High Optical Power Experiments and Parametric Instability ...

High Optical Power Experimentsand Parametric Instability in 80

m Fabry-Perot Cavities

Qi Fang

Submitted in partial fulfilment of the

requirements for the degree of

Doctor of Philosophy

of The University of Western Australia.

School of Physics

2015

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To my parents and my wife:

Shangnian Li, Zuoping Fang and Qi Chu

who give me every piece of their loves

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Abstract

Gravitational waves were predicted almost 100 years ago. Detecting gravita-

tional waves is extremely difficult due to their extraordinarily weak interactions

with matter. Aiming for the first detection of gravitational waves, several large-

scale elaborate laser interferometers have been constructed. However, no direct

detection of gravitational waves has been claimed so far. Advanced gravitational-

wave laser interferometers, such as aLIGO, have achieved strain sensitivity of

∼10−24 Hz−1/2 already. With such sensitive devices, a range of detection rates

between 0.4 to 400 events per year is predicted for binary neutron-star coales-

cences.

In order to reduce the shot noise, which limits the sensitivity at high fre-

quencies in advanced laser interferometers, high optical laser power is required.

However, optomechanical interactions become significant at this high optical

power. As a result, parametric instability, which is a nonlinear optomechanical

effect, may occur. Parametric instability at high amplitudes could saturate the

photodetectors, and thus deteriorate the cavity locking. One goal of my project

is to observe parametric instability, and find approaches to suppress it, in the

High Optical Power Test Facility at Gingin, Western Australia.

An overview of gravitational waves and gravitational-wave detectors, espe-

cially laser interferometers, is given in Chapter 1. In Chapter 2, a theoreti-

cal background of optomechanical interactions and parametric instability is re-

viewed. The Gingin Facility, currently composed of two separate Fabry-Perot

cavities (the South arm and the East arm), is briefly introduced in Chapter 3.

All the experiments presented in this thesis were undertaken in this Facility.

Three-mode optomechanical interactions are the basis of parametric insta-

bility. To investigate the properties of three-mode optomechanical interactions,

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an experiment was conducted in the South arm of the Gingin facility. In this

experiment, mechanical vibrations modulated the fundamental cavity mode and

created two sidebands. The cavity was tuned by CO2 laser heating, such that one

sideband resonated as a first-order cavity mode. As a result, a high-sensitivity

transducer was realized by this strong coupling between cavity modes and the

test mass mechanical mode. This three-mode optomechanical transducer has

the advantage of being immune to laser amplitude and phase noises. This work

is presented in Chapter 4.

In order to observe parametric instability, some components were upgraded

in the East arm of the Gingin facility. Specifically, a 50 W laser source and

low-optical-loss fused silica test masses were installed, to achieve high optical

power. The cavity needed to be characterized prior to starting the parametric

instability experiment. In Chapter 5, the cavity length, finesse and transverse

mode frequency offset were investigated. The cavity was tuned by using a CO2

laser to heat the input test mass. Cavity mode degeneracies were observed

through this thermal tuning, which could account for dropouts of the cavity

finesse.

The test masses in the East arm are hung from a niobium modular suspen-

sion system, which introduces some mechanical losses to the test masses. In

Chapter 6, ring-down measurements of the input test mass were conducted to

determine its quality-factors in different mechanical modes. In order to estimate

the loss angle contributed by the suspension, an ANSYS simulation was created

to calculate the strain energies of the substrate, coating and suspension holes.

The results showed that the suspension contributed only ∼10% of the total ther-

mal noise at 100 Hz, when a Gaussian beam was injected onto the test mass.

A calculation was also undertaken assuming the same suspension system was

installed in aLIGO. A similar result was deduced — that no more than 15% of

the thermal noise was contributed by this suspension system at 100 Hz.

After characterizing the optical and mechanical properties of the cavity, an

experiment to observe parametric instability was conducted. In Chapter 7, an

appropriate mechanical mode with a large spatial overlap with the first-order

cavity mode was selected. A ring-up in amplitude of this mechanical mode was

observed. We demonstrated that the figure errors of the test masses caused a

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dependence of the transverse mode frequency offset on the laser spot position.

Due to residual angular motions of the test masses at low frequencies, and these

figure errors, the transverse mode frequency offset was modulated. The effective

parametric gain was suppressed by this dynamical modulation mechanism. In

addition, by applying thermal modeling to aLIGO test mass data, we showed

that the parametric gain could be suppressed by factors of 10-20 for individual

modes. This suppression scheme could greatly relieve the parametric instability

problem in aLIGO.

One problem during these experiments was that the locking of the laser was

often lost, due to a low-frequency drift of the optical path. This low-frequency

drift was caused by microseismic vibrations from the ground, or thermal expan-

sions of the optical components. Relocking had to be done manually. In order to

avoid losing laser lock, an Auto-Alignment system was investigated in Chapter

8. This system was installed and preliminarily tested in the East arm of our test

facility.

Conclusions and suggestions for future work are summarized in Chapter 9.

In particular, our newly designed suspension system is shown to make only a

minor contribution to the test mass thermal noise. In addition, this thesis has

demonstrated the importance of three-mode optomechanical interactions and

parametric instability in gravitational-wave laser interferometers — an interest-

ing finding is the strong suppression of parametric instability by a dynamical

modulation from the test mass residual angular motions.

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Acknowledgments

Firstly, I would like to express my sincere gratitude to my supervisors: David

G. Blair, Chunnong Zhao and Li Ju, who constantly gave me support, help and

advice. Thanks to David for your guidance over the years, which led me through

my research. I am always impressed by your insightful perspective on every area

of physics. To Zhao, I would like to give my appreciation to your attitude toward

research and life. I remember every moment when we discussed physics on the

way to and from Gingin. I learned from you that no problem is unsolvable. To

Ju, you are like a family member to me. You always gave me help and positive

feedback whenever I needed them. It is you who makes our research group like

a big family.

I would also like to give my appreciation to my colleagues at Gingin. To

Carl Blair, we spent every work day together for two years. You have a great

understanding of experiments and can fix everything. I wish I could spend more

time with you. To Yaohui Fan, you are like an elder brother to me. You know

every aspect of our Gingin facility, from optics to the control system. You led

me through optical experiments at the beginning of my PhD. To Jean-Charles

Dumas, Andrew Wooley and Sunil Susmithan, I appreciate the time working

with you and all the effort you put in the work. You made my lab work enjoyable.

Also thanks to Mark Dickinson for making our lab run smoothly. Special thanks

are due to David Hosken and Nick Chang from Adelaide for building the high

power lasers for us.

I must express my thankfulness to Haixing Miao, who is my mentor in study-

ing the theory of optomechanics, and for being a nice person. I am also grateful

to my colleagues and friends in my group. Thanks to Yiqiu Ma for organizing

the weekly journal club, from which I learned a lot about optomechanics and

quantum physics. Thanks to Xu Chen, Shinkee Chong, Stefan Danilishin, Jian

Liu, Yubo Ma, Jiayi Qin, Andrew Sunderland, Shenghua Yu, Xingjiang Zhu and

others for sharing their numerous interesting ideas, and for coloring my PhD life.

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I would like to address my special thanks to Ruby Chan, who always smiles

and shares her wonderful life experience with me. I would like to salute Ian

McArthur, Paul Abbott and Jay Jay for running a fantastic School of Physics.

I am very grateful to Andre Fletcher who proofread my thesis with great

patience and carefulness.

To my best friends: Kai Zhao, Jing Xu, Lei Yin, Ran Wei and Yiping Shu, I

give my sincere thanks. You make me believe that I will never feel lonely in the

world.

Last but not least, I must give my sincerest thankfulness to my family: my

parents, my wife, my parents-in-law and my little daughter. Without your love

and support, this thesis would never have become true.

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Abbreviations and Acronyms

aLIGO — Advanced LIGO

aVIRGO — Advanced VIRGO

AA — Automatic Alignment

AR — Anti-Reflection

BS — Beam Splitter

CCD — Charge-Coupled Device

CP — Compensation Plate

DSP — Digital Signal Processor

EOM — Electro-Optic Modulator

ETM — End Test Mass

FEM — Finite Element Method

FFT — Fast Fourier Transform

FG — Function Generator

FP — Fabry-Perot

FW — Forward Wave

FWHM — Full Width at Half Maximum

HOM — High-order Optical Mode

HOPTF — High Optical Power Test Facility

HWS — Hartmann Wave-front Sensing

ITM — Input Test Mass

KAGRA — Kamioka Gravitational Wave Detector

LED — Light-emitting Diode

LIGO — Laser Interferometric Gravitational-wave Observatory

Nd:YAG — Neodymium-doped Yttrium Aluminium Garnet

NPRO — Non-Planar Ring Oscillator

OAPA — Opto-Acoustic Parametric Amplifier

PBS — Polarized Beam Splitter

PD — Photodetector

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PDH locking — Pound-Drever-Hall locking

PI — Parametric Instability

PRM — Power Recycling Mirror

PZT — Piezo-electric Transducer

QPD — Quadrant Photodetector

RoC — Radius of Curvature

RF — Radio Frequency

RMS — Root-Mean-Square

RW — Reverse Wave

SA — Spectrum Analyzer

SQL — Standard Quantum Limit

SRM — Signal Recycling Mirror

TMFO — Transverse Mode Frequency Offset

UWA — The University of Western Australia

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Contents

Contents xiii

List of Figures xvii

List of Tables xxix

1 Introduction 1

1.1 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Generation of Gravitational Waves . . . . . . . . . . . . 2

1.1.2 Propagation of Gravitational Waves . . . . . . . . . . . . 4

1.1.3 Polarization of Gravitational Waves . . . . . . . . . . . . 5

1.2 Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Configurations of Laser Interferometers . . . . . . . . . . 9

1.2.2 Noise Budget of aLIGO . . . . . . . . . . . . . . . . . . 14

1.2.3 Nonlinear Effects and Instabilities . . . . . . . . . . . . . 17

1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Theoretical Background 23

2.1 Gaussian Beam in Fabry-Perot Cavities . . . . . . . . . . . . . . 23

2.1.1 Hermite-Gaussian Modes . . . . . . . . . . . . . . . . . . 23

2.1.2 Resonance in Fabry-Perot Cavities . . . . . . . . . . . . 26

2.2 Optomechanical Interactions . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Cavity Response . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Optical Spring Effect . . . . . . . . . . . . . . . . . . . . 31

2.2.3 Quantum Noises and Standard Quantum Limit . . . . . 31

2.3 Parametric Instability . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.1 Three-mode Parametric Interaction . . . . . . . . . . . . 37

2.3.2 Derivation of Parametric Instability . . . . . . . . . . . . 38

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3 Gingin Test Facility 45

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 The South Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Laser Source . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.2 Injection Table and Cavity . . . . . . . . . . . . . . . . . 48

3.2.3 PDH Locking System . . . . . . . . . . . . . . . . . . . . 48

3.3 The East Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 50 W Fiber Laser Amplifier . . . . . . . . . . . . . . . . 51

3.3.2 Injection Table and Cavity . . . . . . . . . . . . . . . . . 52

3.3.3 Advanced Vibration Isolation System . . . . . . . . . . . 54

3.3.4 Feedback Control System . . . . . . . . . . . . . . . . . . 57

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Three-mode Optomechanical Transducer 63

4.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Cavity Degeneracy Losses 75

5.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Characterization Results . . . . . . . . . . . . . . . . . . . . . . 84

5.5 Degeneracy Results . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Test mass Mechanical Loss 95

6.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Model for Test Mass Mechanical Loss . . . . . . . . . . . . . . . 98

6.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5.1 Measurement Results . . . . . . . . . . . . . . . . . . . . 103

6.5.2 Simulation of Mechanical Modes . . . . . . . . . . . . . . 103

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6.5.3 Simulation with a Gaussian Radiation Pressure

Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . 108

6.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.8 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.9 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7 Observation and Suppression of PI 113

7.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.3 Theory of PI and Frequency Modulation . . . . . . . . . . . . . 117

7.4 Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . 122

7.5 High Optical Power Cavity Observations . . . . . . . . . . . . . 125

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 130

8 Automatic Alignment 133

8.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.3.1 Misalignment of an Optical Cavity . . . . . . . . . . . . 135

8.3.2 Obtaining the Misalignment Signals . . . . . . . . . . . . 139

8.3.3 Misalignment Signals in the East Arm Cavity . . . . . . 143

8.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.5 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.5.1 Mode-matching . . . . . . . . . . . . . . . . . . . . . . . 148

8.5.2 Misalignment Signals . . . . . . . . . . . . . . . . . . . . 150

8.6 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . 152

9 Conclusions and Future Work 153

9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Bibliography 157

Bibliography 157

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List of Figures

1.1 Conceptual images of some gravitational wave sources. The Big

Bang (left) [7]; a supernova explosion (right) [8]. . . . . . . . . . 2

1.2 Decaying period P of a binary neutron star system. A set of

cumulative values of the time at periastron for the binary system

PSR B1913+16 was plotted versus observation year (left) [11]. A

shift of the cumulative time is shown clearly in this Figure. A set

of conceptual images of an orbiting binary star system (right) [12]. 4

1.3 A set of points which are initially arranged as a circle in the

x-y plane will experience distortion when a gravitational wave

passes by. The distance between each mass in the circle and

the central mass can be illustrated by introducing new coordi-

nates (X(t), Y (t)), where X(t) = (1 + 12a sin Ωt)x and Y (t) =

(1 − 12a sin Ωt)y. The time-varying position of each mass in the

circle is described by its position vector from the central mass

at the origin of the (x, y) coordinate system. The time interval

between each figure is 18T , where T = 2π

Ωis the period of the am-

plitude oscillation of the gravitational wave. Figure reproduced

from Hartle’s text book [9]. . . . . . . . . . . . . . . . . . . . . 6

1.4 Conceptual and real pictures of some gravitational-wave detec-

tors. 1. Pulsar timing arrays [31]. 2. LISA [32]. 3. DECIGO

[33]. 4. aVIRGO [34]. 5. NAUTILUS [35]. . . . . . . . . . . . . 9

1.5 A conventional laser interferometer with its major subsystems. . 10

1.6 Response of an interferometer to a gravitational wave travel-

ing perpendicularly to the detector plane. The intensity signal

recorded by the photodetector has twice the frequency of the

gravitational wave. . . . . . . . . . . . . . . . . . . . . . . . . . 11

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1.7 Modifications to a simple Michelson-Morley interferometer. An

Fabry-Perot cavity is formed in each arm by adding an input test

mass in the path (left). A power recycling mirror and signal recy-

cling mirror are further added in order to increase the intracavity

power and the signal amplitude, respectively (right). All these

added mirrors need to be suspended. . . . . . . . . . . . . . . . 14

1.8 Expected noise contributions from various different sources. This

Figure was generated from the GWING software. Figure adapted

from Adhikari (2014) [41]. . . . . . . . . . . . . . . . . . . . . . 15

1.9 The thermal lensing effect is induced when an optical beam power

is absorbed by the test mass. The laser heating the test mass

normally has a Gaussian beam profile. Figure reproduced from

Fan (2010) [58]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.10 PI effects observed. PI was observed at 65 kHz in aLIGO (left).

The color in this Figure indicates the mechanical amplitude at 65

kHz [67]. An increase of a mechanical mode amplitude was also

observed in a small-scale device (right). The amplitude reaches a

saturation after an exponential ring-up, which is consistent with

simulations [68]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Simulated intensity of some lowest-order Hermite-Gaussian modes

on a mirror surface. The scales of the beam and the mirror are

proportional to that of the ITMs in aLIGO. . . . . . . . . . . . 25

2.2 Lateral view of a Gaussian beam. Figure reproduced from Chen

(2014) [79]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 A Gaussian beam mode-matched to a Fabry − Perot cavity. . . 27

2.4 Schematic diagram of Optomechanical Interaction. . . . . . . . . 28

2.5 The effective resonance frequency (left) and the effective damping

rate (right) in the optical spring effect. The parameters used

in plotting these two figures are: m=5.8 g, γ=150 kHz, L=74

m, Ωm=600 kHz, γm=6 Hz, ∆ = −Ωm. The intracavity power

I0 = 109 for the blue solid line, I0 = 7 × 108 for the red dashed

line, and I0 = 3 × 108 for the black dot-dashed line. . . . . . . . 32

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2.6 Power spectral densities of quantum shot noise, quantum radia-

tion noise, total quantum noise, and the standard quantum limit

(left) [81]; strain spectra of quantum noises in different interfer-

ometer setups (right) [42]. In the right-hand figure, the red line

denotes the quantum noise in a simple Michelson interferometer,

which is dominated by shot noise (N.B. right-hand plot is of the

strain spectra, which has different units). The brown curve de-

notes the total quantum noise of an interferometer with FP cavi-

ties, where the lowest noise floor is 600 times smaller than that in

a simple Michelson detector. The noise spectrum can be further

reduced by adding power recycling mirrors, as in the cyan curve,

where the frequency of minimum noise shifts to ∼100 Hz. The

bandwidth of this quantum noise trough can be broadened from

∼200 Hz to ∼1 kHz by adding signal recycling mirrors, which is

the blue curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Two sidebands of the light field are generated by a mechanical

vibration (top). A phonon is created in the Stokes process (bot-

tom left), and a phonon is absorbed in the anti-Stokes process

(bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8 Transverse mode frequencies of a near-concentric cavity. The

first-order transverse mode of the qth longitudinal group has a

frequency slightly less than that of the fundamental transverse

mode of the (q + 1)th longitudinal group. The frequency differ-

ence between them is given by Eq. (2.16). . . . . . . . . . . . . 38

2.9 Feedback loop diagram of parametric instability. A higher-order

cavity mode (TEM11 in this diagram) is scattered by the me-

chanical vibration and resonates in the cavity. The beating note

between the fundamental optical mode and the higher-order opti-

cal mode excites the mechanical mode to produce more scattering

of the light field. Figure reproduced from Adhikari (2014) [41]. . 39

3.1 The HOPTF, located in a bush field at Gingin, Western Australia.

Figure provided by Li Ju. . . . . . . . . . . . . . . . . . . . . . 46

3.2 Conceptual diagram of a complete configuration of the Gingin

HOPTF. Currently the two recycling tanks are not installed. Fig-

ure provided by Li Ju. . . . . . . . . . . . . . . . . . . . . . . . 46

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3.3 Laser source system used in the South arm. A commercial 50

mW NPRO laser (left) and a homemade 10 W slave laser (right)

form a master-slave laser system. . . . . . . . . . . . . . . . . . 47

3.4 The optical layout of an injection-locked slave laser. The output

from the NPRO master laser is linearly polarized, and is mode-

matched by a combination of optical elements, such as lenses,

half-wave plates (λ/2), and a quarter-wave plate (λ/4). The re-

verse beam propagating back to the NPRO laser is blocked by a

Faraday Isolator (FI) since the polarization of the beam injecting

into the FI can be tuned by the half-wave plates. The slave laser

is locked to the master laser by a PDH locking scheme, which in-

volves a phase modulation by an ElectroOptic Modulator (EOM).

The Forward-Wave (FW) photodetector is there to collect the er-

ror signal from the PDH locking, while the Reverse-Wave (RW)

photodetector is used to monitor the output of the slave laser.

Figure reproduced from Fan (2010) [58]. . . . . . . . . . . . . . 49

3.5 Setup of the injection table in the HOPTF South arm injection

room. Figure reproduced from Fan (2010) [58]. . . . . . . . . . . 50

3.6 Pound-Drever-Hall (PDH) locking system. . . . . . . . . . . . . 50

3.7 The 50 W fiber laser amplifier is put on the East arm injection

table (left). The output power as a function of the diode current

was measured (right) (provided by David Hosken). . . . . . . . . 52

3.8 The schematic setup on the East arm injection table before im-

plementing the Automatic Alignment (AA) system (see Chap. 8).

The laser is amplified by a fiber laser amplifier, which is not shown

in this Figure. The cavity is not drawn to scale. . . . . . . . . . 53

3.9 Advanced vibration isolation system used in the HOPTF East

arm. All the isolation stages are labeled in this Figure. Figure

reproduced from Dumas (2009) [88]. . . . . . . . . . . . . . . . . 55

3.10 Pre-isolator (left) and Roberts linkage (right). A pre-isolator con-

sists of an inverse pendulum and a LaCoste linkage. The inverse

pendulum moves in the x direction, while the LaCoste linkage

has a degree of freedom in the z direction. Figures adapted from

Dumas (2009) [88]. . . . . . . . . . . . . . . . . . . . . . . . . . 56

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3.11 Euler springs (left) and self-damping pendulum (right). There are

three stages of Euler springs in a vibration isolation system. The

relative motion between the rocker mass and the pendulum frame

is damped through induced eddy currents. Figures reproduced

from Dumas (2009) [88]. . . . . . . . . . . . . . . . . . . . . . . 57

3.12 Shadow sensor (left) and actuator (right) used in the feedback

control system. Figures reproduced from Dumas (2009) [88]. . . 58

3.13 Interface of the digital control system. . . . . . . . . . . . . . . 58

3.14 5 pairs of shadow sensors used for controlling the test mass. 3

pairs are sensing and actuating in the horizontal and yaw direc-

tions (left), and 2 pairs are for the vertical and pitch directions

(right). Figures adapted from Dumas (2009) [88]. . . . . . . . . 59

3.15 Schematic setup of the optical lever sensing system. Figure re-

produced from Dumas (2009) [88]. . . . . . . . . . . . . . . . . . 60

4.1 (a) The scheme for three-mode interactions. The anti-Stokes

mode ω1 has a similar shape to the internal acoustic mode. (b)

The shapes of the TEM00 and TEM01 modes. The TEM00 mode

has a symmetric shape, while the TEM01 mode is antisymmetric.

(c) The frequency structure of the two optical modes and the side-

bands. The laser is tuned to the fundamental TEM00 mode with

frequency ω0. The mechanical motion induces two sidebands, one

of which is scattered into a high-order optical mode at ω1, which

gets amplified. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 (Color online) The displacement noise spectrum density. (a) The

blue dashed line is the spectrum of the quantum noise ∆xq(Ω);

the red solid line is the thermal noise ∆xth(Ω). . . . . . . . . . . 67

xxi

4.3 Plot showing the experimental setup with three-mode interaction.

The Fabry-Perot cavity consists of two sapphire test-mass mirrors

separated by 77 meters (ITM: input test mass; ETM: end test

mass). The radius of curvature (RoC) of the ETM is thermally

tuned by a CO2 laser, enabling the frequency gap between the

TEM00 and TEM01 modes to be tuned to the mechanical-mode

frequency. This coherently amplifies the mechanical sideband sig-

nal, which is then detected by a quadrant photodetector (QPD).

The apparatus also contains a thermally tunable compensation

plate (CP) that enables changing the effective RoC of the ITM. 69

4.4 Plot showing the experimental result of the measured signal. The

blue dots are the measured data, fitted by a Lorentzian curve

(solid red line). The brown dash-dotted line is the calculated

thermal noise spectrum; the green short-dashed line is the calcu-

lated shot noise; and the red long-dashed line is the sum of the

thermal noise and the shot noise. . . . . . . . . . . . . . . . . . 71

4.5 (Color online) Finite element model of the mechanical mode at

181.6 kHz. (a) 3D mode profile of the surface of the test mass. (b)

Mode shape cross-section distribution along the central Y direction. 72

4.6 Plot showing calculated quantum noise limited sensitivity for the

same mechanical parameters, and with the optical parameters

listed in Table 4.2. The green dashed line shows the shot noise;

the brown dash-dotted line shows the quantum radiation pressure

noise; and the red long-dashed line shows the sum of these two

noise contributions. . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1 Setup of the high finesse HOPTF cavity. The two fused silica

test mirrors have transmissivities of 200 ppm (ITM) and 20 ppm

(ETM), respectively. The quadrant photodetector (QPD) at the

output port is for monitoring the beat note signals between the

TEM00 and higher order modes. The lens L1 is for adjusting the

beam size on the QPD. The focal length of L1 is 100 mm. . . . 79

xxii

5.2 The size of the fundamental mode resonating inside the cavity.

D denotes the distance between the waist position and the ITM,

which can be calculated according to the cavity configuration.

The radii of curvature of the ITM and ETM are R1 = 37.4 m and

R2 = 37.3 m, respectively. . . . . . . . . . . . . . . . . . . . . . 80

5.3 Finesse in degeneracy, as a function of the coupling rate A at

different Lnm. The green, blue, red and black lines represent the

finesse curves when Lnm = L00, Lnm = 2L00, Lnm = 3L00, and

Lnm = 5L00, respectively. . . . . . . . . . . . . . . . . . . . . . . 84

5.4 (a) Response function of the PDH error signal versus frequency

deviation measured at fFSR; the FWHM of the fitted curve is

163.6 ± 5.5 Hz. (b) Transmitted light from the cavity measured

at fFSR. The blue dots are the measurement data, the red curve

is fitted. The FWHM of the curve is 138± 13 Hz. Fig. 5.4(b) has

been scaled to have the same frequency range as Fig. 5.4(a). . . 85

5.5 Ring-down curve of the transmitted light from a resonant TEM00

mode; the reference time ‘0’ is the point at which the laser is

turned off. The blue dots are the time series data, the red curve

is an exponential ring-down fit. . . . . . . . . . . . . . . . . . . 87

5.6 Power spectral density plot of the transmission signal measured

by the QPD at the transmission port. Two peaks are measured

with a frequency difference δf1=(3.882 ± 0.017) kHz. Blue dots

are measured data, red curve is fitted. (color online) . . . . . . . 87

5.7 Measurement of the TMFO ∆f versus the CO2 laser output

power. The power values are given as percentages of the max-

imum output power (10 W) of the CO2 laser. The standard devi-

ations of the frequency measurements are plotted as uncertainty

bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.8 An image of a TEM11,6 Hermite-Gaussian mode degenerate with

the fundamental mode, with 5% (0.5 W) of the maximum CO2

laser power heating the ITM. . . . . . . . . . . . . . . . . . . . 89

5.9 Finesses at different TMFOs are determined by the time decay

constant in the ring-down measurements. Each red cross corre-

sponds to a ring-down measurement. The TMFOs were measured

from transmission spectra. They were changed by tuning the CO2

laser power heating the ITM. (color online) . . . . . . . . . . . . 90

xxiii

5.10 Finesse as a function of the coupling rate for each degenerate

mode indicated in Fig. 5.9. The blue dashed lines and red solid

lines represent the most symmetric mode and the most asymmet-

ric mode, respectively, in the corresponding order. (color online) 91

5.11 Finesse as a function of the the total loss in the degenerate HOM.

A coupling rate of 7 × 10−4 is assumed for this curve. . . . . . 93

6.1 A replaceable suspension system. (a) A conceptual plot of the

holes and niobium sticks on one of the two flat sides of a test mass.

(b) Cross-section of a niobium stick suspended by a niobium wire.

(c) A microscopic image of a peg with its flat top, which is used

for contacting and supporting the test mass. . . . . . . . . . . 98

6.2 A schematic of the experimental setup. The donger is for exciting

the test mass mechanical modes. The monitoring laser beam is

reflected from the inner surface of the ITM and received by a

quadrant photodiode (QPD). Only the modes of the ITM were

measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 A typical result of spectrum and ring-down measurement. (a)

Spectrum of a mechanical mode at 42.938 kHz. (b) Ring-down

curve of the same mechanical mode. The local oscillator frequency

used to beat with the mechanical signal in the spectrum analyzer

was set to be ∼0.45 Hz higher than the mechanical frequency. . 103

6.4 Simulation pictures of the test mass with a radiation pressure of

Gaussian profile applied on the coating surface. (a) Total defor-

mation. (b) Strain energy. . . . . . . . . . . . . . . . . . . . . . 107

6.5 Total deformations of mechanical modes whose quality factors

were measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.6 Strain energies of mechanical modes whose quality factors were

measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.1 Acoustic mode amplitude ring-up curves for various detuning am-

plitudes. Here, we assume maximum gain Rmax=6, an acoustic

mode ring-down time τ=6 s, and a dynamic modulation frequency

of 0.1 Hz. For comparison, cases for on resonance (a=0) with

Rmax=6 and Rmax=1.45 are also plotted. . . . . . . . . . . . . . 119

7.2 Effective parametric gain suppression as a function of dynamic

detuning amplitude a. . . . . . . . . . . . . . . . . . . . . . . . 120

xxiv

7.3 As the dynamic detuning frequency increases, the acoustic mode

amplitude excursions are reduced but the effective parametric

gain is unaltered. Here, three detuning frequencies 1 Hz, 0.1 Hz

and 0.01 Hz are shown. The detuning amplitude is fixed at a = 5

(Rmax = 6 and τ = 6s). . . . . . . . . . . . . . . . . . . . . . . 121

7.4 Dynamic detuning frequency limit to prevent amplitude excur-

sions exceeding a predetermined value β. For example, for Rmax =

10 and τ = 6 s, with an amplitude growth requirement of β = 2,

then the minimum dynamic detuning frequencies are limited to

between 0.6 Hz and 0.1 Hz for detuning amplitudes a in the range

of 2 through 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.5 a) Typical aLIGO test mass figure errors (compared to a perfect

sphere of RoC ∼2242 m) showing deformations across a mirror

diameter. b) FFT code model for frequency offset as a function

of test mass angular motion. . . . . . . . . . . . . . . . . . . . . 123

7.6 The maximum thermal deformation when 0.1 Hz sinusoidal heat-

ing power of amplitude 2 W is applied on the front surface of the

test mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.7 Schematic diagram of the experimental setup: the laser light from

a seed laser is amplified by a 50 W fibre laser amplifier. The high

optical power laser beam is injected into the 74 m long optical

cavity. The seed laser is frequency-locked to the long cavity us-

ing PDH locking. The cavity transmitted beam is detected by

a quadrant photodiode (QPD). The differential signal from the

QPD measures the beating between the cavity fundamental mode

and the first-order mode. . . . . . . . . . . . . . . . . . . . . . . 126

7.8 The test acoustic mode amplitude distribution on the test mass

surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.9 The correlation between horizontal spot position on the ITM

alone and the TMFO indicates that the frequency detuning is

caused by the spot position change. The solid line is a linear fit

to the measurement data. The spread of the data is due to the

fact that the laser spot position is also changing at the ETM. . 128

xxv

7.10 The QPD differential output signal at the test mass acoustic mode

frequency (150.28 kHz). The signal was down-converted to ∼0.91

Hz by mixing with a local oscillator signal. The solid line is a

fitting curve of 0.91 Hz, with parametric gain Rmax = 6 and

detuning amplitude a=2. The growing signal envelope (dashed

line) is consistent with suspension modulation at 0.15 Hz. The

effective parametric gain is ∼1.45. . . . . . . . . . . . . . . . . . 129

8.1 Misalignments of an optical cavity. . . . . . . . . . . . . . . . . 135

8.2 The EOM is used to modulate the injection laser beam. The

sidebands created by this modulation will beat with the first-

order mode electric field leaking from the cavity, and thus generate

differential signals on the QPDs. . . . . . . . . . . . . . . . . . . 139

8.3 Schematic figure of the optical setup of the AA system. The

dashed box represents the optical table. The thick solid lines

in red color represent the optical path of the main beam. The

dashed lines represent the optical path in the AA system. Four

galvanometer scanners are used in the auxiliary centering system.

