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A Study of Optical Instabilities for

Advanced Gravitational-Wave

Detectors

This thesis is

presented to the

Department of Physics

for the degree of

Master of Science

By

Lucienne Merrill

February 2011

c© Copyright 2011

by

Lucienne Merrill

iii

Abstract

This thesis reports the experimental observations of radiation pressure induced

optical instabilities in a 80 m suspended high optical power cavity. These

instabilities have the potential to disrupt the operation of the next generation gravi-

tational wave detectors. This thesis provides the theoretical development, as well as

experimental results and solutions to two such optical instabilities. The first exper-

iment describes and presents results on the angular optical instability effect, where,

depending on several parameters, the optical spring effect could disrupt the per-

formance of gravitational wave interferometers. The second experiment presents

two potential solutions for parametric instability, a known threat to high power

Fabry-Perot optical cavities.

In the angular optical instability experiment presented in this thesis, it was found

that the magnitude of the negative optical spring constant per unit power is a

few N · m/W as the result of optical torsional stiffness in the yaw mode of the

suspended mirror south arm Fabry-Perot cavity at AIGO. These results are shown

to be consistent with the theory, reviewed also in this theory, of the optical torque

effect as described by Sidles and Sigg in their paper published in 2006 [56].

The parametric instability experiment described in this thesis provides a preliminary

solution to the opto-acoustic parametric interactions as they arise in high power,

suspended Fabry-Perot cavities. This experiment demonstrated the suppression of

an excited high order in the south arm Fabry-Perot cavity at AIGO, by injecting a

low power, anti-phase TEM01 mode, as part of an optical feedback loop, into the

cavity to destructively interfere with the excited cavity mode. Although preliminary,

the results of this experiment provide a stepping stone to finding a solution using an

optical feedback loop, to suppress parametric instability in advanced gravitational

wave detectors.

iv

Acknowledgements

This thesis would not be possible without the support of many people. First, I

would like to thank my supervisors, David Blair, Li Ju and Chunnong Zhao.

Together, you helped push me to succeed and supported me in finishing this thesis.

Without any of your support, I would never have made it to UWA in the first place,

thank you.

I would like to thank Jean-Charles Dumas, you were a constant support for the

first years of my work at UWA and a very good friend. Additionally, I would

like to thank all of the people in the gravity wave group at UWA who made the

trips between Gingin, the work in the labs and in the office much more enjoyable:

Yaohui Fan, Shaun Hooper, Susmithan Sunil, Andrew Woolley, Pablo Barriga, Hai-

Xing Miao, Zhongyang Zhang, Slawek Gras, Francis Torres, Viet Dang, Hamish

Glenister, Sundae and Andrew. You were all great friends that I hope to see again

someday. Also, I would like to thank two members of the Optics & Photonics group

at the University of Adelaide: David Hosken and Miftar Ganija. Since your visit

to Gingin in 2008 I have learnt more than I knew in the area of laser physics, in

addition to making two really good friends.

Lastly, I would like to thank my family. To my parents, Dan and Noelle Merrill, as

well as my brother Brecht and sister Zora: you have seen me go around the world

more than once, and have always been supportive of my dreams and doings. Thank

you, and I love you all. To my husband, Timo, without your never-ending patience,

love and encouragement we would never be where we are today, and perhaps this

thesis would never have seen the light of day. Dankeschon an mein Lowe und unser

Lowchen, Jacqın.

v

Contents

1 Introduction 3

1.1 A New Wave of Physics . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 A Brief History of Interferometry . . . . . . . . . . . . . . . . . . . 10

1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Theory of Radiation Pressure Effects 16

2.1 The Art of Fabry-Perot Cavities . . . . . . . . . . . . . . . . . . . . 16

2.2 Radiation Pressure Forces . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Optical Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Quasi-Static Motion . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Full Optical Spring Effect . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Parametric Instability . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Experimental Layout 35

3.1 High Power Optical Facility . . . . . . . . . . . . . . . . . . . . . . 35

3.2 The 10-W Adelaide Laser . . . . . . . . . . . . . . . . . . . . . . . 40

vi

3.3 Optical Table Layout . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 The Fabry-Perot Cavity . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.1 The Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.2 The Suspensions . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.3 Thermal Compensation System . . . . . . . . . . . . . . . . 51

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Angular Instability 54

4.1 Sidles-Sigg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Optical Torsional Stiffness Experiment . . . . . . . . . . . . . . . . 63

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Parametric Instability 70

5.1 Parametric Instability in FP cavities . . . . . . . . . . . . . . . . . 71

5.2 Suppressing Parametric Instability . . . . . . . . . . . . . . . . . . 74

5.2.1 Suppressing the Acoustic Mode . . . . . . . . . . . . . . . . 76

5.2.2 Suppressing the Optical Mode . . . . . . . . . . . . . . . . . 80

5.3 Optically Suppressing Parametric Instability . . . . . . . . . . . . . 85

5.3.1 Optical Feedback Results . . . . . . . . . . . . . . . . . . . . 92

5.4 Closed Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Conclusions 101

6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 101

vii

6.2 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2.1 Future Work at AIGO . . . . . . . . . . . . . . . . . . . . . 103

viii

List of Tables

1 Optical and material properties for the ITM and ETM mirrors in the

79-m suspended high optical power cavity. . . . . . . . . . . . . . . 49

2 Parameters of the south arm cavity for the optical torsional stiffness

experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

ix

List of Figures

1 The worldwide array of current and proposed interferometric gravitational-

wave detectors and interferometric test facilities. . . . . . . . . . . . 5

2 An illustration of the two polarizations of a gravitational wave. . . . 9

3 Diagram and picture of the Michelson-Morley experimental set-up. . 10

4 The basic set up for current experimental interferometric gravita-

tional wave detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 An example electromagnetic field arrangement for a Fabry-Perot cav-

ity with moveable mirrors. . . . . . . . . . . . . . . . . . . . . . . . 18

6 Analytical approximations to the cavity power storage P (φ) given by

the Airy formula in Eq. 14, for a Fabry-Perot cavity with identical

mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 The intra-cavity power incident on an end mirror, with T = 1, is

shown as a function of δγ. . . . . . . . . . . . . . . . . . . . . . . . 28

8 The optical spring constant KOS, as a function of the detuning factor

δγ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9 An aerial view of the Australian International Gravitation Observa-

tory (LIGO-Australia), located in Gingin, Western Australia. . . . . 36

10 A photo of the inside of the main lab. . . . . . . . . . . . . . . . . . 38

x

11 An inside look at the components of the 10-W slave laser. . . . . . . 41

12 The optical set-up used for injection locking the master to the slave

laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

13 The rest of the main laser rooms optical table set up, following the

injection optics shown in Figure 12. . . . . . . . . . . . . . . . . . . 45

14 The input optical table set up. . . . . . . . . . . . . . . . . . . . . . 47

15 Example of the hemispherical resonant South Arm cavity at AIGO. 49

16 Small optics suspensions design as provided by LIGO. . . . . . . . . 50

17 Fused silica compensation plate installed in the AIGO south arm

cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

18 A normal high power resonant Fabry-Perot cavity. . . . . . . . . . . 55

19 Stable optical torsional mode in a Fabry-Perot cavity. . . . . . . . . 57

20 Unstable optical torsional mode in a Fabry-Perot cavity. . . . . . . 58

21 Experimental set-up to measure the negative optical spring constant. 65

22 A plot of the negative torsional spring constant KOS as a function of

the cavity g-factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

23 Example of a good spatial overlap between an optical and acoustic

mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

24 The three mode interactions which lead to parametric instability in

an optical cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

25 The electro-static drive patterns as proposed by the MIT LIGO

group: an example and an experimental implementation. . . . . . . 78

26 The acoustic mode mechanical damping system. . . . . . . . . . . . 79

xi

27 Example of the effect of the passive resonant dampers acting together

to damp various acoustic modes. . . . . . . . . . . . . . . . . . . . . 79

28 Theoretical experimental setup to suppress or enhance opto-acoustic

parametric interactions. . . . . . . . . . . . . . . . . . . . . . . . . 84

29 Experimental set up to instigate parametric instability in AIGOs

79-m cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

30 Experimental set up for the excitation of ω1 in a Fabry-Perot cavity

based on the work done by Zhao, et al. in 2008. . . . . . . . . . . . 87

31 Creation and injection of interfering ω1 mode . . . . . . . . . . . . . 88

32 Actual experimental set up used to enhance and suppress the opto-

acoustic parametric interactions. . . . . . . . . . . . . . . . . . . . . 89

33 Contour map of our ETMs acoustic mode at 178 kHz. . . . . . . . . 91

34 Experimental results of the exponential excitation of the high order

mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


mode: suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


mode: suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

37 The experimental set up for the ”closed loop” opto-acoustic para-

metric interactions suppression experiment. . . . . . . . . . . . . . . 97

38 Results of the ”closed loop” experiment. . . . . . . . . . . . . . . . 98

xii

Preface

Thesis Overview

This thesis is mainly comprised of two parts. The first part describes the

theory and experimental results of the Sidles-Sigg instability, and the second

part describes the theory and experimental results of parametric instability, both of

which arise as a result of radiation pressure effects against the mirrors in a Fabry-

Perot cavity.

Chapter 2 provides a review of Fabry-Perot cavities, and an overview of the effects

of radiation pressure inside an optical cavity, and the resulting potential instabilities

(the optical spring effect, and parametric instability).

Chapter 3 presents the optical and electronic equipment used in the experimental

layout of the south arm Fabry-Perot cavity at AIGO.

Chapter 4 gives an extension to the theoretical development provided in Chapter 2

for the optical spring effect. The theoretical development discussed in this chapter

follows closely the model established by John Sidles and Daniel Sigg in 2003, and

expands further to provide the experimental work and results from the experimental

optical spring measurements taken at AIGO.

Similarly, Chapter 5 also provides an extension to the theoretical development pro-

vided in Chapter 2, however alternatively concentrated on parametric instability.

This chapter discusses the theory initially introduced by Braginsky, et al. in 2001,

1

2

and continues with more recent theory developed by the gravity group at the Univer-

sity of Western Australia and their proposed solution to potential parametric insta-

bilities. The experimental procedure and results of two successful optical feedback

loops are presented as initial solutions for future gravitational wave interferometers.

As conclusion, Chapter 6 provides an overview of the important results discussed

in Chapters 4 and 5. This chapter also provides a look at the necessary future

prospects for solutions to the parametric instability problem, including specifically

future experimental work to find a solution to parametric instability at the AIGO

site.

Chapter 1

Introduction

1.1 A New Wave of Physics

In the last century, gravitational wave physics has become a bigger and more

widely researched field in both the experimental and theoretical sciences. In 1916

Albert Einstein predicted the existence of gravitational waves in his General Theory

of Relativity. He showed that the acceleration of masses in space-time generates

time-dependent gravitational fields that propagate away from their sources, at the

speed of light, as distortions of space-time [1, 2, 3].

Gravitational wave physics has been a largely theoretical field since 1916. Exper-

imental efforts to detect gravitational waves have only been underway since the

1960s. Joseph Weber began the search when he built the first gravitational wave

detector, known as a resonant mass gravitational wave detector or also as the Weber

bar, in 1961. Unfortunately, his detectors were never sensitive enough to detect a

gravitational wave and a lot of his work was discredited due to a false claim from

Weber and a minority of scientists that they had detected a gravitational wave

signal [4, 5]. However, despite being discredited, Weber is still thought of today as

the godfather of experimental gravitational wave physics, providing the motivation

3

4 CHAPTER 1. INTRODUCTION

behind current detectors which include the spherical mass detectors (miniGRAIL.

SFERA and Mario SCHENBERG [6]), high frequency detectors, and interferomet-

ric detectors.

A decade or so after Weber’s attempts to detect gravitational waves, the physics

community was given the first indirect evidence for the existence of gravitational

waves. The evidence was found woven into the pulsing radio emissions of binary

pulsar, PSR B1913+16. This pulsar, and the resulting data, was discovered and

monitored by Russell Taylor and Joseph Hulse in 1975 [8]. Hulse and Taylor were

initially only interested in the pulsar emissions, but after measuring these emissions

over a period of time, they discovered that there were variations in the arrival time

of the pulses. These variations they measured were characteristic of a pulsar in

a binary orbit with another star. Einstein’s general theory of relativity predicted

that a binary system of unbalanced masses emits energy in the form of gravitational

waves as the two objects spiral in towards each other. Hulse and Taylor measured

the shift in the period and orbit of the binary pulsar system over several years. Their

measurements of the pulsars emissions (and the loss associated with them) agreed

very closely with Einstein’s predictions, thereby indirectly proving the existence of

gravitational waves. Their findings along with the previous work done by Weber

propelled the pursuit to develop more sensitive gravitational wave detectors on

Earth, including in particular, interferometric gravitational waves detectors.

Currently, there are five land-based operational interferometric detectors around

the world: two 4-km long interferometers in the United States operated by MIT

(Massachusetts Institute of Technology) and Caltech (California Institute of Tech-

nology), known collectively as LIGO (the Laser Interferometer Gravitational-wave

Observatory), one 3-km long interferometer in Italy, known as VIRGO, operated

by Italian and French research teams, one 600-m long interferometer in Germany,

known as GEO600, operated by German and British research teams, and one 300-m

long interferometer in Japan, known as TAMA300, operated by the University of

Tokyo [10, 14, 15, 17].

1.1. A NEW WAVE OF PHYSICS 5

Additionally, there are several planned gravitational wave detectors around the

world: the Australian International Gravitation Observatory (LIGO-Australia) lo-

cated in Australia, operated by the Australian International Gravitational Research

Centre through the University of Western Australia [18]; the Large Scale Cryogenic

Gravitational-wave Telescope (LCGT) located in Japan, operated by the University

of Tokyo (recently approved for funding on 22 July 2010) [19, 20, 21]; and the Laser

Interferometer Space Antenna (LISA), run jointly by NASA and ESA, is proposed

to be launched into space in the early 2020s [23, 22, 24]. Figure 1 shows the location

and name of all the current and proposed land-based interferometric gravitational

wave detectors around the world.

Figure 1: The worldwide array of current and proposed interferometric gravitational-wave de-

tectors and interferometric test facilities.

Despite having only seven gravitational wave observatories around the world, there

are several hundreds of research groups and universities around the world also par-

ticipating in the experimental and theoretical efforts to detect gravitational waves.


The LIGO laboratories, along with the GEO600 and AIGO groups comprise what

is known as the LSC (LIGO Scientific Collaboration) [36]. A beautiful aspect of this

gravitational wave community is the variety of specializations within and between

the different research groups and people that make up the LSC. As more detectors

are being built and others being upgraded, this diverse group of researchers and

engineers comes closer to discovering a new view of our Universe, and hopefully

their common goal will be reached in the near future.

1.2 Gravitational Waves

Absolute, true, and mathematical time flows at a constant rate without relation to

anything external. Absolute space, without relation to anything external, remains

always similar and immovable.

Isaac Newton

In 1687 Isaac Newton introduced what is known as ”Newtons Law of Universal

Gravitation” to the world. At the time, and for two centuries following his

proposal, academics and physicists worldwide accepted and used his laws to describe

events in space. Eventually, experimental results were produced which disagreed

with his predictions, and there was only one conclusion: that Newton’s law was not

an accurate description of space and time. At the beginning of the 20th century,

a new description of space was introduced.The new description was comprised of

two theories, known as the special and general theories of relativity. These theories

were derived between 1905 and 1925 by Albert Einstein and provided the answers

to certain experimental observations that physicists had been looking for, and that

Newton’s law could not suffice.

Newton’s ”Law of Universal Gravitation” suggests that occurrences in space take

place due to the force of gravity. In his general theory of relativity, Einstein proposes

1.2. GRAVITATIONAL WAVES 7

that these occurrences are not the result of gravity, but rather the curvature of space

and time. According to Einstein’s theory, only a heavy body could significantly

change the curvature of space-time, and if a heavy body moved or was moved around

in a certain way in space-time, it would produce ripples across space-time. These

ripples, which can be thought of in the same manner as ripples moving through

a pond after a disturbance in the ponds surface, are known more commonly to

scientists as gravitational waves.

Gravitational waves are produced from fluctuating high-mass energy distributions

in space-time and their strength is dependent on how close they are to the source.

That is to say, these fluctuations are predicted to severely distort space-time near

their source of fluctuation, but far from the source (i.e. a distant planet like Earth),

these ripples are predicted to become only small perturbations in space-time.

Following the work done by Kip Thorne, John Wheeler and Charles Misner in

their book ”Gravitation”, these small gravitational perturbations produced by a

gravitational wave in space-time can be thought of as a small perturbation, hµν ,

on the four-dimensional Minkowski space-time ηµν . This assumption produces a

gravitational wave field that can be expressed by the following equation [1]:

gµν = ηµν + hµν (1)

where |hµν | 1 and ηµν is the Minkowski metric, defined as:

ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(2)


Equation 1 is known as the weak field approximation for small gravitational ripples

across space-time, and it is linear with respect to hµν .

