm, and a reflectivity of 99.84%. Coatings with 20 pairs give a thickness of

6.3 μm, and a reflectivity of 99.9999%. Finally, Coatings with 30 pairs give a thickness of

9.3 μm, which is beyond what the lift-off resin technique can withstand, and a theoretical

transmissivity of 10−8 is expected. For reference, the labs at LMA have made coatings

with a maximum of 21 layers, for other applications.

D.1.4 Laser machining in Taiwan

Collaborators at the Hsinchu University in Taiwan have access to advanced facilities near

their campus for silicon wafer processing, and they perform studies on silicon cantilevers

which bare resemblance to the research presented in this project. They have organised

laser micromachining of silicon resonators at LEGEND Laser Inc, in Taipei, Taiwan. A

UV laser of 355 nm, 2W power and a beam spot of size 15 μm were used to cut through

226 D. Appendix 3

the wafer.

The fabrication method of laser micromachining [99] consists of focusing the beam

waist along the pattern at intervals, to burn holes through the wafer. A protective film

is applied to the surface before the laser cutting, in order to protect the substrate surface

from laser damage [97]. Once multiple holes are created along the pattern profile line, the

laser beam is focused and swept along the pattern lines, until the pattern is cut all the

way through the wafer. The protective film is then removed.

Taiwan collaborators also performed optical coatings, similarly to the process in France.

The laser machining and coating method is summarised in Fig.D.9.

Figure D.9: Steps for laser machining and optical coating of samples made in Taiwan.

This is a very simple method of fabrication, and requires less chemicals and expensive

machinery. The drawback is that the pattern is cut in a way which creates undulating lines

and scorch marks from laser damage to the substrate [97]. Despite these imperfections,

this method produced samples of suitable quality in a cost-effective and prompt manner.

Here is a description of the experimental setups designed to measure the acoustic

spectrum and quality factors of resonators fabricated in this research. These methods

D. Appendix 3 227

were not suitable to produce very high quality factors. They were stepping stones along

the path to the 3-point suspension setup which produced the high quality-factor results

reported in this thesis.These preliminary methods are discussed here for the purpose of

future PhD students who would wish to attempt similar methods, to provide information

on attempts with these setups.

Note that the basic optical lever method which is common to all attempted methods is

discussed in Chapter 3, along with the 3-point suspension method which proved the more

useful method in obtaining high acoustic quality-factors from fabricated resonators.