This figure is not to scale. . . . . . . . . . . . . . . . . . . . . . 145

8.4 Schematic figure (left) and pictures (right) of the electronic setup.

The bottom right picture is a home-made low-pass filter designed

for obtaining low frequency signals. The arrows in the schematic

figure show the directions of the signal flow. . . . . . . . . . . . 147

8.5 Simulation interface for the near-field mode-matching. The resul-

tant Guoy phase is close to 0, while the beam size is only 0.9 mm,

at the position of 48000 mm. . . . . . . . . . . . . . . . . . . . . 149

8.6 Simulation interface for the far-field mode-matching. The resul-

tant Guoy phase is close to π/2, while the beam size is also 0.9

mm, at the position of 46600 mm. . . . . . . . . . . . . . . . . . 150

xxvi

8.7 Testing of misalignment signals. Four degrees of freedom were

all tested. (1) In the top left figure, the ITM pitch mode was

excited — the lower periodic curve in this figure is the ITMX

signal from the A/S. (2) In the top right figure, the upper curve

is the ITMY signal. (3) In the bottom left figure, the upper curve

is the ETMX signal. (4) In the bottom right figure, the lower

curve is the ETMY signal. In all 4 degrees of freedom, only one

channel shows periodic motion with the same frequency as the

excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xxvii

xxviii

List of Tables

1.1 A comparison of different types of gravitational-wave detectors. 8

4.1 Parameters of the two test masses used in the current experiment. 69

4.2 Optical parameters for a future experiment to be undertaken in

the East arm of the Gingin facility. The cavity finesse is an order

of magnitude larger than that in the current experiment. Other

parameters are all reasonable. . . . . . . . . . . . . . . . . . . . 73

5.1 Some cavity parameters of gravitational-wave detectors and HOPTF.

DRFPMI stands for Dual-Recycled Fabry-Perot Michelson Inter-

ferometer, DRMI stands for Dual-Recycled Michelson Interfer-

ometer, FPC stands for Fabry-Perot Cavity. GEO600 has no

Fabry-Perot configuration for its arm cavity. . . . . . . . . . . . 78

5.2 Minimum and maximum clipping losses from the 11th to 20th-

order transverse modes. . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Four observed dropouts of finesse compared with the nearest de-

generacies. In this Table, fa represents the TMFO at which the

corresponding finesse dropout was observed, fb represents the

nearest TMFO at which a degeneracy takes place, and Nd is the

order of the nearest degenerate mode. The finesse and the fre-

quency difference |fa − fb| are also listed. . . . . . . . . . . . . . 90

6.1 Measured mechanical frequencies and quality factors. The letter

“B” signifies that the frequency was measured by the birefrin-

gence method, while “D” indicates the direct reflection method.

Some modes can be measured using both methods. The vacuum

pressure during the measurements was well kept below 10−5 mbar.

However, we did not test if gas damping can be excluded. . . . . 104

xxix

6.2 Measurement of the quality factor with the mode frequency, to-

gether with the simulation results. fm and fs denote the mea-

surement and simulation frequencies of the mechanical modes,

respectively. The simulation results of the total deformations and

strain energies of these modes are given in the Appendix. . . . . 106

6.3 Coefficients calculated from the test mass simulation with radia-

tion pressure applied. . . . . . . . . . . . . . . . . . . . . . . . . 108

xxx

Chapter 1

Introduction

1.1 Gravitational Waves

100 years ago, Einstein discovered general relativity [1, 2, 3]. In this theory,

space and time are united in a four-dimensional spacetime. A heavy mass curves

spacetime in its vicinity, and other masses in its neighborhood all freely fall

along trajectories which are straight lines — called ‘geodesics’ — in the curved

spacetime. Consequently, general relativity is in fact a theory about geometry.

Innumerable followers have been intrigued to either develop his theory fur-

ther, or test its validity. Much of our understanding of many frontier astronom-

ical phenomena is based on general relativity, such as black holes, neutron stars,

active galactic nuclei, and the Big Bang. In this precious heritage left by Ein-

stein, gravitational waves have been one of the most intriguing of his predictions

[4, 5, 6].

Gravitational waves can be thought of as ripples in the curvature of space-

time, which are generated from, e.g. oscillating masses. Gravitational waves

propagate in spacetime with the speed of light and transport energy along with

them. The curvature of spacetime is disturbed by an incoming gravitational

wave. The directions of this disturbance are determined by the polarizations

of the gravitational wave. The energy emitted by gravitational-wave sources

could be extremely high, but the energy transferred to other objects is almost

1

2 CHAPTER 1. INTRODUCTION

negligible. In general, gravitational waves are everywhere in the universe, and

come from every direction, but have not yet been directly detected.

Figure 1.1: Conceptual images of some gravitational wave sources. The BigBang (left) [7]; a supernova explosion (right) [8].

1.1.1 Generation of Gravitational Waves

Gravitational waves are copiously generated by some violent astrophysical

events, whose masses experience nonaxisymmetric motions, such as in the Big

Bang, and in supernova explosions (Fig. 1.1). Even objects of a few solar masses

can generate a large quantity of gravitational waves. For instance, binary-star

systems, each consisting of a pair of stars orbiting each other around a common

axis, are common sources of gravitational waves.

Mathematically, gravitational waves are solutions of Einstein field equation,

which can hardly be solved analytically due to its high nonlinearity [9]. For grav-

itational waves generated from relativistically varying, strong-curvature sources,

the Einstein equation can only be solved by numerical simulations. However,

a weak source approximation appears to be appropriate in many cases, and

simplifies the Einstein equation dramatically.

The linearized Einstein equation in a weak source approximation is given by:

hαβ = −16πT αβ, (1.1)

where = −∂2/∂t2 + ~∇2, hαβ is the amplitude of gravitational waves, and T αβ

is the stress-energy tensor. Obviously, Eq. (1.1) is a wave equation.

1.1. GRAVITATIONAL WAVES 3

In a long-wavelength approximation, viz.:

λ ≫ Rsource, (1.2)

where λ is the wavelength of the generated gravitational waves, and Rsource is

a characteristic source dimension, the solution of Eq. (1.1) at a large distance

from the source is given by:

hij(t, ~x) −→ 2

rI ij(t − r), (r → ∞), (1.3)

where i, j denote spatial components and a dot denotes a time-derivative; the

quantity

I ij(t) ≡∫

d3xµ(t, ~x)xixj (1.4)

is called the second mass moment, and µ(t, ~x) is called the rest-mass density.

Eq. (1.3) indicates that, at a distance far away from the source, and over a small

range of solid angle, gravitational waves can be considered as plane waves. If we

define another tensor:

J ij ≡ I ij − 1

3δijIk

k , (1.5)

which is called the quadrupole moment tensor, then the luminosity emitted in

gravitational radiation is given by:

LGW =1

5(...J ij

...J

ij). (1.6)

For instance, in a binary star system, where the orbital period is P , the

luminosity of gravitational waves emitted by this system at a distance far away

is given by:

LGW = 1.9 × 1033(M

M⊙

1 hr

P)10/3 erg

s, (1.7)

where M and M⊙ are the total mass of the binary system and the mass of the

Sun, respectively.

Consequently, gravitational waves take energy and angular momentum away

from a binary system. The rate of change of the period P while gravitational

waves are emitted can be expressed as:

dP

dt= −3.4 × 10−12(

M

M⊙

1 hr

P)5/3. (1.8)


This period decay rate, P , was observed for a binary pulsar system PSR

B1913+16 [10]. A decay rate of ∼10 µs per year was predicted for this system.

Observations were well consistent with this prediction (see Fig. 1.2). This is an

important piece of evidence for the existence of gravitational waves.

Figure 1.2: Decaying period P of a binary neutron star system. A set of cu-mulative values of the time at periastron for the binary system PSR B1913+16was plotted versus observation year (left) [11]. A shift of the cumulative time isshown clearly in this Figure. A set of conceptual images of an orbiting binarystar system (right) [12].

1.1.2 Propagation of Gravitational Waves

Like electromagnetic waves, gravitational waves are transverse waves. The

amplitude oscillation plane is perpendicular to the direction of propagation. In

1.1. GRAVITATIONAL WAVES 5

(t, x, y, z) coordinates (in units where c=1), the metric of spacetime perturbed

by linearized gravitational waves can be written as:

gαβ(x) = ηαβ + hαβ(x), (1.9)

where ηαβ is the metric of a flat spacetime, and hαβ is the metric perturbation of

the local spacetime. A plane gravitational wave propagating along the z direction

at the speed of light can be described by the following metric perturbation:

hαβ(t, z) =

0 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 0

f(t − z), (1.10)

where f(t−z) ≪ 1, which describes the amplitude and shape of the gravitational

wave. The curvature of spacetime is perturbed, and the line element with this

perturbation is given by:

ds2 = −dt2 + [1 + f(t − z)]dx2 + [1 − f(t − z)]dy2 + dz2. (1.11)

This plane-wave approximation is an appropriate estimate of gravitational

waves produced in a weak source. For instance, the perturbation amplitude is

only 10−21 if the gravitational waves are generated and emitted from a binary

star system, assuming each of the two stars possesses a mass of M⊙ , the orbital

period P is 1 hour, and the distance from the Earth is 100 pc.

1.1.3 Polarization of Gravitational Waves

Unlike in an electromagnetic wave, polarizations of a gravitational wave are

quadrupole, which can be seen in Eqs. (1.3) and (1.4). Imagine that a gravita-

tional wave propagates along the z direction, with an amplitude function f(t−z)

of harmonic form:

f(t − z) = a sin [Ω(t − z)], (1.12)

where a is a constant, and Ω is the harmonic frequency. Initially, a circle of free

masses are located around a central mass in the x-y plane, as illustrated in Fig.


1.3. When the gravitational wave passes by, the distance between each mass

in the circle and the central mass oscillates with the same frequency Ω as the

gravitational wave, according to Eq. (1.11) and (1.12), while the coordinate of

each mass remains unchanged.

Figure 1.3: A set of points which are initially arranged as a circle in the x-y plane will experience distortion when a gravitational wave passes by. Thedistance between each mass in the circle and the central mass can be illustratedby introducing new coordinates (X(t), Y (t)), where X(t) = (1+ 1

2a sin Ωt)x and

Y (t) = (1 − 12a sin Ωt)y. The time-varying position of each mass in the circle

is described by its position vector from the central mass at the origin of the(x, y) coordinate system. The time interval between each figure is 1

8T , where

T = 2πΩ

is the period of the amplitude oscillation of the gravitational wave.Figure reproduced from Hartle’s text book [9].

The polarization depicted above is called the “+” (plus) polarization. There

is another polarization, which is independent of the “+” polarization, called the

“×” (cross) polarization — this is best described by using (x′, y′) axes, which

are obtained by rotating the (x, y) plane by 45 [9].

In the new coordinates (x′, y′), the element distance is given by:

ds2 = −dt2 + dx′2 + dy′2 + f(t − z)dx′dy′ + f(t − z)dy′dx′ + dz2. (1.13)

1.2. LASER INTERFEROMETERS 7

The metric perturbation for the “×” polarization can be deduced from Eq. (1.13)

as:

hαβ(t, z) =

0 0 0 0

0 0 1 0

0 1 0 0

0 0 0 0

f(t − z). (1.14)

In the case where the polarization of a gravitational wave has a general angle

with the (x, y) coordinate axes, the metric perturbation is expressed as:

hαβ(t, z) =

0 0 0 0

0 f+(t − z) f×(t − z) 0

0 f×(t − z) −f+(t − z) 0

0 0 0 0

, (1.15)

where f+(t − z) and f×(t − z) are the amplitudes of the “+” polarization and

“×” polarization, respectively.

1.2 Laser Interferometric Gravitational-Wave

Detectors

Someone may ask why we have not detected gravitational waves yet, since

they are omnipresent in the universe. The answer is that detection of gravita-

tional waves is extremely difficult, due to their extremely weak interaction with

matter. The situation is quite similar to that with the detection of dark matter:

although the total mass of dark matter is about 5 times larger than the visible

matter [13], the scattering cross-section of dark with visible matter is incredibly

small.

The strategies for gravitational-wave detection can be assigned to two cat-

egories. The first one is to utilize the mechanism that acoustic vibrations of a

heavy mass are induced by couplings to gravitational waves, such as in John

Weber’s bar detector [30]. The second strategy is to take advantage of the effect

that spacetime is disturbed by gravitational waves, and some techniques can be

designed to identify this disturbance, such as using pulsar time arrays, or laser


Type Frequencyband (Hz)

Expectedsources

Projects

Pulsar timing 10−9 − 10−6 Massive blackhole binaries

Nanograv [14],Parkes Pulsar TimingArray [15],European Pulsar TimingArray [16],International Pulsar TimingArray [17]

Spacetransponderinterferometer

10−4 − 0.5 Massive blackhole mergers,compact binarysystems

Evolved Laser InterferometricSpace Antenna (eLISA) [18]

Fabry-Perotspace laserinterferometer

0.1 − 10 Inspiralingsources

Deci-hertz InterferometricGravitational WaveObservatory (DECIGO) [19]

Terrestriallaserinterferometer

10 − 103 Binarysystems,supernovae,accretingneutron stars,relicgravitationalwaves

LIGO and aLIGO [20, 21],VIRGO and aVIRGO [22, 23],Kamioka Gravitational WaveDetector (KAGRA) [24],GEO600 [25]

Acousticdetector

103 − 104 Bursts,supernovae,binary systems,stochasticbackgrounds

IGEC [26],IGEC2 [27],MiniGRAIL [28],Schenberg [29]

Table 1.1: A comparison of different types of gravitational-wave detectors.

interferometers. No matter which scheme is chosen, the task is an enormous

challenge. Various types of gravitational-wave detectors, which operate at dif-

ferent detection frequencies, are listed in Table 1.1. In addition, Fig. 1.4 shows

pictures of some of these detectors.

To review all the different types of gravitational-wave detectors would be an

intriguing task, as a wide variety of physics and technologies are involved. In

this thesis, however, I will only focus on one type of gravitational-wave detec-

tor: the terrestrial laser interferometer. The gravitational-wave group in the

University of Western Australia (UWA) is a member of the Laser Interferometer


1 2

3

4 5

Figure 1.4: Conceptual and real pictures of some gravitational-wave detectors.1. Pulsar timing arrays [31]. 2. LISA [32]. 3. DECIGO [33]. 4. aVIRGO [34].5. NAUTILUS [35].

Gravitational-wave Observatory (LIGO) Scientific Collaboration (LSC), which

is an international collaboration in gravitational-wave detection. This thesis was

done as part of that collaboration.

1.2.1 Configurations of Laser Interferometers

The idea of using a laser interferometer to detect gravitational waves was

first proposed by Felix Pirani in 1956 [36, 37]. In the early 1970s, the first

laser interferometric gravitational-wave detector was built at Hughes Research

Laboratories, USA [38]. A few large-scale laser interferometers, namely LIGO

[20], VIRGO [22], GEO600 [25], and TAMA [39], were built during the 1990s,

with some modifications to the simple Michelson-Morley interferometer, such as

Fabry-Perot cavities and a power recycling mirror. After successfully achiev-

ing their designed sensitivities, LIGO and VIRGO were promoted to undergo

technological upgrades — their new versions are called Advanced-LIGO (aLIGO)

[21] and Advanced-VIRGO (aVIRGO) [23], respectively. Another detector, KA-

GRA, with 3-km arms, is also being built in Japan within a cryogenic environ-


ment [24]. The thermal noise of this detector will be largely suppressed owing

to its cryogenic environment.

Simple Michelson-Morley Interferometer

A simplified configuration of a laser interferometer is shown in Fig. 1.5. It

consists of some major components, which are described in the following para-

graphs.

Figure 1.5: A conventional laser interferometer with its major subsystems.

Laser source — this is to generate a strong laser beam which is used to

detect the disturbance of spacetime. In order to introduce minimum noise into

the signal, the laser should be very stable in terms of power, frequency and

pointing of the beam. In addition to a low-noise laser source, a feedback control

loop with sophisticated sensing methods is necessary. Moreover, the output

laser should be linearly polarized to avoid birefringence effects in the optical

components of the detector.

Beam splitter and End Test Masses — a 50/50 beam splitter sends the

laser source beam into two orthogonal directions. These split beams are reflected


by end test masses, combine again at the beam splitter, and then create con-

structive and destructive interferences in the two orthogonal output directions,

namely at the bright port and dark port, respectively. A gravitational-wave sig-

nal can only be registered in the dark port, as the signal-to-noise ratio is much

higher there than in the bright port. The material of the end test masses is care-

fully selected with a requirement of low mechanical loss. A dielectric coating is

normally applied to the test mass surface to provide high reflectivity.

Suspension and Vibration isolation system — to keep the detector from

being disturbed by ground vibrations, the test masses and beam splitter have to

be suspended. The suspension and vibration isolation system are composed of

a few stages, with each stage isolating the vibration in some degrees of freedom.

The intrinsic pendulum frequencies of this system are much smaller than the

frequencies of seismic vibrations, which allows the test masses to be considered

as free masses.

Read-out system — the interference pattern registered at the dark port can

be sensed by a photodetector. Signals are converted into Fourier spectra which

can be analyzed in real-time or afterwards. The data analysis is an intensive

task, as it requires extracting the extremely weak gravitational-wave signals from

a background of a great diversity of noises.

Figure 1.6: Response of an interferometer to a gravitational wave traveling per-pendicularly to the detector plane. The intensity signal recorded by the pho-todetector has twice the frequency of the gravitational wave.


Imagine a gravitational wave with harmonic amplitudes in pure “+” polar-

ization propagating along a direction, denoted as z, which is perpendicular to

the plane of the detector, denoted as x-y. If the x-y coordinates are oriented

along the two interferometer arms, the two arm lengths will vary periodically,

as illustrated in Fig. 1.6. According to Eqs. (1.11) and (1.12), the arm length

variations as functions of time are given by:

δLx(t) =1

2aL sin (Ωt), (1.16)

δLy(t) = −1

2aL sin (Ωt), (1.17)

where L is the arm length (assumed equal here, for the 2 arms). The light

intensity registered by the photodetector at the dark port is then:

SGW ∝ L2a2 sin2 Ωt ∝ [δLx(t) − δLy(t)]2. (1.18)

It is obvious that the gravitational-wave signature is imprinted on the signals of

the photodetector.

Note, in Eq. (1.18), that the signal is stronger when L is larger. However, the

arm length of a ground-based interferometer cannot be too long, due to the finite

radius of curvature of the Earth. In an interferometer in which the arms are too

long, one end of each arm would be below the horizon of the other. Therefore,

a scheme to increase the effective arm lengths without building too-long actual

arms is necessary.

Modifications to a Simple Michelson-Morley Interferometer

A critical modification in the increase of the effective arm length is to use

Fabry-Perot cavities, as shown in Fig. 1.7. An Fabry-Perot cavity is composed

of two high-reflectivity mirrors, namely the Input Test Mass (ITM) and the

End Test Mass (ETM). Each laser beam emerging from the 50/50 beam splitter

enters the corresponding Fabry-Perot cavity, and travels back and forth many

times between the ITM and ETM, before exiting. The number of cycles it travels

inside the cavity is defined as:

N ≡ Fπ

, (1.19)

where F is called the finesse of the cavity, which is determined by the reflec-


tivities of the ITM and ETM.

Assume that the same gravitational wave described in the previous section

is traveling to the detector; the difference between the cavity lengths distorted

by the gravitational wave is given by:

δL(t) =4NLa sin Ωt√

1 + (2ΩNLc

)2

. (1.20)

Note, in Eq. (1.20), that if 2ΩNLc

≪ 1, the sensitivity of the interferometer is

increased by a factor of N by adding Fabry-Perot cavities.

However, the sensitivity cannot be increased arbitrarily by simply increasing

the cavity finesse. It will reach a limit:

δL(t) → 2ca sin Ωt

Ω(2ΩNL

c→ ∞), (1.21)

which is a fixed number for gravitational waves with a certain frequency, and

independent of the cavity length and finesse.

On the other hand, the optical power inside the cavity is increased. This leads

to an increased coupling of the optical field to the gravitational wave. However,

the cavity finesse cannot be too high as the sensitivity to displacement noises

of the test masses is also increased by using Fabry-Perot cavities. Therefore, an

optimal effective arm length needs to be chosen for the detection frequency. For

instance, to detect gravitational waves with a frequency of 100 Hz, the optimal

effective cavity length will be:

Lopt ≡ NL =1

4λ = 750 km, (1.22)

where λ is the wavelength of the gravitational wave.

A problem exists in the Fabry-Perot cavity interferometer configuration: as

the dark port photodetector registers the destructive interference, the laser beam

in constructive interference propagates back to the laser source. This will intro-

duce extra noise into the laser power and frequency. In addition, most of the

laser power generated by the laser source is wasted.

To cope with this problem, a high-reflectivity power recycling mirror (PRM)

is added between the laser source and the beam splitter (BS), as shown in Fig.

1.7. The position of this PRM can be chosen to build up the power in a cavity

between the PRM and the BS, thus further increasing the optical power in the


Figure 1.7: Modifications to a simple Michelson-Morley interferometer. AnFabry-Perot cavity is formed in each arm by adding an input test mass in thepath (left). A power recycling mirror and signal recycling mirror are furtheradded in order to increase the intracavity power and the signal amplitude, re-spectively (right). All these added mirrors need to be suspended.

arm cavities. The detection sensitivity is also increased by adding the PRM.

Another mirror, which is called the signal recycling mirror (SRM), is put

between the BS and the photodetector (see Fig. 1.7). A cavity is formed between

the BS and the SRM at the antisymmetric dark port of the interferometer [40].

In addition, the peak frequency of the detection bandwidth can be tuned by

selecting different reflectivities for the SRM, and by adjusting its position.

1.2.2 Noise Budget of aLIGO

A gravitational-wave laser interferometer is a rather complex system. The

complexity is not in the large number of various components, but in the difficulty

of making each component as quiet as possible. Noises from different sources are

summed in the final detection stage, which could be large enough to wipe out

any gravitational-wave signal. Thus, summarizing different noise sources, and

setting a noise budget, are critical in determining the capability of the detector.

Fig. 1.8 shows the expected noise budget for aLIGO [41, 42, 43]. The solid lines

in this Figure are primary noise contributions. I will briefly describe them in

the following paragraphs.


fluctuations

Figure 1.8: Expected noise contributions from various different sources. ThisFigure was generated from the GWING software. Figure adapted from Adhikari(2014) [41].

Seismic noise — these noises can be classified into two categories with

respect to their frequencies. The first class is noises with frequencies lower

than 10 Hz [44, 45]. The main sources of these noises are from nature, such as

storms, and ocean waves impinging on the continental land masses. Although

these frequencies are normally out of the aLIGO detection frequency band, the

root mean square movements of the test masses are still disturbed by coupling to

the suspension and vibration isolation systems. On the other hand, the second

class is noises with frequencies higher than 10 Hz, which are mainly induced by

human activities [46, 47]. This noise varies randomly from time to time due to

different human activities occurring nearby.

To suppress these seismic noises, the test masses are suspended by several

stages of vibration isolation systems. At frequencies much higher than the in-

trinsic resonant frequency of the suspension system, the motion of the test mass

is much smaller than that of the ground. By applying this principle, the intrin-

sic resonant frequencies of the suspension system are set to be much less than


the detection band and the seismic noise frequencies. Therefore, the seismic

response motion of the test mass can be highly suppressed.

Newtonian Gravity noise — this noise is usually less than 10 Hz, and is di-

rectly caused by environmental density disturbances. The density of the ground

at the detector, for instance, can be disturbed by a seismic wave, or other human

activity. To avoid this noise, an array of accelerometers and/or seismometers

are arranged at the laser interferometer. The data from the measured ground

vibrations are fed back to the signal of the detector instantly, or are recorded

and used later to extract the real signals from the Newtonian noise background

[48]. In future generation gravitational-wave detectors, such as KAGRA and

the Einstein telescope, test masses will be about 100 m below ground level; for

these detectors, the Newtonian gravity noise can be intrinsically reduced by a

factor of ∼10 [49].

Thermal noise — originates from random fluctuations of an environmental

thermal bath with which the mechanical system is in equilibrium. It results

in random displacement fluctuations in surface particles of the test masses. In

general, thermal noise can be divided into three classes: test mass substrate

thermal noise, mirror coating thermal noise, and suspension thermal noise. The

latter two are predominant in the detection band of aLIGO.

The coating of a test mass is made of alternating layers of high refractive

material and low refractive material, in order to provide high reflectivity for the

test mass [50]. The coating thermal noise originates from mechanical loss due to

friction between the coating and substrate, friction between coating layers, and

internal friction within the coating materials. It was found that the latter is the

main noise source [51]. Much effort has been put into finding coating materials

with high reflectivity as well as low mechanical loss [52, 53].

Mechanical vibrations of the suspension system are coupled to the test mass,

and thus introduce a mechanical loss. Fused silica, as an ultra-low dissipation

material, is used for the suspension wires in aLIGO to minimize the suspension

thermal noise. This noise level is much lower than the coating thermal noise at

100 Hz. However, there is a peak at 9 Hz in the suspension thermal noise curve,

which is caused by a coupling to the vertical bounce mode [54].

Quantum noise — this, as a fundamental noise source, arises from the


quantum properties of the photons generated from the laser source and injected

into the detector. It consists of two parts, namely, quantum shot noise and

quantum radiation pressure noise. The former originates from the Poisson dis-

tribution of the photon number over time due to the randomness of the lasing

process. The photodetector at the detection port thus registers amplitude fluc-

tuations over time. The latter originates from the fluctuation of the radiation

pressure impinging on a test mass due to the uncertainty in the photon number.

The quantum shot noise amplitude is inversely proportional to square root

of the circulating power, while the amplitude of the quantum radiation pres-

sure noise is proportional to that power. In aLIGO, the circulating laser power

achieves ∼1 MW, where these two quantum noises are about the same in ampli-

tude. This noise level sets a fundamental limit to the sensitivity of gravitational-

wave detectors, which is called the Standard Quantum Limit (SQL), which un-

avoidably exists even if all other noises were to be suppressed. However, some

schemes have been proposed to surpass this SQL, such as using a double optical

spring [55], a filter cavity [56], and a speedmeter [57].

1.2.3 Nonlinear Effects and Instabilities

Besides the noises introduced above, in which gravitational-wave signals can

be submerged, some other processes can also keep gravitational waves from being

detected. These are the thermal lensing effect, the Sidles-Sigg instability and

parametric instability. Even though all the noises may be suppressed below

the expected gravitational-wave signal level, these nonlinear effects may still be

detrimental to gravitational-wave detection.

Thermal lensing effect — due to absorption of the optical power, the test

masses are heated. Although this absorption is only a tiny part of the total

power, the heat transferred to each test mass can still be very significant, as

the total power is huge. The temperature of the test mass is increased, and a

temperature gradient ∆T is created inside the test mass. Some temperature-

dependent effects are induced, such as an expansion of the test mass, and an

increase in its refractive index. As a result, the total optical path for the test

mass is extended (see Fig. 1.9). This resultant lengthening is equivalent to


adding some new lenses to the original optical path — this is called the thermal

lensing effect.

Figure 1.9: The thermal lensing effect is induced when an optical beam poweris absorbed by the test mass. The laser heating the test mass normally has aGaussian beam profile. Figure reproduced from Fan (2010) [58].

The problems induced by the thermal lensing effect are critical. The laser

beam will not match the cavity mode if the optical path is varied, and the mirror

surfaces are deformed. Therefore, the optical power circulating in the cavity will

be reduced, and the locking system will fail if the mode mismatch is too large.

In addition, a part of the optical power in the fundamental optical mode will be

transferred into higher-order optical modes (HOMs), if degeneracy conditions

are satisfied. These HOMs could introduce extra noises into the detector signal.

Some other nonlinear effects could be induced by the generation of HOMs, such

as parametric instability (see page 19).

In advanced gravitational-wave laser interferometers, several techniques are

used to compensate for the thermal lensing effect [58, 59, 60, 61]. One method

is to use a ring heater around each test mass to maintain a constant radius of

curvature of the mirror surface. Another method is to add a compensation plate

between the beam splitter and each ITM. This plate is heated by a CO2 laser

in order to change its radius of curvature. The advantage of this method is that

the response time of CO2 heating is of the order of 1 second, so that the radius


of curvature can be controlled instantly through a feedback loop.

Sidles-Sigg instability — a significant radiation pressure acts on the test

mass, as the intracavity optical power is huge. Besides this, the longitudinal

position of the test mass is affected by the radiation pressure forces; as the test

mass is suspended by wires, the angular positions can also be shifted when the

beam spot is not exactly at the mirror center. This optical torque effect on a

test mass was first described and analyzed by Solimeno et al. in 1991 [62]. If

this optical torque has the same frequency as one of the angular modes, and

the optical rigidity has an absolute value larger than that of the pendulum, but

with a negative sign, the angular motion will be excited. This effect is called

the Sidles-Sigg instability, first realized by Sidles and Sigg in 2006 [63].

This optical-torque induced rigidity was first observed in a suspended three-

mirror cavity by Driggers [64]. The torsional stiffness of the mirrors are depen-

dent on the circulating power. Fan et al. [65] then observed a negative optically

induced torsional rigidity with a high circulating power, and discovered that the

angular optical spring constant is dependent on the cavity g-factor. Consider-

ing the optical torque in arm cavities, near-concentric cavities are used as arm

cavities in aLIGO to minimize the Sidles-Sigg instability.

Parametric instability — parametric interaction in a gravitational-wave

detector was first realized and analyzed by Braginsky in 2001 [66]. It is described

as follows: due to optomechanical interactions, a beating between the cavity

optical field and the optical field scattered by the motion of the test mass induces

a vibration of the test mass. This mechanical vibration then scatters more of

the optical field, which in turn induces further mechanical vibrations, as in an

active feedback system.

On the other hand, mechanical loss in the test mass damps these induced

vibrations via mechanical dissipation processes. However, if the feedback gain

produced by the parametric interaction exceeds the mechanical damping rate,

the amplitude of the mechanical vibration will keep increasing until the cavity

locking system fails. This effect is called parametric instability (PI), and was

recently observed in aLIGO [67], and in a small-scale membrane-in-the-middle

device [68] (see Fig. 1.10). It was also observed in our 74-meter Fabry-Perot

cavity; details will be discussed in Chapter 7 of this thesis.

Various schemes to suppress this PI were suggested. They are divided into


Figure 1.10: PI effects observed. PI was observed at 65 kHz in aLIGO (left).The color in this Figure indicates the mechanical amplitude at 65 kHz [67].An increase of a mechanical mode amplitude was also observed in a small-scaledevice (right). The amplitude reaches a saturation after an exponential ring-up,which is consistent with simulations [68].

two categories: passive and active control. In passive control schemes, mechani-

cal dampers [69, 70, 71, 72] are used to reduce the mechanical quality factor, and

thus reduce the feedback gain in the parametric interaction. In active control

schemes, the mechanical vibrations are monitored and then suppressed by apply-

ing forces on the test mass [73, 74], or by thermally tuning the cavity [75, 59, 76].