In order to allow for small changes in hµν , it is necessary to introduce the transverse

traceless gauge in which Einstein’s equations of general relativity become a system

of wave equations generalized by

(− 1

c2

δ2

δt2+ ∆2

)hµν = 0. (3)

In Equation 3, hµν is given as a four dimensional field with solutions in the form

of waves propagating in vacuum at the speed of light c. In the transverse traceless

gauge, for waves traveling along in the z-direction, hµν may be expressed as

hµν =

0 0 0 0

0 hxx hxy 0

0 hyx hyy 0

0 0 0 0

(4)

where −hyy = hxx = h+ and hxy = hyz = h×. As seen in the matrix above, there are

no z-components and this is a result of the transverse nature of gravitational waves.

This matrix also provides information on the polarization states for a potential

gravitational wave. As already given, there are only two possible polarization states

for a gravitational wave, denoted by h+ and h×. Figure 2 gives an illustration of

the two polarizations, and as can be noted from the picture, h+ and h× are 45 out

of phase from each other.

Gravitational waves propagate by stretching space in one transverse direction (h+

or h×) and compressing it, at the same time, in the other transverse direction (h+

or h×). The charge on a gravitational wave is always positive, because the mass

1.2. GRAVITATIONAL WAVES 9

of a particle is always positive and the lowest mode of oscillation of a gravitational

wave is quadrupolar.

Figure 2: The graphic representation of the two polarizations of a gravitational wave, following

the description of hµν in Equation 4. The arrows indicate in which way the wave will expand or

contract.

As gravitational waves travel through space-time, they expand and contract space-

time in a manner as shown in Figure 2. These contractions are also referred to as

cross (h×) and plus (h+) polarizations in essence of their motion.

A typical passing gravitational wave from an astronomical event will change objects

in length by 1 part in 1021, which is an extremely small effect, for even the strongest

astrophysical sources present in the universe. In order to measure these small per-

turbations, gravitational wave detectors need a design sensitive enough to detect a

change of length of at least ∆L = 10−18m [26, 28]. In 2005, interferometric detec-

tors reach the design sensitivity of 1 part in 1021 over a 100 Hz bandwidth. Despite

reaching the necessary strain level for some astrophysical events and their potential

gravitational wave emissions, interferometric detectors need to keep pushing their

strain level sensitivity lower and lower.


1.3 A Brief History of Interferometry

Interferometric gravitational wave detectors are based upon the interferometric

design developed by Albert Michelson and Edward Morley in the 19th century,

which was initially created to measure the theorized ”aether” through which light

was thought to propagate [29, 30].

Michelson and Morley assumed that light moving through the aether would be

slower in the direction of motion of the earth, as it would have to fight an aether

drift. They decided to build an apparatus, known as the interferometer, which could

uncover the effect of the aether on the speed of light. Consequently, their results

proved the non-existence of aether, but did however, herald the use of interferometry

in scientific experiments, including gravitational wave detection.

(a) Experimental Design (b) Actual Experimental Layout

Figure 3: Diagram and picture of the Michelson-Morley experimental set-up, where both mirrors

were fixed to a slab, that was rotatable. The light was split by a beam-splitter into two perpendic-

ular directions. The light then went to the two mirrors shown, reflected off them and recombined

at the beam-splitter to be sent to a viewing area. According to Michelson and Morley, if aether

affected the speed of light in the direction of the motion of the earth, than a fringe pattern would

appear at the viewing area [31, 32].

1.3. A BRIEF HISTORY OF INTERFEROMETRY 11

The Michelson-Morley interferometer, shown in Figure 3, depended upon a light

from a spectral line source propagating through a beam-splitter, transversing two

perpendicular arms and reflecting off two orthogonal, fixed mirrors. Michelson and

Morley were trying to detect a change in the fringe pattern at the viewing area of

the interferometer. When an aether drift would affect the light, then after the light

had bounced off the two mirrors and recombined at the beam-splitter, there would

be a distinct interference pattern at the viewing area. However, when nothing

would affect the light traversing the arms of the interferometer, then ideally the

light would constructively interfere and no light would show at the viewing area

(in an ideal case that would mean that the arms would be exactly equal in length,

the whole system would be kept in vacuum to restrict potential disturbances to the

fringe pattern and the laser would be coherent). As was already revealed, Michelson

and Morley found no change in the fringe pattern of light regardless of the position

of the interferometer to the supposed aether drift.

One of the more recent applications of the Michelson-Morley experiment is, as

already mentioned, the interferometric gravitational wave detector. The basic

Michelson-Morley design is used as the backbone of the detectors design, but its

basic operation is buried underneath a plethora of upgrades and extensions.

In Michelson and Morley’s experiment, the mirrors were fixed to a moveable table-

top, however in a gravitational wave interferometer, the mirrors must be suspended

in order for gravity to be the only force to change the optical path length of the

light.

Most interferometric detectors are also designed to be broadband detectors, meaning

there is no one resonant frequency, and gravitational waves have been theorized

to have frequencies ranging from mHz to 104 Hz, depending on the mass-energy

source (Note that signal-recycled interferometers do however have a low Q resonant

frequency). Accordingly, gravitational wave interferometers need to have a storage

time of the light equal to half the period of a potential gravitational wave [27],


τ =1

2fGW (5)

where fGW is the frequency of a gravitational wave. Therefore, the optimal length

of a gravitational wave interferometer can be calculated as

L = cτ =c

2fGW(6)

For example, in order to measure gravitational waves around 10 Hz, then the opti-

mum length for one interferometer arm is calculated to be L ∼ 1500 km long. It is

impossible to build a 1500 km planar interferometer on earth, but it is possible to

extend the travel length of the light inside the interferometer design, by inserting

a second suspended mirror to each arm, just after the beam splitter, creating what

is known as a Fabry-Perot cavity, see Figure 4.

1.3. A BRIEF HISTORY OF INTERFEROMETRY 13

Figure 4: The basic interferometric gravitational wave detector design. All mirrors are suspended

and an additional mirror is placed just after the beam splitter in each arm to increase the travel

time of light.

The US based gravitational wave interferometers, LIGO Hanford and LIGO Liv-

ingston, have opted for 4-km length arms to improve the broadband range of their

detectors.The Fabry-Perot cavities in the LIGO detectors wrap the laser light 75

times, thereby increasing the total travel length of the light to 300 km. All other pre-

viously mentioned interferometric detectors around the world (except the GEO600)

use Fabry-Perot cavities to lengthen the travel time of light in their interferometers.

In addition to suspending the mirrors and installing Fabry-Perot cavities, gravi-

tational wave interferometers inject at least 10 W of laser power into the optical

cavity arms where, once resonant, the power inside the cavity reach several tens of

kW. One fundamental limit in the sensitivity of a gravitational wave interferometer

is parametrized by the laser power.

The last significant upgrade from the basic Michelson-Morley design is that all the

primary optics of the interferometer are maintained inside a vacuum. The vacuum

used in gravitational wave detectors envelops that advanced detectors have included


in their design is a vacuum. The vacuum used in gravitational wave detectors

envelops the whole interferometer, except the laser and its injection optics. The

vacuum systems operated by the LIGO detectors are the largest sustained ultra-

high vacuums in the world. Every mentioned interferometric gravitational wave

detector uses or plans to use a vacuum to isolate their main optics and laser paths

for several reasons: first, the vacuum keeps air currents from disturbing the mirrors.

Second, they help maintain the straight path of the laser within the cavity arms.

Any slight bending of the light within the cavities could cause the laser beam to

hit the inside wall of the vacuum beam tubes (for example, a slight bend could be

caused by temperature differences across the arm) [33, 34, 35, 36].

Though derived from the Michelson-Morley table-top interferometric design, grav-

itational wave interferometers have far surpassed its predecessor with several more

improvements than what are possible to properly describe in this thesis. Addition-

ally, each interferometric detector around the world has their own unique design

and methods, all of which the reader can find more information about in the bibli-

ography.

1.4 Conclusion

Interferometric gravitational wave detectors are on the brink of discovering gravi-

tational waves. From Einstein’s theoretical predictions in 1916, Weber’s resonant

bar detectors in the 1960s, Hulse and Taylor’s indirect proof in the 1970s and the

beginning of interferometric detectors, starting operation at the start of the 21st

century, the science world comes closer and closer to potentially detecting gravi-

tational waves. There are, however, still several obstacles which interfere with the

current interferometric detectors ability to measure for gravitational waves. The

goal of this thesis is to analyze two such obstacles which could potentially upset the

operation of upgraded detectors and thus their ability to detect for gravitational

waves.

1.4. CONCLUSION 15

The United States interferometric detectors plan to inject 100 W laser power into

their interferometers, as an attempt to increase their design sensitivity and poten-

tially go below the quantum limit. This particular upgrade has the potential to

instigate angular instability as well as opto-acoustic parametric interactions in the

Fabry-Perot cavities, both of which could negatively impact the interferometers and

both of which are the main topics of this thesis.

Chapter 2

Theory of Radiation Pressure

Effects

This chapter presents the fundamental theoretical background of the potential ra-

diation pressure effects observed and described in detail later in this thesis. This

chapter begins by reviewing the core attributes of Fabry-Perot cavities, the physics

of radiation pressure forces, and two potential instabilities which occur as a result

of excess radiation pressure in a Fabry-Perot cavity.

2.1 The Art of Fabry-Perot Cavities

This thesis studies the effect of radiation pressure on the mechanical motion of

mirrors in a suspended Fabry-Perot cavity. This section will provide a brief

history and review on the properties of Fabry-Perot cavities.

The Fabry-Perot optical resonator is a widely used research instrument. Its origin

stems from the theory of multi-beam interference which was developed in 1891 by

Charles Fabry, and then later incorporated into the design of the first interferometer

created by Fabry and his colleague, Alfred Perot, in 1897[37].

16

2.1. THE ART OF FABRY-PEROT CAVITIES 17

The Fabry-Perot cavity was a break through for scientists in the 1800s. At that

time, there were only a few devices available to scientists for examining the spectral

content of a light field, whether it be for chemists to discover the atomic properties

of some of the elements in the periodic table or for astronomers to distinguish

between the different chemical compositions of a distant star. The spectroscopy

equipment, and Michelson interferometers were the only available devices for such

inquiries and both devices were limited in how well they could resolve closely-spaced

spectral lines. Thus, the Fabry-Perot cavity was a highly necessary accomplishment

in the scientific community at the time of its unveiling [37].

One beautiful aspect of the Fabry-Perot cavity is its simplicity: light is passed

between parallel, highly reflective mirrors and the interference between the com-

ponents of the light undergoing multiple reflections within these mirrors produces

extremely well-defined interference fringes. These fringes were used to deduce the

spectral properties of light and they provided much more accuracy than what was

available before [37].

Overtime the use of the Fabry-Perot cavity extended far beyond its original inten-

tions. For example, as mentioned in Chapter 1, experimental gravitational wave

groups have employed Fabry-Perot cavities in their Michelson-Morley interferomet-

ric configuration. The Fabry-Perot cavities used in interferometric gravitational

wave detectors make use of suspended mirrors, in order for gravity to be the only

source acting on them. Furthermore, the use of suspended mirrors makes it easier

to tune the cavity to resonance.

18 CHAPTER 2. THEORY OF RADIATION PRESSURE EFFECTS

Figure 5: A suspended Fabry-Perot cavity, consisting of two moveable mirrors, a length L apart,

into which light is injected as E0, resonates in the cavity as E1, E2, E3 and E4, transmitted

through as Etrans, and reflected as Erefl. The amount of reflectance/transmittance out of and

into the cavity are determined by the coefficients of transmittance, t1,2 and reflectance r1,2.

A suspended Fabry-Perot cavity is shown in Figure 5. In this figure coherent light,

E0, incident on a partially transmitting mirror, enters cavity space, and is reflected

off an almost completely reflective mirror. As the power of the light transmitted

through the first mirror is lowered, the light already inside the cavity (assuming its

on resonance in the cavity) continues to bounce between the two mirrors building

up in power and intensity. Following the work done by A.E. Siegman in his book

entitled ”Lasers”, it can be shown that the power inside the cavity is resonantly en-

hanced when the cavity length, L, is a half-integer multiple of the fixed wavelength,

λ0 of the light. Siegman also shows that the power resonances can be determined by

computing all the optical fields in the cavity and from them, calculating the stored

power as a function of the round-trip phase, φ.

In order to determine the power storage as a function of frequency, ω0 (or equally,

the wavelength λ0 , length L or phase φ), the relationships between the light fields

as they interact with the mirror surfaces need to be calculated. The steady state

equations for the reflected field Erefl, transmitted field Etrans and internal cavity

fields E1, E2, E3 and E4, can all be derived as a function of the incident field

on the cavity, E0, the mirror reflectivity’s, r1 and r2, and the mirror amplitude


transmissivities, t1 and t2 [69].

Continuing in Siegman’s footsteps, the steady state equations for a cavity such as

the one shown in Figure 5 can be obtained from the definition for the path of cavity

fields, where light is transmitted through the input mirror, reflected off the end

mirror and the input mirror over and over again. These field definitions are made

under the assumption that the laser field contains monochromatic plane waves,

the Fabry-Perot cavity is mode-matched, the transverse field propagation doesn’t

change inside or outside the cavity, and that all fields are traveling along the z-axis

with the form ei(kz−ω0t) over a beam profile with area A. The resulting steady state

equations for the Fabry-Perot cavity shown in Figure 5 are given by [38]:

E1 = t1E0 − r1E4 (7)

Etrans = t2E1e−iφ (8)

E3 = −r2E3 (9)

Erefl = −r1E0 + t1E4 (10)

where φ includes both the axial and Guoy phase shifts. The resulting fields of

interest in terms of the incident field, E0 are

E1 =t1e

iφ

1− r1r2ei2φE0 (11)

Etrans =t1t2e

iφ

1− r1r2eiφE0 (12)


Erefl = r1 −t21r2e

iφ

1− r1r2eiφE0. (13)

The circulating power inside the cavity can be calculated using Siegmans definition

for the power inside an optical cavity: P (φ) = |E1|2, where P (φ) is the power as

a function of the round trip phase, φ. The result is most commonly referred to as

the Airy Formula,

P (φ) = |E1|2 =T1

(1− r1r2)2 + 4r1r2 sin2(φ)|E0|2, (14)

where T1 = t21 is the energy transmission of the input mirror. The energy transmis-

sivity, reflectivity and losses for the cavity are T1,2 = t21,2, R1,2 = r21,2 and L = l21,2,

respectively, and the relationship between them is given by the equation

T1,2 +R1,2 + L1,2 = 1. (15)

Referring back to the Airy formula in Equation 14, it can be shown that the cavity

is resonant whenever φ = 2πn for any integer n. This expression for φ can be

rewritten in terms of the wavelength and cavity length where,

L =nλ0

2(16)

indicating that the cavity is able to move through two power resonances for each

change in cavity length, ∆L of λ0. The resonances can also be measured as a


function of frequency, where the angular frequency difference between two adjacent

power resonances is known as the free spectral range, ∆ω, of the cavity, where

∆ω =πc

L. (17)

As an example, the P (φ) of a Fabry-Perot cavity is plotted, using Mathematica,

in Figure 6. The free spectral range is the phase (or frequency) difference between

the adjacent power resonances as shown in the plot. The values used to calculate

the power resonances were made from a sample operating Fabry-Perot cavity with

a reflectance R = 0.95 and mirror loss, L = 0.002. These values and calculations

also include the assumption that both mirrors are identical, meaning that r1 = r2,

t1 = t2 and l1 = l2.

Figure 6: Analytical approximations to the cavity power storage P (φ) given by the Airy formula

in Eq. 14, for a Fabry-Perot cavity with identical mirrors, both with a reflectance R = 0.95, and

a loss L = 0.002. The ∆ω between the peaks measures the free spectral range of the cavity, and γ

as shown, gives the full-width, half-maximum measurement of a peak, which can be used to find

the finesse F of a cavity.


