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Instrumentation of the Next Generation Gravitational Wave Detector

Permanent Link: http://ufdc.ufl.edu/UFE0021385/00001

Material Information

Title: Instrumentation of the Next Generation Gravitational Wave Detector Triple Pendulum Suspension and Electro-Optic Modulator
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Wu, Wan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: eo, gw, interferometer, laser, modulator, pendulum, piezo, residual, suspension, thermal
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The Laser Interferometer Gravitational Wave Observatory (LIGO) is now operating at its design sensitivity. To extend the ability to sense gravitational wave signals in a broader range with reasonable rate of detections, the Advanced LIGO has been planned and R & D for it is underway. Advanced LIGO is designed to have an order of magnitude better sensitivity then the current detector. This will be achieved by improving several technical features, including the use of triple pendulum suspension systems and an order of magnitude higher laser power. Going from LIGO to Advanced LIGO requires new designs for some key components or subsystems. As part of the LIGO group at the University of Florida, I was involved in the research work related to two components of Advanced LIGO. These include the triple pendulum suspension system which has been designed and installed in the JIF lab at Glasgow University. My research provides a complete mechanical analysis for the triple pendulum and describe the local control system with feedback forces applied on the top mass via six electro-magnetic actuators. Also presented is the characterization measurement of the transfer function of this suspension system. Another research project I have worked on focuses on the development and test of a novel electro-optic phase modulator (EOM), which is a key component of the input optics subsystem. The experiments include measurements of the physical properties of the EOM crystals (e.g. piezo-resonance and optical absorption at 1064 nm wavelength), characterization of the residual amplitude modulation and investigation of the additional laser amplitude and phase noise imposed by the EOM on a 1064nm laser beam.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Wan Wu.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Tanner, David B.
Local: Co-adviser: Reitze, David H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021385:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021385/00001

Material Information

Title: Instrumentation of the Next Generation Gravitational Wave Detector Triple Pendulum Suspension and Electro-Optic Modulator
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Wu, Wan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: eo, gw, interferometer, laser, modulator, pendulum, piezo, residual, suspension, thermal
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The Laser Interferometer Gravitational Wave Observatory (LIGO) is now operating at its design sensitivity. To extend the ability to sense gravitational wave signals in a broader range with reasonable rate of detections, the Advanced LIGO has been planned and R & D for it is underway. Advanced LIGO is designed to have an order of magnitude better sensitivity then the current detector. This will be achieved by improving several technical features, including the use of triple pendulum suspension systems and an order of magnitude higher laser power. Going from LIGO to Advanced LIGO requires new designs for some key components or subsystems. As part of the LIGO group at the University of Florida, I was involved in the research work related to two components of Advanced LIGO. These include the triple pendulum suspension system which has been designed and installed in the JIF lab at Glasgow University. My research provides a complete mechanical analysis for the triple pendulum and describe the local control system with feedback forces applied on the top mass via six electro-magnetic actuators. Also presented is the characterization measurement of the transfer function of this suspension system. Another research project I have worked on focuses on the development and test of a novel electro-optic phase modulator (EOM), which is a key component of the input optics subsystem. The experiments include measurements of the physical properties of the EOM crystals (e.g. piezo-resonance and optical absorption at 1064 nm wavelength), characterization of the residual amplitude modulation and investigation of the additional laser amplitude and phase noise imposed by the EOM on a 1064nm laser beam.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Wan Wu.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Tanner, David B.
Local: Co-adviser: Reitze, David H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021385:00001


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f81371d459f5a871887404480be94995
72f9a34dfc0d91f5c89240e6670d4f98725d4971







INSTRUMENTATION OF THE NEXT GENERATION GRAVITATIONAL WAVE
DETECTOR: TRIPLE PENDULUM SUSPENSION AND ELECTRO-OPTIC
MODUJLATOR


















By
WAN WU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF THE PHILOSOPHY

UNIVERSITY OF FLORIDA

2007


































02007 Wan Wu




































To people who are persistent with their dreams









ACKENOWLED GMENTS

Thanks for all the help I have received during my graduate studies. My advisor Dr.