In addition, residual angular motions of the test mass can be considered as an

intrinsic dynamic modulation, and therefore as an effective approach to suppress

the PI without introducing any extra components. The details of this approach

will be described in Chapter 7.

1.3 Thesis Structure

This thesis is focused on experimental studies of three-mode optomechanical

interactions and their parametric instability. The physical background of these

subjects is given in Chapter 2.

All the experiments described in this thesis have been undertaken in the

High Optical Power Test Facility at Gingin, Western Australia. This facility,

composed of two separate Fabry-Perot cavities, will be introduced in Chapter 3.

1.3. THESIS STRUCTURE 21

The main scientific object of this thesis is the observation of parametric

instability. However, a few steps are necessary to take prior to implementing

the experiment to observe parametric instability: 1) to demonstrate three-mode

parametric interaction, which is the physical mechanism of parametric instabil-

ity; 2) to characterize the optical cavity in which parametric instability takes

place; 3) to characterize the mechanical property of the test mass on which para-

metric instability might happen, including internal mechanical modes and loss

angles. The next three chapters discuss these three steps in details.

In Chapter 4, a high sensitivity transducer is demonstrated in an experi-

ment, based on a strong optomechanical coupling between the pump laser and

a resonant sideband. This technique is very useful in the optical cooling of the

test mass used in gravitational-wave laser interferometers and in quantum non-

demolition measurements. This work was published in a Physical Review A

paper.

The optical properties of the East arm cavity in the Gingin facility are de-

scribed in Chapter 5. In this Chapter, observation and analysis of optical mode

degeneracy are also presented. A paper based on this work will be submitted to

Optics Express.

In Chapter 6, a modular suspension system is studied. The quality factors of

various mechanical modes were measured in an experiment. A simulation was

created to estimate the mechanical loss angle due to coupling to the suspension

system, and to calculate the thermal noise of the test mass. A paper based on

this work will be submitted to Classical and Quantum Gravity.

Chapter 5 and 6 can be considered as experimental preparations and char-

acterizations for the observation of parametric instability, which is introduced

in Chapter 7. A mechanical mode with a frequency of ∼150.2 kHz was ex-

cited due to three-mode parametric interactions. This Chapter also describes a

suppression mechanism of the parametric instability due to a dynamic modula-

tion. Based on this experiment, a method to combine this dynamic modulation

and a thermal modulation is developed to suppressed parametric instabilities in

aLIGO. A paper based on this work was submitted to Physical Review D.

An automatic alignment (AA) system to align the cavity against slow drifts

due to low-frequency vibrations was built for the East arm cavity. Mode-

matching of this system was achieved, and preliminary tests were successful.

This work is presented in Chapter 8. However, the misalignment signals still

need to be applied to the control system, to make this AA system work.


Finally, conclusions for these experimental investigations, and a discussion

of future work, are given in Chapter 9.

Chapter 2

Theoretical Background of

Three-mode Optomechanical

Interactions

2.1 Gaussian Beam in Fabry-Perot Cavities

Although some non-Gaussian-beam schemes have been suggested for laser

interferometers to achieve higher sensitivity [77], the laser sources used in current

detectors are still Gaussian beams, due to technical issues. The laser sources

used in our experiments are also Gaussian beams. Thus, I will only discuss

Gaussian beams in this Chapter.

2.1.1 Hermite-Gaussian Modes

A propagating Gaussian beam in a uniform space is the simplest solution

of the Helmholtz equation. In the paraxial approximation, assuming that the

beam is traveling along the z direction in (x, y, z) coordinates, a whole set of

solutions of the electric field distributions, with the same longitudinal order, is

23

24 CHAPTER 2. THEORETICAL BACKGROUND

given by [78, 79]:

Emn(x, y, z) = E0ω0

ω(z)Hn(

√2

x

ω(z))Hm(

√2

y

ω(z))e

−x2+y2

ω2(z)

· exp −i[kz − (1 + n + m) arctanz

zR+

k(x2 + y2)

2R(z)], (2.1)

where E0 is a constant, ω0 is the beam waist size, ω(z) is the beam size at

position z, k is the wave number, n and m denote the mode number in the

x and y directions, respectively, Hn and Hm are the Hermite polynomials, zR,

called the Rayleigh range, is defined as the position where ω(zR) =√

2ω0, which

is given by:

zR =kω2

0

2, (2.2)

and R(z) is the radius of curvature of the wave front at position z, which is

given by:

R(z) = z[1 + (zR

z)2]. (2.3)

Note, from Eq. (2.1), that the beam profile of a particular mode is determined

by the Hermite polynomials, and therefore by the transverse mode numbers n

and m. Some lowest-order modes are shown in Fig. 2.1.

A Gaussian beam is obtained when m, n = 0 in Eq. (2.1), which is expressed

as:

E00(x, y, z) = E0ω0

ω(z)e−

x2+y2

ω2(z) exp −i[kz − arctanz

zR+

k(x2 + y2)

2R(z)]. (2.4)

The beam size ω(z) can be expressed as:

ω(z) = ω0

√

1 + (z

zR)2. (2.5)

A lateral view of the Gaussian beam is given in Fig. 2.2. Note that beyond zR

from the beam waist, along the z axis, the wave front is close to being a spherical

wave, with the center of the sphere is always at the waist position. This is a

very important feature in designing mirrors for Fabry-Perot (FP) cavities used

in laser interferometers.

A useful parameter, called the Guoy phase, is defined as:

φ = arctanz

zR

(2.6)

2.1. GAUSSIAN BEAM IN FABRY-PEROT CAVITIES 25

0,0 1,0 0,1

1,1 2,0 0,2

3,3 5,1 2,4

Figure 2.1: Simulated intensity of some lowest-order Hermite-Gaussian modeson a mirror surface. The scales of the beam and the mirror are proportional tothat of the ITMs in aLIGO.

for the Gaussian beam. For higher-order Hermite-Gaussian modes in the same

longitudinal order, the effective Guoy phase can be written as:

φnm = (1 + n + m) arctanz

zR. (2.7)

The Guoy phase is critical in discovering the difference between a plane wave

and a Gaussian beam. It will be discussed later in this Chapter, and also in

Chapter 8.


Figure 2.2: Lateral view of a Gaussian beam. Figure reproduced from Chen(2014) [79].

2.1.2 Resonance in Fabry-Perot Cavities

Fabry-Perot cavities are used in laser interferometers to increase the effective

arm length. This means that the light field needs to resonate in the cavities.

Two requirements have to be satisfied: wave front matching (also known as

mode matching) and frequency matching.

Wave-front Matching

As discussed before, at positions far away from the waist (z ≫ zR), the wave

front of a laser beam is spherical. Therefore, to minimize scattering loss, the

high-reflectivity surfaces of the test masses should also be spherical. For a laser

beam whose wave fronts are matched with the surfaces of the test masses, the

position and size of the beam waist can be calculated in terms of the radii of

curvature of the test masses (see Fig. 2.3). The cavity g-factors are defined as

[78]:

g1, 2 ≡ 1 − L

R1, 2, (2.8)

where R1,2 denote the radii of curvature of the two test masses, respectively.

2.1. GAUSSIAN BEAM IN FABRY-PEROT CAVITIES 27

The waist size and the Rayleigh range are then given by:

ω20 =

2L

k

√

g1g2(1 − g1g2)

(g1 + g2 − 2g1g2)2(2.9)

and

zR = L

√

g1g2(1 − g1g2)

(g1 + g2 − 2g1g2)2, (2.10)

respectively. The distances from the test masses to the waist can be expressed

as:

z1 =g2(1 − g1)

g1 + g2 − 2g1g2L (2.11)

and

z2 =g1(1 − g2)

g1 + g2 − 2g1g2

L. (2.12)

Figure 2.3: A Gaussian beam mode-matched to a Fabry − Perot cavity.

Frequency Matching

In a paraxial approximation, the frequency matching is satisfied when:

2L

cωq, nm + (1 + n + m)(2 arctan

z1

zR

+ 2 arctanz2

zR

) = q · 2π, (2.13)

where L is the cavity length, c is the speed of light, ωq,nm is the circular frequency

of the beam resonant in the cavity, n, m denote the transverse order of this beam,

and q is an integer number. By substituting Eqs. (2.10), (2.11) and (2.12) into


Eq. (2.13), the resonant frequency is given by:

fq, nm =ωq, nm

2π

=c

2Lq +

c

2Lπ(1 + n + m) arccos (±√

g1g2), (2.14)

The sign before√

g1g2 is determined by the value of g1 + g2 − 2g1g2. If g1 + g2 −2g1g2 > 0, the sign is +; if g1 + g2 − 2g1g2 < 0, the sign is −. The difference

between each longitudinal order is called the free spectral range, which is:

fFSR =c

2L. (2.15)

Higher-order Mode Spacing

According to Eqs. (2.14) and (2.15), the frequency difference between a fun-

damental mode and an (n, m) mode in the same longitudinal order is:

∆fnm =fFSR

π(n + m) arccos (±√

g1g2). (2.16)

2.2 Optomechanical Interactions

Figure 2.4: Schematic diagram of Optomechanical Interaction.

2.2. OPTOMECHANICAL INTERACTIONS 29

2.2.1 Cavity Response

A system involving a laser beam in an FP cavity and a mechanical oscillator

is an optomechanical system (see Fig. 2.4). Assuming the input test mass (ITM)

is fixed, and only the end test mass (ETM) has a displacement of x, the equations

of state are given by:

E0(t) = t0Ein(t)e−iω0t + r0E1(t −

L + x

c), (2.17)

E1(t) = r1E0(t −L + x

c), (2.18)

Eref(t) = −r0Ein(t) + t0E1(t −L + x

c), (2.19)

Etrans(t) = t1E0(t −L + x

c) − r1ǫ(t), (2.20)

where r0,1 and t0,1 are the reflectivities and transmissivities of the test masses,

Ein(t) is the input electric field, E0(t) and E1(t) are the intracavity electric

fields, Eref(t) and Etrans(t) are the electric fields reflected and transmitted from

the cavity; x is also a function of time, and ǫ(t) denotes the vacuum fluctuation

injected into the cavity from the cavity transmission port.

In a slowly-varying approximation:

E0(t) = E0(t)e−iω0t + E∗

0(t)eiω0t, (2.21)

where ω0 is the angular frequency of the input light field, E0(t) is the slowly

varying amplitude and E∗0 is its conjugate. We define a detuning as ∆ ≡ ω0−ωc,

where ωc denotes a resonant frequency of the cavity, and let

E0(t) = E0 + E(1)0 (t), (2.22)

where E0 is a zero-order amplitude, which is a time-independent constant, and

E(1)0 (t) is a first-order amplitude. Thus, substituting Eqs. (2.18), (2.21) and

(2.22) into Eq. (2.17), we get the first-order intracavity field:

˙E(1)0 (t) + (γ − i∆)E

(1)0 (t) = i

r0r1ω0

LE0x(t) + ΩcE

(1)in (t), (2.23)

where γ ≡ c(t20+t21)

4Lis called the cavity’s bandwidth, and Ωc is a constant, which


is defined as: Ωc ≡ ct02L

. In the frequency domain, Eq. (2.23) can be written as:

E0(Ω) =−1

Ω + ∆ + iγ[GX(Ω) − iΩcEin(Ω)], (2.24)

where G ≡ r0r1ω0E0

L≈ ω0E0

L.

It is straightforward to write the first-order reflection light field in the fre-

quency domain as:

Eref(Ω) = −r0Ein(Ω) + t0E0(Ω)

=iγ − (Ω + ∆)

iγ + (Ω + ∆)Ein(Ω) − t0ω0E0

L(Ω + ∆ + iγ)X(Ω). (2.25)

Note that the following derivations in this Section are all in the frequency do-

main, unless otherwise specified.

The equation of motion of the displacement X(Ω) is given by:

m(−Ω2 − 2iγmΩ + Ω2m)X(Ω) = Frad(Ω) + ξth(Ω) + Fsig(Ω), (2.26)

where m is the mass of this mechanical oscillator, γm is its mechanical damping

rate, Ωm is its resonance frequency, Frad(Ω), ξth(Ω) and Fsig(Ω) are the radiation

force, thermal force, and force induced by gravitational waves on the test mass,

respectively.

Specifically, the radiation force of the intracavity light field can be expressed

as:

Frad(Ω) = 2E0[E0(Ω) + E0∗(−Ω)]

= −K(Ω)X(Ω) + A(Ω)Ein(Ω) + B(Ω)E∗in(−Ω), (2.27)

K(Ω) = − 8~ω0E0∆G

c(Ω + ∆ + iγ)(Ω − ∆ + iγ), (2.28)

A(Ω) =4i~ω0E0Ωc

c(Ω + ∆ + iγ), (2.29)

B(Ω) = − 4i~ω0E0Ωc

c(−Ω + ∆ − iγ), (2.30)

where E0 ≡√

I0~ω0

is the normalized intracavity optical power. If we ignore the

signal Fsig(Ω) and the thermal force ξth(Ω), and substitute Eq. (2.27) into Eq.


(2.26), the displacement of test mass can be written as:

X(Ω) =Rxx(Ω)

1 + Rxx(Ω)K(Ω)[A(Ω)Ein(Ω) + B(Ω)E∗

in(−Ω)], (2.31)

where

Rxx(Ω) ≡ 1/[m(−Ω2 − 2iγmΩ + Ω2m)]. (2.32)

2.2.2 Optical Spring Effect

In Eq. (2.27), it is clear that the first term works as the restoring force

of a spring. This restoring force is provided by the radiation pressure of the

intracavity light field. Thus, this effective spring is called an optical spring [80],

and the coefficient K(Ω) is called the optical spring coefficient.

The mechanical oscillator is modified by this optical spring effect. The effec-

tive resonance frequency and damping rate are given by:

Ωeffm (Ω) =

√

Ω2m +

Re[K(Ω)]

m

=

√

Ω2m − 4~ω0E0∆G(Ω2 − ∆2 − γ2)

mc[(Ω + ∆)2 + γ2][(Ω − ∆)2 + γ2], (2.33)

and

γeffm (Ω) = γm − Im[K(Ω)]

2mΩ

= γm − 4~ω0E0∆Gγ

mc[(Ω + ∆)2 + γ2][(Ω − ∆)2 + γ2], (2.34)

respectively. These two effective parameters as functions of Ω are illustrated in

Fig. 2.5.

2.2.3 Quantum Noises and Standard Quantum Limit

This subsection will derive the quantum noises in the optomechanical system

described above. We consider the reflected beam as the output to be measured,


Figure 2.5: The effective resonance frequency (left) and the effective dampingrate (right) in the optical spring effect. The parameters used in plotting thesetwo figures are: m=5.8 g, γ=150 kHz, L=74 m, Ωm=600 kHz, γm=6 Hz, ∆ =−Ωm. The intracavity power I0 = 109 for the blue solid line, I0 = 7 × 108 forthe red dashed line, and I0 = 3 × 108 for the black dot-dashed line.

which is:

Eout(Ω) = Eref(Ω)

= Z(Ω) + RZF (Ω)X(Ω), (2.35)

where

Z(Ω) =iγ − (Ω + ∆)

iγ + (Ω + ∆)Ein(Ω), (2.36)

RZF (Ω) = − t0ω0E0

L(Ω + ∆ + iγ). (2.37)

Substituting Eq. (2.31) into Eq. (2.35), we get:

Eout(Ω) = Z(Ω) +Rxx(Ω)F0(Ω)

1 + Rxx(Ω)K(Ω)RZF (Ω), (2.38)

where

F0(Ω) = A(Ω)Ein(Ω) + B(Ω)E∗in(−Ω). (2.39)

The amplitude and phase quadratures of Ein(Ω) and Eout(Ω) are given by:

Ein1(Ω) =Ein(Ω) + E∗

in(−Ω)√2

, (2.40)


Ein2(Ω) =Ein(Ω) − E∗

in(−Ω)

i√

2, (2.41)

Eout1(Ω) =Eout(Ω) + E∗

out(−Ω)√2

, (2.42)

Eout2(Ω) =Eout(Ω) − E∗

out(−Ω)

i√

2, (2.43)

respectively. The quadratures of the output are then derived by substituting

Eq. (2.38) into Eq. (2.42) and (2.43):

Eout1(Ω) = Z1(Ω) +Rxx(Ω)F0(Ω)

1 + Rxx(Ω)K(Ω)RZF1(Ω) (2.44)

and

Eout2(Ω) = Z2(Ω) +Rxx(Ω)F0(Ω)

1 + Rxx(Ω)K(Ω)RZF2(Ω), (2.45)

where

Z1(Ω) =Z(Ω) + Z∗(−Ω)√

2, (2.46)

Z2(Ω) =Z(Ω) − Z∗(−Ω)

i√

2, (2.47)

RZF1(Ω) =RZF (Ω) + R∗

ZF (−Ω)√2

, (2.48)

RZF2(Ω) =RZF (Ω) − R∗

ZF (−Ω)

i√

2, (2.49)

R∗xx(−Ω) = Rxx(Ω), (2.50)

K∗(−Ω) = K(Ω), (2.51)

F ∗0 (−Ω) = F0(Ω). (2.52)

If the detuning ∆ = 0, the output signals are:

Eout1(Ω) =iγ − Ω

iγ + ΩEin1(Ω), (2.53)

Eout2(Ω) =iγ − Ω

iγ + ΩEin2(Ω) − 2~t20Ω

20E

20

L2(Ω + iγ)2Rxx(Ω)Ein1(Ω). (2.54)

Note that only Eout2(Ω) contains both amplitude and phase quadratures. Thus,


the power spectrum density of Eout2(Ω) can be written as:

Sout2(Ω) = S2(Ω) +4~

2t40Ω40E

40

L4(Ω2 + γ2)2m2[(−Ω2 + Ω2m)2 + 4γ2

mΩ2]S1(Ω). (2.55)

The displacement power spectrum density is given by:

Sx(Ω) =|Ω + iγ|2L2

t20Ω20E

20

S2(Ω) +4~

2t20Ω20E

20

L2(Ω2 + γ2)m2[(−Ω2 + Ω2m)2 + 4γ2

mΩ2]S1(Ω)

≥4~

√

S1(Ω)S2(Ω)

m[(−Ω2 + Ω2m)2 + 4γ2

mΩ2]12

. (2.56)

In the quantum ground state, S1(Ω) = S2(Ω) = 1.

The linear spectral density of the displacement is given by:

∆X(Ω) =

√

Sx(Ω)

≥ 2~12

m12 [(−Ω2 + Ω2

m)2 + 4γ2mΩ2]

14

. (2.57)

When Ω = Ωm,

∆X(Ωm) = (2~

mγmΩm)

12

.= 10−19 m/

√Hz, (2.58)

in which m = 0.28 kg, γm = Ωm

2Qm≈ 1.14 Hz, and Ωm = 181.6 kHz.

When Ω ≫ Ωm,

∆X(Ω) = (4~

mΩ2)

12 , (2.59)

which is the Standard Quantum Limit (SQL) for a free mass [81].

In the general case, the quadratures of the output signals are normalized to

displacement, and Eqs. (2.44) and (2.45) become:

O1(Ω) = Z1(Ω) + Rxx(Ω)F1(Ω), (2.60)

and

O2(Ω) = Z2(Ω) + Rxx(Ω)F2(Ω), (2.61)

where

O1, 2(Ω) = Eout1, 2(Ω)1 + Rxx(Ω)K(Ω)

RZF1, 2(Ω), (2.62)


Figure 2.6: Power spectral densities of quantum shot noise, quantum radiationnoise, total quantum noise, and the standard quantum limit (left) [81]; strainspectra of quantum noises in different interferometer setups (right) [42]. In theright-hand figure, the red line denotes the quantum noise in a simple Michelsoninterferometer, which is dominated by shot noise (N.B. right-hand plot is of thestrain spectra, which has different units). The brown curve denotes the totalquantum noise of an interferometer with FP cavities, where the lowest noisefloor is 600 times smaller than that in a simple Michelson detector. The noisespectrum can be further reduced by adding power recycling mirrors, as in thecyan curve, where the frequency of minimum noise shifts to ∼100 Hz. Thebandwidth of this quantum noise trough can be broadened from ∼200 Hz to ∼1kHz by adding signal recycling mirrors, which is the blue curve.

Z1, 2(Ω) =Z1, 2(Ω)

RZF1, 2(Ω), (2.63)

F1, 2(Ω) = F0(Ω) + K(Ω)Z1, 2(Ω)

RZF1, 2(Ω). (2.64)

We assume that the first-order amplitude in the input light field is the noise of

the input laser beam. The quantum noise embedded in Z1, 2(Ω) is the shot noise,

and the quantum noise contained in F1, 2(Ω) is the radiation pressure noise.

The normalized power spectral density of the output field is denoted by:

Sout1, 2(Ω) =< O1, 2(Ω)O∗1, 2(Ω) >, (2.65)

where the normalization rules are:

< Ein1(Ω)E∗in2(Ω

′) >= 0, (2.66)


′) >= S1(Ω)δ(Ω − Ω′), (2.67)


′) >= S2(Ω)δ(Ω − Ω′), (2.68)


where S1(Ω) and S2(Ω) are the power spectrum densities of the two quadratures,

namely, the laser amplitude noise and laser phase noise. When Ω = Ω′, δ(Ω −Ω′) = 1; otherwise δ(Ω − Ω′) = 0. Thus, Eq. (2.65) can be expressed as:

Sout, i(Ω) = SZiZi(Ω) + 2Rxx(Ω)Re[SFiZi

(Ω)] + R2xx(Ω)SFiFi

(Ω), (2.69)

where i = 1, 2, and the terms in Eq. (2.69) are given by:

SZ1Z1 =[(Ω + ∆)2 + γ2][(Ω − ∆)2 + γ2]

t20ω20E

20∆

2, (2.70)

SZ2Z2 =[(Ω + ∆)2 + γ2][(Ω − ∆)2 + γ2]

t20ω20E

20(Ω

2 + γ2), (2.71)

SF1F1 =~

2

4γ2t20ω20E

20

, (2.72)

SF2F2 =~

2(4γ2 + ∆2)

4γ2t20ω20E

20(Ω

2 + γ2), (2.73)

SF1Z1 = ~∆2 − γ2 − Ω2

2∆γ, (2.74)

SF2Z2 = ~∆(∆2 + 3γ2 − Ω2)

2∆(∆2 + Ω2). (2.75)

2.3 Parametric Instability

In aLIGO and aVIRGO, the circulating power will achieve ∼1 MW, in order

to suppress the quantum shot noise. Due to this high optical intracavity power,

the nonlinear effects between optical fields and test masses will not be negligible.

Some other nonlinear effects, such as the thermal lensing effect and the Sidles-

Sigg instability, were briefly introduced in Chapter 1. Here, we will discuss

parametric instability in more detail.

2.3. PARAMETRIC INSTABILITY 37

2.3.1 Three-mode Parametric Interaction

Three-mode optomechanical interaction occurs between two cavity modes

and one mechanical mode of the test masses. It can be understood in the

following way: when a bunch of photons meet a test mass, they are scattered

by mechanical vibrations, and two sidebands are generated (see Fig. 2.7). The

photons in these sidebands have frequencies of

ωs, a = ω0 ± ωm, (2.76)

where ω0 is the frequency of the original beam, and ωm is the mechanical mode

frequency.

This process is similar to Brillouin scattering, where phonons are either gen-

erated or absorbed. We call the scattering process Stokes when the frequency

relation ωs = ω0−ωm is satisfied, in which a phonon is created and the generated

photon has less energy than the original photon. Another process, satisfying the

frequency relation ωa = ω0 + ωm, is called anti-Stokes, where a phonon is ab-

sorbed and the generated photon possesses more energy than the original one.

If the photons generated in the Stokes process resonate in the cavity, the

photon number with the frequency ωs will build up. Similarly, the number

of photons generated in the anti-Stokes process increases if ωa is a resonance

frequency of the cavity. In the East arm cavity of the HOPTF, for instance,

g1, 2 < 0 and g1 ·g2 ≈ 1. This cavity is a near-concentric one, where the first-order

transverse mode belonging to a longitudinal group is close to the fundamental

transverse mode of the next longitudinal group (see Fig. 2.8). The frequency

difference between this first-order mode and the fundamental mode is about

∼100 kHz, determined by g1 and g2 of the cavity. When both the fundamental

mode and the first-order mode are resonating in the cavity, only the Stokes

process is enhanced.

Three-mode parametric instability happens when the two resonating optical

modes couple and excite the mechanical mode, which will in turn induce more

Stokes processes. This feedback process is illustrated in Fig. 2.9. If the optical

power in the fundamental mode is high, and a large spatial overlap between

the higher-order optical mode (HOM) and the mechanical mode is achieved,

the power in the HOM could increase exponentially. This phenomenon is call

parametric instability.


Figure 2.7: Two sidebands of the light field are generated by a mechanicalvibration (top). A phonon is created in the Stokes process (bottom left), and aphonon is absorbed in the anti-Stokes process (bottom right).

Figure 2.8: Transverse mode frequencies of a near-concentric cavity. The first-order transverse mode of the qth longitudinal group has a frequency slightlyless than that of the fundamental transverse mode of the (q + 1)th longitudinalgroup. The frequency difference between them is given by Eq. (2.16).

2.3.2 Derivation of Parametric Instability

To describe parametric instability mathematically, we follow the derivation

already presented by Braginsky et al. in their Appendix A [66]. We first denote


Figure 2.9: Feedback loop diagram of parametric instability. A higher-ordercavity mode (TEM11 in this diagram) is scattered by the mechanical vibrationand resonates in the cavity. The beating note between the fundamental opticalmode and the higher-order optical mode excites the mechanical mode to producemore scattering of the light field. Figure reproduced from Adhikari (2014) [41].

(E0, E1) and (H0, H1) as the electric and magnetic fields of the fundamental and

HOM of the FP cavity. They are given by:

E0(t) = −√

2π

S0L(f0e

ik0z − f ⋆0 e−ik0z)∂tq0(t), (2.77)

E1(t) = −√

2π

S1L(f1e

ik1z − f ⋆1 e−ik1z)∂tq1(t), (2.78)

H0(t) =

√

2π

S0L(f0e

ik0z + f ⋆0 e−ik0z)ω0q0(t), (2.79)

H1(t) =

√

2π

S1L(f1e

ik1z + f ⋆1 e−ik1z)ω1q1(t), (2.80)

where q0(t) and q1(t) are the generalized coordinates of the two optical modes,

respectively; k0, k1 are wave numbers, f0, 1 = f0, 1(~r⊥, z) are functions of the

distribution of the optical modes, where ~r⊥ denotes a vector over the (x, y) plane,

z is the direction of propagation of the optical modes, and S0, 1 =∫

|f0, 1|2~dr⊥.

In addition, we denote x(t) and ~u(~r) as the generalized coordinate and the


displacement spatial distribution of the mechanical mode. The Lagrangian of

the system is given by:

L = L0 + L1 + Lm + Lint, (2.81)

L0 =

∫

L(〈E0〉2 − 〈H0〉2)8π

~dr⊥ =∂tq

20 − ω2

0q20

2, (2.82)

L1 =∂tq

21 − ω2

1q21

2, (2.83)

Lm =M(∂tx)2 − Mω2

mx2

2, (2.84)

Lint = −∫

xuz〈H0 + H1〉28π

|z=0d~r⊥

= −2ω1ω0q1q0Bx

L, (2.85)

where

M = ρ

∫

|~u(~r)|2dV, (2.86)

B =

∫

f0(~r⊥)f1(~r⊥)uzd~r⊥√

∫

|f0|2d~r⊥∫

|f1|2d~r⊥. (2.87)

Thus, the equations of motion can be written as:

∂2t q0 + 2δ0∂tq0 + ω2

0q0 = −2ω0ω1q1Bx

L, (2.88)

∂2t q1 + 2δ1∂tq1 + ω2

1q1 = −2ω0ω1q0Bx

L, (2.89)

∂2t x + 2δm∂tx + ω2

mx = −2ω0ω1q0q1B

ML. (2.90)

Slowly varying amplitudes and the frequency detuning are introduced as:

q0(t) = D0(t)e−iω0t + D⋆

0(t)eiω0t, (2.91)

q1(t) = D1(t)e−iω1t + D⋆

1(t)eiω1t, (2.92)

x(t) = X(t)e−iωmt + X⋆(t)eiωmt, (2.93)

∆ω = ω0 − ω1 − ωm. (2.94)


The equations of motion for the slowly varying amplitudes are:

∂tD0 + δ0D0 = −iBXD1ω1

Lei∆ωt, (2.95)

∂tD1 + δ1D1 = −iBX⋆D0ω0

Le−i∆ωt, (2.96)

∂tX + δmX = −iBD0D⋆1ω0ω1

MωmLe−i∆ωt. (2.97)

To derive the condition for parametric instability in the resonance case ∆ω =

0, we Fourier transform Eqs. (2.95) and (2.96) into the frequency domain and

combine the resulting equations. Then we get:

D1(δ1 − iΩ) = −iBD0ω0

L× iBD⋆

0D1ω0ω1

MωmL(δm + iΩ), (2.98)

where Ω is the Fourier frequency. Parametric instability takes place when the

right-hand side of this equation is larger than the left-hand side at Ω = 0, viz.:

ε0Λ

2mω2mL2

ω1ωm

δ1δm> 1, (2.99)

where

ε0 =∂tq

20 + ω2

0q20

2= 2ω2

0|D0|2 (2.100)

is the energy stored in the fundamental optical field, and

Λ =V (∫

f0(~r⊥)f1(~r⊥)uzd~r⊥)2

∫

| f0 |2 d~r⊥∫

| f1 |2 d~r⊥∫

| ~u |2 dV, (2.101)

is the spatial overlap factor between the two optical modes and the mechanical

mode.

Parametric Gain

A parametric gain [66, 82] can be defined as:

R ≡ ε0Λ

2mω2mL2

ω1ωm

δ1δm, (2.102)

where ε0 can be defined by:

ε0 =2LP0

c, (2.103)


where P0 is the intracavity power.

In Eq. (2.102), if R < 0, energy is extracted from the mechanical mode and its

effective temperature is cooled; if 0 < R < 1, the the mechanical mode amplitude

is increased by a factor of 1/(1 − R); if R > 1, the mechanical mode amplitude

increases exponentially. For instance, in the East arm of the HOPTF at Gingin,

L=74m, ω1 = 2×1015 Hz, ωm ≈ 9×105 Hz, δ1 ≈ 600 Hz, δm ≈ 0.3 Hz. If Λ = 1

and P0 = 1 kW, R ≈ 0.6; this result corresponds to an amplification factor of

1/(1-R)≈ 2.5. However, if P0 = 10 kW, R ≈ 6 and parametric instability will

occur in this cavity.