The ratio of the free spectral range to the full-width, half-maximum of any one of

the transmissions bands, γ, is shown in Figure 6, gives what is known as the finesse,

F , of the cavity, defined by

F =∆ω

γ=

2π

T1

. (18)

In spectroscopy, the finesse of a cavity is a measure of the frequency discrimination

of the cavity, also it can be shown as analogous to the quality factors, or the Q’s,

of mechanical oscillations. Furthermore, the finesse can be used to calculate the

number of effective round trips of light in a given optical cavity Neff , as well as the

storage time of light τsto,

F ≈ πNeff , (19)

τstor ≈Fπ

2L

c. (20)

When the distance between the mirrors and frequency of the laser are arranged so

that the light entering the cavity is in phase with the light already inside the cavity,

then the intra-cavity field is resonantly enhanced. When the cavity is operated

slightly detuned from this resonance, an optical feedback mechanism can occur

where the power fluctuating in the cavity becomes linearly dependent on the position

of the mirrors. This linear dependence can cause the mirror motions to couple to

the intensity fluctuations of the light inside the cavity, which in turn pushes on the

mirrors. This pressure on the mirrors is known as radiation pressure, and it has the

ability to knock the cavity out of resonance, i.e. detuning it. Thus, it is necessary

to introduce a detuning parameter, δ, defined as

2.2. RADIATION PRESSURE FORCES 23

δ = ω0 − ωres = ω0 −πcn

L. (21)

Though this is a useful quantity, it is better to define a dimensionless detuning

parameter for later calculations. This new detuning parameter is normalized by the

line-width, γ, introduced in Figure 6 and Eq. 18 of this chapter. It follows that δ

can be written as follows:

δγ =δ

γ=cT1

4L(ω0 − ωres). (22)

It follows that the power storage formula defined earlier in Eq. 14, can be rewritten

as a function of the normalized detuning parameter, as well as the intensity, I0

where I0 = E20 , giving

P (I0, δγ) =4I0

T1

1

1 + δ2γ

. (23)

When plotted, Eq. 23 produces a Lorentzian distribution with a line-width of 1.

2.2 Radiation Pressure Forces

In electrodynamics, electromagnetic fields in vacuum are shown to have a mo-

mentum density, ~℘, as defined by

~℘ =1

c2~S, (24)


where ~S = 1µ0

( ~E× ~B) is the Poynting vector which describes the flux imparted by the

electromagnetic fields. Assuming the laser beams inside the cavity are monochro-

matic plane waves of the form ~E = E0ei(kz−ω0t)x over the beam area, then the time

averaged momentum density ~℘ is given by

〈~℘〉 =1

2cε0E

20 z. (25)

When a laser beam, with area A, strikes a perfectly reflective mirror, it will impart

a force, ~F defined by the change in momentum ∆~ρ over a time ∆t, such that

~F =∆~ρ

∆t= 2〈~℘〉Ac. (26)

Substituting the previously defined time averaged momentum 〈~℘〉 into Eq. 26, the

force can be further defined in terms of the intra-cavity power, P , as follows:

~F = ε0E20Az =

2P

cz. (27)

Eq. 27 defines what is known as: the radiation pressure force.

2.3 Optical Rigidity

In the cavity used for the experimental results of this thesis, the input and end

mirrors are suspended as pendulums in separate vacuum chambers. The vacuum

reduces the mirrors susceptibility to acoustic noise and air currents, and the pendu-

lum provides isolation to ground noise. The fact that the mirrors are free to move

2.3. OPTICAL RIGIDITY 25

is a crucial element in this and all interferometric gravitational wave experiments.

Therefore, it is necessary to discuss the dynamics of a cavity in terms of a generic

mechanical oscillator.

2.3.1 Quasi-Static Motion

First, assume a cavity where the input mirror is stationary and the end mirror

suspended, and additionally that light is being injected into the cavity, resonating

and as a result, creating a constant DC radiation pressure (also referred to as

ponderomotive) force on the mirrors.

It is easiest to begin describing this particular system in terms of the bulk pendular

motion of the mirrors, where all the relevant physics can be applied to any generic

oscillator. The expression for the radiation pressure force as defined in Eq. 27, can

be rewritten so that the power is a function of the detuning parameter and intensity,

as was defined earlier in Eq. 23, to give

F =2P (I0, δγ = 0)

c. (28)

If the end mirror is slowly moved a distance x away from resonance, then the

mean static detuning δ = ∆(x/L) could cause the stored power in the cavity to

change, resulting in a fluctuating force, F . This fluctuating force is made up of two

components, the restoring force of the pendulum and the radiation pressure force,

defined as

F = −(MΩ2p +

2

c

∂P (I0, δγ

∂δγdδγdx

)x, (29)

where Ωp is the resonant frequency of the pendulum, and x describes the small


amplitude fluctuations about a point (i.e. x = x+ x, where x is the mean displace-

ment and x are the fluctuations about that displacement). Eq. 29 is valid for mirror

motions at frequencies where Ωp γ. This can be better understood by looking at

the definition of γ−1, as γ−1 characterizes the cavity response time, or rather the

amount of time it takes for the cavity power to adjust to a change in its length. As

long as the small amplitude fluctuations, x, are negligible on the γ−1 time scale,

the motion is quasi-static and Eq. 29 holds.

As already stated, Eq. 29 is made up of two forces: the radiation pressure force

and restoring pendulum force. The right-hand term in Eq. 29 describes the ra-

diation pressure force. Within that term lies the opto-mechanical force given by:

dδγ/dx. By differentiating Eq. 22 and treating the length L as a variable, the

opto-mechanical force can be defined as follows:

dδγdx

=4ω0

cT1

. (30)

Furthermore, by partially differentiating Eq. 23,

∂P (I0, δγ)

∂δγ= −8I0

T1

δγ

(1 + δ2γ)

2(31)

and combining it with Eq. 30, it is found that

2

c

∂P (I0, δγ)

∂δγ

dδγdx

= −64ω0I0

c2T 21

δγ

(1 + δ2γ)

2. (32)

For all δγ > 0, the negative sign indicates that the radiation pressure force has

created an additional restoring force. This restoring force is known as the optical

2.3. OPTICAL RIGIDITY 27

spring constant KOS, defined as:

KOS =64ω0I0

c2T 21

δγ

(1 + δ2γ)

2. (33)

When a cavity is detuned away from resonance it will create a power deficit, of which

the suspension restoring force works to correct for. As a result an additional optical

restoring force arises, as well as a resultant characteristic optical spring frequency

Θ, given by:

Θ ≡√KOS

M=

8

T1

√2πI0

cλ0M

√δγ

(1 + δγ2), (34)

where M is the mass of the suspended mirror. Depending on whether the cavity

is blue-shifted or red-shifted, the detuning can assume both positive and negative

values. A blue-shifted cavity (δγ > 0) will produce a Hooke’s Law restoring force

for fluctuating δγ, P (I0, δγ) will have an almost constant negative slope, and Θ is

a real, positive number. A red-shifted cavity (δγ < 0) results in an anti-restoring

force, where Θ is a purely imaginary number and the power is an almost constant

positive slope. Figure 7 illustrates the two different types of detuning, where the

intra-cavity power incident on a mirror is plotted, using Mathematica, as a function

of the detuning.


(a) Optical Spring (b) Optical Anti-Spring

Figure 7: The intra-cavity power incident on an end mirror, with T = 1, is shown as a function

of δγ . These plots illustrate the different opto-mechanical dynamics depending on the sign of the

detuning δγ . In Fig. 7a, δγ is positive, resulting in a restoring force or a blue-shifted cavity. In

Fig. 7b, δγ is negative, resulting in an anti-restoring force or a red-shifted cavity.

When Equation 33 is differentiated with respect to δγ, and set to 0, it is possible

to determine the maximum and minimum values of the detuning which correspond

to the maximum and minimum values of the optical spring. Figure 8 shows the

differential plot, where δγ = ± 1√3

are the defined maximum and minimum points

for the detunings.

Figure 8: The optical spring constant KOS , as a function of the detuning factor δγ . As can be

seen in this plot, when δγ = 0, the spring constant disappears, and when δγ = ± 1√3

the spring

constant reaches its maximum amplitudes.

2.4. FULL OPTICAL SPRING EFFECT 29

Figure 8 also shows the dependence between the sign of the spring constant and

the sign of the detuning. When δγ = 0, the spring constant disappears and likewise

when δγ > 0 or δγ < 0 then the spring constant also increases and decreases,

respectively.

2.4 Full Optical Spring Effect

This section will provide an initial theoretical look into the optical spring effect

due to radiation pressure in a Fabry-Perot cavity. Chapter 4 provides a more

in-depth theoretical analysis of angular instabilities within Fabry-Perot cavities, as

well as the experimental results from an experiment held at the AIGO facility.

As previously mentioned, the mirrors in the Fabry-Perot cavities of interferometric

gravitational wave detectors are suspended. Up to this point in this chapter, all

calculations regarding Fabry-Perot cavities have been made either disregarding the

mirrors suspensions or they have been made under the assumption that only one

mirror was suspended.

Before considering any mechanical mode of the mirrors, it is necessary to develop

an approximate dynamic equation of motion for all mechanical modes of the mir-

ror. This equation should also include terms which signify a potentially delayed

cavity response. In Eq. 29, the fluctuating forces, F , were defined as a result of

a change in the cavity’s stored power, where the power change was the result of

static detuning. Eq. 29 only took into account the radiation pressure and restoring

forces of the pendulum, however when the cavity mirrors are free to oscillate they

are also susceptible to two additional forces: the environmental and applied forces

Fa. Thus, rewriting Eq. 29 to take these two forces into account, F is shown to be:

F = Md2x

dx2= −(MΩ2

p +KOS)x+MΩp

Q

dx

dt+ Fa, (35)


where Q is the quality factor of the mechanical oscillations. Subsequently, Eq. 35

can be transformed into the frequency domain by assuming ddt→ iΩ, giving

−MΩ2x = −(MΩ2p +KOS(Ω)x+

MΩp

Q

dx

dt+ Fa. (36)

From Eq. 36, it is possible to determine the displacement per unit force, S0, of an

oscillator coupled to an optical field, as a function of frequency, such that

S0(Ω) =x

Fa=

1

M

1

−Ω2 + Ω2p + iΩΩp

Q+ KOS(Ω)

M

. (37)

Eq. 37 is also known as the susceptibility of an oscillator. This equation can be

thought of as representing a feedback system, where an input force is transformed

into a displacement by a mechanical oscillator, and the displacement is then con-

verted back into a force by the optical rigidity (KOS(Ω)), which again drives the

oscillator.

Opto-mechanical coupling in Fabry-Perot cavities, is the coupling between the phase

fluctuations induced by the mechanical motions of the mirrors, to the intensity

fluctuations of the intra-cavity field. In their paper, entitled ”Low Quantum Noise

Tranquilizer for Fabry-Perot Interferometers”, Braginsky and Vyatchanin deter-

mined an expression for the frequency dependence of a cavity, K(Ω), effective for

all Ω ∆ω [39]. The frequency dependence defines the non-instantaneous response

of the power build up in the cavity to any changes in the cavities length, and is

defined by:

K(Ω) = −2iω20|Ec|2

cL

[1

l(Ω)− 1

l∗(−Ω)

], (38)

2.4. FULL OPTICAL SPRING EFFECT 31

where the denominator l(Ω) = γ − i(δ + Ω), and l∗ is the complex conjugate.

According to Braginsky, et al. the l(Ω) term represents the addition of sidebands

to the carrier beam inside the cavity. Mirror motions at Ω phase modulate the

carrier beam, which creates sideband fields at the frequencies ω0±Ω. The inclusion

of l places poles in K(Ω) at the sideband frequencies γ ± δγ. Eq. 38 can also be

rewritten as a function of the normalized detuning parameter of the cavity, such

that

K(Ω) = KOS

1 + δ2γ

(1 + iΩγ

)2 + δ2γ

. (39)

As a result, Eq. 39 contains both real and imaginary components. The real part

corresponds to the proper rigidity and the imaginary part corresponds to damping

[39]. The damping force could either be damping or anti-damping, depending on

the sign.

Braginsky, et al.’s definition of frequency dependence for a suspended mirror in a

Fabry-Perot cavity can be applied to any mechanical mode of a suspended mirror.

For example, the pendulum mode of a suspended mirror is found to have a frequency

dependence given by

K(Ω) ≈ KOS

(1− 2iΩ

γ

1

1 + δ2γ

). (40)

which is derived under the assumptions that K(Ω) µΩ2p, where Ωp is the pendu-

lum frequency of a suspended mirror, and all Ω’s are restricted to frequencies where

Ω γ.

Subsequently, it is possible to express the susceptibility of the pendulum mode,

S0(Ω), with a modified resonant frequency as follows:


Ω′2p = Ω2

0 +KOS

µ≈ Θ2. (41)

It follows that the quality factor is given by:

Q′= − γ

2Θ

(1 + δ2

γ

). (42)

One important thing to note is that the Q-factor of the pendulum mode is negative,

which corresponds to an anti-damping viscous optical force. Experimentally this

equation suggests that the optical spring could increase in strength inside the Fabry-

Perot cavity, and the phase of the response will change by 180 across resonance.

2.5 Parametric Instability

This section will provide a brief theoretical analysis of parametric instability, to

sum up this chapters discussion of radiation pressure effects in Fabry-Perot

cavities. A more detailed analysis of parametric instability as well as the initial

experimental results to suppress this instability are presented in Chapter 5.

So far in this thesis, the discussion of radiation pressure induced effects has been

centered around the bulk longitudinal motion of the pendulum mode of the mirror.

These equations can also be applied to any mirror mode which interacts with and

has an effect on the cavity’s length.

The optical spring effect is the primary physical effect which has the ability to alter

the mirrors pendulum resonance at frequencies at or below γ. Mirror motions well

above γ are known as the mirrors acoustic modes. These modes have the ability

to interact with the light inside the cavity, as the light hits the mirrors surface.

2.5. PARAMETRIC INSTABILITY 33

The mirrors in a typical gravitational wave Fabry-Perot interferometer have many

acoustic modes, of which several have the ability to disrupt the signals of advanced

gravitational wave detectors .

All of the acoustic modes discussed in this thesis lie in the kHz frequency range.

In this higher frequency domain, the mechanical restoring force provided by the

acoustic mode is much larger than the optical restoring force, that is to say ω2m

KOS/µ, where ωm is the frequency of the acoustic mode and µ is the reduced mass

of the input and end mirrors, given by the following equation:

µ =M1M2

(M1 +M2), (43)

where M1,2 are the masses of the input and end mirrors, respectively. As a result,

the resonant frequency of the mechanical mode should remain unchanged.

If the quality factor of the mode and optical rigidity are large enough, the Q′ will

become negative and an instability will result. In this case, the instability factor,

also known as the susceptibility, is defined by a gain R.

In a latter paper by the same Braginsky, et al. previously mentioned, they esti-

mated this gain as the product of two factors, one being the response of the mirror

surfaces to the forces at the resonant frequency Q′

µωm, and the other being the viscous

radiation pressure force per unit displacement felt by the optical field as a result

of the motions of the mirrors. The imaginary component Im[K(ωm)] gives this

viscous damping constant. Braginsky, et al. defined R as a function of this viscous

damping constant as follows:

R ≈ Im[K(ωm)]Q′

µω2m

. (44)

Braginsky, et al. also determined what the experimental significance would be in

a Fabry-Perot cavity. The theorized that an R greater than 1, would produce a

negative Q′ and unstable oscillations. Likewise, when R < 1, Q′ would then be


positive and the oscillations damped. This model gives a similar approximation

to that of Braginsky, et al. [70], who were the first to define this dimensionless

parametric susceptibility. A closer look at their theoretical work on parametric

instability, and the work done at the University of Western Australia is discussed

later in this thesis.

The physical process of parametric instability is easier to understand than the

theoretical. When the acoustic mode of a mirror in a Fabry-Perot cavity oscillates

at a frequency ωm, it has the ability to phase modulate the carrier light inside the

cavity, which in turn, creates a pair of sidebands centered around ω0 ± ωm. As a

result, the cavity could develop an asymmetric optical response to these sidebands,

meaning that only one sideband would build up in intensity. In order for the one

sideband to begin resonance in the cavity, it would have to coincide with a high order

mode already present in the cavity. When the sidebands become unbalanced, this

produces fluctuating amplitudes in the light inside the cavity, and thus oscillating

radiation pressure forces on the mirrors.

Two possible scenarios could occur when this imbalance is present, depending on

which sideband is favored in the cavity. If the upper sideband is favored (the anti-

Stokes mode), then it would build up in intensity and dampen the mechanical mode

ωm. If the lower sideband is favored (the Stokes mode), and builds up in intensity

then the acoustic mode becomes resonantly driven by the amplitude fluctuations at

a rate faster than the damping effect of the anti-Stokes mode.

The second scenario produces what is known as parametric instability. Analogous

to the optical spring, parametric instability only occurs when δγ is positive. The

strength of the instability is dependent on the cavity power and detuning, and

it should only occur on one side of resonance, that is, where the Stokes mode is

favored.

Chapter 3

Experimental Layout

This chapter describes the layout and equipment of the Australian Interna-

tional Gravitation Observatory research site. Particularly, this chapter pro-

vides an in-depth look at the specific experimental layout and equipment used to

attain the results presented in Chapters 4 and 5.

3.1 High Power Optical Facility

Currently , the Australian International Gravitation Observatory (LIGO-Australia)

[40], located in Gingin, Western Australia, is a prospective interferometric

gravitational wave observatory. It houses two ∼ 80-m long operational high power

Fabry-Perot cavities, as well as the facilities and personnel to maintain them. The

Fabry-Perot cavities are used as test beds for a variety of research, including but

not limited to, research for advanced gravitational wave detectors.