Tanner and co-advisor Dr. Reitze offered me the opportunity to join the LIGO group at

UF and supported me with full assistantship until the end of my graduate study. Without

their encouragement and help, this thesis would never have been completed. I am also

indebted to Dr. Mueller, who often acted as a replacement advisor and kept on helping me

with detailed instructions. Without his guidance, I would have been lost in the darkness

a lot of times. I have to appreciate the tutoring I received front Rupal Antin when I came

to the LIGO lah for the first time. He had great patience in showing me the fundamental

lah skills even if my response is ak- .1-< dumb. His hard-working style scared me away front

being lazy. I also learned a lot front Malik Rakhnmanov. He is the best physics tutor I

have ever met. The physics that he taught me covered a wide scope front the fundamental

niechanics to the advanced optics. His detail-oriented teaching style made the learning

process really enjoi- 11.'~. Alany thanks to Volker Quetschke for his help on electronics.

He is a computer genius and an experienced control engineer. Thanks also goes to Klate

Dooley, ?- i.1. a Margankunte, Rodrigo Delgadillo, and Hsin-jung Lin for their kind help of

correcting the graninatical errors in my thesis. Rod is also my lunch time partner along

with Alike Hartman and Daniel Arenas. They just kept on cheering me up when I was

frustrated by the tedious thesis writing work.









TABLE OF CONTENTS


pagfe


ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES

ABSTRACT

CHAPTER

1 INTRODUCTION


1.1 Gravitational Waves
1.1.1 Introduction.
1.1.2 Gravitational Wave Sources
1.1.3 Detection of Gravitational Waves
1.2 Gravitational Wave Detection Usingf Michelson


Interferometric


Detectors


1.3 Signals For The Ground Based Interferometric Detectors
1.3.1 Chirp Signals
1.3.2 Periodic Signals.
1.3.3 Burst Signals
1.3.4 Stochastic Signals.
1.4 Noise In The Ground based Interferometric Detectors.
1.5 LIGO & Advanced LIGO .
1.5.1 Optical Configfuration.
1.5.2 Readout Scheme.
1.6 Overview Of The Thesis

2 TRIPLE PENDULUM SUSPENSION SYSTEM .

2.1 Seismic Isolation Suspension Systems.
2.2 Introduction to the Triple Pendulum Suspension.
2.3 Mechanical Analysis of the Triple Pendulum.
2.3.1 Variables and Parameters.
2.3.2 Vertical Response
2.3.3 Longitudinal and Pitch Dynamics
2.3.4 Yaw Motion
2.3.5 Sider-i--li- and Roll Motion
2.4 Local Control of the Triple Pendulum.
2.5 Measurement Result

3 ELECTRO-OPTIC MODULATOR (EOM)

3.1 Input Optics S- I -i--rh il (IO) of Advanced LIGO
3.2 Application of EOMs.










3.3 RTP Crystal EOMs.
3.3.1Physcal properties of RTP Crystals.
3.3.2 EOM Configfuration.
3.4 Technical Features
3.4.1 Piezo-resonances.
3.4.1.1 Theory .
3.4.1.2 Measurement
3.4.2 Thermal Lensing
3.4.2.1 Thermal effects in crystals.
3.4.2.2 Experiment
3.4.3 Residual Amplitude Modulation (RAM).
3.4.3.1 Generation mechanism.
3.4.3.2 RAM in Advanced LIGO .
3.4.3.3 C'!I. II .:terization
3.4.4 Laser Amplitude and Phase Noise Produced
3.4.4.1 Generation mechanism.
3.4.4.2 C'!I. II .:terization


EOMs


4 CONCLUSION.


4.1 Triple Pendulum Suspension Model.
4.2 EOM.


APPENDIX


Close Loop Transfer Function (clp.m) ..
Parameters (parameter.m) .....
Model (model.m) .........
Transformation Matrix (matrixT1T2.m)
Local Control Servo (LocalCtr.m) ....


133
134
136i
141
142

143


B ELECTRONICS .....

REFERENCES ......... .

BIOGRAPHICAL SKETCH ..........