Ring-up Time

In the general case, the HOM and the mechanical mode can be expressed as:

D1(t) = D1eλ−t, X⋆(t) = X⋆eλ+t,

λ− = λ − i∆ω

2, λ+ = λ +

i∆ω

2. (2.104)

The characteristic equation of λ− and λ+ is written as:

(λ+ + δ1)(λ− + δm) − A = 0, (2.105)

where

A =D2

0ω20ω1Λ

mωmL2. (2.106)

The solutions of λ in Eq. (2.105) are:

λ1,2 = −δ1 + δm

2±

√Det, (2.107)

Det = (δ1 − δm

2− i∆ω

2)2 + A. (2.108)


At resonance, ∆ω = 0, and the solution for λ1 is:

λ1 = −δ1 + δm

2+

√

(δ1 − δm

2)2 + δ1δmR

= −δ1 + δm

2+

√

(δ1 + δm

2)2 + δ1δm(R − 1)

= −δ1 + δm

2+

δ1 + δm

2

√

1 +4δ1δm(R − 1)

(δ1 + δm)2

≈ −δ1 + δm

2+

δ1 + δm

2(1 +

2δ1δm(R − 1)

(δ1 + δm)2)

≈ δm(R − 1). (2.109)

Note that the λ2 solution does not lead to an exponential amplification (i.e.

parametric instability).

The ring-up time constant τ of the exponentially increasing HOM is given

by:

τ =1

λ=

1

δm(R − 1). (2.110)

For instance, if R = 1, τ = ∞; if R = 18, τ ≈ 0.2s.


Chapter 3

Gingin High Optical Power Test

Facility

3.1 Overview

The High Optical Power Test Facility (HOPTF) was proposed as a prototype

gravitational-wave detector located at Gingin, Western Australia (see Fig. 3.1).

It consists of two arms, namely the South arm and the East arm, which are

perpendicular to each other. At the current time, the two arms work separately.

The setups in the East arm are upgraded versions of those in the South arm.

The South arm was assembled in 2006-2007, while the upgraded setups were

assembled in the East arm in 2011-2012. These two arms will be introduced in

Sects. 3.2 and 3.3, respectively.

Both arms are ∼80 meters long, and enclosed in vacuum pipes and tanks.

To work properly, each arm is composed of several systems, as follows: a stable,

low-noise laser source (in the laser source system) generates a high power laser

beam, which is mode-matched and modulated by several optical components

(in the beam-injection system) before entering the Fabry-Perot (FP) cavity (in

the FP cavity system). The laser beam is frequency-locked to the cavity via a

Pound-Drever-Hall (PDH) locking system. The two test masses of the cavity

are suspended by a vibration isolation system, and controlled by a feedback

45

46 CHAPTER 3. GINGIN TEST FACILITY

Figure 3.1: The HOPTF, located in a bush field at Gingin, Western Australia.Figure provided by Li Ju.

Figure 3.2: Conceptual diagram of a complete configuration of the GinginHOPTF. Currently the two recycling tanks are not installed. Figure providedby Li Ju.

3.2. THE SOUTH ARM 47

control system. Eventually, the reflection or transmission beam from the cavity

is measured by a readout system. When it is fully assembled, the HOPTF will

work as a laser interferometer similar to aLIGO and aVIRGO (see Fig. 3.2).

3.2 The South Arm

3.2.1 Laser Source

Figure 3.3: Laser source system used in the South arm. A commercial 50 mWNPRO laser (left) and a homemade 10 W slave laser (right) form a master-slavelaser system.

The laser sources in gravitational-wave detectors are required to be stable,

and low-noise in frequency, amplitude and phase. A 10 W slave laser was devel-

oped by one of our collaborators — the Optics and Photonics Research Group

in the University of Adelaide [83]. A diode array, which is illuminated by a

master laser, generates a collimated planar pumping light beam onto a gain

medium, which is made of a Nd:YAG (neodymium-doped yttrium aluminium

garnet) crystal slab. The master laser beam is generated by a commercial 500-

mW Non-Planar Ring Oscillator (NPRO) [84] laser source (InnoLight, Model

Mephisto 500NE). This NPRO laser is ultra-stable and low-noise in terms of fre-

quency and amplitude, satisfying the requirements of a master laser. The 10 W


master-slave laser system is mounted in a compound cooling system which con-

sists of a layer of indium, a chunk of copper and a temperature control system.

The whole system is air-cooled via thermal conduction (see Fig. 3.3).

To obtain the maximum output power from this master-slave laser system,

an injection-locking scheme is required. Specifically, the laser beam generated

from the master laser source is incident on the slave laser crystal, and mode-

matched with one of the free-running modes of the crystal. The laser beam is

then resonant in the crystal, and produces a single-frequency, low-noise laser

output. The wavelength of the output laser is exactly the same as that of the

master laser, which is 1064 nm. The frequency noise of this output laser is also

the same as that of the NPRO laser. The layout of this injection-locking scheme

is shown in Fig. 3.4.

3.2.2 Injection Table and Cavity

The beam produced by the laser source needs to be mode-matched before

entering the cavity. On the injection table, a set of optical components is used to

mode-match the beam to the cavity (Fig. 3.5). A photodetector (PD) receiving

the reflected beam is used for PDH-locking of the laser frequency to the cavity

length. A Hartmann wavefront sensing (HWS) system is applied here to monitor

the radius of curvature of the input test mass (ITM).

After the injection table, the laser beam enters the South arm cavity, whose

mirrors are made of sapphire. The ITM is a flat mirror which is 100 mm in

diameter and 46 mm thick. The end test mass (ETM) has a radius of curvature

of 720 m, and is 150 mm in diameter and 80 mm thick. The cavity g-factor is

∼0.89. In order to investigate the thermal lensing effect of the test mass, the

ITM was reversed so that the coating face was toward the outside of the cavity.

If a laser beam is resonant inside the cavity, the power absorbed by the ITM

substrate will heat the test mass and make it a convex lens.

3.2.3 PDH Locking System

3.2. THE SOUTH ARM 49

Figure 3.4: The optical layout of an injection-locked slave laser. The outputfrom the NPRO master laser is linearly polarized, and is mode-matched by acombination of optical elements, such as lenses, half-wave plates (λ/2), and aquarter-wave plate (λ/4). The reverse beam propagating back to the NPROlaser is blocked by a Faraday Isolator (FI) since the polarization of the beaminjecting into the FI can be tuned by the half-wave plates. The slave laser islocked to the master laser by a PDH locking scheme, which involves a phasemodulation by an ElectroOptic Modulator (EOM). The Forward-Wave (FW)photodetector is there to collect the error signal from the PDH locking, whilethe Reverse-Wave (RW) photodetector is used to monitor the output of the slavelaser. Figure reproduced from Fan (2010) [58].

Because the cavity is suspended, and the laser has frequency and amplitude

noises, the resonant conditions cannot be always satisfied without an appropriate


Figure 3.5: Setup of the injection table in the HOPTF South arm injectionroom. Figure reproduced from Fan (2010) [58].

slave laserFaraday

IsolatorBS Cavity

photodetector

Figure 3.6: Pound-Drever-Hall (PDH) locking system.

3.3. THE EAST ARM 51

locking system. With the advantages of simplicity, stability, and high dynamic

range, the PDH locking system [85, 86] is widely used in gravitational-wave

detection, and in microscopy. Both the South and East arms use PDH systems

to lock the laser frequency to the cavity length.

The idea of this locking scheme is to measure the slope of the cavity response

function at resonance, and then apply feedback control to the laser frequency

according to this measurement. A typical setup of a PDH locking system is

shown in Fig. 3.6. An incident laser beam is reflected back from the cavity.

This reflected beam is then received by a photodetector (PD). Two sidebands

of the incident beam are created by an EOM, and are also received by the PD.

The radio frequency (RF) signals from the PD are demodulated in a mixer by

a local oscillator (LO). The DC signals from the mixer are then extracted via

a low-pass filter, and then amplified by a servo-amplifier. The amplified signals

are sent to a piezo-electric transducer (PZT) behind one of the cavity mirrors,

and inside the laser source. As a result, the laser frequency is adjusted to always

conform to the FP cavity length.

3.3 The East Arm

3.3.1 50 W Fiber Laser Amplifier

A 50 W fiber laser amplifier (NuFern, NUA-1064-PV-0050-D0) is used as the

laser source for the East arm. A seed laser is injected into this fiber amplifier

through an optical fiber, and amplified by doped optical fibers as active gain

medium. The output of the fiber amplifier is highly collimated and linearly

polarized. In our experiment, the seed laser is produced by the 500 mW NPRO

laser, and the input power of the fiber amplifier is ∼120 mW. The output power

as a function of diode current was tested (see Fig. 3.7). Because the output

optical mode is dependent on the diode current, the output power should be

fixed in an experiment. The power incident on the arm cavity is adjusted by

inserting a polarizer plate into the optical path.


Figure 3.7: The 50 W fiber laser amplifier is put on the East arm injection table(left). The output power as a function of the diode current was measured (right)(provided by David Hosken).

One of the benefits of this fiber amplifier is that it produces a clean beam pro-

file, which is very important in mode-matching the laser to the cavity. This fiber

amplifier also maintains the frequency and ultra-narrow frequency linewidth of

the seed laser. Therefore, the frequency noise is the same as in the seed laser.

The input, output and back-reflection power are monitored by the operating

system. It shuts down automatically if there is a back-reflection.

3.3.2 Injection Table and Cavity

Several optical components, such as lenses and concave mirrors, are used in

the East arm injection table to mode-match the beam to the East arm cavity (see

Fig. 3.8). The reflection beam from the cavity is split by a 95% reflective beam

splitter. The transmitted beam from this splitter is received by a PD, whose

radio frequency signal is used to PDH-lock this cavity. The same reflection

beam is also used in an Automatic Alignment (AA) system for the East arm.

The design and setup of this AA system will be introduced in Chapter 8.

The two cavity mirrors are made of fused silica, and are almost identical.

Each mirror is cylindrical, 100 mm in diameter and 50 mm thick. Similar to

the arm cavities in aLIGO, the East arm cavity in HOPTF is a near-concentric

one. The nominal radii of curvature of the ITM and ETM are 37.4 m and 37.3

m, respectively. The nominal cavity g-product is 0.96. The transverse mode


Laser source

Faraday

IsolatorC

avity

PD for

PDH locking

f=10 m

f=4 m

BS

BS (R=95%)

Table

Figure 3.8: The schematic setup on the East arm injection table before im-plementing the Automatic Alignment (AA) system (see Chap. 8). The laseris amplified by a fiber laser amplifier, which is not shown in this Figure. Thecavity is not drawn to scale.

frequency offset (TMFO) is small, and can be easily tuned by using a CO2 laser

to heat the mirror. This is a key point of the optical tuning in the following

Chapters. Two cameras are installed inside the cavity to monitor the test masses.

By monitoring the resonant beam spots on the test masses, the cavity alignment,

circulating power and resonance optical mode can be determined. Therefore, the

cavity resonance can be optimized by controlling the angular positions of the

test masses. The feedback control system will be described later in Sect. 3.3.4.


3.3.3 Advanced Vibration Isolation System

The vibration isolation system is used to isolate the test masses from environ-

mental vibrations, such as seismic noise and human activities. The strategy of

this system is to reduce the eigenmode frequencies of the pendulums from which

the test masses are suspended. Each vibration isolation system is installed within

a 3 m stack. It consists of 3 main parts: an ultra-low-frequency pre-isolator, a

Roberts linkage and a 3-stage cascaded isolator. The pre-isolator is composed of

an inverse pendulum as a horizontal isolator and a LaCoste linkage as a vertical

isolator. The 3-stage isolator is composed of Euler springs as vertical isolators

and self-damping pendulums as horizontal isolators. A schematic setup of the

whole system is shown in Fig. 3.9 [87, 88].

Inverse pendulum

The inverse pendulum stage contains 4 inverse pendulum legs with a square

table mounted on the tops of these legs (see Fig. 3.10). The motion has two

degrees of freedom: x and y. The frequency of this stage can be tuned to well

below 100 mHz by adjusting the load mass. The dynamic range of this stage is

±10 mm in x and y. It allows the isolators to effectively buffer tidal effects and

temperature drifts. Due to the softness of inverse pendulums, this stage can be

easily controlled by adding magnetic actuators. The sensing and controlling of

the inverse pendulums is similar to the feedback control system, which will be

described in Sect. 3.3.4.

LaCoste linkage

The LaCoste linkage stage (see Fig. 3.10) has an anti-spring geometry, where

the restoring force is larger than 0 when the spring is at its original length. A

few vertical springs near each leg of the inverse pendulum stage are connected to

the load. Similar to the inverse pendulums, the LaCoste linkage also has a large

dynamic range. It can be controlled vertically not only by magnetic actuators,

but also by running a current in the spring to change its coefficient of thermal

expansion.


Figure 3.9: Advanced vibration isolation system used in the HOPTF East arm.All the isolation stages are labeled in this Figure. Figure reproduced from Dumas(2009) [88].

Roberts linkage

The Roberts linkage stage is another horizontal isolator, as shown in Fig.

3.10. It contains a frame and four wires hung off the LaCoste linkage. By using

a geometry of solid joints, the mass suspension point can be restricted to move in

a horizontal plane. The gravitational potential is nearly independent of the load


Figure 3.10: Pre-isolator (left) and Roberts linkage (right). A pre-isolator con-sists of an inverse pendulum and a LaCoste linkage. The inverse pendulummoves in the x direction, while the LaCoste linkage has a degree of freedom inthe z direction. Figures adapted from Dumas (2009) [88].

displacement. Therefore, the restoring force is minimized. The geometry can

be adjusted to optimize the flatness of the suspension point motion, in order

to minimize the resonant frequency of this stage. A control method can be

realized by sensing the motions of the frame, and applying the sensed signals

to the electric currents running in the suspension wires. The lengths of these

wires will be changed by the currents. This method is only valid for controlling

low-frequency vibrations. The electric current noises are automatically avoided

in the feedback control.

Euler springs

Three stages of Euler springs are suspended from the Roberts linkage. An

Euler spring (see Fig. 3.11) has a small buckling load, beyond which it becomes

an ideal spring. Euler springs are used in the isolation system due to the fol-

lowing advantages. First, the mass of an Euler spring is minimized to meet the

requirement of supporting the load. This results in internal resonances with

high frequencies, which are far away from the vibration frequencies to be iso-

lated. Second, an Euler spring only needs half the length of a “zero-length” coil

spring to have the same internal resonant frequency. This benefits in a compact

isolator design. Each of the three stages consists of 8 Euler spring blades, each

of which is 200 mm long and 10 mm wide. The thicknesses of the Euler springs


are different at different stages, because the loads of these three stages are not

the same. The blades are made of maraging steel to avoid length creep, which

is detrimental to the isolation system.

Figure 3.11: Euler springs (left) and self-damping pendulum (right). There arethree stages of Euler springs in a vibration isolation system. The relative motionbetween the rocker mass and the pendulum frame is damped through inducededdy currents. Figures reproduced from Dumas (2009) [88].

Self-damping pendulum

The frame of the self-damping pendulum supports a rocker mass via two-

dimensional (2D) flexures. Powerful magnets are attached to the edges of the

rocker mass, and copper plates are attached to the corresponding surfaces of

the frame (see Fig. 3.11). When the pendulum experiences a swing motion, the

rocker mass will have an angular motion relative to the frame. This angular

motion then induces an eddy current in the copper plates. The mechanical

energies of the pendulum are then transferred to electrical and thermal energies,

which will dissipate into the environment. Therefore, the pendulum is damped

by this energy dissipation process.

3.3.4 Feedback Control System


Shadow sensors and magnet-coil actuators

Figure 3.12: Shadow sensor (left) and actuator (right) used in the feedbackcontrol system. Figures reproduced from Dumas (2009) [88].

Figure 3.13: Interface of the digital control system.

Other than being isolated from environmental vibrations by the vibration

isolation system, the test mass can be feedback controlled by a control mass


(see Fig. 3.9). The basic elements of this control system are shadow sensors

and magnet-coil actuators [88], as shown in Fig. 3.12. In a shadow sensor, an

infrared LED shines a beam onto two PDs. A shadow card, which is attached to

the control mass, is positioned perpendicularly to the PDs and casts a shadow

onto them. When the control mass has a displacement, the difference between

the signals of the two PDs will have a linear response to that displacement.

These signals are amplified and sent to a digital control system (see Fig. 3.13),

via a digital signal processor (DSP). The dynamic range of this shadow sensor is

±5 mm, with a displacement sensitivity of 10−10 m/√

Hz [89]. In an actuator, a

magnet is attached to the control mass, and inserted between two coils which are

mounted on a fixed frame (see Fig. 3.12). The magnet is driven by a magnetic

force induced by the running currents in the coils. The magnetic force is nearly

uniform over the central 10 mm range between the coils.

Figure 3.14: 5 pairs of shadow sensors used for controlling the test mass. 3 pairsare sensing and actuating in the horizontal and yaw directions (left), and 2 pairsare for the vertical and pitch directions (right). Figures adapted from Dumas(2009) [88].

The two angular degrees of freedom of the test mass, namely ‘pitch’ and

‘yaw’, are controlled by 5 pairs of shadow sensors and actuators. Three pairs

are arranged 120 to each other, to convert horizontal displacements to signals

in yaw; the other two pairs are arranged along the propagation direction of the

laser beam to convert vertical displacements to signals in pitch (see Fig. 3.14).

The displacements in the x, y and z directions can also be deduced from these

shadow sensor signals.

The shadow sensors and magnet-coil actuators are also used in controlling

the vibration isolation stages. For instance, the inverse pendulum is sensed and

actuated in x, y and yaw by using 4 pairs of shadow sensors and actuators. The


LaCoste linkage and Roberts linkage use actuators, as well as heated suspension

springs or wires, as their control method.

Optical lever

The angular displacements of the test mass in the yaw and pitch directions

are directly sensed by an optical lever system, as illustrated in Fig. 3.15. A laser

beam is directed onto the test mass, and reflected to a quadrant photodetector

(QPD) 5 m away. The angular motions of the test masses in yaw and pitch are

registered as the differential signals of the QPD in x and y, respectively. The

arm length of the optical lever (5 m) is much longer than that of the shadow

sensors (200 mm). Therefore, the sensitivity to the angular motions of the test

mass is much better in the optical lever than in the shadow sensors. However,

the dynamic range of the optical lever is small, due to the limited size of the

QPD. In experiments, the angular motions of the test mass are initially damped

by the feedback signals from the shadow sensors. The optical lever is used while

the angular motions are damped down to within its more limited dynamic range.

Figure 3.15: Schematic setup of the optical lever sensing system. Figure repro-duced from Dumas (2009) [88].

3.4 Conclusion

3.4. CONCLUSION 61

This Chapter briefly introduces the Gingin HOPTF, which consists of the

South arm and the East arm. Some major systems are described in this Chapter.

In the next 5 Chapters, some of the experiments undertaken in this Facility

will be introduced. The three-mode optomechanical transducer experiment (see

Chap. 4) was undertaken in the South arm, while the experiments in Chaps. 5

through 8 were undertaken in the East arm.


Chapter 4

High sensitivity transducer using

three-mode optomechanical

interactions

4.1 Preface

In the South arm of the Gingin High Optical Power Test Facility, a high-

sensitivity optomechanical transducer was demonstrated, by using three-mode

optomechanical interactions. A displacement sensitivity of ∼1 × 10−17 m/√

Hz

with a signal-to-noise ratio of at least 20 dB was achieved in this experiment.

This Chapter is based on a pre-publication paper draft which was mainly

written by the author. The co-authors of this paper have since modified the

introduction section, and made minor improvements to the language. The paper

was published in Physical Review A by C. Zhao, Q. Fang et al. [90]. The author’s

contribution is 80% of the theoretical work under the supervision of Dr. Haixing

Miao, 30% of the experimental work, 50% of the data analysis, and 70% of the

writing of the paper. The author built the theoretical framework of three-mode

interaction, calculated the displacement spectral density for quantum noise and

thermal noise. The author also helped set up the experiment and measure the

three-mode interaction. All the data analysis and curve fitting for the measured

spectrum was done by this author.

63

64 CHAPTER 4. THREE-MODE OPTOMECHANICAL TRANSDUCER

4.2 Introduction

Currently there is great interest in the coupling of mechanical oscillators to

optical fields. Optomechanical coupling provides a means of studying the quan-

tum behavior of macroscopic mechanical degrees of freedom, and also enables

high-sensitivity probes for quantum-noise-limited measurements [91]. Many ex-

periments have demonstrated optical cooling of mechanical oscillators [92, 93],

paving the way for creating quantum entanglements involving mechanical de-

grees of freedom for quantum information [94, 95, 96, 97], and for probing me-

chanical energy quantization [98, 99]. Recently a GHz micro-mechanical os-

cillator has been thermodynamically cooled to its quantum ground state, and

prepared in a single quantum state by coupling it to a superconducting qubit

[100]. On a larger scale, laser interferometer gravitational-wave detectors LIGO

[101] and VIRGO [102] have achieved a displacement sensitivity of 10−19 m/√

Hz

at around 100 Hz, while on a small scale the high-frequency thermal noise of

a mechanical oscillator has also been measured with similar sensitivity [103].

Using this high displacement sensitivity and feedback cooling, a LIGO interfer-

ometer test mass has been cooled to an effective temperature of 1.4 µK [104],

corresponding to an occupation number of ∼200 quanta.

In most opto-mechanical interaction experiments to date, a single TEM00

optical mode is coupled to a mechanical oscillator mode. For such two-mode

interactions, the optical quality factor is often sufficiently low that the me-

chanical sideband frequencies occur within the linewidth of the optical mode.

Three-mode optomechanical interactions which include two optical modes were

first investigated theoretically by Braginsky et al. [66] in the context of long

baseline gravitational wave detectors. He showed that such interactions could

induce parametric instability in the high optical power cavities of advanced

gravitational-wave detectors through an interaction that can inject acoustic en-

ergy into selected acoustic modes to the point of instability. Zhao et al. [82]

and many others [105, 106, 107] extended this analysis to include realistic mode

shape modeling. In 2009, Zhao et al. [108] pointed out that three-mode interac-

tions can be harnessed to create a general Opto-Acoustic Parametric Amplifier

(OAPA), which can function as a high-sensitivity transducer with the capability

of cooling a mechanical mode down to the quantum ground state. The high

sensitivity arises because the single sideband signal is coherently amplified when

4.2. INTRODUCTION 65

the frequency gap is equal to the frequency of the mechanical mode, and this

occurs without compromising the optical power that defines the optomechanical

coupling strength. Dobrindt and Kippenberg [109] confirmed this analysis in

the context of a 4-mode transducer using 3 optical modes, and discussed the

experimental challenge of engineering appropriate mode gaps. The single side-

band 3-mode transducer system discussed here has similar sensitivity, solves the

engineering using thermal tuning, and is simpler to implement.

In Fig. 4.1, we show the mode and frequency structure of the three-mode

interaction. Basically, the fundamental optical mode at ω0 is scattered by the

mechanical motion at ωm. This creates two sideband modes: one at ω0−ωm (also

called the Stokes mode), and the other at ω0 +ωm (the anti-Stokes mode). With

the correct frequency gap between the high-order mode at ω1 and the funda-

mental mode, namely |ω1−ω0| = ωm, one sideband mode (the anti-Stokes mode

in the case shown in Fig. 4.1) becomes resonant and gets coherently amplified.

If the mode shape of the high-order mode is similar to the mode shape of the

acoustic mode, the optomechanical coupling can be large. The high-order mode

Input Output

(a)

(c)

pumpingsidebandsideband

(b)

Figure 4.1: (a) The scheme for three-mode interactions. The anti-Stokes modeω1 has a similar shape to the internal acoustic mode. (b) The shapes of theTEM00 and TEM01 modes. The TEM00 mode has a symmetric shape, while theTEM01 mode is antisymmetric. (c) The frequency structure of the two opticalmodes and the sidebands. The laser is tuned to the fundamental TEM00 modewith frequency ω0. The mechanical motion induces two sidebands, one of whichis scattered into a high-order optical mode at ω1, which gets amplified.


carries the resonantly enhanced signal sideband. It achieves a high displacement

sensitivity with relatively low optical power, as we elaborate below. The higher

the cavity finesse, the larger the amplification, and the greater the sensitivity.

In addition, the high finesse cavity also acts as an efficient low pass filter that

reduces the effect of classical laser noise. It is interesting that the three-mode

transducer is equivalent to the signal recycling configuration used for amplifying

the signal sidebands in gravitational wave detectors [110].

4.3 Three-mode optomechanical transducer

theory

The formalism for the three-mode interaction has been presented previously

in general form in Refs. [110, 109]. We shall summarize some of the results

in this section. The three-mode interaction can be described by the following

Hamiltonian:

H =1

2~ωm(q2

m + p2m) + ~ω0a

†a + ~ω1b†b

+ ~G0qm(a†b + b†a) + Hext. (4.1)

Here, qm and pm are the position and momentum of the mechanical mode; a

and b are the annihilation operators for the fundamental optical cavity mode and

the high-order optical cavity mode (the TEM00 and TEM01 modes, respectively,

in this experiment); G0 ≡√

Λ~ω0ω1/(mωmL2) is the optomechanical coupling

constant, with Λ representing the spatial overlap between the TEM01 mode and

the mechanical mode; Hext is the coupling between the cavity modes and the

external continuous mode ain and bin (where Hext = i~(√

2γ0a†ain +

√2γ1b

†bin −H.C.), with γ0 and γ1 being the decay rates of the cavity modes and H.C. being

the Hermitian conjugate).

From the above Hamiltonian, we can derive the equations of motion for the

linearized dynamics (in the rotating frame at ω0):

4.3. THEORY 67

150. 160. 170. 180. 190. 200. 210. 220.

1020

1018

1016

Figure 4.2: (Color online) The displacement noise spectrum density. (a) Theblue dashed line is the spectrum of the quantum noise ∆xq(Ω); the red solid lineis the thermal noise ∆xth(Ω).

¨qm + 2γm˙qm + ω2

mqm = G0(b + b†) + Fth + Fsig, (4.2)

˙b + (γ1 + i∆)b = −iG0qm +

√

2γ1bin, (4.3)

where γm denotes the mechanical mode half linewidth, ∆ ≡ ω1 − ω0, G0 ≡ G0a

and a is the zero-order intracavity intensity of the TEM00 mode (we have also

included the fluctuation and dissipation mechanism of the mechanical mode and

the optical mode into the above equation); bin is the injection of the TEM01 mode

that is the vacuum fluctuation only, Fth is the thermal Langevin force, Fsig is

the signal that we seek to probe, and we neglected the intensity change of the

TEM00 mode, as it is almost constant and only determines the optomechanical

interaction strength. Such linear dynamics can be easily solved, and from the

standard input-output relation [111]:

bin(t) + bout(t) =√

2γ1b(t), (4.4)

we can obtain the TEM01 mode output bout(t) that we detect. In our experiment,


of which details will be given in the next Section, we detect the beating at the

mechanical frequency in the cavity transmitted signal. Because the scattering

sideband is always in phase with the cavity TEM00 mode, we detect the ampli-

tude quadrature b1 = (bout + b†out)/√

2 of the high-order mode. In the frequency

domain, we can rewrite the output as:

b1(Ω) = Z1(Ω) + χ(Ω)[FBA(Ω) + Fth(Ω) + Fsig(Ω)], (4.5)

where Z1 is the measurement shot noise, while FBA and Fth are the back action

force noise and thermal force noise, with χ(Ω) = [−m(Ω2 −ω2m)− iγmΩ]−1 being

the mechanical response function. Referred to the displacement, the correspond-

ing noise power spectral density of the output is:

Sx(Ω) =~L2η[(Ω + ∆)2 + γ2

1 ][(Ω − ∆)2 + γ21 ]

ω0I0T1∆2

+4~ω0I0

T1c2η|χ|2 + 4mγmkBT |χ|2. (4.6)

The first and second terms are the power spectral densities of the shot noise and

back action noise, respectively, with T1 being the power transmissivity of the end

test mass, I0 being the intracavity power for the TEM00 mode, and η being the

quantum efficiency of the quadrature photodiode. The third term is the power

spectral density of thermal noise, in which m is the effective test mass, kB is the

Boltzmann constant, and T represents the environmental temperature.

To estimate the quantum noise-limited displacement noise spectrum and the

thermal noise level, we used a finite element analysis package, ANSYS, to cal-

culate the mechanical mode frequency, mode effective mass and the overlap

factors, which are shown in Table 4.1. We measured a mechanical Q-factor of

∼1.2±0.2×106, as explained in the next Section. Figure 4.2 shows the quantum

noise-limited noise spectral density (blue dashed line) and the thermal noise at

room temperature (red solid line). In this Figure, we can see that the quantum

noise-limited spectrum reaches its minimum when the TEM01 mode is resonant

inside the cavity, where the TEM00 mode is also resonant. This is because the

measurement shot noise at the quadrant photodetector (QPD) is constant at all

frequencies, and the signal, which is proportional to the scattering sideband, is

coherently amplified by the cavity resonance. This increases the signal to shot

noise ratio by the cavity resonance factor. At the optical power level used in the

4.3. THEORY 69

experiment described here, the quantum back action noise is still much smaller

than the shot noise.

Parameter ITM ETM

Radius of Curvature (m) R1 = ∞ R2 = 720Materials Sapphire SapphireDiameter (mm) 100 150Thickness (mm) 50 80HR transmission (ppm) 1840 ± 100 20 ± 20AR reflectivity (ppm) 29 ± 20 12 ± 12Cavity length (m) 77Mass (kg) 1.5 5.6Mode frequency (kHz) 181.6Mode effective mass (kg) 0.28Mode overlap 0.4QPD Quantum efficiency 0.2

Table 4.1: Parameters of the two test masses used in the current experiment.

Laser

QPD

ITM ETM

Figure 4.3: Plot showing the experimental setup with three-mode interaction.The Fabry-Perot cavity consists of two sapphire test-mass mirrors separated by77 meters (ITM: input test mass; ETM: end test mass). The radius of curvature(RoC) of the ETM is thermally tuned by a CO2 laser, enabling the frequencygap between the TEM00 and TEM01 modes to be tuned to the mechanical-modefrequency. This coherently amplifies the mechanical sideband signal, which isthen detected by a quadrant photodetector (QPD). The apparatus also containsa thermally tunable compensation plate (CP) that enables changing the effectiveRoC of the ITM.


4.4 Experimental Results

A 77 m high optical power cavity is used to investigate the three-mode in-

teractions. The experimental setup is shown in Fig. 4.3. We use a 10-Watt

Nd:YAG laser at a wavelength of 1046 nm. The laser is frequency stabilized to

the fundamental TEM00 mode of the 77-meter Fabry-Perot cavity. After pass-

ing through the mode-matching optics, the remaining optical power that enters

the cavity is about 3 ± 0.3 W. With a cavity finesse of (1.3 ± 0.1) × 103, the

intracavity power is 310±50 times as much as the input power, achieving almost

1 kW of circulating power.