LIGO-Australia is currently operated by several universities around Australia, all of

which are a part of the the Australian Consortium for Interferometric Gravitational

35

36 CHAPTER 3. EXPERIMENTAL LAYOUT

Astronomy which is a member of the worldwide interferometric gravitational wave

group, the LIGO Scientific Collaboration (LSC) [41]. The LIGO-Australia High Op-

tical Power Facility is primarily maintained by the Australian International Grav-

itational Research Centre, based in the University of Western Australia’s physics

department, in Perth, Western Australia [42].

LIGO-Australia’s research site is made up of four main buildings, as shown in the

aerial photo in Figure 9.

Figure 9: An aerial view of the Australian International Gravitation Observatory, located in

Gingin, Western Australia. The research site contains four main buildings which house and are

used to operate the two ∼80-m Fabry-Perot cavities of the LIGO-Australia interferometer [42].

3.1. HIGH POWER OPTICAL FACILITY 37

In the middle of Figure 9 are several buildings. The accommodation building,

angled towards the lower right part of the photograph is, as the name infers, mainly

a sleeping/resting facility for researchers and operators who spend long days and

nights at the site working on experiments and/or maintaining the site. However, this

is not its only function as a building, the building also houses the metal and wood

working workshop used by facility members to manufacture pieces for experiments

and/or for the site.

Clockwise from the accommodation block in Figure 9 is the main lab. The main

lab houses the majority of the experiments and the equipment currently being

undertaken by the research group from the University of Western Australia. The

main lab also contains an electronics workshop, kitchen and meeting area, as well

as several offices for work, meetings, and/or if necessary further accommodation.

The most important and interesting part of the main lab is the research area. The

main lab is reached by passing through two doors and one clean room area (where

clean room gear is required before entrance). The main lab research area is home

to several two meter tall vacuum chambers, as well as 700 mm in diameter vacuum

tubes, which extend away from the vacuum chambers and the main lab. As shown

in Figure 9, these tubes connect the main lab and the two end stations, the east

end and the south end stations. A small portion of the equipment in the main lab

is shown in Figure 10.


Figure 10: A photo of the south arm components of the main lab. In the bottom left corner of

the photo is a clear tent-like structure. This clear tenting houses the Class 1000 clean room used

to clean and prepare parts going into vacuum. To the left of the tent, closer to the ground is a

beam tube extending from the input test mass (ITM) vacuum chamber. This chamber houses the

ITM used in the south arm Fabry-Perot cavity. To the right of the ITMs vacuum chamber is the

input optics table enclosure, and the main laser room [42].

A few of the important pieces of the main lab are labeled in Figure 10. The vacuum

chambers inside the main lab house a variety of optics. In Figure 10 the input test

mass (ITM) vacuum chamber shown houses the ITM of the south arm Fabry-Perot

cavity at the site. As mentioned, there are two operational Fabry-Perot cavities at

the AIGO site. Each Fabry-Perot cavity is labeled according to its geographical

position: the east arm and south arm Fabry-Perot cavities. The equipment for

the east arm cavity is not shown in Figure 10 however it also occupies part of the

research area in the main lab.

To the right of the vacuum tank shown in Figure 10 are two smaller adjoined

enclosures. These enclosures contain the laser and input optics required to stabilize

3.1. HIGH POWER OPTICAL FACILITY 39

and steer the laser to the ITM vacuum chamber, and into the Fabry-Perot cavity.

The laser and optical components used in these two enclosures are described in

more detail in the following sections of this chapter.

The last important structure to note in Figure 10 is the clear encasement, shown

in the lower left of this picture. This clear encasement is the main lab’s class 1000

clean room and it is used to clean all parts that will be or have already been installed

in vacuum.

Figure 10 is an important photo for the work done in this thesis, as it shows the

setup inside the main lab of what is known as the south arm research area. The

remainder of this chapter and thesis focuses on the experimental set up as was

developed and used in the south arm Fabry-Perot cavity, as that is the cavity in

which both highlighted experiments took place.

As a side note, the east arm Fabry-Perot cavity has only recently been operating

with the addition of brand new vibration isolation systems, developed theoretically

and physically by John Winterflood, Jean-Charles Dumas, Andrew Wooley and

many others for use in future high power experiments [43, 44, 45, 46].

Returning to Figure 9, the last two buildings shown are known as the south end

and east end stations. As previously mentioned, these end stations are connected

to the main lab by the vacuum tubes which extend ∼80 m beyond their respective

ITM vacuum chambers. Both end stations are fairly similar with the exception

of the experiments being conducted inside of them. Each station is a Class 1000

clean room, and each contain an end test mass vacuum chamber and end test mass

(ETM) to complete the Fabry-Perot cavity.

In the south end station, there is an optics table set up to monitor the transmitted

beam from the ETM, as well as a thermal compensation experiment set up with

the prospect of developing a new compensation system to replace the one used for

the experiments detailed in this thesis.


3.2 The 10-W Adelaide Laser

In the main lab’s laser room, shown in Figure 10 as the bigger of the two enclosures

to the right of the ITM vacuum chamber, there are two lasers coupled together

to create a stable, high power light source. Together these two lasers are known as

a coupled master and slave laser. The master laser (which controls the stability and

frequency of the output beam) is a 500-mW Innolight NPRO Nd:YAG crystal laser,

and the slave laser (which controls the high output power) is a 10-W Nd:YAG crystal

laser, developed by the Optics and Photonics group at the University of Adelaide

[47].

The two lasers are situated on one end of the laser table with locking and steering

optics in between them. The main laser room itself is a Class 1000 clean room,

where both a clean jacket and booties are required before entering.

The optics, electronics and slave laser used in the coupling set up were designed

and developed by David Hosken and Damien Mudge from Adelaide University [48,

49, 50]1. The slave laser itself appears at first glance to be only a box, outputting

some amount of light. Despite its seeming simplicity from the outside, inside this

box is an intricate electronic and optical set up. The most important part of

this laser is the side pumped Nd:YAG crystal with Brewster angle windows. This

crystal is surrounded by thermo-electric cooling plates on all sides and faces two

semi-transmitting mirrors known as the output coupler and maximum reflectivity

(MaxR) mirrors. The MaxR and output coupler are PZT controlled mirrors used

to lock the master lasers beam to the slave lasers beam. Figure 11 is a photo of the

inside of the slave laser box.

1The author herself spent a lot of time working closely with David Hosken to maintain thecurrent slave laser.

3.2. THE 10-W ADELAIDE LASER 41

Figure 11: An inside look at the main components of the University of Adelaide’s 10-W slave

laser. Here the top of the crystal is covered by a blue thermo-electric cooling (TEC) plate. The

diodes used to side pump the crystal sit at the edge of the copper block, where two white coated

wires extend out from it. The MaxR PZT mirror sits to the right of the copper block, and is held

by a black mount. The output coupler sits across from the MaxR mirror and is held by a silver

mount.

In Figure 11 the crystal itself is hidden underneath a blue thermo-electric cooling

(TEC) plate (in the shape of a fat T). The diodes used to side pump the Nd:YAG

crystal are sitting on the copper block, with their electronic drivers leading away

in the red and blue labeled, white coated wires. The MaxR mirror is shown to the

right of the copper block, where it is sitting on a black mount. The output coupler is

sitting just above it in the photo, in the silver mount. All of the components shown

here are normally concealed underneath a grey cover, where all that is important

is the input and output ports for the laser beam.

In addition to the locking optics inside the slave laser box, there is also a series of


optics devoted to coupling the lasers outside of both laser boxes. Figure 12 shows

the optical path for the injection locking optics which couple the master and slave

laser beams together.

Figure 12: The optical set-up to couple the 500-mW stable master laser to the 10-W high power

slave laser. As the beam leaves the master laser it travels at a height of 76.2mm, until it reaches

the first periscope and is projected up to 130 mm, where it maintains this height through the

slave laser and back out again until it reaches the next periscope and is reprojected down to 76.2

mm. Every lens and 3 wedge are labeled in the diagram, where the lens are labeled according to

their focal point. The 3 wedges divert a small portion of the beam to the photodetectors shown.

These detectors feedback information is fed to the PZTs in the slave laser and keep the master

and slave locked.

3.2. THE 10-W ADELAIDE LASER 43

The pathway of light through the coupling optics in Figure 12 is best understood

by starting from the output of the master laser. As shown, directly after leaving

the master laser, the beam is redirected by a mirror down the table, through a

λ/4-wave-plate and a λ/2 wave-plate, to adjust the beams polarization.

The beam is then passed through a beamsplitter (which is not used at the moment,

the beam is only sent to a black beam stop as shown), and thereafter to a Faraday

Isolator (OFR, Model # IO-5-1064-VHP) which protects the master laser from the

reverse wave (RW) beam of the slave laser and any reflected light from its own

output. The first lens, f = 300mm, adjusts the position of the beams waist so

that it is near the middle of the electro-optic modulator aperture. The electro-optic

modulator (New Focus Model # 4003M) creates sidebands on the master laser beam

at 150 MHz, used in a Pound-Drever-Hall frequency control system with the slave

laser [93]. The beam travels further across the table, untouched, guided by steering

mirrors until it is transmitted through a f = 500mm lens. This lens together with

the f = 200mm AR coated cylindrical lens near the head of the slave laser box are

used to couple the master laser beam to optical axis of the slave laser’s crystal.

If the beams are correctly aligned, all of the power is emitted from the slave laser’s

crystal in the forward wave direction, that is, outbound through the second f =

200mm cylindrical lens and outwards to the second periscope.

In this optical set up there are two photodetectors arranged to measure the power

in the forward and reverse wave beams. The photodetectors used these signals to

generate a Pound-Drever-Hall signal, which measures the suppression of the reverse

wave. When the reverse wave signal is zero, the two beams can be easily stabilized

and locked together using only small motions of the PZTs inside the slave laser

head, resulting in a combined high power, frequency stabilized laser beam.

The periscopes shown in Figure 12 are used to transmit the beam from a height

of 76.2 mm to 130 mm into the slave laser, and back down to 76.2 mm after the

coupled beam leaves the slave laser. The beam remains at this height until it reaches


the end of the optics table.

The cylindrical lenses used in the coupling optics set up also shape the beam profile

of the laser beam from the master laser. The master lasers beam is gaussian in

profile before it reaches the cylindrical lenses, where once passed through, its beam

profile is transformed into an elongated beam profile. In order for the beam to pass

through the crystal, it needs to be elongated, but once the locked beam is output

from the slave laser box, another cylindrical lens is set up to transform the beam

back into a circular (TEM00) mode.

3.3 Optical Table Layout

After leaving the injection coupling optics end of the laser table, the beam is

sent onto another series of optics on the opposite end of the laser table, and

then further onto another optics table, known as the input optics table.

The rest of the laser table is used primarily for beam stabilization, sideband addi-

tion, and mode cleaning. A diagram of the rest of the optics table in the main laser

room is shown in Figure 13.

In Figure 13, as the beam leaves the periscope (the same periscope shown previously

in Figure 12, it is steered down the table (or rather up the diagram) through a beam

splitter, two λ/2 wave-plates and into an electro-optic modulator. This electro-optic

modulator adds sidebands at 18 MHz to the beam which lock the beam to the pre-

mode cleaner. After the electro-optic modulator and a few more optics, the beam

is split at the second beamsplitter into a Mach-Zehnder interferometer. The Mach-

Zehnder interferometer is currently set up as a part of the parametric instability

experiment, for which the optical set up is explained in more detail in Chapter 5.

3.3. OPTICAL TABLE LAYOUT 45

Figure 13: The optical layout of the laser table after the beam has left the injection locking

optics, shown in Figure 12. The beam is sent through a series of optics before it is modulated upon

at 18MHz by the first electro-optic modulator shown. This modulation is used to lock the beam

to the pre-mode cleaner (PMC). The PMC cleans the beam profile to whichever beam profile is

necessary in the current experiment. Typically it is locked to produce a TEM00 mode. After the

PMC the beam is sent through several more optics which are explained in more detail in Chapter

5 and then down a periscope off the table to the input optics table.


Normally the Mach-Zehnder interferometer is not installed on the table and the

beam passes only through the upper path shown where there is a pre-mode cleaner.

This pre-mode cleaner is typically used to lock the beam to a TEM00 gaussian

mode. The pre-mode cleaner uses a double pass tilt locking mechanism and, once

locked, outputs ∼85% of the power that was input. Just after the pre-mode cleaner

a Faraday Isolator is used to block any optical feedback into the pre-mode cleaner.

Each Faraday Isolator shown on the optical table has a transmissivity of 91%.

The beam is then modulated at 10 MHz by another electro-optic modulator before

being transmitted downwards off the table by another a periscope onto the input

optics table.

The input optics table enclosure was shown in Figure 10 as the smaller of the two

enclosures to the right of the ITM vacuum chamber. This optics table is used to

telescope and mode match the beam from the main laser room into the Fabry-Perot

cavity. All of the primary optics used on the input optics table are shown in the

diagram in Figure 14. Each mirror shown has a different radius of curvature, and

each slowly increases the waist size and position of the beam as it is steered into

the cavity.

3.3. OPTICAL TABLE LAYOUT 47

Figure 14: The input optics table optical layout, where the input beam comes from the main

laser rooms optical table, shown in Figure 13. Here, the beam is passed between several larger

mirrors all with larger diameters and radii of curvature with the purpose to project a larger beam

onto the ITM, with an appropriately sized waist for the Fabry-Perot cavity.

The input optics table, as shown in Figure 14, is an arrangement of mode-matching

telescopes steering the beam from the laser table into the Fabry-Perot cavity. The

first mirror, M0, that the beam interacts with is a flat 4” mirror, used to steer the

beam into the mode-matching telescopes. The next three mirrors, MMT1, MMT2

and MMT3 are the Mode-Matching Telescope (MMT) mirrors, where MMT1, is a

2” mirror with a radius of curvature of -2m, MMT2 is a 3” mirror with a radius of

curvature of 8m and MMT3 is a 4” mirror with a 20m radius of curvature. These

mirrors mode-match and telescope the beams input and output from the south arm

Fabry-Perot cavity. The last mirror shown on the optical diagram is a steering

mirror, MST. This mirror is used as a directional for both beams going into and

coming out of the cavity.


The beam transmitted back through the ITM and the MMT’s is sent to a photode-

tector (shown in Figure 14 as the beam extending to the left past M0), whose signal

is used to lock the cavity.

3.4 The Fabry-Perot Cavity

After leaving the input optics table, the beam is transmitted through the ITM

and into the south arm Fabry-Perot cavity. This section details the innards of

the south arms Fabry-Perot cavity at AIGO, including the optics, their suspensions

and the thermal compensation system installed to compensate for a strong thermal

lens generated between the two mirrors.

3.4.1 The Optics

The ITM and ETM in the south arm Fabry-Perot cavity are both sapphire mirrors

with differing diameters and radii of curvature. Each are coated with an anti-

reflection and high reflection coating, depending on which side is meant for reflection

or transmittance. The ITM is a flat, 100 mm in diameter A-axis mirror, with a

radius of curvature greater than 5.5 km. The ETM is an M-axis Sapphire mirror

with a larger diameter (150 mm) and a radius of curvature of 720 m. The different

diameters of the mirrors is irrelevant to any experimental reasons, they just happen

to be the size that were available for the site. The optical characteristics for the

ITM and ETM are given in Table 1 [51, 62].

3.4. THE FABRY-PEROT CAVITY 49

ITM ETM

Material Sapphire Sapphire

Diameter (mm) 100 150

Thickness (mm) 46 80

Radius of Curvature (m) R1 =∞ R2 = 790

AR Coating Reflection (ppm) 29±20 12±12

HR Coating Transmission (ppm) 1840±100 20±20

Table 1: Optical and material properties for the ITM and ETM mirrors in the south arm

Fabry-Perot cavity.

When the beam is resonant inside the cavity, a hemispherical optical cavity is

formed, where the waist is found to be approximately on the ITM itself and ∼ 8.6

mm in size. When the cavity is locked, the g-factor is ∼0.89 and the circulating

power (assuming that ∼7 W is injected into the cavity) is ∼5.5 kW. Figure 15 shows

the resonant hemispherical cavity.

Figure 15: Shown is an example of the hemispherical resonant optical cavity operated in the

south arm of the AIGO observatory. The ITM has a radius of curvature greater than measurable

(∞) and the ETM has a radius of curvature of 720 m. The waist of the south arm Fabry-Perot

cavity is ∼ 8.6 mm and located on the inside of the ITM.


3.4.2 The Suspensions

The ITM and ETM are suspended using a system based on the LIGO small op-

tics suspensions design, shown in Figure 16. The small optics suspensions support

structure is a rectangular metal frame which sits on top of a 900 × 600 mm bread-

board in the vacuum chamber. A single loop of wire wraps around each test mass

and is attached to the metal frame by a suspension block.