. .. 144


A MATLAB MODEL FOR THE TRIPLE PENDULUM









LIST OF TABLES
Table pagfe

2-1 Input and output points for different transfer function measurements. .. .. 88

3-1 Physical properties of RTP crystals. . ..... .. 99










LIST OF FIGURES

Figure page

1-1 Effect of the incident GWs with two polarizations on test particles arranged
in a circle. The GWs propagate in a direction perpendicular to the page. ocw
refers to the phase of the GWs. .. ... .. 17

1-2 A Michelson interferometer with two orthogonal arms lying on the plane. The
propagation direction of GWs in two polarization direction is vertical to the
plane. BS beam splitter, M mirror, PD photo detector. .. .. .. 20

1-3 Optical configuration of the LIGO detector. BS beam splitter, PBS- polarizing
beam splitter, EOM electro-optic modulator, IT:\i input test mass, ET:\i -
end test mass, PRM power recycling mirror, AOM acoustic optical modulator. 32

1-4 Optical configuration of the Advanced LIGO detector. PSL laser pre-stabilization
subsystem, IO input optics subsystem, SRM signal recycling mirror. .. .. 35

2-1 A simple spring stack and a pendulum. . ..... 39

2-2 Transfer function of a simple spring stack. ...... .. 39

2-3 Three dimensional view of the triple pendulum suspension system. .. .. .. 41

2-4 Schematic view of the triple pendulum. The stacks and the damping arm have
been omitted for clarity. This a modified drawing which originates from [26]. 42

2-5 'T-shape' top mass with six magnets attached. .... ... .. 44

2-6 Photograph of the top mass. ......... .. .. 45

2-7 Parameters of a triple pendulum. ........ ... .. 47

2-8 Photograph of a triple pendulum. ........ ... .. 48

2-9 Bode plot of Hzv. .. ... .. .. 50

2-10 Side view of a single pendulum's longitudinal and pitch motion. .. .. .. .. 51

2-11 Longitudinal displacement introduced by the pitch motion. .. .. .. .. 52

2-12 Components of restoring forces which act to tilt the mass. .. .. .. .. 54

2-13 Side view of the top mass suspended with two wires. ... .. .. 56

2-14 Bode plot of Hz. ......... .. .. 60

2-15 Bode plot of Hp. .. ... .. .. 61

2-16 Bode plot of Hol. ......... . .. .. 61

2-17 Bode plot of Hep. .. ... . .. 62










2-18 Yaw motion of a single pendulum. The upper part is the view from above. And
the low part is the geometric plot of the effect on one wire when the mass is
rotated through an angle 4. ......... .. .. 63

2-19 Projection of the tension onto A-y plane which produces the restoring torque.
The upper plot and the lower plot are associated with two different effects on
wires when the mass rotates. ......... .. .. 64

2-20 Bode plot of H4,. ... . .. 67

2-21 Face on view of the -;I .1. li-- and the roll motion of a single pendulum. .. .. 68

2-22 Expanded to show various lengths. . .. .. 69

2-23 The components of the tension in two wires act to roll the mass. .. .. .. .. 71

2-24 Relative motion between two .Il11 Il:ent masses. .... .. .. 73

2-25 Bode plot of H,,. ... . .. 83

2-26 Bode plot of Hr. ... . .. 83

2-27 Bode plot of H,,. ... . .. 84

2-28 Bode plot of Hr. ... . .. 84

2-29 Feedback control block diagram of the triple pendulum. .. .. . .. 85

2-30 Close loop transfer function measurement for the triple suspension system. .. 87

2-31 Close loop transfer function with the input signal being injected to coil 1 and
coil 2 and the output signal from channel 1 ... .... . 89

2-32 Close loop transfer function with the input signal being injected to coils 3, 4
and 5 and the output signal from channel 3. .... .. .. 90

2-33 Close loop transfer function with the input signal being injected to coil 6 and
the output signal from channel 6. . ... ... .. 91

3-1 Overall IO schematic. .. ... . .. 93

3-2 Configuration of an EOM. ......... .. .. 95

3-3 Possible locations of photodetectors in Advanced LIGO. .. .. .. 96

3-4 The optical configuration of using a Mach-Zehnder interferometer to eliminate
the sidebands on sidebands problem. . ...... .. .. 97