In order to satisfy the resonant condition for the three-mode interaction, we

need to tune the frequency gap between the TEM00 and TEM01 modes to match

the mechanical mode frequency. This is achieved by thermally tuning the RoC

of the ETM with CO2 laser heating, which uses the fact that the frequency gap

depends on the RoC of the mirrors through the following relation:

ω0 − ω1 =c

Larccos

[√

(

1 − L

R1

)(

1 − L

R2

)

]

, (4.7)

where R1 and R2 are the RoCs of the input test mass and end test mass mirrors

(ITM and ETM here), and L is the cavity length. As the sapphire test mass has

a high thermal conductivity, this thermal tuning is relatively fast, and we can

tune the RoC by a few percent within seconds. A nominally flat anti-reflection

(AR)-coated fused silica compensation plate (CP) with a circumferential heating

element was previously used for compensating thermal lensing in the ITM [112].

This provides an independent tuning mechanism of the effective RoC of the

ITM. Because of the low thermal conductivity of the CP, the process of tuning

is slow and has large thermal inertia. This enables slow sweeping of the effective

RoC, but stabilization to the optimum RoC is difficult.

When the resonant conditions are satisfied, the sideband signal from the

mechanical mode coincides with the TEM01 mode. After mixing with the TEM00

mode, it gives a beating signal which can be observed in the transmitted light

from the cavity. As the spatial profile of the signal is antisymmetric (due to the

TEM01 mode), it cannot be detected by a normal single-element photodiode.

Instead we use a QPD to detect it. When we vary the RoC by adjusting the

CO2 laser heating power, we find a high Q-factor acoustic mode at ∼181.6

4.4. EXPERIMENTAL RESULTS 71

181.56 181.58 181.6 181.62 181.64 181.661 10

18

2 1018

5 1018

1 1017

2 1017

5 1017

1 1016

2 1016

Figure 4.4: Plot showing the experimental result of the measured signal. Theblue dots are the measured data, fitted by a Lorentzian curve (solid red line).The brown dash-dotted line is the calculated thermal noise spectrum; the greenshort-dashed line is the calculated shot noise; and the red long-dashed line isthe sum of the thermal noise and the shot noise.

kHz. The amplitude of this signal on the spectrum analyzer is a function of

the CO2 heating power. We optimize the heating power to obtain maximum

signal amplitude. Figure 4.4 shows the signal spectrum at optimum heating

power together with the theoretical prediction (Fig. 4.2). To prove that the

signal we detected at ∼181.6 kHz is not from the CO2 laser noise, we blocked

the CO2 laser and used the CP to thermally tune the ITM effective RoC. The

signal amplitude at ∼181.6 kHz is a function of the CP heating power, and

reaches maximum amplitude after ∼10 minutes applying ∼9 W heating power

to the CP. To estimate the level of thermal noise, we first measured the Q-factor

of the mechanical mode by resonantly exciting the mode using an electrostatic

actuator, and recording the ring-down. By fitting the recorded ring-down curve,

we obtained the Q-factor for this particular mode as ∼(1.2 ± 0.2) × 106

To identify the ETM acoustic modes, we used ANSYS to calculate the reso-

nance frequencies, the mode effective mass, and the overlap between the optical

modes and the acoustic mode. The simulation results show several acoustic

modes around 181 kHz, but with one particular mode that has high overlap


Y-position (mm)

0 25 50 75-25-50-75

Am

plit

ude (

a.u

.)

(a) (b)

Figure 4.5: (Color online) Finite element model of the mechanical mode at181.6 kHz. (a) 3D mode profile of the surface of the test mass. (b) Mode shapecross-section distribution along the central Y direction.

with the TEM01 mode, and very small vibration amplitude at the suspension

point, implying low losses into the suspension wires and high Q-factor. This

mode is shown in Fig. 4.5. The effective mass is 0.28 kg, and the overlap with

the TEM01 mode is 0.4, according to the simulation.

The total transmitted power at the QPD was measured to be ∼4± 0.2 mW.

To estimate the noise contribution to the QPD output, we blocked the cavity

transmitted laser light, and then illuminated it with a white light to create

the same photocurrent. By comparing the QPD differential outputs with and

without white light, we calculated that the shot noise level is about half of

the QPD electronic noise. Since the noise spectra are the same for both the

cavity transmitted light and the white light on the QPD, we concluded that the

transmitted laser light technical noise is negligible, and is shot noise limited at

∼181.6 kHz. The estimated radiation pressure force that could drive the test

mass internal mode motion is much smaller than the thermal noise. For this

reason, we identify the resonance peak we measured as the thermal noise peak.

Using the calculated thermal noise as a calibrator, we converted the measured

data to the displacement noise spectrum shown in Fig. 4.4.

We can see in Fig. 4.4 that the noise level off resonance is ∼10−17 m/√

Hz,

limited mainly by the sum of photodetector electronic noise and quantum shot

noise. The sensitivity of this experiment can be improved in a straightforward

fashion, by increasing the cavity finesse, the ETM transmissivity and the pho-

4.4. EXPERIMENTAL RESULTS 73

Cavity finesse ETM trans-missivity

Input power QPD quantumefficiency

15000 100 ppm 5W 0.8

Table 4.2: Optical parameters for a future experiment to be undertaken in theEast arm of the Gingin facility. The cavity finesse is an order of magnitude largerthan that in the current experiment. Other parameters are all reasonable.

181.56 181.58 181.6 181.62 181.64 181.66

5´ 10-22

1´ 10-21

5´ 10-21

1´ 10-20

5´ 10-20

1´ 10-19

Figure 4.6: Plot showing calculated quantum noise limited sensitivity for thesame mechanical parameters, and with the optical parameters listed in Table4.2. The green dashed line shows the shot noise; the brown dash-dotted lineshows the quantum radiation pressure noise; and the red long-dashed line showsthe sum of these two noise contributions.

todetector quantum efficiency. With the optical parameters shown in Table 4.2,

and the same mechanical parameters as in the current experiment, we would

expect to achieve a displacement sensitivity of ∼ 2 × 10−19 m/√

Hz at the me-

chanical resonance frequency, limited by the quantum radiation pressure noise,

and ∼ 1 × 10−20 m/√

Hz outside the linewidth limited by the quantum shot

noise, as shown in Fig. 4.6. If the test mass were thermodynamically pre-cooled

to 1 K, it would be possible to cool the kilogram test mass internal mode to

its quantum ground state using the intrinsic self-cooling of the cavity combined

with radiation pressure feedback cooling [113].


4.5 Conclusion

We have demonstrated for the first time the intrinsic high sensitivity of a 3-

mode opto-acoustic parametric amplifier that uses relatively low optical power.

The measurement scheme has intrinsic immunity to laser amplitude and phase

noise, and could be used to achieve ground state cooling in kilogram-scale test

masses. The technique has applications to quantum non-demolition, measure-

ment of radiation pressure noise, and to the precision monitoring of test mass

modes in gravitational wave detectors.

Chapter 5

Degeneracy Losses in a 74m

Suspended Fabry-Perot Cavity

5.1 Preface

Three-mode optomechanical interactions were investigated in the previous

Chapter. In order to observe parametric instability (PI), by applying three-

mode optomechanical interactions, we need a high circulating optical power

inside the laser cavity.

A 50 W fiber laser amplifier and low-optical-loss test masses were installed

in the East arm of the Gingin High Optical Power Test Facility, with the goal

of making an observation of PI. Before conducting this experiment, the optical

properties of the cavity needed to be characterized. In this Chapter, we accu-

rately measure the cavity length, linewidth, finesse and g-factor. While tuning

the cavity g-factor through a wide range, we observed some significant reductions

(up to ∼30%) of the cavity finesse. Based on our theoretical model developed in

this Chapter, these dropouts of finesse correspond to different transverse mode

degeneracies. The optical losses in these degenerate higher-order optical modes

(HOMs) are also estimated, according to our model.

This work led to a paper written by the author. The author’s contribution

is 50% of the experiment, and 100% of the theoretical work and data analysis.

The experiments were undertaken together by this author and Carl Blair, in-

cluding characterizing the cavity, observing the optical mode degeneracy. The

theoretical model of the mode degeneracy was built by this author. And all the

75

76 CHAPTER 5. CAVITY DEGENERACY LOSSES

calculation of the finesse and optical loss was done by this author. The paper

will be submitted to Optics Express. The version of this paper presented here is

current as of the thesis submission date (April 2015).


5.2 Introduction

Advanced laser interferometer gravitational-wave observatories, such as Ad-

vanced LIGO (aLIGO) and Advanced VIRGO (aVIRGO), are currently being

commissioned [114, 115]. These detectors are some of the most sensitive instru-

ments on Earth. To achieve such high sensitivities requires many sub-systems

for noise suppression. An example is the complex vibration isolation systems

that suspend the mirrors so that, in the detection band, they behave like free

masses. Another example that is of interest here is that the optical cavities in

these interferometers are very close to being concentric, allowing for large laser

spot sizes on the mirrors, which average out the thermal noise effects from the

mirrors and mirror coatings. An important subsystem for noise suppression, of

particular interest in this paper, is one where the laser power inside the arms

of these interferometers will be close to 1 MW, in order to suppress quantum

fluctuations or shot noise from the laser. Such high laser power is stored within

a power recycling cavity encompassing the Fabry-Perot cavities that make each

arm of an advanced laser interferometer gravitational-wave detector.

The High Optical Power Test Facility (HOPTF) at Gingin, Western Aus-

tralia, was designed to investigate optomechanical effects under a high laser

power in a high finesse near-concentric cavity. In particular, a phenomenon

called three-mode parametric instability (PI) [66], which is predicted, and now

observed [68, 67, 116], to limit the contained power and hence the sensitivity

of advanced gravitational-wave detectors. With the UWA advanced vibration

isolation system [88], the facility has been instrumental in understanding many

aspects of the operation of such suspended cavities. With the ability to tune

the cavity g-factor, we can investigate effects relating to the limited stability

of near-concentric cavities. The particular effect of interest in this paper is the

losses from degenerate coupling to high-order transverse optical modes that have

high clipping losses.

To demonstrate that our cavity is a good model for advanced detector arm

cavities, a comparison of useful parameters [117, 118, 119, 120] is presented in

Table 5.1. The magnitudes of the opto-mechanical effects produced by the laser-

cavity system are proportional to the intra-cavity power, and to the test mass

ratio; degeneracy is determined by the cavity g-factor.


aLIGO aVIRGO GEO600 KAGRA HOPTFArm length 4 km 3 km 600 m 3 km 74 mCavityfinesse

450 443 1550 N/A 14500

Arm power 800 kW 700 kW 20 kW 400 kW 100 kWMirrormass

40 kg 42 kg 6 kg 23 kg 0.88 kg

Mirrormaterial

fusedsilica

fusedsilica

fusedsilica

sapphire fusedsilica

Topology DRFPMI DRFPMI DRMI DRFPMI FPC

Table 5.1: Some cavity parameters of gravitational-wave detectors and HOPTF.DRFPMI stands for Dual-Recycled Fabry-Perot Michelson Interferometer,DRMI stands for Dual-Recycled Michelson Interferometer, FPC stands forFabry-Perot Cavity. GEO600 has no Fabry-Perot configuration for its arm cav-ity.

In the HOPTF, a high finesse Fabry-Perot (FP) cavity has been installed.

This 74m cavity is composed of two fused silica mirrors: the input test mass

(ITM) and the end test mass (ETM). The nominal radii of curvature of the ITM

and ETM are 37.4 m and 37.3 m, respectively. Their nominal transmissions are

200 ppm and 20 ppm, respectively. The dimensions of each of the cylindrical

test masses are 10.0 cm in diameter, and 5.0 cm thick, with modifications of

2 flats and 4 holes for a high Q-factor niobium suspension [121]. Each mirror

weighs 0.88 kg and is suspended freely by a vibration isolation system [88]. A

non-planar ring oscillator (NPRO) laser is locked to the cavity using the Pound-

Drever-Hall (PDH) locking technique [86]. To tune the cavity g-factor, a CO2

laser is used to heat the ITM. The setup of our cavity is shown in Fig. 5.1.

Other than being a testing apparatus for PI, our cavity is also expected to

play a key role in researching other high optical power opto-mechanical phe-

nomena, such as the double optical spring effect for modifying the test mass

dynamics [122, 123, 124], and in monitoring the thermal distortion of the test

masses [125].

During the characterization of our cavity, in which the cavity length, the

cavity line-width, and the transverse mode frequency offset (TMFO) were de-

termined, anomalies in the cavity finesse were discovered. We investigated the

degeneracies between different modes which resulted in these finesse anomalies.

This paper is arranged as follows: Section 5.3 introduces a theoretical model for

5.3. MODELING 79

Faraday

IsolatorBS

Cavity

4 2

photodetector

500mW laser

He-Ne laser

EOM

local

oscillator

lowpass

filter

servo

amp

spectrum

analyzer

PC

CO2 laser

quadrant

photodetector

ITM ETM

L1

Figure 5.1: Setup of the high finesse HOPTF cavity. The two fused silica testmirrors have transmissivities of 200 ppm (ITM) and 20 ppm (ETM), respectively.The quadrant photodetector (QPD) at the output port is for monitoring the beatnote signals between the TEM00 and higher order modes. The lens L1 is foradjusting the beam size on the QPD. The focal length of L1 is 100 mm.

estimating the clipping losses of the cavity higher-order optical mode (HOM),

and accounting for the effect that the degeneracy has on the cavity finesse.

Section 5.4 gives the characterization results of the cavity, followed by the de-

generacy results in Section 5.5. The conclusions of this study are given in Section

5.6.

5.3 Modeling the Optical Cavity Finesse

in Degeneracy

The model used for our cavity is a spherical-mirror FP cavity (see Fig. 5.2). The

cavity length is roughly 74 m, which will be measured accurately in the next

Section. The ITM and ETM parameters were given in the previous Section.

The radius of curvature of the ITM can be tuned by CO2 laser heating; without

this heating, the radii of curvature of the ITM and ETM should remain at their

nominal values. For a Gaussian beam resonant in the cavity, the waist size, waist

position, and laser spot sizes on the test masses can all be derived in terms of

the cavity length and the radii of curvature of the test masses. The radii of


Figure 5.2: The size of the fundamental mode resonating inside the cavity. Ddenotes the distance between the waist position and the ITM, which can becalculated according to the cavity configuration. The radii of curvature of theITM and ETM are R1 = 37.4 m and R2 = 37.3 m, respectively.

curvature of the wave fronts incident on the test masses are given by:

R(D) = D(1 +z2

R

D2) = R1 = 37.4 m, (5.1)

R(L − D) = (L − D)[1 +z2

R

(L − D)2] = R2 = 37.3 m, (5.2)

where D is the distance from the beam waist to the ITM, zR =πω2

0

λis the

Rayleigh range of the resonant beam, with ω0 being the waist size, and λ = 1064

nm is the laser wavelength. D and ω0 can be solved from Eq. (5.1) and (5.2).

The results are:

D = 37.05 m, ω0 = 1.10 mm. (5.3)

The spot sizes on the ITM and ETM can be easily calculated as:

ω(D) = ω0

√

1 + (D

zR)2 = 11.46 mm, (5.4)

and

ω(L − D) = ω0

√

1 + (L − D

zR)2 = 11.43 mm. (5.5)

For a higher-order transverse mode TEMn, m, where n, m denote the trans-

verse mode numbers, the spot sizes are approximately√

n + m + 1 times larger

5.3. MODELING 81

than the fundamental mode on the test masses. For instance, a 15th-order mode

has a beam size of 45.84 mm on the ITM and 45.72 mm on the ETM. Due to

the limited sizes of the test masses, there are clipping losses for the HOMs. The

radius of each mirror is 50 mm, thus in general the clipping loss is not significant

until the 19th-order mode. However, it is worth noting that the clipping losses

are larger in asymmetric modes than in symmetric ones with the same n + m.

For instance, TEM1, 15 has more clipping loss than TEM8, 8. Table 5.2 gives a

list of clipping losses from the 11th to the 20th-order transverse modes. The

minimum clipping loss of a certain order occurs for the most symmetric mode,

while the maximum clipping loss occurs for the most asymmetric mode in that

order.

Order Lmin Lmax Order Lmin Lmax

11 3.0 × 10−6 5.0 × 10−6 16 1.3 × 10−3 2.6 × 10−3

12 1.1 × 10−5 2.1 × 10−5 17 3.5 × 10−3 7.1 × 10−3

13 4.5 × 10−5 9.1 × 10−5 18 8.1 × 10−3 1.5 × 10−2

14 1.6 × 10−4 3.0 × 10−4 19 1.6 × 10−2 3.1 × 10−2

15 5.0 × 10−4 1.0 × 10−3 20 2.8 × 10−2 5.0 × 10−2

Table 5.2: Minimum and maximum clipping losses from the 11th to 20th-ordertransverse modes.

Degeneracy will occur when the frequencies of the fundamental mode and a

HOM are within one linewidth of each other [126, 127]. In our cavity, which is

near-concentric, the transverse mode frequency offset (TMFO) is given by:

∆f ≡ f1 − f0 = fFSR − fFSR

πarccos(

√g1 · g2), (5.6)

where f0, f1 are the frequencies of the fundamental and the first-order modes,

fFSR ≡ c2L

is called the free spectral range, g1, 2 ≡ 1 − LR1, 2

and g1 · g2 is called

the cavity g-factor. The frequency condition required for degeneracy is:

Nd · ∆f ≈ fFSR, (5.7)

where Nd = (n + m) denotes the order of the degenerate mode.

In degeneracy, the fundamental mode couples to some HOMs due to surface

scattering on the test masses. The coupling rate of the fundamental mode to

the degenerate mode is determined by their spatial and frequency overlap, and

by the mirror surface quality [128, 129]. In our model, we assume only one


degenerate mode is coupled to the fundamental mode in each degeneracy. The

effect this mode degeneracy has on the measured finesse of the cavity depends

on the total loss of the degenerate mode and the coupling rate. For example, a

degenerate mode with the same finesse as the fundamental mode will not affect

the cavity finesse, if all the transmitted light is collected by the QPD.

Finesse in the non-degeneracy case

When the cavity is tuned far away from degeneracy, only the fundamental

mode is resonant in the cavity. The equilibrium state equation of the fundamen-

tal mode is given by:√

T1Ein + r00E00 = E00, (5.8)

where T1 is the power transmissivity of the ITM, Ein and E00 are the electric

field amplitudes of the input light and the fundamental mode inside the cavity,

respectively; r00 is the total amplitude reflectivity of the cavity, which is given

by:

r00 ≈√

1 − T1 − T2 − L00 ≈ 1 − T1 + T2 + L00

2, (5.9)

where T2 is the power transmissivity of the ETM, and L00 denotes the total

optical loss of the fundamental mode. L00 is mainly contributed by optical

absorption and scattering loss. Clipping loss is negligible for the fundamental

mode. The finesse of the cavity is expressed as [?]:

F = π

2 arcsin1−r002√

r00

≈ 2πT1+T2+L00

. (5.10)

From Eq. (5.8), the intracavity power of the fundamental mode can be derived

as:

P00 =T1

(1 − r00)2Pin ≈ 4T1

(T1 + T2 + L00)2Pin, (5.11)

where Pin denotes the input optical power, which is a constant in our experiment.

Thus, the finesse can be expressed in terms of P00 as:

F = 2π

√

P00

4T1Pin

. (5.12)

5.3. MODELING 83

Finesse in the degeneracy case

On the other hand, when the fundamental mode is degenerate with a TEMn, m

HOM, the equilibrium state equations of the light fields resonating inside the

cavity are:√

T1Ein + (1 − A)r00E00 + Ar00Enm = E00, (5.13)

ArnmE00 + (1 − A)rnmEnm = Enm, (5.14)

where E00 and Enm are the electric field amplitudes of the fundamental mode

and the degenerate HOM, respectively; the parameter A denotes the coupling

rate between the two degenerate modes within a round trip, rnm represents

the amplitude reflectivity of the cavity for the degenerate mode — assuming

the transmissions of the test masses are constants for any transverse mode, the

amplitude reflectivity for the degenerate mode is given by:

rnm ≈√

1 − T1 − T2 − Lnm ≈ 1 − T1 + T2 + Lnm

2, (5.15)

where Lnm denotes the total optical loss of the HOM. Lnm is contributed not only

by optical absorption and scattering loss, but also by clipping loss. Specifically,

Lnm can be expressed as:

Lnm = L0nm + Lc

nm, (5.16)

where L0nm denotes the absorption and scattering loss of the TEMn, m HOM,

and Lcnm denotes the clipping loss of this mode. Normally, HOMs have larger

surface scattering than the fundamental mode, due to the surface roughness of

the test mass. Therefore, Lnm > L0nm ≥ L00.

The optical powers of the fundamental mode and the degenerate mode can

be obtained by solving Eqs. (5.13) and (5.14) as:

P00 =T1

[1 − (1 − A)r00 − A2r00rnm

1−(1−A)rnm]2

Pin, (5.17)

and

Pnm =T1A

2r2nm

[(1 − r00)(1 − rnm) + A(r00 + rnm) − 2Ar00rnm]2Pin. (5.18)


As with Eq. (5.10), the finesse in degeneracy can be expressed as:

F ′ ≈ 2π

T1 + T2 + P00

P00+PnmL00 + Pnm

P00+PnmLnm

, (5.19)

where P00 and Pnm are given by Eqs. (5.17) and (5.18), respectively.

From Eq. (5.19), it is clear that F ′ is dependent on the coupling rate A and

the total loss in the degenerate mode. Fig. 5.3 illustrates the effect that A and

Lnm have on the finesse F ′.

Figure 5.3: Finesse in degeneracy, as a function of the coupling rate A at differentLnm. The green, blue, red and black lines represent the finesse curves whenLnm = L00, Lnm = 2L00, Lnm = 3L00, and Lnm = 5L00, respectively.

5.4 Characterization Results

The free spectral range fFSR and cavity length L are inferred from the re-

sponse function of the cavity to a deviation of the laser frequency [130]. The

cavity is aligned such that the TEM0, 0 mode is resonant in the cavity through

a PDH locking system. This resonant condition is satisfied at every multiple of

5.4. CHARACTERIZATION RESULTS 85

the free spectral range frequency fFSR. The response function of the PDH error

signal to a frequency deviation of the laser source can be expressed as [131]:

Hω(s) =1 − e−2sT

2sT

1 − rarb

1 − rarbe−2sT, (5.20)

where ra, rb are the amplitude reflectivities of the ITM and ETM, respectively,

T is the transit time (L/c) of the light inside the cavity, s ≡ iω is the Laplace

transform parameter, and ω is the angular frequency. Conceptually, this can be

understood as the cavity having a higher transmission for the laser noise that is

resonant in the cavity, and hence there are dips in the spectrum of the reflected

light.

2.0282 2.0284 2.0286 2.0288 2.029 2.0292 2.0294 2.0296 2.02980

0.5

1

1.5

2

Measurement frequency (MHz)(b)

Magnitude

2.028 2.0282 2.0284 2.0286 2.0288 2.029 2.0292 2.0294 2.0296 2.0298

0

2

4

6

8

10

12

14

16

Measurement frequency (MHz)(a)

Magnitude

Figure 5.4: (a) Response function of the PDH error signal versus frequencydeviation measured at fFSR; the FWHM of the fitted curve is 163.6 ± 5.5 Hz.(b) Transmitted light from the cavity measured at fFSR. The blue dots are themeasurement data, the red curve is fitted. The FWHM of the curve is 138± 13Hz. Fig. 5.4(b) has been scaled to have the same frequency range as Fig. 5.4(a).

Figure 5.4(a) shows a response function of the PDH error signal. There is a

dip at 2.028965 MHz ± 2 Hz, which is the first integer multiple of fFSR. The

cavity length can be calculated as 73.92930 m, with a precision of 80 µm. The

full width at half maximum (FWHM) of the dip is expected to be the linewidth

of the cavity, which is fitted as 163.6± 5.5 Hz. A more accurate measure of this

value can be inferred from the cavity ring-down (presented later).

In addition, the peaks in the spectrum of the transmitted light can also be

used to identify fFSR. A peak was found at a frequency of 2.028974 MHz ± 1

Hz in the transmitted signal (see Fig. 5.4(b)). The linewidth fitted from this


measurement is 138 ± 13 Hz. Note that these results do not agree with those

derived from the PDH error signal, as the cavity is freely swinging with the laser

locked to the cavity. The discrepancy between the linewidth measurements is

presumed to be from extra laser noise in the PDH error signal that was not

accounted for in Eq. (5.19). Also note the improved uncertainty in the peak

frequency in the second method, corresponding to a length precision of 40 µm.

While the cavity finesse can be deduced from linewidth measurements, a ring-

down measurement from the transmission port gives a more accurate result. The

definition of the finesse is:

F =fFSR

γ=

πcτ

L, (5.21)

where γ is the linewidth of the cavity, and τ is the ring-down time constant.

A typical ring-down measurement is shown in Fig. 5.5, where τ is fitted as

1.14±0.02 ms. The finesse is thus deduced to be 14500±300, and the linewidth

is 140 ± 2 Hz.

In addition to the finesse evaluation, we can further determine the total loss

L00 of the fundamental mode in the cavity. Recalling Eq. (5.10), L00 is given

by:

L00 ≈2π

F − T1 − T2. (5.22)

For instance, the value of L00 for the measurement shown in Fig. 5.5 is ∼210

ppm.

Higher-order transverse modes also circulate inside the cavity at different fre-

quencies. The frequencies of these HOMs are critical to both PI and degeneracy

within the cavity. In order to determine the HOM frequencies, the TMFO was

measured. The differential signals of the QPD were used to isolate the odd-order

modes, especially the first-order modes. The first-order modes in the laser noise

are identified as peaks in the transmitted spectrum (see Fig. 5.6). Two peaks

spaced by δf1 = 3.882 kHz ± 17 Hz can be seen in this Figure; the multiple

peaks could be due to astigmatism in our cavity.

In addition, the cavity g-factor g1·g2 can be determined by this measurement.

The mean value of the two peaks in Fig. 6 is 99.289 kHz ± 9 Hz. Thus, g1 · g2

is derived to be 0.976551 ± 1.4 × 10−5. The radii of curvature are nominally

specified to be 37.4 m and 37.3 m, thus the nominal g-factor is 0.96. This

measurement of the TMFO indicates that the radii of curvature of the ITM and

5.4. CHARACTERIZATION RESULTS 87

Figure 5.5: Ring-down curve of the transmitted light from a resonant TEM00

mode; the reference time ‘0’ is the point at which the laser is turned off. Theblue dots are the time series data, the red curve is an exponential ring-down fit.

0 . 95 0 . 9 6

0 . 9 70 . 9 8

0 . 9 91 1 . 01 1 . 02 1 . 0 3 1 . 04

0 . 511 . 522 . 53

3 . 5

×

105

Measuremen

tude

Figure 5.6: Power spectral density plot of the transmission signal measured bythe QPD at the transmission port. Two peaks are measured with a frequencydifference δf1=(3.882 ± 0.017) kHz. Blue dots are measured data, red curve isfitted. (color online)


ETM are slightly different from their nominal values.

5.5 Degeneracy Results

In a near-concentric cavity, the TMFO is very sensitive to the cavity g-factor,

which in our cavity can be adjusted by CO2 laser heating, and is significantly

modulated by the laser spot position due to figure errors in the mirrors [116].

Fig. 5.7 shows a measurement of the TMFO versus varying CO2 laser power.

Due to the residual angular motions of the suspended test masses and the mirror

surface figure errors, the standard deviations of the frequency measurements are

quite large.

Figure 5.7: Measurement of the TMFO ∆f versus the CO2 laser output power.The power values are given as percentages of the maximum output power (10W) of the CO2 laser. The standard deviations of the frequency measurementsare plotted as uncertainty bars.

The degeneracy of HOMs in our 74m cavity is evident at certain CO2 tunings.

Fig. 5.8 shows an image of the ITM where the fundamental mode is degenerate

with a 17th-order mode. It is clear that the observed HOM is most similar to

5.5. DEGENERACY RESULTS 89

TEM11, 6. This mode was observed with an output CO2 laser power of 0.5 W,

which corresponds to 5% of its maximum power.

Figure 5.8: An image of a TEM11,6 Hermite-Gaussian mode degenerate with thefundamental mode, with 5% (0.5 W) of the maximum CO2 laser power heatingthe ITM.

Losses in degeneracy are contributed by both of the degenerate modes. Based

on the model introduced in Section 5.3, losses in the fundamental mode consist

only of absorption and scattering losses, while clipping loss has to be taken into

account for HOMs. According to our model and Fig. 5.3, the finesse measured

in degeneracy is expected to drop compared to that in non-degeneracy cases.

In order to verify this prediction, measurements of finesse across a wide range

of TMFOs were investigated in this cavity. Fig. 5.9 is a series of ring-down

measurements with varying CO2 heating power. In an effort to minimize uncer-

tainty, the differential signals from the QPD were monitored concurrently with

the ring-down measurements. The TMFOs were determined from the spectra of

the QPD signals. The CO2 laser power was varied in 0.05 W increments, and

measurements were taken continuously. The settling time at each CO2 laser

power is ∼3 minutes. We did not wait for the settling time in our attempt to

cover the spectrum of ∆f in detail. However, due to problems with synchroniza-

tion and integration times, our estimates of ∆f still contained relatively large

uncertainties.

We are convinced that the dips in Fig. 5.9 represent reductions in finesse due


to degeneracies at given TMFOs. The minimum measured value of the finesse is

10,950 while the maximum is 16,700, and the average is 15,300. A comparison

between these drops in finesse and the corresponding expected degenerate modes

is given in Table 5.3.

Transverse mode frequency offset (kHz)

Fin

esse

Figure 5.9: Finesses at different TMFOs are determined by the time decayconstant in the ring-down measurements. Each red cross corresponds to a ring-down measurement. The TMFOs were measured from transmission spectra.They were changed by tuning the CO2 laser power heating the ITM. (coloronline)

1 2 3 4Finesse 13490 11740 11620 10950fa (kHz) 104.8 144.6 164.2 185.4fb (kHz) 106.8 144.9 169.1 184.5|fa − fb| (kHz) 2.0 0.3 4.9 0.9Nd 19 14 12 11

Table 5.3: Four observed dropouts of finesse compared with the nearest de-generacies. In this Table, fa represents the TMFO at which the correspondingfinesse dropout was observed, fb represents the nearest TMFO at which a de-generacy takes place, and Nd is the order of the nearest degenerate mode. Thefinesse and the frequency difference |fa − fb| are also listed.

According to the average value of finesse, and recalling Eq. (5.10), the total

5.5. DEGENERACY RESULTS 91

loss in the fundamental mode can be calculated as L00 = 190 ppm. In degen-

eracy, the finesse expected to be measured is given by Eq. (5.19). For the four

degenerate modes indicated in Fig. 5.9, we can investigate whether their losses

are mainly contributed to by clipping losses. Here, we assume that the absorp-

tion and scattering of every mode are the same, i.e. L0nm = L00. Thus, the only

difference between these degenerate modes is the clipping loss. The maximum

and minimum clipping losses of these four modes can be found in Table 5.2.