Figure 16: The small optics suspensions systems design by LIGO. This design is similar to the

suspensions holding the ITM and ETM in the south arm Fabry-Perot cavity at AIGO. As can be

seen, the mirror is surrounded by sensors, used to monitor the test mass’ position. These sensors

work in conjunction with the magnets shown to adjust the test mass positions for use in locking

the cavity [53].

3.4. THE FABRY-PEROT CAVITY 51

Surrounding the mirrors in the small optic suspension structure are electro-magnetic

actuators, magnets and standoffs, which provide local damping control (against

seismic noise, etc.). Shadow sensors are installed to monitor the motion of the test

masses. Standoffs are installed on both sides of each test mass as a means to reduce

noise which may arise when the wire rubs against the mirrors’ surface.

The pitch and yaw motions of the mirrors are electronically controlled by digital

signal processor computer boards, and LabView 6.1 programs (these programs were

created and installed by members of the Australian Consortium for Interferometric

Gravitational Astronomy group)[54].

3.4.3 Thermal Compensation System

Inside the ITMs vacuum chamber in the south arm, there is a thermal compensa-

tion system. The thermal compensation system control the thermal lensing issues

that were found associate with the ITM. In 2006, Jerome Degallaix, et al. found

that the high power circulating in the south arm cavity created a thermal lensing

induced radius of curvature of ∼ 230m on the ITM. Thermal lensing typically arises

whenever absorbed light in an optical substrate generates a temperature gradient,

which as a result of thermal expansion and thermo-optic coefficient, has the ability

to distort the wave-front profile of the optical modes in the cavity [62, 55].

In 2006 a thermal compensation system was designed and installed in the south

arm Fabry-Perot cavity to compensate this thermal lens.

This thermal compensation system designed and installed by Jerome Degallaix,

et al. is comprised of a fused silica plate situated on a bread board, surrounded

by a radiative heating ring. Fused silica was selected for its low absorption, high

homogeneity and very good isotropy [62, 55]. The fused silica plate is 160 mm in

diameter and 17 mm thick, with an anti-reflective (AR) coating of 150 mm across

its front. A heating ring made from a hollow copper tube surrounds the fused silica


plate. This copper ring is 12 mm in diameter and bent to form a 130 mm diameter

ring. Inside the copper ring is a series of ceramic beads, all ∼7 mm in length, with a

length of tungsten wire threading them together. Figure 17 shows the south arm’s

thermal compensation plate, drawn based on the explanation provided in Jerome

Degallaix’s thesis submitted in 2006 [62].

Figure 17: The fused silica thermal compensation plate installed in the south arm Fabry-Perot

cavity. The compensation plate is situated ∼15 cm from the ITM and screwed down to the same

bread board that the ITMs suspension system sits on. The compensation plate is comprised of

an anti-reflecting coated fused silica plate, surrounded by a hollow copper fibre filled by a twisted

tungsten wire which is surrounded by non-conducting ceramic beads [62].

The plate and copper ring combination are clamped together inside an aluminum

ring with the same thickness as the plate. Five grooves are cut into and around the

aluminum ring, and woven inside these grooves is a layer of teflon and on top of

it a length of nichrome heating wire. The aluminum ring works together with the

copper ring to heat the fused silica plate. Together they are capable of delivering

80 W heating power, with a maximum current of 10 Amps. When the fused silica

plate is heated, the plate begins to act as a divergent lens, increasing the cavity

waist size as the power is increased.

3.5. SUMMARY 53

3.5 Summary

This chapter has provided an overview of the AIGO facility, as well as an in-depth

look at the experimental layout of the south arm Fabry-Perot cavity installed at the

site. The experiments detailed in the next two chapters of this thesis rely completely

on the experimental set up of the optics and equipment described.

Chapter 4

Angular Instability

Angular instability, as previously discussed in Chapter 2, is just one possible

result of radiation pressure on the mirrors inside a high power optical cav-

ity. An increase in optical power within an optical cavity increases the radiation

pressure, which thereby increases the chances of opto-mechanical coupling. This

coupling can act in both longitudinal and torsional degrees of freedom. Higher

power within the cavity can change both the suspended mirror pendulum stiffness

as well as the torsional mode stiffness, causing them to act like two strongly cou-

pled oscillators, or better, an optical spring. This chapter will present the theory

on optical angular instability, following the work done by Sidles and Sigg in 2006,

as well as the experimental design and results from the optical torsional stiffness

experiment conducted in the south arm Fabry-Perot cavity at AIGO [68].

4.1 Sidles-Sigg Theory

In 2006, John Sidles and Daniel Sigg published a paper which detailed the theo-

retical geometric effects of optical torques in a two mirror suspended Fabry-Perot

cavity, the optical torques being a potential result of radiation pressure in a high

optical power cavity [56, 57]. Their research followed in the steps of Solimeno et

54

4.1. SIDLES-SIGG THEORY 55

al. who, in 1991, analyzed optical torsional stiffness in a cavity with one suspended

and one fixed mirror [59].

Sidles and Sigg predicted that optically induced, negative torsional stiffness could

potentially be large enough to overcome the stiffness of the mirrors suspensions

within the Fabry-Perot cavities of gravitational wave detectors. As discussed in

Chapter 2, these optical instabilities depend upon the circulating power, cavity

finesse and line-width, and the detuning between the laser frequency and the cavity

resonant frequency.

The geometry and positioning of a suspended optical cavity, with high circulating

power inside, is shown in Figure 18.

Figure 18: A normal high power optical resonator. The high power circulating in the cavity

could potentially upset the normal centered optical axis and cause an optical spring reaction.

If the mirrors are tilted by any means in Figure 18, i.e. by an increase in power

(to some critical value) inside the cavity, then the radiation pressure force, already

pushing on the mirrors surface, would start to tilt the mirrors more. The radiation

pressure force is defined as a function of the cavity power, P and the speed of light

56 CHAPTER 4. ANGULAR INSTABILITY

c as: Frp = 2Pc

. If the resonant mode inside the cavity is tilted off-center from the

normal optical axis, defined by the line connecting the centers of curvature of the

mirrors, the mirror will experience a torque τ due to the radiation pressure defined

by:

τ1,2 =2P

cx1,2 (45)

where x1 and x2 define the distance of the mode spot from the center of each mirror,

respectively.

There are three possible optical resonator scenarios of interest when analyzing a

suspended Fabry-Perot cavity. The first is very simply, the stable resonant opti-

cal cavity scenario, where there are no torques and no radiation pressure effects

disrupting the cavity performance, shown in Figure 18.

The next two scenarios are the more interesting, and are a result of radiation pres-

sure forces causing an angular instability inside the cavity. The first scenario is

known as the symmetric cavity yaw (see Figure 19). If both of the mirrors in an

optical cavity are tilted in such a way that the resonant mode walks off from the

normal optical axis, but still crosses it, then the radiation pressure will enhance the

restoring force created by the wire that suspends the mirrors to overcome the tilt.

If the tilt angle of the mirrors is large enough, the mode will wander off the mirrors

surface, allowing the radiation pressure force to exert an even larger torque on the

mirrors and as a result, cause the restoring force of the mirrors to become even

larger as an attempt to reduce the angle. This scenario is also referred to as the

stable optical torsional mode, stable because the restoring force is always trying to

restore the natural resonance of the light inside the cavity [61].


Figure 19: The stable optical torsional mode in a suspended Fabry-Perot cavity, where radiation

pressure enhances the mechanical restoring force of the mirrors suspension. The larger the tilt,

the greater the torque due to radiation pressure. Here α1,2 are the angles at which the mirrors

radii of curvature are off-centered, I1,2 are the mirrors moments of inertia and c1,2 are the points

at which the center of each mirrors radius of curvature interacts with the optical axis [56].

The second scenario, also known is one in which the mirrors are tilted in such a

way that the resonant mode becomes off-centered but no longer passes through the

original optical axis. In this situation the radiation pressure force works against

the mechanical restoring force of the mirrors suspensions. If the tilt angle becomes

large enough, the mode will walk outward, and the torque will become larger. In

contrast to the previous scenario, the restoring force will drop to zero or a negative

value (meaning it is no longer a restoring force!). Above a certain level of circulating

power this optical torsional mode is unstable. Once the cavity power level reaches

this critical power level and the mirrors are tilted, the beam will start to walk

outwards and away from the cavity.


Figure 20: The unstable optical torsional mode in a Fabry-Perot cavity. Here the radiation

pressure force on the mirror works against the restoring force of the mirror. When the power

inside the cavity reaches a certain value, the mode will walk away from the cavity.

The two scenarios shown in Figures 19 and 20 can be further explained theoretically.

Following some of the derivations presented by Sidles and Sigg, A.E. Siegman in

his book ”Lasers”, E. Hirose, et al., T. Corbitt, et al. in several papers regarding

angular instabilities in advanced gravitational wave detectors, as well as many other

authors work, it is possible to find the quantitative dynamics of the suspended

mirrors by first considering the equations of motion of the mirrors themselves [56,

57, 59, 60, 66, 69].

The displacement of the modes position on each mirror, ∆x1 and ∆x2, can be

written as a function of the misalignment angles α1 and α2 such that

∆x1 =g2

1− g1g2

Lα1 +1

1− g1g2

Lα2, (46)

and


∆x2 =1

1− g1g2

Lα1 +g1

1− g1g2

Lα2, (47)

where g1,2 = 1− LR1,2

are known as the g-parameters, R1,2 are the radii of curvature

for each mirror and L is the length of the cavity. The angular differential equations

for the undamped torsional pendulums are given by:

d2

dt2α1,2 = ω2

1,2α1,2 (48)

where ω1,2 are the angular frequencies of the mirrors, respectively. Combining Eq.

46 and 47 with Eq. 48, the differential equations for the suspended mirrors can be

written as follows:

d2

dt2α1 = −ω2

1α1 +2P

cI1

(g2

1− g1g2

Lα1 +1

1− g1g2

Lα2) (49)

d2

dt2α2 = −ω2

2α2 +2P

cI2

(1

1− g1g2

Lα1 +g1

1− g1g2

Lα2) (50)

where I1,2 are the mirrors’ moment of inertia. As theorized by Sidles and Sigg, the

kinetic and potential energies for the given suspended mirror system are described

by the following equations:

K =1

2I1α2

1 +1

2I2α2

2, (51)

and


U =1

2k1α

21 +

1

2k2α

22 −

∫τ1dα1 −

∫τ2dα2, (52)

where k1,2 are the spring constants for each torsional mirror. The last two terms

given in the potential energy equation can be rewritten using the previous definition

for the torque as induced by radiation pressure, Eq. 45, as follows:

∫τ1dα1 =

∫(2P

cx1)dα1 =

2P

c

∫(

g2

1− g1g2

Lα1 +1

1− g1g2

Lα2)dα1 (53)

∫τ2dα2 =

∫(2P

cx2)dα2 =

2P

c

∫(

1

1− g1g2

Lα1 +g1

1− g1g2

Lα2)dα2. (54)

When Eq. 53 and 54 are further integrated the potential energy equation becomes:

U =1

2k1α

21 +

1

2k2α

22 −

PL

c(1− g1g2)(g2α

21 + g1α

22 + 4α1α2). (55)

The Langrangian, L, is an expression which summarizes the dynamics of the system

by combining the potential and kinetic energies into one equation. Thus, L is given

by:

L = (1

2I1α2

1 +1

2I2α2

2 −1

2k1α

21 −

1

2k2α

22) +

PL

c(1− g1g2)(g2α

21 + g1α

22 + 4α1α2) (56)

The first four terms of the Lagrangian in Eq. 56 summarize the mechanical aspect

of the two mirrors, and the last three terms summarize the radiation pressure ef-

fects from the light inside the cavity. The equations of motion can be determined


by differentiating Equation 56 with respect to each misalignment angle, α1,2, and

solving for the system at equilibrium, that is dLdα1,2

= 0, giving:

dLdα1

= I1α1 + (2PL

c

g2

1− g1g2

− k1)α1 + (4PL

c

1

1− g1g2

)α2 = 0, (57)

dLdα2

= I2α2 + (2PL

c

g1

1− g1g2

− k2)α2 + (4PL

c

1

1− g1g2

)α1 = 0. (58)

Eq. 57 and 58 make it possible to consider the effect of small oscillations around

the equilibrium position. The expressions that were defined to describe the motion

of the test masses, and their amplitudes of vibration are known as the normal

modes. Each of these normal modes corresponds to a frequency of vibration, the

eigenfrequency. The eigenfrequencies can be determined by solving for the det(U −

ω2K) = 0. Sidles and Sigg rewrote the potential and kinetic energy terms in the

form of vector tensors as follows [56, 57],

K =1

2K1,2Θ1Θ2, (59)

K1,2 =

I1 0

0 I2

, (60)

U =1

2U1,2Θ1Θ2, (61)

U1,2 =

2PLc

g21−g1g2 − k1

4PLc

11−g1g2

4PLc

11−g1g2

2PLc

g11−g1g2 − k2

. (62)


Using the newly defined energies, the equation to determine the eigenfrequencies is

defined as follows:

0 = I1I2ω4 + I1ω

2

(−2PL

c

g1

1− g1g2

+ k2

)+ I2ω

2

(−2PL

c

g2

1− g1g2

+ k1

)− 2PL

c(1− g1g2)(k1g1 + k2g2) + k1k2 +

4P 2L2

c2(1− g1g2)2(g1g2 + 4). (63)

The eigenfrequencies for a particular system can be determined when Eq. 63 is

solved for ω. For example, in advanced gravitational wave interferometers, it is

possible to approximate that each test mass has roughly the same moment of inertia

and spring constant, that is: I1 = I2 = I and k1 = k2 = k [61]. When Eq. 63 is

rewritten with these approximations, the equation for the eigenfrequencies for the

advanced gravitational wave detectors is found to be:

I2ω4 − 2I

(k − PL

c

g1 + g2

1− g1g2

)ω2 +

(k2 − 2PL

c

g1 + g2

1− g1g2

k − 4P 2L2

c2

1

1− g1g2

)= 0,

(64)

which, after applying the quadratic formula to solve for the eigenfrequencies, sup-

plies two possible solutions for ω:

ω2± = ω2

0 +PL

Ic

(−(g1 + g2)±

√(g1 + g2)2 + 4

1− g1g2

), (65)

where ω20 = k

I, is the normal angular frequency for a torsional pendulum.

Equation 65 gives the theoretical predictions that were illustrated and described

in Figures 19 and 20 earlier in this section. Eq. 65 indicates that as the power

4.2. OPTICAL TORSIONAL STIFFNESS EXPERIMENT 63

inside the cavity increases, ω+ increases and as a result, ω− decreases. The mode

corresponding to ω+ was introduced in Figure 19 as the stable optical torsional

mode, and ω−, correspondingly, was introduced in Figure 20 as the unstable optical

torsional mode.

4.2 Optical Torsional Stiffness Experiment

This section provides the experimental set up and results of the optical tor-

sional stiffness experiment conducted in the south arm Fabry-Perot cavity at

AIGO.

In this experiment there are a few parameters which vary in comparison to the

previously used parameters for advanced gravitational wave detectors. First, the

ETM used in this experiment has a moment of inertia more than 10 times greater

than the ITM. The ETM is also heavily damped. Thus, the ETM can be considered

a well-aligned fixed mirror, where α2 = 0.

Accordingly, Eq. 49 and 50 introduced in section 4.1 become:

d2

dt2α1 = −ω2

1α1 +2P

cI1

(g2

1− g1g2

)Lα1. (66)

where all the α2 terms have dropped out. Furthermore, Chapter 2 introduced the

optical spring constant, KOS, in terms of the optical spring frequency of the cavity,

Θ. Now it is possible to rewrite that equation in terms of the cavity g-factors instead

of the cavity detunings, giving:

KOS = −IΘ2 = −I 2P

cI1

g2

1− g1g2

. (67)


As indicated by this Eq. 67, KOS is strongly dependent on the g-factor g1g2. Thus

making the manner in which the KOS measurements are attained much more flex-

ible. As an example, when the g-factor of the cavity is greater than 1 ( g1g2 > 1)

the optical spring constant increases, making the cavity inherently unstable, and

vice versa for g1g2 < 1.

The critical power level at which the system approaches this optical instability can

be found when the resonant frequency of the ITM is equivalent to the resonant

frequency of the optical spring, ω1 = Θ. Thus, by redefining the critical power as a

function of the cavity g-factors, the power can also be rewritten as:

Pcritical =cI1ω

21

2PL

1− g1g2

g2

. (68)

When the optical spring frequency of the cavity exceeds the mirror’s natural fre-

quency, the system becomes unstable. The critical power levels for the ITMs pitch

and yaw in the south arm Fabry-Perot cavity are found to be P yawcrit = 4.5 kW and

P pitchcrit = 7.0 kW. These values are determined using the parameters of the south

arm cavity as given in Table 2, below:

ITM ETM Compensation Plate

Radius of Curvature (m) R1 =∞ R2 = 790 Flat

Materials Sapphire Sapphire Fused Silica

Diameter (mm) 100 150 160

Thickness (mm) 46 80 17

ω1pitch (Hz) 0.81 - -

ω1yaw 0.65 - -

Cavity length (m) 80

Table 2: Parameters of the south arm cavity for the optical torsional stiffness experiment.