3-5 RTP crystals (wedged and non-wedged) mounted between two electrodes. .. 100

3-6 Circuit diagram for the EOM. ......... ... .. 101










:3-7 Experimental arrangement for the piezo resonance measurement of an RTP < li -r ,1 10:3

:3-8 RTP crystal piezo-resonances. ......... ... .. 104

:3-9 Experimental arrangement for thermal lensing measurement. .. .. .. .. 106

:3-10 The measured laser beam divergence with and without the RTP beam path. ...... ...... ........... 107

:3-11 C'!s lIly.-: of the wave front at the beam waist. ..... .. . 108

:3-12 Residual amplitude modulation due to the misaligfnment of an EOM. .. .. 109

:3-13 C'!s lIl,.- of the orientation of principal axes with respect to the light field when
a voltage is applied across the crystal. ...... .. . 111

:3-14 Fabry-Perot cavity effect due to the back reflection between the front and end
surfaces of the crystal. ......... . .. 115

:3-15 The existence of R AM changes phase relationship between the leaked carrier
light and the RF sidebands at the dark port of LIGO. ... .. .. .. 115

:3-16 Experimental setup to characterize the residual amplitude modulation created
by an EOM heated by a 105:3nm Nd:YLF laser. .... .. .. 117

:3-17 Correlation between the heating power and R AM. .. . .. 118

:3-18 Deformation of end surfaces of a < l i--r I1 when an acoustic wave passes by. ... 121

:3-19 Experimental setup to characterize the laser amplitude and noise imposed by
an EOM. .... ........ ........... 122

:3-20 Loop transfer function of PLL. ......... ... .. 124

:3-21 The fields of two phase-locked lasers. . ..... .. 125

:3-22 Block diagram of a I/Q measurement system inside a phase meter. .. .. .. 125

:3-2:3 Amplitude noise spectra of the laser heat notes and the modulation signal from
the EOM driver as measured by the phase meter. ... ... .. 126

:3-24 Linear spectral density of phase noise in laser heat notes and the modulation
signal from the EOM driver. ......... .. .. 128

:3-25 Linear spectral density of phase noise in laser heat notes and the modulation
signal from the EOM driver. ......... .. .. 129

B-1 Circuit schematic of the phase locking servo. ... .. .. 14:3









LIST OF ABBREVIATIONS

AOM Acoustic optic modulator

ASC Alignment sensing and control

BS Beam splitter

BH Black hole

EOM Electro-optic modulator

ETM End test mass

FI Faradayl~i Isolator

FSR Fr-ee spectral range

GW Gravitational Wave

IFO Interferometer

ITM Input test mass

IO Input optics

LIGO Laser interferometer gravitational wave observatory

LSC Length sensing and control

MMT Mode matching telescope

MSPRC Marginally stable power recycling cavity

NS Neutron Star

SPRC Stable power recycling cavity

PD Photo detector

PDH Pounder-Drever-Hall

PM Phase modulation

PRM Power recycling mirror

RAM Residual amplitude modulation

RF Radio frequency

RS E Re sonant -si deb and-ext ract ion

SAW Surface acoustic wave










SQL Standard quantum limit

SRM Signal recycling mirror

TT Transverse-traceless









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of the Philosophy

INSTRUMENTATION OF THE NEXT GENER ATION GR AVITATIONAL WAVE
DETECTOR: TRIPLE PENDITLIM SUSPENSION AND ELECTRO-OPTIC
MODITLATOR

By

Wan Wu

December 2007

Cl.! I!1-: David B. Tanner
Co('l! I!1-: David H. Reitze
Major: Physics

The Laser Interferometer Gravitational Wave Observatory (LIGO) is now operating

at its design sensitivity. To extend the ability to sense gravitational wave signals in a

broader range with reasonable rate of detections, the Advanced LIGO has been planned

and R&D for it is underway. Advanced LIGO is designed to have an order of magnitude

better sensitivity then the current detector. This will be achieved by improving several

technical features, including the use of triple pendulum suspension systems and an order of

magnitude higher laser power.