Based on the estimated value of L00, the finesse as a function of the coupling

rate A for each degenerate mode is given in Fig. 5.10. In each plot of Fig. 5.10,

the two curves correspond to finesses for the most symmetric mode (minimum

clipping loss) and the most asymmetric mode (maximum clipping loss), in the

corresponding order.

Figure 5.10: Finesse as a function of the coupling rate for each degeneratemode indicated in Fig. 5.9. The blue dashed lines and red solid lines representthe most symmetric mode and the most asymmetric mode, respectively, in thecorresponding order. (color online)

For the 11th and 12th-order degenerate modes, it is clear in Fig. 5.10 that the

expected finesses do not drop to the measured values even though the coupling

rate is as high as 0.1. In other words, the drops of finesse in these two modes are

caused by absorption and scattering losses higher than that in the fundamental

mode (i.e. L0nm > L00) — the clipping losses are negligible in these two modes.


For the 14th-order degenerate mode, the finesse drops to the measured value

only in highly asymmetric modes, and the coupling rate also has to be as high

as 0.01. The clipping loss in the 19th-order mode is at least 100 times higher

than in the lower-order degenerate modes. Thus, clipping loss can be considered

as the main loss in this mode. According to Fig. 5.10, the finesse drops to the

measured value when the coupling rate is ∼ 7 × 10−4.

By carefully comparing Fig. 5.3 with Fig. 5.10, we note that the finesse of

the 19th-order mode in Fig. 5.10 does not drop as quickly as the lowest curve

in Fig. 5.3, for coupling rates . 10−3. This is despite the fact that the total

loss (∼2%) — which is dominated by clipping for this 19th-order mode — is

much higher in Fig. 5.10 than in Fig. 5.3, which only shows finesse curves for

relatively small total losses (. 0.1%). The physical reason is that when the

total loss increases, less power in the HOM resonates in the cavity. Thus, the

cavity can be considered to be far away from degeneracy. If the coupling rate is

low, the circulating power of the fundamental mode is not dramatically reduced.

Therefore, the finesse does not drop significantly. This mechanism explains the

phenomenon that the measured finesse in degeneracy in Fig. 5.9 is higher for

the 19th-order mode than for the other 3 modes.

Fig. 5.11 shows a curve of finesse as a function of the total loss, when the

coupling rate is a constant (A = 7×10−4). It is evident that the finesse increases

when the total loss in the degenerate HOM exceeds a certain value. There are

two solutions for the total loss at each value of finesse. Assuming the coupling

rates for the four degenerate modes in Fig. 5.9 are same, and that the total loss

is larger in a higher-order than in a lower-order mode, then the total loss in

each mode can be deduced from Fig. 5.11. Lnm is estimated to be 7.8 × 10−3,

1.1 × 10−2, 1.2 × 10−2 and 3.0 × 10−2 for the 11th, 12th, 14th and 19th-order

modes, respectively. As discussed before, the main contribution to the total

loss in the 19th-order mode is clipping loss, while the total losses are mainly

contributed to by absorption and scattering in the other 3 degenerate modes.

Though we have not observed all the degenerate modes, and there is a rela-

tively large discrepancy in TMFO between the measurements and the expected

degenerate modes, we have no better explanation for the dramatic reduction

in finesse at some TMFOs, and we can anticipate that the uncertainties in fre-

quency measurements are very high. We have measurements showing that the

TMFO can change by as much as 10 kHz/s. With an integration time for the

spectrum analyzer of ∼ 0.1 second, and an estimated synchronization of ± 0.4

5.6. CONCLUSION 93

Figure 5.11: Finesse as a function of the the total loss in the degenerate HOM.A coupling rate of 7 × 10−4 is assumed for this curve.

seconds due to the necessity to manually switch the servo amplifier and stop

the spectrum analyzer, ± 5 kHz could be taken as an optimistic uncertainty,

and would be consistent with the difference values listed in Table 5.3. Further-

more, the cavity linewidth (∼ 140 Hz) is very small compared to the changing

rate of the TMFO during CO2 heating transitions. As each measurement took

∼30 seconds, it is reasonable to expect that some degenerate frequencies could

have been missed. This could explain why we did not observe all the degenerate

modes.

5.6 Conclusion

A high optical laser power cavity, with a length of 73.92930 m ± 80 µm, a

linewidth of 140 ± 2 Hz, a g-factor of 0.976551 ± 1.4 × 10−5, and a finesse of

14500 ± 300, was measured. On tuning the g-factor from 0.976551 to 0.915764

by varying the CO2 laser power heating of the ITM, several degeneracies were

observed. At four TMFOs, roughly corresponding to the 11th, 12th, 14th and

19th-order modes, significant losses were observed which resulted in as much


as a 28.4% reduction of the cavity finesse. Based on our model, the inferred

coupling rate is as high as 7 × 10−4, and the total losses of these 4 modes were

estimated. Investigation of this loss mechanism showed that the clipping loss

is responsible for the dropout of the finesse in the 19th-order mode, and that

absorption and scattering are responsible for the finesse dropouts of the other 3

modes.

A ring-down signal in degeneracy measures the cavity loss of the fundamental

mode, which in turn indicates the scattering of the HOM on the mirror surfaces.

In addition, degeneracy mode patterns and spot positions can be monitored by

using the images of the ITM and ETM taken by the cameras. Using this method,

the figure errors at different positions on the mirrors can be deduced — this

method could be used as a tool for monitoring these figure errors. However, to

make this method useful and precise, a more accurate synchronization technique

has to be applied. A program which simultaneously shuts down the locking

system, and starts recording the spectrum and ring-down needs to be developed,

and the time delay also has to be taken into account. In addition, a more

continuous varying method of the CO2 heating power is also needed, as this

would tune the TMFO with higher accuracy. A thermal diffusion simulation of

the ITM would also help in analyzing the shape change of the mirror surface

during CO2 heating.

5.7 Acknowledgments

The authors would like to thank Dr. Jean-Charles Dumas and Mr. Andrew

Wooley for setting up the optical layout on the injection table, and to Dr. Andre

Fletcher for proof-reading a draft this paper. This work was supported by a grant

from the Australian Research Council.

Chapter 6

Mechanical loss in a suspended

test mass with a modular

suspension system

6.1 Preface

In last chapter, the optical properties of the Fabry-Perot cavity were char-

acterized. However, before testing the parametric instability in the Gingin High

Optical Power Test Facility, the mechanical properties of the suspended test

masses must also be characterized.

In the East arm of the Gingin facility, test masses are hung from a remov-

able modular suspension system. The total mechanical loss of the suspended

test mass comprises three parts: coating loss, substrate loss, and loss due to

interaction with the suspension system. In this Chapter, the loss due to such

suspension system is investigated. The quality factors of some lowest-order me-

chanical modes of the ITM are measured. The total loss angle of the test mass

is estimated by combining the measurements with an ANSYS simulation which

calculates the strain energy for the test mass. The result shows that the me-

chanical loss of the test mass due to interaction with the suspension system is

negligible.

This Chapter is based on a preprint written by the author. The author’s

contribution is 100% of the theoretical work, 100% of the data analysis, and 50%

of the experimental work. The experiments were done together by this author

95

96 CHAPTER 6. TEST MASS MECHANICAL LOSS

and Carl Blair. The author did all the ANSYS simulation and calculation. The

data was also fitted by this author. This paper will be submitted to Classical

and Quantum Gravity. The version of this paper presented here is current as of

the thesis submission date (April 2015).


6.2 Introduction

Gravitational waves are likely to be detected by laser interferometers within

the next decade. Advanced laser interferometers, such as aLIGO and aVIRGO,

are being commissioned [21, 23]. The current sensitivity achieved by aLIGO

is equivalent to a detection range of 60 Mpc for a binary neutron star merger

system, which is already 4 times better than the initial LIGO. Almost every

aspect in aLIGO is improved compared to initial LIGO, including the laser source

system, test masses, read-out system, suspension system, etc. The dominant

factors which limit the sensitivity in the 10−100 Hz detection band are thermal

noise and quantum noise [21]. The latter can be suppressed by applying squeezed

light sources [132], while the thermal noise remains limited by the mechanical

losses in the coatings, and by the suspension system for the test masses [54].

In the first-generation laser interferometers, such as LIGO, VIRGO and

TAMA, steel wires were used in their suspension systems [133, 134, 135]. In

the GEO600 system, silica fibres were used, as they introduced much lower me-

chanical losses to the test masses [136]. In the advanced (second-generation)

laser interferometers, 4 fused silica fibres are welded to “ears” which are silicate

bounded to the flat regions on the right and left sides of the test masses [137].

Other suspension systems, such as rectangular ribbons or flexures [138, 139,

140], were studied for reducing thermal noise in pendulum modes. At the Univer-

sity of Western Australia (UWA), a detachable modular suspension system using

a niobium flexure was designed and tested [141, 142]. Due to high thermal con-

ductivity [143] and very low mechanical loss [144, 145] at cryogenic temperatures,

niobium suspension systems could be suggested to be used in third-generation

laser interferometers, such as the KAGRA [24] and the Einstein Telescope [49].

The current suspension system in our 74-m High Optical Power Test Facility

at Gingin, Western Australia uses niobium and also takes the detachability of the

test masses into account. The construction is shown in Fig. 6.1. Two cylindrical

holes are drilled into the right and left flat sides of the test masses, and the

interior surfaces of the holes are not polished. A small niobium stick with two

small pegs is inserted into each hole (see Fig. 6.1(a)). The stick is suspended by

a niobium wire from a control mass, which is in turn suspended from a vibration

isolation system. The design of the stick and pegs is as shown in Fig. 6.1(b). The

contacting tip of each peg forms a flat square area which is ∼150 µm in length

98 CHAPTER 6. TEST MASS MECHANICAL LOSSN i o b i um

wireP e g s

Figure 6.1: A replaceable suspension system. (a) A conceptual plot of the holesand niobium sticks on one of the two flat sides of a test mass. (b) Cross-sectionof a niobium stick suspended by a niobium wire. (c) A microscopic image of apeg with its flat top, which is used for contacting and supporting the test mass.

(see Fig. 6.1(c)). The test mass is fully supported by the contacts between pegs

and hole surfaces. The suspension wire can be readily replaced by removing the

stick from the hole and inserting a new one, if any defect or breakage occurs.

The mechanical loss introduced to the test mass by this modular suspension

system is studied. The quality factors of different mechanical modes of the

suspended test mass are measured. A simulation is undertaken to estimate the

strain energy of the coating and the hole surfaces, as well as the displacements of

the contact areas between pegs and hole surfaces. Thus, the loss angle introduced

to the test mass by the suspension system can be calculated.

This paper is arranged as follows: the next Section gives a model to analyze

the loss angle of the test mass due to interaction with the suspension system. In

Sect. 6.4 the experimental setup for measuring the quality-factors is described.

Section 6.5 presents measurement results as well as ANSYS simulations. The

loss angle due to the suspension system is estimated.

6.3 Model for Test Mass Mechanical Loss

6.3. MODEL FOR TEST MASS MECHANICAL LOSS 99

Quality factors of intrinsic mechanical modes of the test mass, which can

be deduced from ring-down measurements, are given by the test mass total

mechanical loss angle φtotal: Q = 1/φtotal. In our test mass in Gingin facility,

φtotal is mainly contributed by three parts: substrate loss, coating loss, and loss

due to the suspension holes. In principle, for any mechanical mode, the relation

can be given by the following equation:

φtotal =Usubstrate

Utotalφsubstrate +

Ucoating

Utotalφcoating + φholes, (6.1)

where φsubstrate and φcoating denote the loss angle of the substrate and coating ma-

terials, respectively. Utotal, Usubstrate, and Ucoating denote the total strain energy,

strain energy stored in the substrate material, and that stored in the coating ma-

terial, respectively. φholes is the loss angle introduced by the suspension holes. It

contains a few contributions which will be discussed later. For now, it is treated

as a single parameter. The strain energy is mainly stored in the substrate, as

the coating is very thin and the surface area of the holes is small; therefore

Utotal ≈ Usubstrate. Eq. (6.1) can then be approximated by:

φtotal ≈ φsubstrate +Ucoating

Usubstrate

φcoating + φholes. (6.2)

The substrate we used is a type Suprasil 1 fused silica manufactured by

Heraeus Inc. The quality factors of this material were measured in [146], and

are independent of the acoustic frequency. We use the value quoted from this

reference for φsubstrate, which is 9 × 10−8.

The coatings used for our test masses are multi-layer Ta2O5/SiO2 (Advanced

Thin Films, USA). An effective value of φcoating can be deduced by knowing the

thickness of each coating layer, and the loss angle of each coating material.

Specifically, the coating for our input test mass (ITM) consists of 27 alternating

Ta2O5/SiO2 layers, with λ/4 as the optical thickness of each layer. Our end

test mass (ETM) comprises 33 layers with a similar arrangement as for the ITM.

The effective loss angle of a coating is given by [51]:

φcoating =Y1D1

YcDcφ1 +

Y2D2

YcDcφ2, (6.3)

where Y1,2, D1,2 and φ1,2 represent the Young’s modulus, total physical thickness

and loss angle of the Ta2O5 and SiO2 layers, respectively. For thin coating layers,


where the surface can be considered as stress-free, the total Young’s modulus is

given by [51]:

YcDc = Y1D1 + Y2D2. (6.4)

The effective loss angle of this thin coating is simply:

φcoating ≈ Y1D1φ1 + Y2D2φ2

Y1D1 + Y2D2. (6.5)

The parameters used here are: Y1 = 1.4 × 1011 Pa, Y2 = 7.2 × 1010 Pa; D1 =

1.82 µm, D2 = 2.39 µm for the ITM, and D1 = 2.21 µm, D2 = 2.94 µm

for the ETM. Accordingly, YcDc is 4.3 × 105 (Pa · m) for the ITM and 5.2 ×105 (Pa · m) for the ETM. The coatings we used were fabricated by ion-beam

sputtering deposition. However, the manufactural details are unknown as they

are commercial products. A good estimation of the loss angles of Ta2O5 and

SiO2 are given by [52]:

φ1(f) = (3.8 ± 0.2) × 10−4 + f(1.8 ± 0.5) × 10−9, (6.6)

φ2(f) = (1.0 ± 0.2) × 10−4 + f(1.1 ± 0.5) × 10−9. (6.7)

For instance, if f = 100 kHz, φcoating ≈ 4.2 × 10−4 for both the ITM and ETM.

The value of Ucoating/Utotal can be estimated from an ANSYS simulation, which

will be discussed in detail in Sect. 6.5.

The loss angle introduced by the suspension holes, φholes, has two compo-

nents: the loss due to defects or cracks on the surfaces of the holes, and the

loss due to energy coupling from the test mass to the suspension violin modes,

which is expected to be proportional to the square of displacements of the con-

tact points between the hole surfaces and niobium pegs. Thus, φholes can be

expressed as:

φholes =Usurfaces

Usubstrateφsurfaces +

Upoints

Usubstrateφpoints, (6.8)

where φsurfaces represents the loss angle due to the internal friction in the surface

materials of the four holes, and Usurfaces is the strain energy stored in this surface

material. φpoints represents the loss angle due to energy coupling from the test

mass to the suspension through the four contacting points. Upoints is the strain

energy stored in the contacting points; this is proportional to S2, which is the

sum of squared displacements of the contacting points. The coefficient of S2 is

6.4. EXPERIMENTAL SETUP 101

denoted as k, there is:

Upoints = kS2. (6.9)

Thus, Eq. (6.8) can be written as:

φholes =Uh0

UsubstrateDhφsurfaces +

S2

Usubstratekηφpoints, (6.10)

where Uh0 is the strain energy stored in the hole surfaces per unit length of

the material, Dh is the unknown thickness of this lossy material; η denotes the

coupling rate of the strain energy of the contacting points to the suspension

violin modes, which is a constant. In this equation, the fractions Uh0

Usubstrateand

S2

Usubstratecan be calculated by the ANSYS simulation. Consequently, the terms

Dhφholes and kηφcouple can be considered as single unknown constants to fit. One

of our goals in Sect. 6.5 is to find the values of these constants.

6.4 Experimental Setup

Let us first consider measuring the Q-factors, which correspond to φtotal in

Eq. (6.2). In our facility, a Fabry-Perot cavity is formed by two test masses,

with their high reflectivity coatings facing inward toward the cavity. Each test

mass is 100 mm in diameter and 50 mm thick, with a mass of 880 g. The cavity

is 74 meters long and the radii of curvature are 37.4 m and 37.3 m for the ITM

and ETM, respectively. In this paper, we will discuss only the mechanical modes

of the ITM, which is illustrated in Fig. 6.2.

A donger was used to excite mechanical modes by electrically rapping it on

top of the test mass. The excited motion of the test mass is a composition of all

different modes, which can be distinguished in spectra. In order to measure the

quality factors of these mechanical modes, a 1064 nm NPRO laser was used. The

linearly polarized laser propagated through a half-wave plate, which was used to

rotate the polarization direction. Without exciting the mirror, the polarization

direction of the laser beam was tuned such that the beam was fully reflected

at the diagonal surface of a polarized beam splitter (PBS) (see Fig. 6.2). By

contrast, when the mirror was excited, an anisotropic stress distribution of the


Figure 6.2: A schematic of the experimental setup. The donger is for excitingthe test mass mechanical modes. The monitoring laser beam is reflected fromthe inner surface of the ITM and received by a quadrant photodiode (QPD).Only the modes of the ITM were measured.

mirror was induced, and thus a birefringence effect occurred in this mirror. In

other words, the local refractive index was varying with time due to the time-

varying stress distribution inside the test mass. The beam polarization was also

changing, whose Fourier frequencies were determined by the mechanical modes.

Thus, the laser beam was no longer fully reflected by the PBS. A part of the

beam was transmitted by the PBS, and then detected by the QPD.

Another method to obtain mechanical mode signals is to measure the surface

vibrations of the test mass by using the same setup, after simply removing

the half-wave plate and the PBS. A differential vibration of the surface can be

registered as quadrant signals in the QPD. Clearly, this method is limited to

those modes with differential surface vibrations.

The spectra in these two methods was measured by a spectrum analyzer,

which is connected to the QPD. Ring-down measurements were undertaken by

setting a reference frequency near each mechanical frequency in the spectrum

analyzer, and then recording time series data. The results will be discussed in

the next Section.

6.5 Results

6.5. RESULTS 103

6.5.1 Measurement Results

Mechanical modes with frequencies less than 57 kHz were measured using

the spectrum analyzer. A typical spectrum of a mechanical mode at 42.938 kHz

is shown in Fig. 6.3(a). A ring-down curve of this mode is shown in Fig. 6.3(b).

A decay time constant was fitted to this curve. The mechanical quality factor

was then calculated from:

Q = πfτ, (6.11)

where f is the mechanical frequency, and τ is the decay time constant. We noted

that a clear ring-down curve was not obtained for every mechanical mode, as the

amplitudes of some of them were relatively small, and thus the signal-to-noise

ratios were too low. The results of the frequency and ring-down measurements

are listed in Table 6.1. Note in this Table that the measured Q-factors vary

from 1.46 × 106 to 4.45 × 106. A possible reason is that the deformations and

strain energies are different for all these modes. Thus, the mechanical losses

introduced by the coating, the hole surfaces and the coupling to the suspension

system are also different. In the next subsection, a simulation will be introduced

to examine this assumption.4 2 . 9 3 8 k H zFigure 6.3: A typical result of spectrum and ring-down measurement. (a) Spec-trum of a mechanical mode at 42.938 kHz. (b) Ring-down curve of the samemechanical mode. The local oscillator frequency used to beat with the mechan-ical signal in the spectrum analyzer was set to be ∼0.45 Hz higher than themechanical frequency.

6.5.2 Simulation of Mechanical Modes


fmeasured (kHz) Q (106) fmeasured (kHz) Q (106)19.049 (D) 47.810 (D)25.963 (D) 1.91 ± 0.04 51.974 (D)28.102 (B, D) 1.78 ± 0.03 54.007 (B, D) 3.83 ± 0.1228.211 (B, D) 2.06 ± 0.06 54.162 (B)31.655 (B, D) 54.258 (D) 2.79 ± 0.1533.245 (D) 54.275 (D) 3.17 ± 0.1536.211 (D) 54.644 (B) 2.49 ± 0.1136.294 (D) 54.927 (B)42.938 (B) 4.09 ± 0.03 55.347 (B) 3.52 ± 0.2343.097 (B, D) 55.745 (B) 1.46 ± 0.0547.545 (D) 56.560 (B) 4.45 ± 0.0747.780 (D)

Table 6.1: Measured mechanical frequencies and quality factors. The letter “B”signifies that the frequency was measured by the birefringence method, while“D” indicates the direct reflection method. Some modes can be measured usingboth methods. The vacuum pressure during the measurements was well keptbelow 10−5 mbar. However, we did not test if gas damping can be excluded.

An ANSYS simulation was built to calculate intrinsic mode frequencies, de-

formations and strain energies. The thickness and radius of the cylindrical sub-

strate are given in Sect. 6.4. The distance between the two flat sides is 98 mm.

The two pin holes on each flat side are 10 mm apart, with a cylindrical shape of

3 mm diameter and 4 mm depth for each hole.

In the simulation, the test mass was considered as a single solid body made

of fused silica. The coating layer was not simulated as its thickness is only 4.2

µm. Meshing for such a thin layer on top of the test mass would require a

huge computational power and be time consuming. In principle, it is assumed

that the coating is isotropic and not much stiffer than the substrate — thus the

coating follows the surface motion of the substrate [147]. As the coating does

not dramatically affect the simulation results, and we were only concerned about

the strain energy of the coating layer, we can simply simulate the substrate and

calculate the strain energy of the front surface to estimate the strain energy of

the coating [147]. As discussed in Sect. 6.3, YcDc = 4.3×105 Pa ·m for the ITM

coating. The Young’s modulus for the fused silica substrate is Y2 = 7.2 × 1010

Pa. Therefore, the effective thickness of the original multi-layered coating on

6.5. RESULTS 105

the substrate surface is:

D′c =

YcDc

Y2

= 5.9 µm. (6.12)

A layer with such a thickness can be considered as containing the same strain

energy as the coating.

A thin piece of meshing on the substrate surface was chosen in estimating the

strain energy of the coating. The volume of this thin piece was calculated, and

thus its thickness was obtained. The strain energy of all elements in this thin

piece was then calculated, and the strain energy within the effective thickness

D′c can then be derived. The meshing was fine enough that the deformation of

this thin piece represents that of the surface.

The same method was applied to estimate the strain energy of the hole

surfaces. A thin layer of meshing was chosen for each hole, and the volume,

thickness and strain energy were calculated for this layer. The strain energy Uh0

per unit length on the hole surfaces was deduced. For calculating the displace-

ment square sum S2, meshing nodes were chosen nearest to the contact points

between the niobium pegs and the hole surfaces; squares of displacements of

these nodes were summed as S2. The fractions Uh0

Usubstrateand S2

Usubstratewere then

readily calculated.

The parameters Dhφsurfaces and kηφpoints can be fitted from the measurements

of the quality factors and this simulation. We first simply Eq. (6.10) as:

φholes = αA + βB, (6.13)

where

α =Uh0

Usubstrate

, (6.14)

β =S2

Usubstrate

, (6.15)

A = Dhφsurfaces, (6.16)

and

B = kηφpoints. (6.17)

We also denote:

δ =Ucoating

Usubstrate

, (6.18)


then Eq. (6.2) becomes:

φtotal = φsubstrate + δφcoating + αA + βB. (6.19)

The measurement and simulation results are summarized in Table 6.2, where

the contributions to the total mechanical loss due to different factors are indi-

cated by the values of δ, α and β, respectively — the larger these values are, the

higher are the losses introduced to the test mass. These results are consistent

with the simulation pictures of total deformations (Appendix A) and strain en-

ergies (Appendix B), thus verifying our assumptions and model. According to

these results, the parameters A and B can be fitted as:

A = (7.1 ± 5.1) × 10−8 m, (6.20)

B = (9.6 ± 3.0) × 102 J · m−2. (6.21)

fm (kHz) fs (kHz) Q (106) δ (10−4) α (10−2/m) β (10−12 m2/J)25.963 26.212 1.91 2.7 5.0 280.028.102 27.934 1.78 1.2 1.5 90.028.211 28.012 2.06 1.2 5.0 580.042.940 42.829 4.09 1.3 4.0 6.054.007 54.065 3.83 1.0 5.0 20.054.258 54.102 2.79 1.3 7.0 4.054.275 54.161 3.17 1.3 33.0 70.054.644 54.541 2.49 0.6 400.0 90.055.347 55.673 3.52 1.6 8.0 9.055.745 55.793 1.46 1.6 83.0 190.056.560 56.249 4.45 0.5 3.0 0.4

Table 6.2: Measurement of the quality factor with the mode frequency, togetherwith the simulation results. fm and fs denote the measurement and simulationfrequencies of the mechanical modes, respectively. The simulation results of thetotal deformations and strain energies of these modes are given in the Appendix.

6.5. RESULTS 107

6.5.3 Simulation with a Gaussian Radiation Pressure

Profile

To further calculate the total loss angle of the test mass at gravitational wave

detection frequency, a simulation with a radiation pressure of Gaussian profile

applied to one circular surface was constructed. A data set of a Gaussian profile

was first generated by a Mathematica program. These data were then imported

into ANSYS to mimic the radiation pressure provided by a Gaussian laser beam

impacting on the surface of the test mass. Because the detection frequency of

gravitational waves in aLIGO and aVIRGO is around 100 Hz, which is much

less than the test mass mechanical frequencies, the radiation pressure can be

considered as constant in the simulation [148]. Therefore, the coefficients α, β

and δ can be considered as constants:

α = α0, β = β0, δ = δ0. (6.22)

The amplitude of this pressure does not affect our results, as we are only con-

cerned with energy ratios.

The total deformation and strain energy are shown in Fig. 6.4. From this

simulation, the coefficients α0, β0 and δ0 can be calculated. The results are listed

in Table 6.3.

Figure 6.4: Simulation pictures of the test mass with a radiation pressure ofGaussian profile applied on the coating surface. (a) Total deformation. (b)Strain energy.


δ0 (10−4) α0 (10−2 m−1) β0 (10−12 m2 · J−1)3.8 1.8 10.1

Table 6.3: Coefficients calculated from the test mass simulation with radiationpressure applied.

Finally, the total loss angle of the test mass in the gravitational wave de-

tection band can be calculated. At 100 Hz, for instance, the total loss angle

is:

φtotal(100 Hz) = δ0φcoating(100 Hz) + α0A + β0B

= 1.0 × 10−7 + 1.3 × 10−9 + 9.6 × 10−9

≈ 1.1 × 10−7, (6.23)

We note that from this equation, the loss angle of the substrate is negligible

compared to the coating loss angle. The loss angle due to the suspension system,

including the holes and niobium pegs, sticks and wires, is ∼ 10% of the coating

loss angle.

6.6 Conclusions and Future Work

We measured the eigenfrequencies and quality factors of mechanical modes

for the test mass hung from a replaceable suspension system. We investigated

the mechanical loss angle due to this suspension system by combining the mea-

surement results and an ANSYS simulation for the test mass. The total loss

angle at 100 Hz, which is a typical detection frequency in aLIGO and aVIRGO,

were estimated. The result is comparable to that in aLIGO and aVIRGO. It was

indicated that the replaceable suspension system will not introduce too much

extra loss angle to the test mass (∼ 10% of the total loss angle).

Future work can be addressed towards further reducing the loss angle due

to this modular suspension system. A polishing of the hole surfaces can be

considered, and violin modes of the suspension wires can be investigated to

avoid strong coupling to the test mass. In our model, we assumed that the loss

6.7. ACKNOWLEDGMENTS 109

angle due to the coupling to the violin modes is proportional to the sum of the

squared displacements of the contact points on the hole surfaces. A more detailed

model accounting for the mechanical loss related to the suspension system will

be helpful in estimating the loss angle more accurately.

6.7 Acknowledgments

We gratefully thank Dr. Jean-Charles Dumas and Mr. Andrew Wooley for in-

stalling and testing the suspension system for our test mass, and Dr. Andre B.

Fletcher for proofreading a draft of this paper. This work is supported by the

Australian Research Council.


6.8 Appendix A. Pictures of total deformations

of mechanical modes

Figure 6.5: Total deformations of mechanical modes whose quality factors weremeasured.

6.9. APPENDIX B 111

6.9 Appendix B. Pictures of strain energies of

mechanical modes

Figure 6.6: Strain energies of mechanical modes whose quality factors weremeasured.


Chapter 7

Observation and Suppression of

Parametric Instability

7.1 Preface

Parametric instability was observed in the East arm of the Gingin facility.

In this experiment, we observed that the transverse mode frequency offset is

modulated, due to residual angular motions of the test masses and figure errors

of the mirror surfaces. As a result, the observed instability effect is much smaller

than expected, due to this dynamical modulation mechanism, which can be

considered as a suppression technique for parametric instability.

This chapter is based on a paper written by A/Prof. Zhao, which has been

accepted by Physical Review D. The author’s contribution is 50% of the exper-

imental work and 50% of the data analysis of this paper. The author aligned

the optical cavity and tuned the test mass to observe the parametric instabil-

ity. The author also analyzed the experimental data and fitted to exponential

ring-up curves, which directly demonstrated the parametric instability. The ex-

periments were done together by the author, Carl Blair, Jiayi Qin and Chunnong

Zhao.

113

114 CHAPTER 7. OBSERVATION AND SUPPRESSION OF PI

Parametric Instability in Long Optical Cavities and

Suppression by Dynamic Transverse Mode FrequencyModulation

Chunnong Zhao, Li Ju, Qi Fang, Carl Blair, Jiayi Qin, David Blair

School of Physics, University of Western Australia, WA 6009, Australia

Jerome Degallaix

Laboratoire des Materiaux Avances, IN2P3/CNRS, Universite de Lyon, Villeur-

banne, France

Hiroaki Yamamoto

Theoretical Physics & Simulation Group, LIGO Caltech, MC 100-36,Pasadena

CA 91125, USA

Abstract: Three mode parametric instability has been predicted in Advanced

gravitational wave detectors. Here we present the first observation of this phe-

nomenon in a large scale suspended optical cavity designed to be comparable to

those of advanced gravitational wave detectors. Our results show that previous

modelling assumptions that transverse optical modes are stable in frequency ex-

cept for frequency drifts on a thermal deformation time scale is unlikely to be

valid for suspended mass optical cavities. We demonstrate that mirror figure

errors cause a dependence of transverse mode offset frequency on spot position.

Combined with low frequency residual motion of suspended mirrors, this leads

to transverse mode frequency modulation which suppresses the effective para-

metric gain. We show that this gain suppression mechanism can be enhanced

by laser spot dithering or fast thermal modulation. Using Advanced LIGO test

mass data and thermal modelling we show that gain suppression factors of 10-

20 could be achieved for individual modes, sufficient to greatly ameliorate the

parametric instability problem.