The experiment undertaken at AIGO studied the stable optical torsional mode in an

80 m optical cavity. This experiment is meant to complement the work previously

done on angular instabilities in gravitational wave interferometers [56, 57, 59, 60,

61, 62]. This particular experiment presents results on the g-factor dependence of

optical torsional stiffness in a Fabry-Perot cavity.

The experimental design is shown in Figure 21. The input optical set up does

not vary much from the layout described in Chapter 3, thus only the necessary

components of the experiment are shown in Figure 21.

Figure 21: The experimental set-up to measure the negative optical spring constant of the

80 m south arm optical cavity. The spectrum analyzer measured the resonant frequency of the

ITM, from which the value of the negative optical spring constant was calculated. The thermal

compensation plate was used to thermally tune the cavity g-factor.

As shown in Figure 21, the laser is injected and locked to the cavity. The transmitted

beam through the ETM is analyzed by a CCD camera while the resonance of the

ITM is monitored by a spectrum analyzer. By varying the focal length of the

thermal compensation plate, it was possible to measure the ITM yaw degree of

freedom as a function of the cavity g-factor.

There were two possible methods to determine the g-factor dependence. The first

measured the transmitted beam size through the ETM, using the CCD camera,


as shown in Figure 21. In Siegman’s book on lasers, he introduces a relationship

between the beam size and the g-parameters of the cavity as follows [69]:

ω21 =

λL

π

[g2

(1− g1g2)

]1/2

, (69)

ω22 =

λL

π

[g1

(1− g1g2)

]1/2

, (70)

where ω is the beam spot size, λ is the wavelength of light and L is the cavity

length. This equation can be inverted to solve for the g-parameters as a function

of the beam spot sizes, giving [56]:

g1 = ±ω2

ω1

[1− ω4

0

ω21ω

22

]1/2

, (71)

g2 = ±ω1

ω2

[1− ω4

0

ω21ω

22

]1/2

, (72)

where,

ω0 =

√Lλ

π. (73)

The g-factor dependence of the cavity can also be determined by measuring the

mode spacing ∆f , between the first order and fundamental modes of the cavity.

This relationship is presented also by Siegman in his book as the following:

g1g2 = g = cos2

(π∆f

FSR

), (74)

where FSR is the Free Spectral Range of the cavity.


Though it is possible to use both methods to find the g-factors, the results presented

here were attained by measuring the ∆f between the first and the fundamental

modes of the cavity. The incident laser beam was phase modulated with a swept

sine signal, and the resulting cavity g-factors were obtained by measuring the cavity

mode spacing with the spectrum analyzer. When the cavity was locked and the local

control system on, the cavity mode spacing was measured to be approximately

190kHz at various heating levels of the thermal compensation plate. At certain

heating levels the local control system of the ITM was turned off and the yaw mode

resonant frequency shift due to the optical torsional stiffness was measured.

The frequency shift measurements took ∼400 seconds each. In this amount of

time, it is possible to assume that the fluctuations found in each measurement were

averaged out. As the cavity g-factor was increased, the cavity alignment fluctuated

a great deal at the yaw mode resonant frequency (∼0.6 Hz). The results are plotted

in Figure 22.

Figure 22: The negative torsional spring constant KOS as a function of the cavity g-factor. Here

the spring constant is normalized by the circulating power inside the cavity. The error bars shown

here are a result of the power fluctuations as observed in the cavity, where the average power for

each measurement is given as the middle point of these bars.


As shown in Figure 22, the experimental data averages very closely with the theo-

retical predictions. The error bars shown represent the statistical errors from mea-

surements taken under the same conditions.There were several obstacles involved

in gathering the data shown in Figure 22. First, the thermal compensation plate

required at least one hour to reach thermal equilibrium. As mentioned in Chap-

ter 3, this is a result of the compensation plates material properties. Secondly, to

achieve a measurement meant that the local control system of the ITM had to be

turned off. Otherwise, when the control system was left on, in order to experience

some angular instability in the cavity, a signal larger than seismic noise had to be

injected. Regardless, the ITM would swing in large amplitudes, either caused by

the large injection signal or by seismic noise. The ITMs fluctuations had the ability

to drop the amount of power circulating inside the cavity, thus making the optical

spring effect obsolete. These power fluctuations are present in the results as shown

in Figure 22.

In addition to the obstacles, the long integration times required for precise measure-

ments sometimes knocked the cavity out of resonance, especially while operating

the cavity at higher g-factors.

The error bars shown in the results in Figure 22 are the result of instantaneous

power fluctuations in the cavity. These fluctuations shifted the average power level

during each measurement. Thus, the power used to calculate KOS during each

measurement may have been under or over-estimated, particularly at higher g-

factors.

4.3 Conclusion

This chapter has provided the results of optical angular instabilities, as wit-

nessed in an 80 m Fabry-Perot cavity. The data proves that the measured

effect of an optical spring in a high power optical cavity, agrees with the theoretical

4.3. CONCLUSION 69

data as presented by several researchers and several research groups over the past

years.

The magnitude of the optical spring effect is found to depend a lot on the type

of optical cavity the high input power is acting within. The effect of an optically

unstable torsional mode inside a Fabry-Perot cavity may vary according to the

parameters of the cavity, however it is always a potential threat to any high power

operating Fabry-Perot cavity, as witnessed in the results presented at the end of

Section 4.2.

Concentric cavities, that is cavities with negative g-factors, have been found to

alleviate the threat of angular instabilities within Fabry-Perot cavities. They are

more stable and more capable of controlling the beam, should it start to walk off

the center of the optical axis.

Interferometric gravitational wave groups have already taken steps to prevent this

potential instability from occurring, by modifying their interferometric Fabry-Perot

cavities from nearly planar to nearly concentric.

Chapter 5

Parametric Instability

Analagous to the angular instabilities presented in Chapter 4, parametric in-

stability is also a possible result from radiation pressure force effects on the

suspended test masses in a Fabry-Perot cavity. Advanced GW detectors are plan-

ning to increase the power in their Fabry-Perot arm cavities, as an attempt to

suppress the quantum shot noise present in the current interferometers output sig-

nal. Shot noise disrupts the sensitivity of the detectors in the frequency range just

above a few hundred Hertz. However, despite fixing one problem, gravitational wave

groups have raised the potential for others, including angular optical instabilities

and parametric instabilities.

This chapter discusses the theory of parametric instability based on the work com-

pleted by V.B. Braginsky, et al., reviews two different theoretical methods designed

to suppress parametric instability, and lastly presents the experimental results of

an optical suppression experiment developed for the south arm Fabry-Perot cavity

at the AIGO facility.

70

5.1. PARAMETRIC INSTABILITY IN FP CAVITIES 71

5.1 Parametric Instability in FP cavities

Afew years ago, gravitational wave groups around the world decided to upgrade

the current interferometric detector design to increase sensitivity to potential

gravitational wave signals across a broad frequency range. One of the upgrades

proposes to increase the amount of circulating power in the Fabry-Perot cavity

arms (proposed to increase to ∼830 kW in the LIGO detectors). This increase in

optical power in the optical cavities is meant to suppress the level of quantum noise

found in the detectors output signal. However, it has been found that such high

values of circulating power have the ability to raise non-linear effects inside the

cavity [70, 71, 72, 75, 74].

Parametric instability is defined as the non-linear coupling of acoustic and optical

waves at the test mass mirror interface. As discussed in Chapter 2, the physical

mechanism for this coupling arises from radiation pressure force acting on the sus-

pended optic in a suspended Fabry-Perot optical cavity. The acoustic modes in the

test mass (which are always naturally resonating due to thermal noise) have the

ability to scatter light from the resonant cavity mode (TEM00), to lower and higher

frequency sidebands, known as Stokes and anti-Stokes modes, respectively.

In the case of the Stokes mode, the laser pumped mode at frequency ω0 loses energy

to some existing acoustic mode of frequency ωm. In terms of energy, this interaction

creates a scattered optical sideband at a lower frequency, ω0−ωm. Contrastingly, in

the anti-Stokes process, the optical wave, ω0 incident on the test mass absorbs the

acoustic mode energy, creating a scattered optical sideband at a higher frequency,

ω0 +ωm. It should be noted, however, that neither sideband is favored in the cavity

unless one of them coincides with a high order optical mode already resonant in the

cavity.

The acoustic modes of kilometer scale detectors are far outside the arm cavity

bandwidth, thus making the coupling between the TEM00 mode and the acoustic

72 CHAPTER 5. PARAMETRIC INSTABILITY

modes obsolete. The acoustic modes in advanced detectors couple instead to the

higher order transverse modes, TEMmn, where the frequency differences between

the TEM00 and TEMmn modes is given by [74],

∆− = ω0 − ω1 =πc

L

(k1 −

m+ 2n

πarccos

√(1− L

R1

)(1− L

R2

))(75)

∆+ = ω1a − ω0 =πc

L

(k1 −

m+ 2n

πarccos

√(1− L

R1

)(1− L

R2

)), (76)

where ω0 is the fundamental mode frequency, ω1 is the Stokes mode frequency and

ω1a is the anti-Stokes mode frequency. L is the length of the cavity, R1 and R2 are

the radii of curvature of the mirrors, k1 and k1a are longitudinal mode indices, and

m and n are transverse mode indices. Due to the anti-symmetry of the excited high

order modes with the cavity mode, it is most often found that both the Stokes and

anti-Stokes are not resonant together inside a cavity.

Regardless of whether the Stokes mode or anti-Stokes mode is resonant, there is

always a spatial overlap of three modes, each with a different frequency, inside the

Fabry-Perot cavity. The main optical mode, ω0, a sideband, either ω0 + ωm or

ω0 − ωm and the mechanical mode ωm. These three fields, depending upon the

power, could produce a force on the surface of the mirror, and the mirror surface

thereafter could produce a force, or rather a modulation, on the light beam. If this

modulation yields Stokes scattering, then it has the right frequency and phase to do

positive work on the acoustic mode, thereby enhancing the acoustic vibrations. If

the modulation produces anti-Stokes scattering, then the system will, in exchange,

do negative work on the acoustic vibrations, thereby dampening the mode and

restoring normality in the cavity.

Due to the low bandwidth of long Fabry-Perot cavities, like the ones installed in

5.1. PARAMETRIC INSTABILITY IN FP CAVITIES 73

gravitational wave interferometers, the main TEM00 mode does not couple to the

acoustic modes under normal conditions, but the beat signal between the TEM00

and a high order transverse mode will, depending on several parameters. First, the

frequency difference between the high order mode and the fundamental mode must

match the frequency of the acoustic mode, such that ω0 − ω1 = ωm. Second, the

spatial profile of the high order optical mode must substantially overlap with the

spatial profile of the acoustic mode. An example of an acoustic and optical mode

with the potential for substantial overlap is shown in Figure 23.

(a) Optical Mode (b) Acoustic Mode

Figure 23: An example electromagnetic profile of the TEM10 optical mode, and a sample

acoustic mode. As can be seen here, overlaying the two profiles would give a high spatial overlap

value. The value for the spatial overlap of any two modes is given by Equation 78.

Braginsky, et al. developed a dimensionless parameter gain, R, to define the cou-

pling parameters necessary for this effect to occur [70]:

R = ±4PcavQ1Qm

mLcω2m

Λ

1 + (∆ω/δ1)2. (77)

In the first term, Pcav is the optical power as stored in the cavity fundamental mode,

m is the mass of the mirror, c is the speed of light, and Q1,m are the quality factors

for the high order and acoustic modes, respectively. In the second term, Λ describes

the spatial overlap between the optical and acoustic modes, and is defined by


Λ =V (∫

f0 ( ~r⊥)f1 ( ~r⊥)µzd ~r⊥)2∫|f0 |2d ~r⊥

∫|f1 |2d ~r⊥

∫|µ|2dV

, (78)

where f0 and f1 describe the optical field distribution over the mirror surface for the

fundamental and Stokes mode, respectively, µ is the spatial displacement vector for

the mechanical mode, µz is the component of µ normal to the mirrors surface, and

the integrals∫d ~r⊥ and

∫dV correspond to the integration across the mirrors surface

and volume, respectively. Figure 23 is a visual representation of two modes (acoustic

and optical) which have the potential for a high spatial overlap value. Returning

again to the second term in Eq. 77, ∆ω = |ω0 − ω1| − ωm and δ1 = ω1/2Q1, which

gives the half line-width of the high order optical mode.

The possibility for R to assume both positive and negative values is the possibility

for the two opposite coupling modes, Stokes and anti-Stokes, respectively, to occur

in an optical cavity. When R > 1, producing Stokes scattering, the amplitudes

of the acoustic and high order optical modes increase exponentially together with

time, absorb the power from the fundamental mode in the cavity, and limit the

amount of power that is able to build-up inside the cavity.

5.2 Suppressing Parametric Instability

The research surrounding parametric instabilities, initially starting at the State

University of Moscow by Braginsky, et al, has been taken on by several in-

ternational gravitational wave groups over the years, including Caltech, MIT and

the University of Western Australia. Collectively, their analyses of potential para-

metric instabilities in advanced gravitational wave detectors indicate that there are

∼ 5−10 possible unstable modes associated with each test mass in the 10−100kHz

5.2. SUPPRESSING PARAMETRIC INSTABILITY 75

frequency band.

Their research demonstrates that the potential for parametric instability to occur in

advanced detectors is large enough that research, both analytical and experimental,

to find a solution to suppress this instability is essential.

Parametric instability can be simply thought of as an unstable interaction of three

modes. Figure 24 shows the sequence of events which gives rise to parametric

instability in a Fabry-Perot optical cavity.

Figure 24: An illustration of the three mode interactions inside a generic Fabry-Perot optical

cavity. Here the first mode, ω0, enters the cavity, resonates, building up in power in the cavity

over time. Simultaneously, the acoustic modes of the end test mass are excited (due to the thermal

motions of the test mass) and thereafter scatters the main mode into a high order optical mode.

This excited high order optical mode, ω1 begins to ring up inside the cavity, doing positive work

on the acoustic mode, ωm and the acoustic mode on the excited mode. Together, they ring up

exponentially and the cavity becomes unstable.

Figure 24 shows the three mode interaction which give rise to parametric instability

in a Fabry-Perot optical cavity. First, the fundamental mode, ω0 is injected into

the cavity, where it resonates and builds up in power over time. Simultaneously,

the thermal motions of the test mass excite the acoustic modes, ωm, of the end test

mass. Depending upon how much power is in the ω0 mode, one of its sidebands

may couple to one of the excited acoustic modes. When this occurs, the high order


optical mode, ω1 begins to build up in power in the cavity, taking power from the

ω0 mode and ringing up exponentially in time with the ωm acoustic mode. The

two modes do positive work on each other, thus when one increases in power and

amplitude, the second is also further excited. Depending on whether or not the

frequency and spatial overlap of the two modes is significant, as mentioned earlier

in this chapter, these three modes could interact to produce an unstable cavity.

This three mode interaction, referred to primarily as parametric instability, is also

known as three mode opto-acoustic parametric interactions.

When considering the three mode interactions which lead to parametric instability,

as shown in Figure 24, it may be logic that there are two target methods to sup-

press parametric instability being analyzed. The first proposed methods intend to

suppress the high order optical mode ω1 and the second method, the acoustic mode

of the test mass ωm.

5.2.1 Suppressing the Acoustic Mode

Between 2007 and 2009, researchers at the University of Western Australia pub-

lished several papers which studied methods to suppress parametric instabilities by

suppressing the acoustic mode excitation. The research concentrated on dampening

the acoustic mode by installing a broadband damper on the test mass. The damper

was designed as a uniform coating around the barrel of the test mass, where the

coating would act as the dampener. Unfortunately, the overall effectiveness of this

barrel damper was not satisfying, and in the end would only produce a potential

increase in the overall thermal noise budget for gravitational wave interferometers

by 10% at 100Hz [79, 84].


Years later, the LIGO group at MIT suggested two other methods to suppress

parametric interactions by suppressing the acoustic mode of the test mass. Their

first proposal suggests installing an electrostatic drive (ESD) coating over the face of

the reaction test mass, to interact with the already in-place actuators surrounding

the reaction test masses, to dampen acoustic vibrations. The concept is designed

to use the interferometers output signal to detect the excitation of the mechanical

mode of the mirror and apply a damping force, inherently as part of a feedback

system setup from the detectors signal to the electrostatic coating. M. Evans, et

al. developed an expression to determine the amount of required force needed from

the actuators to dampen the potential mechanical modes [80, 82]. This expression

is given by:

Fact =ω2mMm

ΓmQm

xrmsm =ωm

ΓmQm

√MmkBT , (79)

where ωm is the acoustic mode frequency, Mm is the modal mass, Γm is the mode

overlap factor between the acoustic and actuator mode, Qm is the quality factor for

the acoustice mode, xrmsm is the root mean square of the modal amplitude and kBT

is the thermal excitation.