Going from LIGO to Advanced LIGO requires new designs for some key components

or subsystems. As part of the LIGO group at the University of Florida, I was involved

in the research work related to two components of Advanced LIGO. These include the

triple pendulum suspension system which has been designed and installed in the JIF

lah at Glasgow University. My research provides a complete mechanical analysis for the

triple pendulum and describe the local control system with feedback forces applied on

the top mass via six electro-nlagnetic actuators. Also presented is the characterization

measurement of the transfer function of this suspension system.

Another research project I have worked on focuses on the development and test of

a novel electro-optic phase modulator (EOM), which is a key component of the input

optics subsystem. The experiments include measurements of the physical properties of










the EOM crystals (e.g. piezo-resonance and optical absorption at 1064 nm wavelength),

characterization of the residual amplitude modulation and investigation of the additional

laser amplitude and phase noise imposed by the EOM on a 1064nm laser beam.









CHAPTER 1
INTRODUCTION

1.1 Gravitational waves

1.1.1 Introduction

Albert Einstein described how matter and energy change the geometry of space time

using his famous Einstein field equation (EFE).

1 8xrG


The curved geometry, being interpreted as the gravitational field of the matter source, is

characterized by the Ricci tensor R,,, the Ricci scalar R and the metric tensor g,, in the

left part of the equation. T,, is the stress-energy tensor, and the constant is given in terms

of c (the speed of light) and G (the gravitational constant). The EFE is used to determine

the metric tensor g,,.

The general definition of the space-time interval in General Relativity is:


ds2 __ p vXLd1 (1-2)


Because gravitational waves (GWs) are from astronomical objects very far away from us

and thus very weak, they can be seen as small disturbances to the flat metric (11cu!l:0wski

space), defined as



g = 0 + h, (1-3)

where
-1 0 0 0

0 100
0= = (1-4)
0 0 10

0 00 1

The small disturbance h,, can be simplified into a useful and simple form when the

transverse-traceless (TT) gauge is applied to solve for GWs in vacuum,











00 0 0



0 b y,, h,, 0

00 0 0

General relativity (GR) predicts that GWs travel with the speed of light c and can he

treated as plane waves, being seen to be non-dispersive in the theoretical treatment.

The quantity h,,, is the waveform which can he interpreted as the superposition of two

orthogonal polarization states for GWs,

0 0 0 0 0 0 0 0

0 h, 0 0 0 0 hx 0
h =+(1-6)
0 0 -h, 0 0 hx 0 0

0 0 0 0 0 0 0 0

where h+ = b.,:, = -h,, and hx = .,., = hy,, are the strain amplitude of GWs

with two distinct polarizations. A passing GW will impose a compressing or stretching

effect on 'free falling' test masses depending on the phase and polarization of the wave.

For example, the effect on test particles arranged in a circular ring by GWs of two

polarizations coming into the plane is shown in Figure 1-1. If L is the original diameter

of the ring and two GWs have equal strain amplitude b+ = b x = h, the change of the

diameter as indicated in Figure 1-1 can he given as


aL = hL. (1-7)


In other word, the strain amplitude of a GW can he characterized as

aL
b = (1-8)


Intuitively speaking, GWs are emitted when the mass distribution of a object or

system changes in an oscillatory way. This is similar to the way that a change in charge











OE OS-










Figure 1-1. Effect of the incident GWs with two polarizations on test particles arranged
in a circle. The GWs propagate in a direction perpendicular to the page. ocw
refers to the phase of the GWs.

distribution creates an electrical dipole moment and thus radiates electro-magnetic
waves. However, the lowest order radiation multiple that the mass distribution variation

can produce is quadrupole radiation. Gravitational monopoles exist and appear as
Newtonian gravity, but oscillatory changes in the monopole moment of mass distribution
are forbidden due to mass conservation. And changes in the gravitational dipole moment
are forbidden due to conservation of momentum. Specifically, GWs are emitted by physical

objects with a changing quadrupole moment. The strain of the quadrupole radiation h can
be evaluated as
G 1 d2
hy,, R & ",,. (1-9)

Here R is the distance to the source and Qj is the reduced quadrupole moment tensor of
the source mass, defined as

Q,,,3 = r xx, p,2 dV, (1-10)


where p(r) is the mass density, the r is the distance from the center of the mass, and the

integral is over the volume of the source. The amplitude of the strain is very small, which
makes the direct detection of GWs a very challenging task. Take a typical coalescing










binary neutron star system to be about 20 1\pc away and have a mass on the order of that

of the sun. The resulting amplitude strain on earth is of the order 10-21

1.1.2 Gravitational wave Sources

GW sources can he classified as four types according to the frequencies of the

radiation [1]. The first type of sources emit waves in the 'extremely low' frequency hand

(10-15 to 10-ls Hz), and it is believed that initial density perturbations associated with

inflation of the universe introduce gravitational-field perturbation in this frequency hand.