7.2 Introduction

Advanced laser interferometer gravitational wave detectors are currently being

commissioned [149, 150]. Once they reach target sensitivity, they have a high

probability of observing gravitational waves, especially from the coalescence of

binary neutron stars. Target sensitivity requires very high optical power in the

detector optical cavities, which can allow radiation pressure induced instabilities.

In 2001, Braginsky et al. [66, 151] predicted that opto-acoustic interactions

in such detectors could lead to a new form of instability called a three-mode

parametric instability. This could arise from optical transitions between cavity

modes mediated by test mass acoustic modes. Specifically, photons from the

main interferometer pump mode are scattered from thermally excited acoustic

modes in the test masses. The pump photon creates a phonon-photon pair. If

the phonon is resonant in a test mass acoustic mode, and the photon is resonant

in an interferometer cavity transverse mode, this scattering process will occur

resonantly. Assuming that the scattered photons have lower frequency than

the pump photons, energy conservation requires the phonon to increase the

occupation number of the acoustic mode. If the acoustic energy injection by this

mechanism exceeds the characteristic losses of the acoustic mode, the scattering

will lead to an exponential growth of the acoustic mode occupation number.

Braginsky showed that the amplitude of this scattering process could be large if

the spatial 2D surface amplitude distribution of the acoustic mode overlapped

the spatial intensity distribution of the optical mode, thereby causing three-mode

parametric instability (PI).

Subsequently, Zhao et al. [82] demonstrated that for realistic interferometer

designs there was a substantial risk of instability, because the high acoustic

mode density in the 50-150 kHz range led to numerous accidental overlaps of

both mode shape and frequency. Such instabilities could not be completely

avoided through optical design. This led to research focused on observation

and study of three-mode interactions [152], and on methods for suppressing

instability [153, 154, 69, 72, 155].

Strigin et al. [156] extended the theory to a dual recycling interferometer

detector, and showed that the multi-cavity coupling could reduce the effective

linewidth to a sub-Hz range. If the high-order cavity mode involved in PI is reso-

nant in both the arm cavities and the recycling cavity, extremely high three-mode

parametric gain could occur. Detailed analysis of a dual recycling interferometer


with realistic test masses by Gras et al. [105] showed that the highest gain could

reach ∼1000 corresponding to acoustic ring-up times of ∼seconds.

Recently modelling, that takes into account large acoustic amplitudes, and

using parameters close to those of Advanced LIGO (aLIGO), has shown that

the growth of instability saturates. Danilishin et al. showed that PI is likely to

grow on a time scale of minutes for realistic parameters [68, 157].

Three-mode parametric interactions are extremely sensitive to the test mass

mirror parameters. This extreme sensitivity was emphasised by Ju et al. [125],

who showed that mirror radius of curvature (RoC) changes corresponding to

wavefront deformations of 10−6λ could easily be observed by monitoring three-

mode interactions in Advanced interferometers.

To date, three-mode instability has been reported in one free space cavity

experiment using a picogram membrane in a 10 cm cavity [68], and in aLIGO

[67].

At the Gingin High Optical Power Test Facility [158], a 74 m optical cavity

has been set up to be comparable to the conditions of Advanced interferometers.

This paper is based on observations using this facility which, while demonstrat-

ing instability, have revealed a phenomenon that suppresses the exponential

growth of instability at low amplitudes.

Previous modelling has ignored two real world aspects of practical suspended

mass interferometers: a) that mirrors after coating have figure errors ∼ 1 nm

RMS over the central diameter of 160 mm, and b) that the laser spot position

on the mirrors fluctuates due to residual low frequency seismic motion. The

presence of figure errors means that the average RoC of the region of the mirror

intercepted by the laser beam depends on the beam location. This RoC de-

termines the transverse mode frequency offset (TMFO). Because low-frequency

fluctuations of the spot position cause the laser spot to intercept different re-

gions of the mirror surface, it follows that there will be dynamical modulation of

the optical TMFO. The frequency modulation causes the parametric gain to be

time-dependent, and if the modulation amplitude exceeds the transverse mode

optical linewidth, the gain can be strongly modulated. This can create a situ-

ation where PI does not have time to develop because it is only on-resonance

intermittently, and for too short a time for instability to grow to problematic

levels.

In this paper, we will show that the above phenomenon is likely to reduce the

average parametric gain of the candidate modes most likely to become unstable,

7.3. THEORY OF PI AND FREQUENCY MODULATION 117

thereby significantly reducing the risk of instability. Results are confirmed by

modelling and by measurements on the 74 m optical cavity at the Gingin facility.

Recognition of this frequency modulation suppression mechanism also leads to

methods by which suppression can be enhanced, either by modulated thermal

actuation, or by spot position dithering at frequencies below the gravitational

wave sensitivity band.

In Section 7.3, we summarise the theory of PI, and present modelling results

showing how individual unstable modes can be suppressed by seismic-induced

frequency modulation. In Section 7.4, we use aLIGO test mass mirror metrol-

ogy data to estimate the frequency modulation expected for small spot position

motions in aLIGO. In Section 7.5, we present results obtained with the Gin-

gin high optical power cavity: both the observation of PI, and the frequency

modulation that greatly reduces the risk of instability. We discuss the results

obtained, and their implications for aLIGO. We also present thermal actuation

modelling results to estimate the suppression factors achievable.

7.3 Theory of PI and Effect of Transverse Mode

Frequency Modulation

Three-mode opto-acoustic interactions occur when the frequency difference ∆ω

between an optical cavity pump mode at frequency ω0 and a transverse mode

at frequency ω1 is appropriately tuned to the frequency of an acoustic mode

at frequency ωm. This three-mode interaction resonance is defined by ∆m =

(ω0 − ω1) − ωm = ∆ω − ωm = 0. The parametric gain R characterises the ratio

of acoustic energy input compared to mirror acoustic mode losses. If R > 1, the

system is acoustically unstable, and the acoustic mode will grow exponentially

until either non-linearities cause saturation [157], or else the cavity loses lock.

In this paper, we are concerned only with small amplitude excitation, and so

can ignore non-linearities. The magnitude of R depends on the cavity input

power, on acoustic and optical mode losses, and on the spatial overlap between

the relevant modes. For any pair of acoustic and optical modes, the gain R can

be expressed as [66]:

R =PΛω1

MωmL2γmγ0γ1

1

1 + (∆m/γ1)2, (γm ≪ γ1). (7.1)


Here, P is the input power to the cavity, γ0, γ1 and γm are the half-linewidths

of the two optical modes and the acoustic mode of the test mass, respectively,

M is the mass of the test mass, L is the length of the cavity, and Λ is the

overlap factor including the mass to the effective mass ratio as defined in [66].

The optical mode spacing ∆ω is a function of the RoCs of the mirrors of the

optical cavity, and is given by:

∆ω =c

L(m + n) cos−1(±

√

(1 − L

R1)(1 − L

R2)), (7.2)

where R1 and R2 are the RoCs of the end mirrors of the cavity, and m and n

are integers describing the order of the optical mode. The ± sign depends on

the cavity configuration. Equation (7.2) assumes perfect spherical mirrors, but

we will assume that, in the case of figure errors, the mode spacing is defined by

the average RoC at the laser spot position, averaged over the effective spot size.

Equation (7.1) considers only the Stokes process, where parametric ampli-

fication or instability processes occur due to a single high-order optical mode.

Here, we want to focus particularly on the case where dynamic detuning causes

∆m to be time-dependent. We consider the case of harmonic detuning given by

∆m(t) = ∆m0 cos ωdt, (7.3)

where ωd is a dynamic tuning frequency. In suspended mass interferometers, the

test mass-mirrors are supported by low frequency pendula which isolate against

vibration. The test mass positions are controlled by feedback, but finite residual

motion is inevitable because of the requirement that the test masses be inertial

within the gravitational wave signal band.

Thus, in practice, test masses can be expected to have significant motion at

pendulum normal mode frequencies of 0.1-1 Hz. This gives rise to a modula-

tion in the laser spot position. If the mirrors are imperfect, the mirror RoCs

(averaged over the laser spot size) will vary smoothly with spot position. In

this case, modulation in spot position can modulate the TMFO, thus causing

time-dependent detuning fluctuations.

Spot position motion will also modulate the modal overlap parameter. How-

ever, for millimetre scale motions, the overlap parameter modulation is small

compared with the effect of detuning, and is ignored in the following analysis.

Assuming that ∆m changes according to Eq. (7.3), the parametric gain is

given by


0 10 20 30 40 5010

0

101

102

Time (s)

A mplit

ude (

a

. u.)a=10, Rmax=6

a=4, Rmax=6

a=2, Rmax=6

a=0, Rmax=6

a=0, Rmax=1.45

fd=0.1 Hz = 6s

a=1, Rmax=6

Figure 7.1: Acoustic mode amplitude ring-up curves for various detuning am-plitudes. Here, we assume maximum gain Rmax=6, an acoustic mode ring-downtime τ=6 s, and a dynamic modulation frequency of 0.1 Hz. For comparison,cases for on resonance (a=0) with Rmax=6 and Rmax=1.45 are also plotted.

R(t) =Rmax

1 + (a cos ωdt)2, (7.4)

where a = ∆m0/γ1 is the normalised frequency detuning modulation amplitude.

Equation 7.4 allows estimation of the effects of modulation on the growth of PI.

As discussed above, modelling has shown that the characteristic ring-up time

scale for PI in a detector similar to aLIGO is likely to be ∼ 102 s [157]. Since ωd is

fast compared with such ring-up times, one would expect to observe modulated

signal growth.

Figure 7.1 shows examples of possible acoustic mode ring-up signatures. We

assume parameters comparable to those of the experiment reported in this paper:

fd = ωd/2π = 0.1 Hz, Rmax = 6, and normalised detuning amplitudes of a =

1, 2, 4 and 10. It is sufficient to choose a typical acoustic mode decay time

τ without needing to specify the acoustic mode frequency. We chose to use

τ = 6 s, corresponding to an acoustic quality factor Qm = 106 and 4 × 106

for frequencies 50 kHz and 200 kHz, respectively. Results are compared with

acoustic mode ring-up curves in the absence of dynamic detuning (a = 0) for

R = 6. In the case of a = 4, the ring-up slope is equivalent to that of a system


with a = 0 and R = 1.45, as indicated in the figure. This represents a gain

suppression factor of ∼4. Clearly, in all cases, frequency modulation suppresses

the effective parametric gain as determined by the average slope of the ring-up

curves. For a = 10, we see that instability has been replaced by a modulated

acoustic mode amplitude which, while not harmonic, is stable in time. The

equivalent parametric gain Ra in the presence of harmonic dynamic detuning is

given by:

Ra =Rmax√1 + a2

. (7.5)

The suppression of effective parametric gain as a function of modulation

amplitude is shown in Fig. 7.2. Parametric gain can be suppressed by an order

of magnitude for a = 10.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

a

RaR

max

Figure 7.2: Effective parametric gain suppression as a function of dynamic de-tuning amplitude a.

The mechanism discussed above occurs because the dynamic detuning mod-

ulation frequency is fast compared with the acoustic mode ring-up time scale.

The observed acoustic mode amplitude modulation occurs at double the dynamic

detuning frequency ωd. While the effective parametric gain is independent of

ωd, the peak-to-peak acoustic mode amplitude within one cycle is inversely de-

pendent on ωd. Figure 7.3 shows some examples for three different dynamic


0 50 100 150 20010

0

101

102

103

Time (s)

A mplit

ude (

a

. u.)fd = 1Hz

fd = 0 . 1Hz

fd = 0 . 01Hz

τ=6a=5 Rmax=6

Figure 7.3: As the dynamic detuning frequency increases, the acoustic modeamplitude excursions are reduced but the effective parametric gain is unaltered.Here, three detuning frequencies 1 Hz, 0.1 Hz and 0.01 Hz are shown. Thedetuning amplitude is fixed at a = 5 (Rmax = 6 and τ = 6s).

detuning frequencies. The amplitude modulation waveform is highly non-linear

since it is due to a Lorentzian modulation acting on the exponential rise of the

ring-up curve.

It can be seen that if the detuning frequency is too slow, the acoustic mode

amplitude can grow to a very large value within half a detuning period. It

is possible to define a lower limit for the dynamic detuning frequency, fd lim,

to prevent the acoustic mode amplitude excursion from exceeding β times its

original value within one cycle. Figure 7.4 shows three curves plotting this lower

limit of the dynamic detuning frequency as a function of detuning amplitude a,

for two values of Rmax, and two values of the acoustic amplitude excursion limit

β. For example, if Rmax = 10, and τ = 6 s, with a requirement β = 2, then the

dynamic detuning frequency is limited within the range 0.1-0.6 Hz, assuming

the detuning modulation amplitude a is between 2 and 16. We see that higher

detuning frequencies or larger detuning amplitudes both act to reduce amplitude

excursions. This defines the parameter space for suppressing PI by the dynamic

detuning mechanism.

In the following sections, we will see that the above mechanism can occur


Figure 7.4: Dynamic detuning frequency limit to prevent amplitude excursionsexceeding a predetermined value β. For example, for Rmax = 10 and τ = 6 s,with an amplitude growth requirement of β = 2, then the minimum dynamicdetuning frequencies are limited to between 0.6 Hz and 0.1 Hz for detuningamplitudes a in the range of 2 through 16.

naturally as a result of residual motion in the presence of mirror figure errors,

which we consider in the context of aLIGO and the 74 m cavity at Gingin.

7.4 Frequency modulation by mirror figure

errors

The test mass mirrors in aLIGO have RoCs of ∼2000 m. Figure errors mean

that the effective RoC depends on the spot position. For example, using Eq.

7.2, with aLIGO arm cavities, if the ETM RoC changes 1 m from its nominal

value of 2242m (corresponding to a sagitta change within a beam diameter ∼0.3

nm), the TEM10 cavity mode frequency will change by ∼13 Hz.

The residual motion of the test masses in interferometer arm cavities there-

fore cause high-order mode frequency modulation. Residual angular motion

creates beam residual motion on the test mass surface of ∼millimetres [159].

Depending on the test mass figure errors, this residual motion causes dynamic

detuning of the cavity high-order mode frequency, at the frequencies of test mass

7.4. FREQUENCY MODULATION 123

pitch and yaw motions.

−80 −60 −40 −20 0 20 40 60 80−5

−4

−3

−2

−1

0

1

Test mass diameter (mm)

Def

orm

atio

n (

nm

)

( a )

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

10

12

ETM rotation angle (microradians)

Mode

spac

ing c

han

ges

(H

z)

(b)

Figure 7.5: a) Typical aLIGO test mass figure errors (compared to a perfectsphere of RoC ∼2242 m) showing deformations across a mirror diameter. b)FFT code model for frequency offset as a function of test mass angular motion.

To estimate the aLIGO arm cavity high-order mode frequency changes as a

function of the residual test mass angular motion, we used measured test mass

surface data [160] in interferometer simulation codes (OSCAR [161] and FOPG

[162]) to simulate the cavity transverse mode detuning. For the simulation, we

fixed the ITM and modelled the ETM misalignment at various angles from zero

to 0.25 microradians. Figure 7.5(a) shows an example of the input data in the

form of a cross-section across a test mass diameter, showing how the figure errors


increase with radius. Figure 7.5(b) shows the calculated data for real 2D surface

profiles.

Figure 7.5(b) shows that the cavity mode spacing increases roughly quadrat-

ically with ETM misalignment angle. Note that 0.1 microradians corresponds

to a ∼2 mm beam position displacement on the test mass in a typical aLIGO

arm cavity. We extended the simulation to the test mass rotation corresponding

to a spot displacement of ∼6 mm. We note that, in reality, both test masses

move independently of each other, thereby creating somewhat larger detuning

amplitudes.

Cavity high-order mode frequency modulation can also be artificially created

by applying modulated heating to the test mass. We used the ANSYS FEM

software package to simulate the transient thermal deformation of the test mass

surface under sinusoidal heating power. Figure 7.6 shows the maximum thermal

deformation when a 0.1 Hz modulated heating beam of 50 mm radius, with 2 W

peak-to-peak power amplitude, is applied to the test mass front surface. This

deformation corresponds to a cavity mode spacing frequency change of ∼40 Hz

in an aLIGO arm cavity simulated using FFT code [161].

Figure 7.6: The maximum thermal deformation when 0.1 Hz sinusoidal heatingpower of amplitude 2 W is applied on the front surface of the test mass.

The above results indicate that passive detuning frequency modulation in

aLIGO would be expected to be ∼few Hz for the TEM10 transverse mode,

7.5. HIGH OPTICAL POWER CAVITY OBSERVATIONS 125

which is much smaller than the arm cavity linewidth, and does not have sig-

nificant effect on parametric gain. However, this could be increased to ∼40 Hz

modulation using CO2 laser heating. It is important to note that the high-

est predicted parametric gains in aLIGO are for modes up to 4th-order [154].

From Eq. 7.2, detuning scales with mode order. Thus the above estimates would

correspond to at least 4 times larger modulation (∼160 Hz) for 4th-order insta-

bilities. The aLIGO arm cavity half-linewidth is ∼40 Hz. Thus the parametric

gain associated with arm cavity optical modes would be suppressed by factors

∼few, and for low-order cavity modes the above modulations could be negligi-

ble. M. Evans et al. [154] show that the highest parametric gain instabilities are

associated with modes that are resonant in the power recycling cavity. For these

modes, the coupled cavity linewidth is about 0.3 Hz [105], and the normalised

dynamic detuning amplitude can exceed 50 times the coupled cavity linewidth.

Thus it is most likely that intrinsic passive detuning in aLIGO will lead to a

parametric gain suppression factor > 100 for those modes resonant inside the re-

cycling cavity. A detailed simulation to explore how the cavity high-order mode

frequency modulation affects the broad spectrum of PI in aLIGO is beyond the

scope of this paper. However, the experimental observation of PI presented in

the next section largely confirms the above theory.

7.5 High Optical Power Cavity Observations

We studied three-mode PIs at the Gingin facility. The experimental setup is

shown in Fig. 7.7. A 74 m long optical cavity with fused silica test masses is

suspended from high performance vibration isolators [89, 87] by a modular 4-wire

test mass suspension system developed at UWA. The test masses are installed

in two large vacuum chambers connected by a 400 mm diameter vacuum pipe.

The system was assembled in clean room conditions, and uses a hydrocarbon-free

vacuum system to enable high optical power densities to be achieved.

Both test masses are 100 mm in diameter and 50mm thick, with mass ∼0.8

kg. The nominal RoCs of the two test masses are 37.4 m and 37.3 m. The test

masses have a very sparse mode spectrum compared to aLIGO test masses, so

that three-mode parametric interactions need to be tuned to specific candidate

acoustic modes. This is achieved by using a power-stabilised CO2 laser to ther-

mally tune the ITM RoC to create three-mode tuning for the specific candidate

acoustic modes [76]. The dominant test mass residual angular motions are at


frequencies of ∼0.15 Hz.

The measured cavity finesse is 14500±300. The light source is a 50 W fibre

laser amplifier fed by a 400 mW Nd:YAG NPRO seed laser, which is frequency-

locked to the long cavity using PDH locking [85, 86]. The cavity transmission is

detected by a quadrant photodiode (QPD). The differential output of this QPD

measures the beating between the cavity fundamental mode and the first-order

mode, while the sum of the QPD output measures the total cavity transmitted

power. A spectrum analyser (Agilent 89410A) and a PC are used to analyse

and record the signal.

Figure 7.7: Schematic diagram of the experimental setup: the laser light froma seed laser is amplified by a 50 W fibre laser amplifier. The high opticalpower laser beam is injected into the 74 m long optical cavity. The seed laser isfrequency-locked to the long cavity using PDH locking. The cavity transmittedbeam is detected by a quadrant photodiode (QPD). The differential signal fromthe QPD measures the beating between the cavity fundamental mode and thefirst-order mode.

Using the ANSYS software package, we first analysed the test mass acoustic

mode structure and frequencies. Based on the simulation, we then identified one

particular acoustic mode that has good overlap with the cavity first-order mode,

and minimum vibration amplitude at the suspension point — to minimise the

mechanical loss introduced by the suspension. Our target mode, with simulation

frequency 150.49 kHz, is in the range for easy CO2 laser thermal tuning. The

mode amplitude distribution on the test mass surface is shown in Fig. 7.8.

The overlap factor, taking into account the mode effective mass, is ∼16. The


measured mode frequency is ∼150.2 kHz (depending on the temperature). The

measured mechanical Q-factor using the ring-down method is ∼ 3.4 × 106.

Figure 7.8: The test acoustic mode amplitude distribution on the test masssurface.

Three-mode interaction conditions are achieved by tuning the TEM00 and

TEM10 mode spacing close to 150.2 kHz using CO2 laser thermal tuning. Mea-

surement of the tuning is relatively easy because residual laser beam-jitter noise

gives rise to a small amount of TEM10 mode power inside the cavity which beats

with the TEM00 at the QPD, allowing the TEM10 offset frequency to be mon-

itored as a beat note. This provides a means for monitoring the mode spacing

by measuring the cavity transmitted power on the QPD where the two modes

are mixed.

The mode spacing was observed to fluctuate with a typical peak-to-peak am-

plitude ∼few kHz. To confirm that these fluctuations were associated with the

beam spot position on the test masses, we recorded the cavity mode spacing and

the beam position simultaneously for the ITM. Figure 7.9 shows the mode spac-

ing as a function of the beam position on the ITM in the horizontal direction.

The beam position was determined by recording the video of the CCD camera,

and then analysed by referencing it to the test mass diameter. In the horizon-

tal direction, there is a linear correlation between increased mode spacing with

increased beam displacement. The solid line in Fig. 7.9 is a linear least squares

fit to the measurement data. The relative large scatter is due to the fact that

we recorded only the ITM beam position, while the ETM beam position is also

not stable. The effect is more difficult to measure in the other axis because the


suspensions introduce much smaller vertical beam position fluctuations. How-

ever, the single axis correlation is sufficient to confirm our conjecture that mirror

figure errors translate into dynamic detuning.

We do not have precise metrology of our test mass mirror profiles. However,

the observed fluctuations are consistent with the mirror figure error specification

of ∼1 nm. It is interesting to note that, in principle, simultaneous measurement

of spot position on both test masses, and of the TMFO, could be used to deter-

mine the precise metrology of both test masses.

Figure 7.9: The correlation between horizontal spot position on the ITM aloneand the TMFO indicates that the frequency detuning is caused by the spotposition change. The solid line is a linear fit to the measurement data. Thespread of the data is due to the fact that the laser spot position is also changingat the ETM.

When the cavity is correctly tuned, the three-mode interaction occurs, and

the signal at the QPD becomes dominated by the beating between the TEM00

and TEM10 modes at the acoustic mode frequency. The signal is proportional

to the acoustic mode amplitude, the TEM00 mode power and the TEM10 mode

detuning. The signal is normally most easily observed by mixing the acoustic

frequency with a local oscillator, combined with a low pass filter, so as to reduce

the signal frequency to < 10 Hz.

As discussed above, residual motion causes cavity detuning. The residual

motion amplitude depends on environmental noise, which excites the suspension


normal modes. Most of the time, we observe dynamic detuning with a frequency

amplitude of 1-5 kHz. Even under these circumstances, the acoustic mode signal

at frequency ∼150.2 kHz can normally be clearly observed.

Wind forces on the laboratory, microseismic activity and human activity all

contribute to degrading the residual motion. During quiet times, the residual

motion is reduced and for times of ∼30 seconds the detuning amplitudes can be

less than a few cavity linewidths. In these short periods of time, conditions are

suitable for observing three-mode PI.

To observe the signature of PI, we increased the cavity circulating power to

∼ 30 kW. For periods of time ∼10 to 30 seconds, when the dynamic detuning is

low, the acoustic signal can be observed ringing up with time, as shown in Fig.

7.10. In this case, the acoustic signal frequency was down-converted to 0.91 Hz,

as discussed above.

6 8 10 12 14 16 18−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−6

Time (s)

Sig

nal

Am

pli

tude

(a.u

)

Figure 7.10: The QPD differential output signal at the test mass acoustic modefrequency (150.28 kHz). The signal was down-converted to ∼0.91 Hz by mixingwith a local oscillator signal. The solid line is a fitting curve of 0.91 Hz, withparametric gain Rmax = 6 and detuning amplitude a=2. The growing signalenvelope (dashed line) is consistent with suspension modulation at 0.15 Hz.The effective parametric gain is ∼1.45.

Observations under best-tuned quiet conditions show the acoustic signal

growing for times ∼14 seconds. This amplitude growth is modulated but more

complex than the single modulation frequency model used in Section 7.3, due to

the presence of several low-frequency modulations associated with the angular

motions of both test masses. Beating also occurs, due to the fact that the two


test masses have closely spaced suspension normal modes. This beating causes

the detuning amplitude to vary periodically over time scales ∼30 seconds. The

effective parametric gain, based on observed ring-ups during times of minimum

detuning amplitude, such as shown in Fig. 7.10, is R ∼1.45.

In Fig. 7.10, we have fitted to a double frequency of a single 0.15 Hz suspen-

sion mode to model the dynamic detuning. This gives a modest fit to the data,

but a complete fit is not possible due to the stochastic nature of the seismic

excitation of the normal modes.

7.6 Conclusions

We have created conditions in which three-mode parametric instability (PI) can

occur in a suspended high power optical cavity designed to mimic conditions

comparable to those expected in Advanced gravitational wave detectors. We

have observed time-dependent growth of a 150.2 kHz acoustic mode, consistent

with a new model of PI for suspended mass optical cavities. The gain in the

PI regime is lower than previously expected, and is modulated by low-frequency

residual motion. Results are consistent with a new model for the build up of

instability, in which transverse mode frequency fluctuations act to reduce the

PI power build-up through dynamic detuning, which itself is caused by residual

motion in the presence of nm-level mirror figure errors. Data on aLIGO optical

cavities indicate that the same phenomenon will act to reduce the risk of PI for

the highest parametric gain modes. Mirror imperfections have beneficial effects

in this regard. Results also point to simple methods for reducing parametric

gain by thermal modulation, or by low-frequency dithering of the test masses.

Further studies, on full-scale detectors to quantify the dynamic detuning and

linewidths of transverse modes, are needed to quantify these effects.

7.7 Acknowledgements

We wish to thank the Gingin Advisory Committee of the LIGO Scientific Col-

laboration and the LIGO Scientific Collaboration Optics Working Group for

encouragement. Thanks to our collaborators Jesper Munch, Peter Veitch and

7.7. ACKNOWLEDGEMENTS 131

David Hosken for useful advice. We wish especially to thank Slawek Gras for

his careful review of the manuscript, and the LIGO MIT group for their encour-

agement. This research was supported by the Australian Research Council.


Chapter 8

Investigation of an automatic

alignment system for the Gingin

facility

8.1 Preface

In the high-optical-power experiments undertaken in our facility, maintaining

the circulating power stable inside the cavity is critical. However, due to a

low-frequency drift of the optical path, the circulating power deviates from its

maximum value. Furthermore, the laser locking to the cavity is often lost, as

the cavity drifts too far away from its stable state. Relocking has to be done

manually.

In this Chapter, an Automatic Alignment (AA) system is investigated for the

East arm cavity of the Gingin facility. This system will be able to apply feedback

control to the cavity against the low-frequency drift, thus keeping the circulating

power at its maximum. This Chapter includes a theoretical study of the AA

system, and a preliminary test on the East arm cavity. More work needs to be

undertaken to apply the measured misalignment signals to the control system

of the test masses.

133

134 CHAPTER 8. AUTOMATIC ALIGNMENT

8.2 Introduction

A Fabry-Perot (FP) cavity is suspended by a vibration isolation system in

the East arm of the Gingin facility. Along the propagation direction of the laser

beam, the test masses are free. The laser is locked to resonate in the cavity

through a PDH locking system. The test masses have other degrees of freedom

in angular motions. These angular motions (pitches and yaws) of the test masses

are manually controlled at certain points, where the cavity and laser are best

aligned. This works by monitoring control masses hung from the test masses; a

feedback control system is used, which is introduced in Chapter 3.

However, this alignment system is vulnerable to low-frequency noises, such

as micro-seismic vibrations from the ground. In addition, due to temperature

changes, thermal expansions of the optical components cause variations in the

optical path. As a result, the alignment of the cavity will slowly deviate from

its optimal state. Thus, the circulating power will be reduced due to this mis-

alignment. In high-optical-power experiments, such as in the observation of

parametric instability (PI), maintaining maximum optical power is required. In

gravitational wave detection, a reduction of the optical power causes an increase

of the shot noise. Therefore, finding a way to keep a suspended cavity aligned

with the optical path is critical to gravitational-wave detection, and to our ex-

periments.

Other AA systems have been installed in various gravitational-wave detec-

tors, including LIGO [163], GEO600 [164], TAMA300 [165]. Good results were

achieved in all these setups. In this Chapter, a similar setup is introduced. This

Chapter is arranged as follows — the theoretical model of how AA systems work

is first studied in Section 8.3. The experimental setup is introduced in Section

8.4, followed by some preliminary results in Section 8.5. Conclusions and future

work are discussed in Section 8.6.

8.3. THEORY 135

8.3 Theory

8.3.1 Misalignment of an Optical Cavity

Figure 8.1: Misalignments of an optical cavity.

We first study the optics inside the cavity. In general, as shown in Fig. 8.1,

a stable cavity consists of two curved mirrors. The radii of curvature of these

mirrors are denoted as R1 and R2, respectively, and the g-factors of these two

mirrors are g1, 2 = 1 − LR1, 2

. The cavity g-factor is defined as g1 · g2. The East

arm cavity in the Gingin facility is a near-concentric cavity whose nominal cavity

g-factor is ∼0.96. As discussed in Chapter 2, the distances of the beam waist

from these two mirrors are given by Eqs. (2.11) and (2.12) [78]:

z1 =g2(1 − g1)

g1 + g2 − 2g1g2L, (8.1)

z2 =g1(1 − g2)

g1 + g2 − 2g1g2

L. (8.2)

If the angular displacements of the mirrors are θ1 and θ2, the translations of the

beam position from the center of each mirror are:

∆x1 =g2

1 − g1g2Lθ1 +

1

1 − g1g2Lθ2, (8.3)


∆x2 =1

1 − g1g2Lθ1 +

g1

1 − g1g2Lθ2. (8.4)

Then the angular displacement (i.e. tilt) of the cavity optical axis is given by:

∆θ =∆x2 − ∆x1

L=

(1 − g2)θ1 − (1 − g1)θ2

1 − g1g2. (8.5)

Combining Eqs. (8.1), (8.3) and (8.4), the translation at the beam waist is:

∆x = ∆x1 + ∆θ · z1 =L(g1θ2 + g2θ1)

g1 + g2 − 2g1g2. (8.6)

The misalignment of the cavity is denoted by these two values, ∆θ and ∆x. ∆x

is called the translation misalignment, and ∆θ is called the tilt misalignment.

Now we look at the electric field inside the cavity when it has misalignments.