The physical interpretation of this active damping scheme involves coating an elec-

trostatic drive pattern over the face of each reaction test mass, for each potentially

affected test mass of the interferometer. Depending upon the excited acoustic mode

the actuators surrounding the face of the test mass will produce a force on the elec-

trostatic drive pattern that will ideally dampen the acoustic mode. An example

diagram of the electrostatic drive pattern on a reaction test mass, as well as one of

the actual electrostatic drive coated reaction test masses from the LIGO MIT lab

are shown in Figure 25. These images are taken from a powerpoint presentation

by M. Evans, et al. as presented to the Gravitational Wave Advanced Detector

Workshop (GWADW) in Ft. Lauderdale, Florida 2009 [80, 83].


(a) ESD Pattern for ETM (b) Actual Trial of ESD

Figure 25: The electrostatic drive pattern as planned and analyzed by the MIT LIGO group

to install on the faces of the reaction test masses potentially affected by parametric instability.

Figure 25a shows the planned pattern for the LIGO ETM test masses and Figure 25b shows the

actual experimental example installed on a free test mass at the MIT LIGO lab [80, 82].

Despite the fact that the electrostatic drive active damping technique is theoretically

capable of dampening the excited acoustic modes in gravitational wave interferom-

eters, the LIGO group at MIT found that if these patterns were to be installed on

each test mass, there would be a 200µN peak during the interferometers acquisition

mode (when the detector is operating at full laser power).

Thus, they developed a second method to suppress parametric instabilities by the

acoustic mode excitation. The second method they developed is referred to as

the passive damping method. The passive damping method involves strategically

placing resonant mass dampers across the body of the test masses. The dampers

are modeled after a standard mass-spring-damper system. The proposed model for

the mechanical dampers is shown in Figure 26. This model was presented in the

same powerpoint by M. Evans, et al [80].


Figure 26: A model test mass with strategically placed mechanical dampers. The dampers are

theorized to work as a standard mass-spring-damper system would, shown in the bubble diagram

to the right of the test mass [80].

The mass spring damper system shown in Figure 26, shows a test mass with sev-

eral smaller masses attached to its body. These small masses are meant to act as

dampening resonators, where the effect of each dynamic damper can be thought of

as the coupling between a pair of dampers (or more) to dampen the critical modes

which may develop within the test mass. An illustration of this coupling effect is

shown in Figure 27. This image was also taken from the same powerpoint by M.

Evans, et al. for the GWADW workshop in 2009 [80].

Figure 27: In this illustration several acoustic modes are shown, all of which require different

coupling patterns of mechanical dampers (which would theoretically be sitting on the surface of

the test mass) to stabilize. M. Evans et al, determined that the addition of these small mechanical

dampers, shown above the test masses here, strategically placed around the barrel of each test

mass could act as resonant mass dampers against excited acoustic modes [80].


The mechanical dampers can also be thought of as resistively shunted piezos. At

MIT, they determined that they actually need only two of these piezos to effectively

dampen all the potentially critical acoustic modes in advanced gravitational wave

interferometers. The critical modes were determined by calculating the parametric

gain for a single Fabry-Perot arm cavity, and focusing on the modes between 10−

100kHz. Using FEM analysis, they found there were approximately 675 critical

acoustic modes which could produce harmful effects in advanced detectors.

Full experimental analysis on the effect of these methods has yet to be completed.

The electrostatic drivers will already be in place in the control system design of

advanced gravitational wave interferometers, however whether they are used is still

up in the air. The mechanical dampers would already be attached in advanced

detectors, but these dampers are not able to be manipulated or modified once inside

vacuum. Recent analysis has shown that the dampers introduce an undesirable noise

source in the output of the detectors system, thought to be due to the glue used to

attach the dampers to the test mass. The electrostatic drive system can be installed

without perturbing the design or operation of the overall interferometric system, as

long as it is capable of coupling to all optical modes. Further, sufficient testing of

both methods has not been completed, and thus neither one of these methods is

exactly ideal for suppressing potential parametric instabilities.

5.2.2 Suppressing the Optical Mode

The other proposed method to suppress parametric instability involves suppressing

the excited optical mode in the cavity, using an optical feedback system. Similarly to

the electrostatic driver method, optical feedback can be installed without perturbing

the design or operation of the interferometric gravitational wave detectors system.

This section provides a look at the theoretically proposed feedback system developed

by Z. Zhang, et al.


In 2009, Z. Zhang, et al. at the University of Western Australia analyzed the dy-

namics of three mode parametric interactions in a Fabry-Perot cavity, and theorized

a method to suppress these interactions [85]. In their paper, they suggested creat-

ing an optical feedback loop capable of creating a replicant excited optical mode,

ω1, and injecting it into the cavity with the input ω0 mode to suppress the ex-

cited cavity ω1 mode. Depending on the phase, frequency and amplitude chosen

for the replicant ω1 mode, Zhang, et al. theorized that the replicant mode could

be manipulated such that it destructively interfered, suppressing the excitation, or

constructively interfered, enhancing the excitation, with the excited cavity mode

ω1.

Zhang, et al. theoretically proved that an optical feedback system would produce

significant results if installed in a high power Fabry-Perot cavity. In their calcula-

tions, they begin by assuming the mirror oscillations occur only on the end mirror,

at a frequency ωm with amplitude x(t). The expression they developed for the

oscillations of the end mirror is given by the following:

x(t) = ψxχ(t)e−iωmt + ψxχ∗(t)eiωmt (80)

where χ(t) is the slowly changing complex amplitude, χ∗(t) its conjugate and ψx is

the normalized spatial distribution of the mechanical mode. If and when parametric

instability has the opportunity to arise (given the right conditions), an extra optical

field ωin, with the same frequency and spatial mode as the cavity mode ω1, but with

a phase difference can be injected into the cavity. This injection has the ability to

destructively or constructively interfere with the excited mode ω1. Zhang, et al.

continue in their paper to describe the mechanism of the mechanical oscillations as

follows:

χ∗(t) + pχ∗(t) = qωin(t), (81)


p =δ1δmδ1 + δm

(1−R), (82)

q =δ1δmδ1 + δm

√T1R

2ikB√Pcirc

, (83)

where p is the effective damping factor and q is proportional to the fundamental

mode amplitude inside the cavity, R is the parametric gain as defined earlier in

Chapter 4.1, Eq. 77, δ1 and δm are the cavity bandwidths, T1 is the transmission of

the ITM, k is a wave vector and Pcirc is the circulating power stored in the cavity.

According to Zhang, et al. the factor B can be further defined as:

B =

∫ψ0ψ

∗1~u(~r) · d~s√A0A1

, (84)

where ψ0 and ψ∗1 are dimensionless functions for the fundamental mode and the

complex conjugate of the high order cavity mode distributions, respectively. Simi-

larly, the terms A0 and A1 can also defined by these dimensionless functions, such

that

A0 =

∫|ψ0|2d ~s⊥ (85)

and

A1 =

∫|ψ1|2d ~s⊥. (86)


As shown, both A0 and A1 are integrals defined as a function of the displacement,

d ~s⊥. If the feedback field ωin can be controlled as a function of the mechanical

oscillations, then Zhang, et al. proposed that:

ω0(t) = iαχ∗0(t), (87)

where α is the coefficient to control so that the parametric instability is observable.

It is possible to derive a solution for Eq. 83, using Eq. 87:

χ∗(t) = χ∗0eβt, (88)

where β = −(p − iαq). The sign of the index β determines whether the system

is stable or not. If parametric interactions are already instigated (R > 1), then

it is possible to choose an α which makes β negative, thereby suppressing the

interactions. Simply, the phase of the feedback field (the beam to be re-injected) is

reversed, thus giving ω0(t) = −iαχ∗0(t), and β = −(p+ iαq).

Zhang, et al. also developed an experimental design to demonstrate their theoretical

findings. Their proposed experimental set up is shown in Figure 28.


Figure 28: The experimental set-up suggested by Zhang, et al. to optically enhance or suppress

opto-acoustic parametric interactions. The first beam splitter shown (BS1) sends a small portion of

the main beam to an arbitrary number of Mach-Zehnder interferometers, designed to re-inject the

appropriate high order mode into the cavity, ωin. EOM1 together with PD1, a mixer and amplifier

to phase lock the pick off beam to the main beam. EOM2 creates sidebands at the acoustic mode

frequency, ωm while the phasemask converts the TEM00 to the TEMmn of interest. PD2 detects

the high order mode amplitude and frequency from the cavity and feeds it back to EOM2. The

sideband signals are the injected into the cavity through BS2 and, depending on their phase, they

have the ability to enhance or suppress the excited high order mode [85].

As mentioned at the beginning of this section, the optical suppression/enhancement

experiment is developed like a feedback system, where a second beam is injected

into the cavity with the same frequency and spatial mode as the excited high order

beam, ω1. Depending on the phase of the replicated ω1 mode being injected into

the cavity, the amplitude of the excited cavity mode should theoretically be either

amplified or suppressed.

In Figure 28 the main beam from the laser passes through a beam splitter, which

diverts a small portion through a series of optics and electronics to match it to the

5.3. OPTICALLY SUPPRESSING PARAMETRIC INSTABILITY 85

excited mode. The second photodetector (PD2) senses the frequency and amplitude

of the high order mode from the cavity transmittance. This information is filtered

and amplified before it is sent to an electro-optic modulator (EOM2), which uses

the signal to generate sidebands at the given frequency. However, before the light is

passed through EOM2, it is first phase locked to the main beam with the photode-

tector (PD1), mixer, amplifier and EOM1. Whether or not this pick-off beam will

enhance or suppress the instability depends on the phase produced by the phase

mask installed in the lower path of the Mach-Zehnder interferometer. The phase

mask can convert the fundamental mode to the high order mode of interest. The

beam is then re-joined with the main beam through the second beam splitter (BS2)

and sent into the cavity to constructively or destructively interfere with the excited

optical mode.

This feedback system, like the acoustic mode dampers, is not an ideal solution

for advanced gravitational wave detectors. An ideal solution would be a similar

optical feedback system, but implemented using a pick off beam from the ETMs

transmittance port. Nevertheless, Zhang, et al.’s paper inspired the experimental

procedure presented in the next section, where practice is shown to agree with

theory.

5.3 Optically Suppressing Parametric Instability

This section presents the experimental results of an optical feedback system

modeled after Zhang, et al.’s theoretical analyses and proposed experimental

set up as described in the last section.

The experimental goals were to first create the opto-acoustic interaction signal in

the south arm Fabry-Perot cavity, and suppress them using a similar design to the

one shown in Figure 28.

The circulating power inside the south arm cavity was found to be only ∼1.2 kW


before running the feedback experiment. This is far too little power to generate

any instance of natural parametric instability in the south arm cavity at AIGO.

Thus, parametric instability had to be induced in the south arm cavity before

it could be suppressed. In 2008, Zhao, et al. published a paper detailing the

experimental observations of parametric instability in the south arm Fabry-Perot

cavity. To attain their results, Zhao, et al. had to experimentally manipulate the

south arm cavity to develop parametric interactions. The experimental procedure

they developed to create parametric instability was reproduced for the experimental

results presented in this section. A diagram of Zhao, et al.’s experimental set up is

shown in Figure 29 [78].

Figure 29: Zhao, et al.’s experimental set up to observe parametric interactions in the south arm

Fabry-Perot cavity at AIGO. Here the fundamental mode from a laser source (ω0) is injected into

and resonates inside a Fabry-Perot cavity. Simultaneously, a force is exerted on the ETM which

excites one of the acoustic mode frequencies, ωm. The oscillating ETM pushes the ω0 fundamental

mode into a higher order mode sideband of frequency ω1. When the cavity is adjusted correctly,

the excited ω1 mode is observed to ring up exponentially and the instability is created.

In Figure 29, the fundamental mode (ω0) from some laser source enters a Fabry-

Perot cavity. The light resonates inside the cavity while simultaneously being dis-

turbed by some force from an acoustic mode of the ETM at frequency ωm. The


oscillation of the ETMs surface causes light from the ω0 fundamental mode resonat-

ing inside the cavity to scatter into a higher order mode sideband of frequency ω1.

If the cavity is manipulated using the thermal compensation plate, then both the

ω0 and ω1 mode can resonate in the cavity. At a certain heating power of the CP,

the ω1 mode will substantially overcome the ω0 mode, and ring up exponentially in

the cavity. Zhao, et al. observed and measured the amplification of the excited ω1

mode by monitoring the pattern of the transmitted beam through the ETM with a

CCD camera [78].

The optical feedback experiment presented here, employed the method developed

by Zhao, et al. to instigate parametric instability, but with a few modifications to

the input optical path which would eventually suppress the parametric instability

that was created. Figures 30 and 31 show two simplified, but modified experimental

schematics from Figure 29.

Figure 30: The experimental set up for the optical feedback experiment, based on the one

developed by Zhao, et al. in 2008 and shown in Figure 29. The only difference between this

diagram and Figure 29 is the addition of a few unused optics to the input path. This diagram can

be thought of as step one in the feedback experiment, where the important feature is the creation

of the parametric interactions.


Figure 31: This diagram gives the second step in the optical feedback experiment. Here a pre-

mode cleaner is used to mode select the ω1 mode and an EOM is used to adjust its frequency to

the mode detected in the transmitted beam from the ETM port. This new mode is then injected,

in resonance with the ω0 mode, into the cavity where it optically interferes with the excited ω1

cavity mode [86].

Figures 30 and 31 represent the two steps used in the optical feedback experiment

presented in this section. Figure 30 gives the first step of the optical feedback

experiment, where the layout shown is essentially the same as the layout given in

Figure 29 earlier, but with a few unused optics added to the input path of the laser

into the cavity. Parametric instability is created in this step of the experiment,

based on the same method as used by Zhao, et al. in 2008, where an acoustic

mode of the ETM is excited, pushing power from the resonant input beam ω0

into a high order sideband, ω1. This sideband is then amplified and observed to

exponentially increase over time by adjusting the g-factor of the cavity with the

thermal compensate plate.

The next step in the feedback experiment is shown in Figure 31. Once parametric

instability is created, all of the unused optics and light paths shown in Figure 30 are

enabled. In Figure 31, light propagates through both the lower and upper paths of


the Mach-Zehnder, where ω0 remains in the lower path, and the upper path is used

to generate a replicate to the excited cavity mode, ω1. Initially, ω0 travels through

the upper path as well, until it passes through the pre-mode cleaner. The pre-mode

cleaner is used to mode select the excited high order mode, ω1, as observed from the

transmitted beam of the cavity. After the pre-mode cleaner, the beam is then passed

through an EOM which is used to select the appropriate frequency and phase. The

replicate ω1 beam is then recombined with the ω0, and together injected into the

cavity. Depending on the phase of the injected ω1 beam, it can either constructively

or destructively interfere with the excited ω1 beam in the cavity.

The actual experimental set up is not as simple as Figures 30 and 31 indicate. A

more realistic diagram of the experimental design is shown in Figure 32.

Figure 32: Schematic of the opto-acoustic suppression experiment. Here a Mach-Zehnder inter-

ferometer is added to the input path of the beam into the cavity as a way to create an ω1 mode

to inject into the cavity as well. The pre-mode cleaner mode selects the high order mode, ω1,

and a phase modulator (EOM1) generates sidebands at the ω0 ± ωm frequency. The TEM00 and

generated TEM01 are locked to each other by the EOM2 and the PZT in the lower path of the

MZ. Later, they are recombined at BS2 and injected together into the cavity [86].

Although Figure 32 looks complicated, there are only a few important things to

note from the diagram. The first, is the path of the transmitted beam through


the ETM. As shown, the transmitted beam is split into two paths after leaving the

ETM. One path leads to a CCD camera, making it possible to monitor the output

beam profile. The second path goes to a quadrant photodiode (QPD), whose signal

was monitored on a spectrum analyzer. This signal showed the beating signal

between the two modes ω0 and ω1. When the appropriate power was applied to the

thermal compensation plate, the exponential growth of the ω1 mode was seen in

this beating signal. Additionally as shown, a web camera was set up in front of the

Spectrum Analyzer so the excitation could be observed remotely, from the Main

Lab. In Figure 32 this remote connection is shown as a dotted line between the

computer and the spectrum analyzer. The remote connection between the main lab

and the end station is what made the feedback system possible. The measured data

from the spectrum analyzer indicated which phase and frequency EOM1 needed to

be adjusted to, in order to achieve suppression or enhancement between the two ω1

modes in the cavity.

In the experimental design used for the feedback system shown in Figure 32, it is

only possible to excite one kind of acoustic mode using Zhao, et al.’s excitation

method. A capacitive actuator is situated just behind the back surface of the ETM

in the east end stations vacuum chamber. This capacitive actuator is used to excite

the acoustic mode ωm, by applying a signal ωm/2 to the ETMs surface.