The second type of sources are 'very low' frequency ones (10-7 to 10-' Hz). The expected

sources include super-massive binary black hole (BH) systems. The third type sources, the

'low' frequency ones (10-6 to 1 Hz), include the inspiral of the stellar-mass binary objects,

short period stellar-mass binaries, the merger of super-massive binaries, the gravitational

collapse of super-massive stars, and a cosmic gravitational wave background. And sources

of the fourth type, the 'high' frequency sources (1 Hz to 10 K(Hz), consist of the merger of

stellar-mass binaries, supernovae, a stochastic background and non-axisymmetric spinning

neutron stars (NSs).

In general, heavier sources correspond to lower frequencies. And the same physical

objects may generate GW radiations at different frequencies as they evolve. For example,

the inspiral of a stellar-mass binary at early stage radiates in the 'low' frequency hand.

The radiation period decreases as two components of the binary approaches each other.

Hence the GW radiation frequency increases gradually. In the last few seconds of the

binary's inspiral, the frequency has already been in the 'high' frequency hand. The

frequency of the GW radiation keeps on increasing as the binary merges and rings down.

1.1.3 Detection of Gravitational waves

Although the existence of gravitational waves has been predicted since the early

twentieth century, there were no experimental efforts to prove it until in the 1960's

when Weber so__~---- II. to use resonant hars to sense the passing GWs and carried out

subsequent experimental work [2]. In 1974, Russell Hulse and Joseph Taylor provide









substantial evidence for the existence of GWs when they discovered a binary pulsar

(known as PSR 1913+16) with a decreasing orbit. The observed orbital decay was in

agreement with the prediction by the theory of GR that the change of the orbit is due to

the energy loss through the emission of gravitational waves [3, 4]. However, this indirect

detection of the GWs generated by a binary pulsar did not provide us the information

carried by the waves themselves. Hence the direct measurement of the GW strain h(t)

is needed. Direct detection of GWs will open up a way for examining the theoretical

prediction of h(t), testing the el-10 ph~li--;cal models, and characterizing the astronomical

systems that generate GWs.

Several techniques for measuring GWs directly, besides the resonant bar detectors,

have been proposed or implemented. These techniques include spacecraft Doppler-trackingf

[5], pulsar timing searches [6], ground-based Michelson laser interferometric detectors

[7-10], and the laser interferometer space antenna (LISA) [11]. The ground-based laser

interferometric detectors, as the most promising GW detectors that are currently in

operation, are going to be discussed in Section 1.2.

1.2 Gravitational Wave Detection Using Michelson Interferometric Detectors

A simple laser Michelson interferometer is shown in Figure 1-2. An ideal laser

interferometric gravitational wave detector is built up by mirrors which are free of forces.

The laser beam is split into two beams at the beam splitter (BS). The distance between

the BS and the end mirrors (j1), the arm length, can be altered by a passing GW.

The laser interferometer defines its own coordinates which are different from the TT

coordinates. In this coordinate system, the x and y axes are determined by the directions

of the two arms with the z axis pointing perpendicularly to them (see Figure 1-2).