Later on, we will explain how to measure this field as a probe of the cavity mis-

alignment. At the waist location, the normalized amplitudes of the fundamental

(TEM00) mode and the first-order (TEM01) mode can be denoted as [166]:

U0(x) = (2

πω20

)1/4exp[−(x

ω0)2], (8.7)

U1(x) = (2

πω20

)1/4(2x

ω0)exp[−(

x

ω0)2], (8.8)

where ω0 is the waist size. Here, for simplicity, we only deal with one transverse

dimension, as the same discussion can be applied to the other transverse di-

mension. With a pure translation ∆x, the amplitude of the fundamental mode

becomes [167]:

E(x) = AU0(x − ∆x) ≈ A[U0(x) +∆x

ω0U1(x)]. (8.9)

Whereas with a pure tilt misalignment ∆θ, this amplitude becomes [167]:

E(x) = AU0(x)exp(i2π∆θx

λ) ≈ A[U0(x) + iπ

ω0∆θ

λU1(x)]. (8.10)

Combining Eqs. (8.9) and (8.10), the beam field at the waist position inside the

cavity is:

E(x) ≈ A[U0(x) +∆x

ω0U1(x) + iπ

ω0∆θ

λU1(x)]. (8.11)

We note that both types of misalignments are embedded in this expression of the

8.3. THEORY 137

electric field. The next step is to identify this information on the misalignments

in the field.

If there were a photodetector at the waist position, it would be easy to extract

the misalignment signals. However, there is no way to put a photodetector inside

the cavity without blocking the laser beam. The only other option is to measure

the electric field somewhere outside the cavity.

The propagation of the electric field, with pure translation and pure tilt

misalignments, can be expressed as:

U0(x − ∆x, z) ≈ U0(x, z) +1

ω(z)∆xU1(x, z), (8.12)

and

U0(x, z)exp[i2π

λγ(z)x] ≈ U0(x, z) + i

πω(z)

λγ(z)U1(x, z), (8.13)

where the fundamental and first-order modes are given by:

U0(x, z) = (2

πω2(z))1/4exp[−(

x

ω(z))2], (8.14)

U1(x, z) = (2

πω2(z))1/4 2x

ω(z)exp[−(

x

ω(z))2], (8.15)

and where ω(z) = ω0

√

1 + (z/zR)2 is the laser beam spot size at position z,

zR =πω2

0

λis the Rayleigh range of the beam, and γ(z) = ∆θz2

R/(z2 + z2R) is the

angle between the wavefronts of the two modes at z. These two equations are

valid when the following small misalignment conditions are satisfied: ∆x . ω(z),

γ(z) . λ/[πω(z)].

At the waist position (z = 0), Eq. (8.11) can be rewritten as [168]:

E(x, 0) ≈A[U0(x, 0) + χ1U1(x, 0) + χ2iU1(x, 0)]

=A[U0(x, 0) + χexp(iθW )U1(x, 0)], (8.16)

where

χ1 =∆x

ω0

, (8.17)

χ2 =πω0∆θ

λ, (8.18)

χ =√

χ21 + χ2

2, (8.19)


θW = arctan(χ2/χ1). (8.20)

From Eqs. (8.16) through (8.20), it is clear that the amplitudes of both the trans-

lation and tilt misalignments are determined only by the coefficient (χexp(iθW ))

of the second term in the expression for the E(x, 0) field. This discovery is the

key point of our scheme for the AA system, as will be seen later.

In one transverse dimension, an nth-order Hermite-Gaussian mode can be

expressed as:

Vn(x, z) = NnHn(√

2x

ω(z))

ω0

ω(z)× exp[−i(kz − ζn) − x2(

1

ω2(z)+

ik

2R)], (8.21)

where Hn is the nth-order Hermite polynomial, k = 2π/λ is the wave number,

and the normalizing factor Nn is a ratio of Γ functions:

Nn =Γ(n

2+ 1)

Γ(n + 1). (8.22)

The additional phase term ζn, which is the phase difference between the Gaus-

sian beam and an ideal plane wave propagating in the same direction, can be

expressed as:

ζn = (n + 1) arctan(λz

πω20

). (8.23)

Here, we only deal with the Hermite-Gaussian modes, because first-order

Laguerre-Gaussian modes are equivalent to second-order Hermite-Gaussian modes,

which are negligible in our first-order approximations for the translation and tilt

misalignments — the misalignments caused by a longitudinal shift of the waist

position, or a change of the waist size, can be converted to modulations of first-

order Laguerre-Gaussian modes. In our study, we only take into account the

translation and tilt misalignments; the other two misalignments of second-order

are negligible in our cavity.

Note, in Eq. (8.23), that there is a Guoy phase difference between the fun-

damental mode and the first-order mode, which is given by:

ζ(z) = arctan(z

zR

). (8.24)

It can be shown that the electric field of the laser beam propagating to a position

8.3. THEORY 139

z can be expressed as:

E(x, z) = AU0(x, z) + χei[θW +ζ(z)]U1(x, z). (8.25)

From Eq. (8.25), we can conclude that the phase of the coefficient of the

first-order mode is determined by the Guoy phase difference ζ(z). For instance,

when ζ(z) = 0, the real part of the electric field of the first-order mode repre-

sents the translation misalignment, and the imaginary part of it represents the

tilt misalignment; When ζ(z) = π/2, the real part of this coefficient comes to

represent the tilt misalignment, while the imaginary part represents the transla-

tion misalignment. We will use this property in the next subsection to separate

the translation and tilt misalignments.

8.3.2 Obtaining the Misalignment Signals

As mentioned above, the misalignment signals are embedded in the first-order

mode in the propagating electric field. Here, we use a modulation-demodulation

method to extract the misalignment signals [163]. This method is described

below.

50W fibre amplifierFaraday

IsolatorBS

Cavity

4 2

photodetector

Figure 8.2: The EOM is used to modulate the injection laser beam. The side-bands created by this modulation will beat with the first-order mode electricfield leaking from the cavity, and thus generate differential signals on the QPDs.


A conceptual diagram of the setup is shown in Fig. 8.2. The injection laser

beam is modulated by an electro-optic modulator (EOM), which is the same

modulator used in the PDH locking system for this cavity. The EOM dithers

the phase of the injection laser beam and creates two sidebands, which can be

expressed as:

Deiωt+iβ sin(Ωt) = Deiωt[J0(β) +∑

k

Jk(β)eikΩt +∑

k

(−1)kJk(β)e−ikΩt], (8.26)

where D is the amplitude of the injection electric field, ω is its angular frequency,

and β and Ω are the EOM modulation depth and frequency, respectively; Jk(β)

is the kth-order Bessel function in the Jacobi-Anger expansion, where k runs

from 1 to ∞. With a small β, we only expand Eq. (8.26) to first order:

Deiωt+iβ sin(Ωt) ≈ Deiωt(1 + iβsin(Ωt))

= Deiωt(1 +β

2(eiΩt − e−iΩt))

= D(eiωt +β

2ei(ω+Ω)t − β

2ei(ω−Ω)t). (8.27)

As a result, there are created two sidebands with frequency differences of ±Ω

from the carrier light field of frequency ω. In the model discussed in the previous

subsection, D = AU0(x, z).

The light reflected from the cavity consists of two parts: the light directly

reflected from the input test mass (ITM), and the light leaked out from the

cavity. According to Eqs. (8.25) and (8.27), they can be expressed as:

Ereflect(x, z) ≈ −AU0(x, z)[eiωt +β

2ei(ω+Ω)t − β

2ei(ω−Ω)t], (8.28)

Eleak(x, z) ≈ AκeiωtU0(x, z) + χexpi[θW + ζ(z)]U1(x, z), (8.29)

where κ is the coefficient of the leaked light, which can be given by:

κ ≈ 2T1

T1 + T2 + Lt, (8.30)

where T1 and T2 are the respective reflectivities of the ITM and the output

test mass (ETM), and Lt is the total optical loss of the cavity. Note that the

direction of the directly reflected light is determined by the angular motions of

the ITM. By using an auxiliary centering system, the reflected beam is always

8.3. THEORY 141

centered on the quadrant photodetectors (QPDs) used in the AA system.

The total light field reflected from the cavity is:

ER(x, z) =Ereflect(x, z) + Eleak(x, z)

≈− A(1 − κ)U0(x, z)eiωt

− Aβ

2U0(x, z)ei(ω−Ω)t +

Aβ

2U0(x, z)ei(ω+Ω)t

+ AU1(x, z)eiωtκχexpi[θW + ζ(z)]. (8.31)

We only need to consider the electric fields where ζ(z) = nπ and (n + 12)π.

The positions where ζ(z) = nπ are called near-field, while those where ζ(z) =

(n + 12)π are called far-field.

In the experiment, a QPD is put at a near-field position, and another one is

put at a far-field one, to measure the light signals. Specifically, the light intensity

at a near-field position is:

INF(x, z) ≈−A(1 − κ)U0(x, z)eiωt − Aβ

2U0(x, z)ei(ω−Ω)t

+Aβ

2U0(x, z)ei(ω+Ω)t + AU1(x, z)eiωtκχei[θW +ζ(z)]

× −A(1 − κ)U0(x, z)e−iωt − Aβ

2U0(x, z)e−i(ω−Ω)t

+Aβ

2U0(x, z)e−i(ω+Ω)t + AU1(x, z)e−iωtκχe−i[θW +ζ(z)]. (8.32)

The result consists of four terms, which are written as:

INF(x, z) = TNFDC + TNF

2Ω + TNFΩ/S + TNF

Ω/AS, (8.33)

where the 1st term is the DC signal; the 2nd term has a frequency of 2Ω (which

is not of interest); the 3rd term has a frequency of Ω and is symmetric in the

x direction (also not of interest); and the only term we are interested in is the

4th term, which has a frequency of Ω and is antisymmetric in the x direction.

This last term is registered by the QPDs as differential signals, which will be

discussed later.


Specifically, this 4th term in Eq. (8.33) is:

TNFΩ/AS =Ce−iωtκχe−iθW

[−ei(ω−Ω)t + ei(ω+Ω)t]

+ eiωtκχeiθW

[−e−i(ω−Ω)t + e−i(ω+Ω)t]=C[κχe−iθW

(eiΩt − e−iΩt) + κχeiθW

(e−iΩt − eiΩt)]

=2iC sin(Ωt) × (κχe−iθW − κχeiθW

)

=2iC sin(Ωt) × (−2iκχ sin θW )

=4κCχ2 · sin(Ωt), (8.34)

where

C =A2β

2U0(x, z)U1(x, z). (8.35)

Recalling Eq. (8.18), the differential signals in the QPD at a near-field posi-

tion contains only the amplitude of the tilt misalignment.

Similarly, a QPD is put at a far-field position; the intensity of light received

by this QPD is:

IFF(x, z) ≈−A(1 − κ)U0(x, z)eiωt − Aβ

2U0(x, z)ei(ω−Ω)t

+Aβ

2U0(x, z)ei(ω+Ω)t + AU1(x, z)eiωtκχei[θW +ζ(z)]

× −A(1 − κ)U0(x, z)e−iωt − Aβ

2U0(x, z)e−i(ω−Ω)t

+Aβ

2U0(x, z)e−i(ω+Ω)t + AU1(x, z)e−iωtκχe−i[θW +ζ(z)]. (8.36)

It also has four terms:

IFF(x, z) = T FFDC + T FF

2Ω + T FFΩ/S + T FF

Ω/AS, (8.37)

where only the 4th term is of interest to us. Specifically, this term is given by:

T FFΩ/AS =Ce−iωtκχe−i(θW + π

2)[−ei(ω−Ω)t + ei(ω+Ω)t]

+ eiωtκχei(θW + π2)[−e−i(ω−Ω)t + e−i(ω+Ω)t]

=C[κχe−i(θW + π2)(eiΩt − e−iΩt) + κχei(θW + π

2)(e−iΩt − eiΩt)]

=2iC sin(Ωt) × (κχe−i(θW + π2) − κχei(θW + π

2))

=2iC sin(Ωt) × (−2iκχ cos θW )

=4κCχ1 · sin(Ωt). (8.38)

8.3. THEORY 143

Recalling Eq. (8.17), the differential signals in the QPD at a far-field position

contains only the amplitude of the translation misalignment.

8.3.3 Misalignment Signals in the East Arm Cavity

The length of the East arm cavity is L = 74 m. The radii of curvature of

the ITM and ETM are R1 = 37.4 m and R2 = 37.3 m, respectively.

Recalling Eqs. (8.5) and (8.6), the differential signals in the QPDs at the

near-field and the far-field positions can be written as:

SNF(t) =πω0H

λsin(Ωt)

(1 − g2)θ1 − (1 − g1)θ2

1 − g1g2, (8.39)

and

SFF(t) =LH

ω0sin(Ωt)

g2θ1 + g1θ2

g1 + g2 − 2g1g2, (8.40)

where H is a constant given by:

H =8A2βκ

π2. (8.41)

By mixing these signals with a local oscillator which has a frequency of Ω

(the EOM modulation frequency), the misalignment signals can be demodulated

to give:

SNF(t) =πω0H

2λ

(1 − g2)θ1 − (1 − g1)θ2

1 − g1g2, (8.42)

SFF(t) =LH

2ω0

g2θ1 + g1θ2

g1 + g2 − 2g1g2. (8.43)

These two demodulated signals are called the error signals.

The angular displacements of the test masses can be derived from Eqs. (8.5)

and (8.6) as:

θ1 =1 − g1

L∆x +

g1(1 − g1g2)

g1 + g2 − 2g1g2

∆θ, (8.44)

θ2 =1 − g2

L∆x − g2(1 − g1g2)

g1 + g2 − 2g1g2∆θ, (8.45)

where

∆x =2ω0

HSFF(t), (8.46)


∆θ =2λ

πHω0SNF(t). (8.47)

Consequently, θ1 and θ2 can be determined by measuring the error signals

at the near-field and far-field positions. Substituting Eqs. (8.46) and (8.47) into

(8.44) and (8.45), the angular displacements become:

θ1(t) =1 − g1

L

2ω0

HSFF (t) +

g1(1 − g1g2)

g1 + g2 − 2g1g2

2λ

πHω0SNF(t)

= C1 · SFF(t) + C2 · SNF(t), (8.48)

θ2(t) =1 − g2

L

2ω0

HSFF (t) − g2(1 − g1g2)

g1 + g2 − 2g1g2

2λ

πHω0SNF(t)

= C3 · SFF(t) + C4 · SNF(t), (8.49)

where C1, C2, C3 and C4 are the coefficients to be determined in order to control

the two test masses independently.

8.4 Experimental Setup

The setup of the AA system is composed of three major parts: optical setup,

electronic setup and feedback control setup. The optical setup is to sense the

misalignment signals, while the electronic setup is used to obtain the angular

displacements of the test masses in 4 degrees of freedom (2 pitches and 2 yaws).

Eventually, the signals are fed back to the control system to align the cavity

back to its optimal position. These three parts are described in the following

subsections.

The optical setup

The optical setup of the AA system is arranged at the reflection port of the

cavity (see Fig. 8.3). The reflected beam from the cavity transmits through

a 95% reflection beam splitter. This beam is then split by a beam splitter.

One of the split beams is used in PDH locking. The setup of PDH locking


Laser F a r a d ay

Isolator4 2C

avity

NF QPD

FF QPD

PD for

PDH locking

f=10 m

f=4 m

BS

BS

BS (R=95%)

f=100 mm

Table

Figure 8.3: Schematic figure of the optical setup of the AA system. The dashedbox represents the optical table. The thick solid lines in red color represent theoptical path of the main beam. The dashed lines represent the optical path inthe AA system. Four galvanometer scanners are used in the auxiliary centeringsystem. This figure is not to scale.

is not shown in Fig. 8.3. The other split beam is again split into two beams.

One is for near-field misalignment detection, while the other one is for far-field

misalignment detection. A QPD is used at the end of each optical path to receive

the misalignment signals. The signals from the QPDs are sent to the electronic

setup, which will be introduced later. In addition, there are two galvanometer

scanners in each of the near-field and far-field optical paths. The galvanometer

scanners are driven by signals from servo amplifiers, where the DC parts of

the QPD differential signals are injected as inputs. This subsystem, called the

auxiliary centering system [164], is able to center the beams on the QPDs all

the time, and thus dramatically increase the dynamic range of the AA system.

In order to get correct misalignment signals from the QPDs, they need to


be put at near-field and far-field positions, respectively. In other words, the

Guoy phase of the beam at the near-field QPD should be nπ, while the Guoy

phase should be (n + 12)π at the far-field QPD. Therefore, a mode-matching

should be undertaken for each optical path. Because there is only limited space

on the optical table, lenses are required in the mode-matchings. However, the

use of lenses should be minimized, because too many lenses could distort the

laser beams, and thus introduce some extra noise to the AA system. After

an investigation of mode-matching, and several trials, it was found that only a

100 mm lens is needed in the near-field optical path. The results of the mode-

matchings are given in Sec. 8.5.

The electronic setup

The differential signals from the QPDs are used to obtain the information on

the angular displacements of the test masses, through an electronic setup (see

Fig. 8.4). As discussed in Sect. 8.3, the QPD signals are mixed and demodulated

with local oscillators in mixers. NFX, NFY, FFX and FFY represent differential

signals from the near-field and far-field QPDs in the x and y directions, respec-

tively. The local oscillators are phase-matched to the QPD signals by phase

shifters. The demodulated signals are then filtered by low-pass filters. What is

left after the low-pass filters are the signals of the translation and tilt misalign-

ments. These signals are then amplified by generic amplifiers, and then sent to

add/subtractors (A/Ss). According to the discussion in Sect. 8.3, correct gains

in the A/Ss can be chosen to separate the 4 signals of angular displacements.

The signals output from the A/Ss are: ITM pitch (ITMX), ITM yaw (ITMY),

ETM pitch (ETMX) and ETM yaw (ETMY), respectively. They are fed back

to the control system to guide the test masses back to their optimal positions.

The input-output relationship in the electronic setup can be expressed as:

G1NFX + G2FFX → ITMX,

G3NFX + G4FFX → ETMX,

G5NFY + G6FFY → ITMY,

G7NFY + G8FFY → ETMY, (8.50)

where the coefficients G1,...,8 are determined by the gains in the A/Ss.

In addition, four servo amplifiers are used to monitor the DC parts of the


lowpass filter g e n e r ic amplifierm i x e r phase shifter servo amplifier

G+-a d d / s u b t r a c t o r

G G G G

N F X N F Y F F X F F YG N F X G N F Y G F F X G F F YL O L O L O L ON F X N F Y F F X F F Y

+ + + +I T M X I T M Y E T M X E T M Y

Figure 8.4: Schematic figure (left) and pictures (right) of the electronic setup.The bottom right picture is a home-made low-pass filter designed for obtaininglow frequency signals. The arrows in the schematic figure show the directions ofthe signal flow.

QPD differential signals, and send signals back to control the galvanometer

scanners. The signals sent back are denoted as: GNFX, GNFY, GFFX, GFFY,

respectively.


The control setup

As described in Chapter 3, the suspended mirrors are controlled by the con-

trol masses. In a complete AA system, the misalignment signals ITMX, ITMY,

ETMX and ETMY should be sent to the actuators of the control masses via a

digital-signal-processor (DSP). A Labview program needs to be constructed to

ensure that these signals are compatible with the original PID control signals of

the control masses. A switch is also needed in the DSP system for AA. When

the switch is off, the system has only normal feedback control; when it is on, the

AA signals are added. This switch is important because it can avoid putting

the system into a chaotic state, if some accident occurs in the AA system (e.g.

arbitrary signals could be sent to the control system to destroy the stability of

the cavity). This control setup will be completed in the near future.

8.5 Preliminary Results

8.5.1 Mode-matching

As discussed in the previous section, mode-matchings are required for the

near-field and far-field optical paths. A MATLAB program is used to simulate

these mode-matchings. In this program, parameters such as the laser wave-

length, waist size, waist position, positions and focal lengths of the optical ele-

ments are input. The program automatically simulates the propagation of the

laser beam. Various choices and arrangements of optical elements can be tested

in this program. Practical limits, such as the size of the optical table, availability

of some particular lenses, and the size of the charge-coupled devices (CCDs) of

the QPDs, have to be taken into account. As a result of these mode-matching

simulations, it was found that a lens with a focal length of 100 mm is needed in

the near-field path; the far-field path does not need any lens in the simulation.

Figures 8.5 and 8.6 show the simulation setups and results for the near-field and

far-field, respectively. If the position of the laser source in Fig. 8.3 is set as “0”

8.5. PRELIMINARY RESULTS 149

Figure 8.5: Simulation interface for the near-field mode-matching. The resultantGuoy phase is close to 0, while the beam size is only 0.9 mm, at the position of48000 mm.

in the simulation, the position of the f=100 mm lens is then 46963 mm. The

position of the QPD in the near-field is 48000 mm, and the QPD in the far-field

is put at 46600 mm. Note that the two QPDs are actually put in different op-

tical paths because the beam is split into two paths by the beam splitter. The

Guoy phase and the beam size at the near-field QPD are -0.04 radian and 0.9

mm, respectively. The Guoy phase is (π2

+ 0.002) radian and the beam size is

0.9 mm at the far-field QPD. The beam sizes are smaller than the diameter of

the CCD in each QPD, which is ∼5 mm. These simulation results satisfy the

requirements of the AA system.


Figure 8.6: Simulation interface for the far-field mode-matching. The resultantGuoy phase is close to π/2, while the beam size is also 0.9 mm, at the positionof 46600 mm.

8.5.2 Misalignment Signals

The coefficients in Eq. (8.50) were first determined according to Eqs. (8.48)

and (8.49). For the East arm cavity, Eqs. (8.48) and (8.49) can be written as:

(

θ1

θ2

)

≈(

3 1

3 −1

)(

∆x

∆θ

)

, (8.51)

where ∆x and ∆θ can be determined from the amplitudes of the QPD signals,

according to Eqs. (8.46) and (8.47). The gains in the A/Ss can be determined

accordingly. In our experiment, the intensity of the far-field signal was about

two times larger than that of the near-field. Thus, the ratio between far-field

gains and near-field gains was calculated to be 6:1.

8.5. PRELIMINARY RESULTS 151

ITM pitch ITM yaw

ETM pitch ETM yaw

Figure 8.7: Testing of misalignment signals. Four degrees of freedom were alltested. (1) In the top left figure, the ITM pitch mode was excited — the lowerperiodic curve in this figure is the ITMX signal from the A/S. (2) In the topright figure, the upper curve is the ITMY signal. (3) In the bottom left figure,the upper curve is the ETMX signal. (4) In the bottom right figure, the lowercurve is the ETMY signal. In all 4 degrees of freedom, only one channel showsperiodic motion with the same frequency as the excitation.

After setting the gains at the calculated ratio, the misalignment signals

ITMX, ITMY, ETMX and ETMY were then tested. The testing strategy is

as follows: periodic angular motions were electrically excited by electrostatic

plates at the backs of the test masses. Each angular degree of freedom was suc-

cessively excited, one at a time. For instance, the ITM pitch was first excited.

Then the ITMX and ETMX signals from the corresponding A/Ss were mea-

sured. A periodic signal profile should only be measured in ITMX, according to

the theory.

A function generator (FG) was used to provide sinusoidal excitations to the

test masses. The frequencies of these excitations were chosen to be the resonance

frequencies of the angular modes. The resonance frequency of the yaw mode of

the test masses is 1.7 Hz, while the pitch mode resonance frequency is 5.7 Hz.


Fig. 8.7 shows the excitation of these four degrees of freedom. In each of these

figures, only one channel shows periodic motion with the same frequency as the

excitation. This demonstrates that our design for the AA system is working

properly.

8.6 Conclusion and Future Work

An AA system was investigated for the East arm cavity in the Gingin facility,

in order to avoid low-frequency misalignments of the cavity. In this Chapter,

a strategy to extract misalignment signals was first studied in theory, and then

realized in experiment. Mode-matchings of the near-field and the far-field were

undertaken with the help of a simulation program. Good positions of the QPDs

were determined, and a minimum lens was used in the optical setup. A prelimi-

nary test was undertaken. The result shows that the misalignment signals were

successfully extracted by the optical and electronic setups.

To make the AA system work properly, more work needs to be done. First,

further testing may help fine tune the gains of the A/Ss. Second, a Labview

program with a switch is required to add the misalignment signals to the original

control signals, so as to avoid exciting the cavity during malfunctions of the AA

system.

Chapter 9

Conclusions and Future Work

9.1 Conclusions

The experimental studies in this thesis were undertaken in two separate

Fabry-Perot cavities: the South arm and the East arm of the Gingin High Opti-

cal Power Test Facility. In the South arm, three-mode optomechanical interac-

tions were investigated. By tuning this cavity with a CO2 laser heating the end

test mass (ETM), a high-sensitivity three-mode optomechanical transducer was

demonstrated. A displacement sensitivity of ∼ 1 × 10−17 m/√

Hz was achieved,

which was sufficient to observe a thermally excited acoustic mode. This trans-

ducer is intrinsically immune to laser amplitude and phase noise, due to the

properties of three-mode optomechanical interactions. A calculation showed

that a displacement sensitivity of ∼ 2 × 10−20 m/√

Hz could be achieved, by

applying the same scheme to the East arm of the Gingin facility.

In the East arm, several experiments were undertaken. The optical properties

of the cavity were first investigated. The cavity length was precisely measured

with an uncertainty of 40 µm. A linewidth of 140± 2 Hz, a g-factor of 0.976551

± 1.4×10−5 and a finesse of 14500 ± 300 were measured for this cavity. In addi-

tion, through tuning the cavity g-factor over a wide range by using a CO2 laser

to heat the ITM, degeneracies of the optical transverse modes and reductions of

finesse were observed. Based on our model, the reductions in finesse were likely

caused by mode degeneracies. A coupling rate between the fundamental mode

and the 19th-order degeneracy mode was estimated. By applying this coupling

rate to all the observed degeneracy modes, their total optical losses could be

153

154 CHAPTER 9. CONCLUSIONS AND FUTURE WORK

estimated.

The mechanical properties of the test masses in the East arm were also inves-

tigated. The eigenfrequencies and quality factors of some lowest-order mechani-

cal modes were measured for the input test mass (ITM). An ANSYS simulation

was created to calculate the strain energies of the substrate, coating and suspen-

sion points of the test mass. The mechanical loss angle of the ITM due to the

modular suspension system was estimated by combining the measurement data

and this simulation. The thermal noise of the ITM was calculated at 100 Hz,

assuming that a Gaussian beam was incident on the mirror. This result shows

that the modular suspension system contributes only ∼10% of the total thermal

noise. In another ANSYS simulation, the same suspension system was applied

to the test mass used in aLIGO. The result shows that no more than 15% of the

total thermal noise is contributed by this suspension system — this could be

suggested as a possible design to be used in third-generation gravitational-wave

detectors.

After the optical and mechanical properties of the cavity were studied, an ex-

periment to observe parametric instability (PI) was conducted. PI was observed

in a large-scale suspended optical cavity for the first time. The result shows that

the measured parametric gain was smaller than expected for a configuration in

which the suspended cavity is assumed stable. In fact, due to residual angular

motions of the suspended test masses and figure errors in the mirror surfaces,

the transverse mode frequency offset (TMFO) was found to drift periodically.

As a result, the parametric gain was suppressed by this dynamical modulation,

which could be considered as an approach to intrinsically suppress the PI. It was

demonstrated that suppression factors of 10-20 could be achieved for aLIGO, by

using our proposed scheme.

To avoid low-frequency drift of the optical path away from the cavity, an au-

tomatic alignment (AA) system was investigated and preliminarily implemented

in the East arm. The optical path of this system was mode-matched using re-

sults from a MATLAB simulation. Based on these mode-matchings, 2 QPDs

were inserted in the optical paths — one at a ‘near field’ position, the other at

a ‘far-field’ position. The misalignment signals from the QPDs were configured

to generate four independent signals which represent the angular displacements

of the 2 test masses, respectively — the results of the testing were successful.

In this thesis, three-mode optomechanical systems, such as the three-mode

transducer and PI, were introduced, and which are applicable to gravitational-

9.2. FUTURE WORK 155

wave detection and quantum nondemolition measurement; by applying these sys-

tems in a gravitational-wave detector, the mechanical modes of the test masses

could be monitored, their effective temperatures could be cooled down to the

ground-state, and quantum radiation pressure noise could be measured.

9.2 Future Work

There is plenty of work to do to continue the projects presented in this thesis.

The three-mode optomechanical transducer could be demonstrated in the East

arm, where the sensitivity is expected to be limited by the quantum shot noise.

It can also be used in a tilt gravitational-wave detector, which is susceptible to

gravitational waves at mechanical mode frequencies. The idea here is that a

gravitational wave at a mechanical mode frequency could excite mechanical vi-

brations, which would be registered in the transducer. A preliminary calculation

will be implemented to estimate the sensitivity of this tilt detector.

As for the transverse mode degeneracy in the East arm, a finer tuning of

the TMFO could be achieved by increasing the CO2 beam spot size on the test

mass, or by using some neutral density filters. The optical losses of different

modes can be determined with more accuracy. A useful tool can be made to

monitor the roughness, or the figure errors, of the test masses, by measuring

the cavity finesse at degeneracies — this could be a diagnostic technique in the

manufacturing of test masses with smaller roughness and figure errors.

In the same way that mechanical losses in the ITM were investigated in the

East arm, those of the ETM could also be studied, by using the same method

introduced in Chapter 6. A similar result is expected to be attained, as the two

test masses have the same size, and are attached to identical suspension systems.

The coating on the ETM has more layers than the ITM — so we expect that the

modular suspension system should contribute less to the thermal noise on the

ETM. A more careful study could be implemented for the suspension system,

in order to avoid couplings of the test masses to the pendulum violin modes.

As discussed in Chapter 7, the cavity became unstable when the circulating

power was increased. A possible reason is that some higher-order optical modes

were coupled to the laser beam, and scattered to the shadow sensors of the

156 CHAPTER 9. CONCLUSIONS AND FUTURE WORK

control system. As a result, the control system could not work properly as the

signal-to-noise ratio was low. Therefore, a purer laser beam is required. As a

solution, a pre-mode cleaner will be put into the main optical path, to allow only

the fundamental mode to be injected into the cavity. This should dramatically

reduce the scattering noise for the control system. PI with more parametric gain

is expected to be seen in this cavity. In addition, methods to suppress PI, such

as thermal tuning and laser dithering, could be tested.

Apart from PI, a double optical spring could be demonstrated in the East

arm. In this setup, two laser beams would be injected into the cavity. A negative

optical inertia could be produced by finely tuning the powers and frequencies

of the two beams. As a result, the effective test masses would be reduced, and

the response to gravitational waves would be reinforced over a broad frequency

band.

In addition, the design and construction of the AA system should be finished;

it should help prevent the loss of both circulating optical power and cavity lock.

This system may also help stabilize the test masses against residual motions, as it

has higher sensitivity than the optical lever. The electronic setup in this system

should also be modified — an increase in the frequency coverage of the low-pass

filter is needed, in order to accommodate the 1.7 Hz and 5.7 Hz pendulum mode

frequencies.

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