In Zhao, et al.’s experiment, they were able to determine several excitable test mass

acoustic modes for the south arm ETM. One of the modes in particular, the one

used to excite the parametric interactions for the optical feedback experiment, was

found to resonate at ∼178 kHz. Figure 33 shows the contour map of the acoustic

mode for our ETM at 178 kHz.


Figure 33: The contour map of the ∼178 kHz acoustic mode of the south arm ETM. Though

not perfectly vertical or harmonious in shape, the resultant ω1 mode from this ωm was able to be

suppressed and additionally excited by injecting a TEM01 mode into the cavity [87].

Based on the profile of the 178 kHz acoustic mode, shown in Figure 33, it is possible

to estimate the mode shape of the excited ω1 mode in the cavity. It was assumed

in the experiment that ω1 should have a strong electromagnetic profile along the

vertical direction, like the acoustic mode shown. Thus, under these assumptions,

the pre-mode cleaner shown in Figure 32 mode selected the TEM01 mode as the

replicate ω1 mode for the experiment. The TEM01 mode was chosen for its verti-

cal direction and its strong spatial profile overlap with the majority of potentially

excited vertical electromagnetic modes.

Another important part of the experimental design is the use of the thermal com-

pensation plate. The ω1 signal in the cavity is inherently too weak to take power

from the ω0 mode. Subsequently, the thermal compensation plate is tuned to the

necessary cavity g-factor, forcing the excited ω1 sideband into resonance. Recall

that by adjusting the heating level of the thermal compensation plate, it is possible

to adjust the g-factor of the cavity, which in turn can favor one mode of resonance

in the cavity over the other. As mentioned in Chapter 4, the cavity g-factor is


defined as g = (1 − L/R1)(1 − L/R2), where L is the cavity length, R1 is the ef-

fective radius of curvature of the ITM and thermal compensation plate, and R2 is

the radius of curvature of the ETM. When our cavity g-factor is thermally tuned to

g ∼0.913 (which is approximately equivalent to ∼5 W heating power applied to the

thermal compensation plate for ∼30 min), the upper sideband of the TEM01 mode

becomes resonant in the cavity, and its power is amplified. Without the thermal

compensation plate, no high order mode could exponentially ring up in the south

arm cavity at AIGO.

The last important nose on the experimental set up is how the signals were attained

that were fed back to the Mach-Zehnder in the main lab. The amplitude of the

excited TEM01 mode was determined by measuring the heterodyne beat signal

between the TEM00 and TEM01 modes with a QPD. The output of the QPD is

known to be proportional to the square of the amplitude of the cavity optical modes,

giving:

IQPD ∝ |E0eiω0t + E1e

i(ω1t+φ0)|2 = 2E0E1 cos(ωmt+ φ0) +D.C. (89)

where E0 and E1 are the amplitudes of the TEM00 and TEM01 modes, respectively,

φ0 is the relative phase difference between these modes and ωm = ω1 − ω0. The

heterodyne signal was obtained by filtering the D.C. component out of the mea-

surements. The resulting filtered QPD differential output was sent to the spectrum

analyzer, which, as previously mentioned, was monitored remotely from the main

lab. This signal was used to adjust the parameters of EOM1 for each experimental

feedback observation.

5.3.1 Optical Feedback Results

This section provides the results of the optical feedback loop experiment. Using the

experimental design showed and described in Section 5.3, both amplification and


suppression of the excited ω1 have been observed.

Figure 34 shows the results of the first step of the optical feedback experiment, where

the excited ω1 mode of the cavity is amplified by heating the thermal compensation

plate. In this figure, there is no additional ω1 mode being injected into the cavity.

Figure 34: The resonant TEM01 mode in the cavity. Created through electrostatic excitation

of a known acoustic mode of the ETM, and brought into resonance and amplified by thermally

tuning the g-factor of the cavity. In this measurement there is no additional amplification or

suppression by the MZs’ TEM01 beam [86].

Once introducing the replicate ω1 mode, which in this experiment was the TEM01

mode, from the upper path of the Mach-Zehnder interferometer to the cavity, it

was possible to generate the amplification or suppression of the excited ω1 cavity

mode. The amplitude and phase of the TEM01 sidebands were modified by tuning


the amplitude and phase of the first local oscillator signal, LO1. When the am-

plitude was matched in anti-phase with the excited TEM01 mode, it was found to

destructively interfere with the excited cavity mode and consequently suppress its

optical heterodyne signal. Figure 35 presents the results of destructive interference,

where the heterodyne signal was observed to drop from ∼ 42 mV to ∼ 5.35 mV,

with only ∼ 0.4 mW input power applied to the sideband signal.

Figure 35: The results of the suppressed ω1 mode in the cavity. This plot demonstrates the

successful destructive interference, or suppression, of the excited ω1 mode in the cavity with the

injected TEM01 mode. As seen in comparison with Figure 34, the excitation of the cavity ω1

mode was suppressed by approximately 5 times its initial excitation [86].

When the TEM01 sideband signal was injected from the Mach-Zehnder in-phase

with the excited cavity ω1 mode, the optical heterodyne signal doubled in ampli-

tude, implying that the TEM01 mode and the cavity ω1 mode were constructively

interfering with each other. Figure 36 presents the results of the amplification of

the excited cavity ω1 with the TEM01 mode.

5.4. CLOSED LOOP CONTROL 95

Figure 36: The results of the enhanced ω1 mode in the cavity. In comparison with the results of

the initial excitation of ω1 shown in Figure 34, the amplitude of the excited ω1 cavity mode was

enhanced by approximately 8 times more than the original excitation. The observations presented

here prove the successful constructive interference, or amplification, of the excited ω1 mode with

the injected TEM01 mode [86].

As shown in Figures 34, 35 and 36, the optical enhancement and suppression of

the excited ω1 mode is possible using the optical feedback methods described in

this section. To further prove the success of these positive results, it was found that

when injected sideband signal TEM01 mode was blocked from entering the cavity,

the heterodyne beating signal between the ω1 and ω0 modes returned to its original

excitation level, shown in Figure 34.

5.4 Closed Loop Control

The last section described the first experimental procedure and results for an

optical feedback loop to suppress parametric instability. This section de-

scribes a second experimental procedure to suppress parametric interactions, where

the feedback process becomes less hands on. The optical feedback strategy described


in Section 5.3 required local control of the EOM1 to adjust the phase, frequency

and amplitude of the injected TEM01 mode. The second experiment illustrated in

this section requires less local control, upgrading the optical feedback design from

the last section to a ”close loop control” feedback loop.

In the first optical feedback experimental procedure, the amplitude of the excited ω1

mode in the cavity was determined by measuring the heterodyne beat signal between

the two optical cavity modes. After the frequency and phase of this beat signal was

determined, the EOM in the upper path of the Mach-Zehnder interferometer was

adjusted (by hand) to the same frequency and phase (or anti-phase) as the excited

ω1 in the cavity, and then injected into the cavity. In the closed optical feedback loop

experiment, the heterodyne beat signal from the cavity is used to create an error

signal which controls the TEM01 sideband signal. The upgraded optical feedback

experimental layout is shown in Figure 37.

5.4. CLOSED LOOP CONTROL 97

Figure 37: The experimental set-up of the ’closed loop’ system. Here the amplitude of the

TEM01 mode was measured by the QPD. The addition of the mixer after the QPD readout made

it possible to demodulate the differential read-out of the QPD at the acoustic frequency ωm, and

extract the beat signal at ∼ 178 kHz. This signal is then used to control the amplitude sent to

EOM1 in the upper path of the Mach-Zehnder interferometer [86].

As can be seen, the experimental design given in Figure 37 is very similar to the

design shown in Figure 32 in Section 5.3. Despite their similarity, there are some

very important differences between the two layouts. First, instead of measuring the

differential QPD output on a spectrum analyzer, the signal was mixed it with a

signal from a local oscillator (LO3) at the ωm frequency. The output of the mixer

contains a D.C. component which is proportional to the amplitude of the excited ω1

cavity mode, and high frequency signals (e.g. at 2ωm). The output is then passed

through a low pass filter with a cutoff frequency much lower than that which the

acoustic mode attenuates these high frequency signals at, but still proportional to,

the ω1 mode amplitude inside the cavity. This signal is then sent to a computer

SIMULINK program in the main lab which observes the data trend. The output

measured on the computer in the main lab is used to control the amplitude of the

signal which is sent to EOM1. The results from the closed optical feedback loop


experiment are shown in Figure 38.

Figure 38: The results of the ’closed loop’ experiment. Here the top plot gives the output from

the mixer, showing the amplitude evolution of the TEM01 mode. The bottom plot gives the error

signal which controls the feedback signal. As shown, 3 seconds after turning the feedback system

controls on, the TEM01 mode was effectively suppressed and stabilized [87].

In Figure 38, the top curve shows the output signal from the mixer as the amplitude

of the ω1 mode in the cavity grows. The second plot is a time series of the error

signal which controls the feedback signal. The ω1 mode is initially at resonance

inside the cavity, where the beating signal reads ∼1.4 V. Three seconds after the

’closed loop’ control system is turned on (the TEM01 mode is then injected into

the cavity), the power of the excited mode inside the cavity drops and stabilizes

around 0.5 V, indicating that the ω1 mode in the cavity had been suppressed.

In this feedback system only the amplitude and frequency of the signals are con-

sidered as control variables, and the phase of the local oscillators are all set in

advance. It was found that the system operated properly when the phases of the

5.5. CONCLUSION 99

local oscillators were kept constant.

Despite finding positive results for both optical feedback loops, environmental tem-

perature shifts created some problems during measurements. The temperature

shifts would induce a couple of Hz shifts of the ETM acoustic mode, ωm, which

would affect the phase. These shifts had the potential to cause the closed loop con-

trol system to fail when they were large enough. As a solution, the real phase of the

excited ω1 mode was measured and used as a control variable, like the amplitude

and frequency, for all measurements.

5.5 Conclusion

Following much international theoretical study of parametric instabilities, the

results reported in this chapter represent the first successful demonstrations

of optically suppressed parametric instability.

Since the early 2000s, parametric instability has been ruled a potentially harmful

instability to the operation of high power advanced gravitational wave interferom-

eters around the world. Thus, theoretical and experimental solutions to suppress

parametric instability has become an important research topic in the gravitational

wave physics community.

Section 5.2 reviewed two proposed solutions to suppress the harmful effects of para-

metric instabilities in interferometric optical cavities. Section 5.2.1 reviewed the

research being conducted by the LIGO group at MIT. They proposed two different

strategies of suppressing the excited acoustic mode ωm, which could possibly lead

to parametric instabilities. Section 5.2.2 reviewed the work being conducted by the

gravitational-wave group at UWA, where they have proposed to suppress potential

parametric instabilities by suppressing the excited optical mode, ω1, in the cavity.

Sections 5.3-5.4 presented the effective suppression (and likewise the enhancement)


of opto-acoustic parametric interactions by dampening the excited optical mode.

The two experiments described demonstrate the first look at a destructive interfer-

ence optical suppression solution for parametric instability.

Chapter 6

Conclusions

6.1 Summary of Results

As previously mentioned, gravitational wave detector groups around the world

are planning to upgrade the interferometric gravitational wave detector de-

sign, as a way to diminish, or lessen, as many of the disruptive noise sources in the

detectors output signal as possible. In an attempt to improve sensitivity in the shot

noise dominated region, the gravitational wave community has proposed increasing

the circulating laser power inside their Fabry-Perot optical cavities. Although this

power increase would eliminate some of the negative effects of shot noise, it would

in fact give rise to several possible negative radiation pressure effects, including

angular optical instabilities and parametric instabilities.

This thesis has provided a review and the experimental results of both angular

and parametric instabilities in gravitational wave interferometric optical cavities.

Chapter 4 demonstrated the strength of the negative optical spring effect in the 80

m south arm cavity at AIGO. The experiment and its results provided proof that

angular instabilities are still a potential threat to interferometric gravitational wave

detectors even at low power levels. As discussed in Chapter 4, gravitational wave

101

102 CHAPTER 6. CONCLUSIONS

groups have already taken this potential instability into account and have modified

their optical cavities from nearly planar to nearly concentric. Nearly concentric

cavities are known to have negative g-factor cavities, this attribute has been proven

to significantly reduce the possibility for angular instabilities to negatively effect

the operation of detectors.

Chapter 5 demonstrated the creation and suppression of parametric interactions in

the 80 m south arm cavity at AIGO. The experimental results provide an initial

solution to the potentially hazardous opto-acoustic interactions in advanced gravi-

tational wave interferometers, using optical suppression. Although other solutions

have been proposed, the optical feedback suppression experiment is the first to illus-

trate the suppression of parametric instability in a high power Fabry-Perot optical

cavity.

6.2 Future Prospects

There is a lot of work, experimentally and theoretically, left to be completed

before any solution for parametric instability is approved for use in interfer-

ometric gravitational wave detectors. All of the research to suppress parametric

instabilities in interferometric gravitational wave detectors, including the solutions

presented in Chapter 5, are only the preliminary experiments to finding a solution

against parametric interactions. Much further research needs to be completed on

all potential solutions, as detailed in this thesis, before any one solution is chosen.

Chapter 5.2 reviewed the proposed solutions for dampening parametric instabilities

by suppressing the excited acoustic modes of the test mass. There were two sug-

gested methods: the first method suggested installing an electrostatic drive pattern

across the face of the potentially affected test masses. Depending on the excited

acoustic mode (sensed in the detectors output signal), this electrostatic drive pat-

tern would interact with the magnetic actuators surrounding the test mass, to

6.2. FUTURE PROSPECTS 103

actively damp the excited acoustic mode. The second method suggested placing

tiny resonant mass dampers around the test mass to passively damp any potential

excited acoustic modes. Both proposed methods are potential solutions for grav-

itational wave interferometers, however further testing needs to be done to see if

the dampers can operate at a low noise level, while still sufficiently dampening the

acoustic mode. The electrostatic drivers will already be installed on the test masses

by the time the upgrades are complete for the advanced detectors, however whether

they will be used to suppress parametric instabilities is also dependent on whether

or not they can provide the sufficient damping gain at a low noise level.

Chapter 5.3 presented the results from a potential solution for advanced gravita-

tional wave detectors which involved injecting a third optical mode into the cavity

that could optically interfere with the excited ω1 mode, and suppress it. Similarly

to the acoustic mode solutions, the optical suppression experiment needs further

investigation as well. The results presented in this thesis demonstrates the five

fold suppression of an excited mode in a Fabry-Perot cavity. However, advanced

interferometric detectors require an optical suppression of at least 100 fold, if not

greater. Thus, more work also needs to be completed before this is deemed a real-

istic solution for gravitational wave detectors.

6.2.1 Future Work at AIGO

The gravity wave group at UWA plans further work on optically suppressing the

excited cavity mode, with plans for a less intrusive solution for advanced detectors.

The next steps for the experiment at AIGO involves transferring the previous work

from the south arm to the east arm cavity.

For the past years, the vibration isolation system in the east arm Fabry-Perot cavity

at AIGO has been under construction. In the beginning of 2009, the researchers and

students involved in the east arm’s vibration isolation project managed to control

and lock the Fabry-Perot cavity using new optical suspensions for the first time.

104 CHAPTER 6. CONCLUSIONS

In this newly regenerated cavity, the gravity group wishes to install a 100-W fi-

bre laser, provided and developed by the Optics & Photonics group at Adelaide

University [54, 88]. In principle, a 100-W laser could potentially produce natural

opto-acoustic parametric interactions, and thus providing UWA with the ability to

measure naturally occurring PI for the first time.

Additional upgrades to the east arm cavity for the future PI experiment include a

new thermal compensation system and new optics. The new thermal compensation

system will be one based on the advanced GW detector design, where a CO2 laser

is used to provide compensation for thermal lensing problems in the FP cavity (it

will replace the thermal compensation plate that we described earlier in Chapter

3). [91]. Furthermore, new optics made from fused silica, the same material that is

also being used in advanced detectors design are planned to be installed.

After all the new equipment is installed into the easy arm cavity, there is a higher

probability that the natural amplification of parametric instability will be measured

for the first time. Once parametric instability is naturally created in the cavity,

then it will be possible to demonstrate a more realistic suppression of parametric

instability (around 100 fold suppression) and a potential realizable solution for

advanced interferometric gravitational wave detectors.

Initially, the gravitational wave group at the University of Western Australia will

try to reproduce the experimental results from the optical feedback suppression

experiment detailed in Chapter 5.3 in the east arm cavity, and then eventually

upgrade this experiment to one more suitable for the design and layout of advanced

gravitational wave detectors.

One such experimental layout has already been suggested by Zhao et al. in 2010

[89]. In their paper, they suggest another design for an optical feedback solution

6.2. FUTURE PROSPECTS 105

which focuses primarily on using the ETMs transmitted beam to feedback into the

system. [89, 90]. If Zhao et al’s proposed solution works in practice, then it would

provide the optimal solution for advanced gravitational wave detectors.

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