It is the light that is used as the ruler to measure the change of the arm length, which

is the signature by the passing GWs on the interferometer. We know that the proper

























BS

',Dark port



Figure 1-2: A Michelson interferometer with two orthogonal arms lying on the plane. The
propagation direction of GWs in two polarization direction is vertical to the plane. BS -
beam splitter, M mirror, PD photo detector.


length between two space-time events linked by a light is zero, which can be expressed as


ds = J g',dxPdxj = 0,


(1-11)


where gl, is the metric tensor defined in the coordinates associated with the detector. For

light in one of the arms (e.g. the arm along the x-axis),


C2 t2 / (1 hzd2 = 0,


(1-12)


where him is the strain along the x-arm of the interferometer. The distance that light

travels from the BS to the end of the x-arm and back can be calculated as


v 1+ch'z: -c 1


(1-13)


2Le = L. dx dx


himz (t)I d' )









where L, is the length of the x-arm and -r is the round-trip travel time. If we can
approximate the wrave~form him as


him (t) = ho cos (w~t) (1-14)


where Lc/ is the frequency of a GW signal, Equation 1-13 can then be written as

2Le t ,sin (Lclt) sin (Lc/ (t a))


Equation 1-15 can be rewritten as

2L, sin (Lc/t) sin (Lc/ (t + a)) 2L,9 sins ( --L 1
= + ho + ho ~ cos g q
c 2LC/ c Lc 2
(1-16)

Therefore, the phase that light accumulates during a round trip in the x-arm is

2L, sin ( a1Ws) ~>Wt-1 Y
(t) = Loas = o-0 + Lo'ho cos 4t-w (1-17)
c w

A similar expression can be written for the light travels in the y-arm. This is given as

2L, sin ~ (O L4) 1 -
,w (t) = "o00r = Lo0 -- WohLo co 4t + ) (1-18)
c: w 2

The motion of the end mirrors in each arms introduced by GWs will be in a differential

mode. This introduces variations of the phase difference A~ = '. 4, between the two

beams when they are recombined at the BS, changing the interference pattern at the dark

port. This change can be precisely sensed by a photodetector (PD). Thus the information

of GWs is encoded in the readout signal from the PD.

When the round trip time is short compared to the GW period such that Lc/g < 1

and Lc
L, L, 1 o0107 7 CO ig I/ L j o (Z+Ly 1 .S~
a4 m 2 o+ ooa )cs0a2 (e (e+L) ocst
c 2 c c
(1-19)










In this case, the signal simply scales with the length of the arms. However, if the length of

the arms become too long such that

2Li xr
Lclgn = L/ '> -, (1-20)
c 2

where i = x, y, will decrease as L, increases while 4, will increase with L,, as can be

seen from Equation 1-17 and Equation 1-18. Hence the signal A4 will degrade. So there

is a limit on the arm length, which is

xrc
Li (1-21)


Once the length of the arms become longer than the limit, the GW strain starts to

oscillate as the light propagates, shifting the round trip phase back towards zero. This

deteriorates the response of the interferometer to GWs.

The coordinate defined by the interferometer and the TT coordinate can be related by

a rotational transformation,

x'= RTX'T, (1-22)

where XMy represents the coordinates associated with the TT gauge and represents the

coordinates associated with the detector. The rotational transformation induces the

transformation of the metric. The strain tensor in the detector frame h', can be found

from the TT tensor h by means of the induced transformation,


h' = R~hR. (1-23)


Euler angles are used to specify the orientation of the gravitational-wave frame with

respect to the detector frame,


RT = Rz(cp) Rw(0) Rz( ), (1-24)


where cp, 8, 4 are the angles defined as shown in Figure 1-2. The rotations around the x,

y, and z axes are thus given by











cos sin 0

Rz (0) = sin~ cos~ 0 (-5

0 0 1

cos 0 0 sin 8

R,(0) = 0 1 0 (-6

sin 8 0 cos 8


cos c sin c 0

R,( p) = sincp cosc 0 .(-7

0 0 1

These rotations are the transformations within :$-dimensional space. The strain tensor h'

can thus he represented by a :3x:3 matrix. Hence we can describe A4 as the sum of two

parts each originating from an independent polarization,


A=Fh b +Fxhx. (1-28)


The information of the strain amplitude b+ h x is thus encoded in A4.

F+ and Fx are called the 'beam pattern.' Their explicit expressions are


F+ = ~(1 + COS2 H) cos 2~ cos 2(p cos 8 sin 2~ sin 2(p, (1-29)


Fx = ~(1 + COS2 H) cos 2~ sin 2(p + cos 8 sin 2~ cos 2(p. (1-30)

cp is the polarization angle of the wave. 8 and keep on changing as the earth orbits

around the sun and rotates daily. This motion also generates a Doppler effect of the

gravitational wave signals. The beam patterns are nearly omnidirectional except for

certain orientations. Hence the gravitational wave interferometers are sensitive to a very

large area of the sky at all times.










1.3 Signals For The Ground Based Interferometric Detectors

Ground based laser interferometric gravitational wave detectors are expected to detect

GWs in the frequency range of a few tens of Hz to several K(Hz. These GW signals can be

summarized into four categories.

1.3.1 Chirp Signals

Chirp signals are emitted by coalescing binaries. A compact binary system can consist

of two compact objects (NS/NS, NS/BH, BH/BH) orbiting around their common center

of mass. Two binary components may be widely separated and move along an eccentric

orbit when the binary system is formed. A coalescing binary system can be characterized

by the masses radial separation, the inclination of the orbit relative to the plane of the

sky, the starting orbital phase, and eccentricity of the two orbiting bodies. Gravitational

radiation causes the system to lose energy so that the stars will draw closer to each other

and the orbit circularizes eventually. This effect of drawing closer is called an inspiral.

The orbiting period decreases as the orbit shrinks. The GW frequency, which is twice

the orbiting frequency, moves from the low frequency region to the high frequency region,

sweeping through the detection bandwidth of the ground based interferometric detectors

eventually. The amplitude of gravitational radiation also increases as the orbit shrinks, as

can be seen from Equation 1-9.

The waveforms of chirp signals from coalescing binaries have been known to a high

degree of precision. Hence the classical matched filtering technique can be applied on the

output data of the detectors to enable a reliable detection. However, only a few Coalescing

NS/NS systems have been observed while no NS/BH or BH/BH have yet been discovered,

which implies that the distribution density of coalescing binaries in the universe could be

small. Hence the detection rate of those signals using GW detectors is very uncertain.

1.3.2 Periodic Signals

The most important sources of continuous periodic gravitational waves are pulsars.

Pulsars are highly magnetized NSs which rapidly rotates in a non-axisymmetric way. A










pulsar emits a beam of electromagnetic radiation along its magnetic axis. If the magnetic

axis is offset from the rotational axis, the electromagnetic beam will sweep out circular

paths as the star rotates. The radiation can only be detected by the ground based radio

telescopes when the beam points towards the Earth. The rotation period of pulsars are

very stable. The regularity of the time interval between observed electromagnetic radiation

from some pulsars could be as precise as an atomic clock.

The characteristic amplitude of gravitational waves from pulsars can be given as

16;,2G f2
he= IE, (1-81)
c47

where I is the moment of inertia of the pulsar around the rotation axis, f is the GW

frequency, E, the equatorial ellipticity, is a measure of the .-i-mmetry of the pulsar, and

r is the distance between the observer and the source. For example, The characteristic

frequency of gravitational wave generated by the Crab pulsar is 59.6 Hz, an upper limit

of strain amplitude of the GW signal from the Crab pulsar is calculated to be h e 10-24

with a E 7 x 10-4 and r = 1.8 kpc. Weak periodic signals like this will be buried in the

noise of the output signals of the GW detectors. However, the signal to noise ratio can be

increased by integrating the output signal over a long observation time.

The detection of periodic GW signals will be realized by applying fast Fourier

transform (FFT) analysis on the detector output and construct the power spectrum. If

we could keep on searching the sky with long enough observation time T, the signals

with nearly fixed frequencies peaks will show up in the power spectrum since the signal

to noise ration grows as T. The statistically important peaks in the spectrum could be

associated with these periodic GW signals. However, such data analysis algorithm just

imposes a big computational burden on the signal processing procedure. This is because

that the integration required time in order to achieve a reasonable signal to noise ratio

will likely be so long that a huge number of data points need to be handled in the FFT










analysis. The required processing speed goes beyond the limits of even the most powerful

computers .

Pulsars can lose rotational energy by electromagnetic radiation, the emission of

particles, and gravitational wave dissipation. That is why the rotational frequency varies

over a time scale in the order of the pulsar age, which is called the 'spindown'. This leads

to the frequency drift of the GW signals. The other mechanism that causes the frequency

change of the observed GWs from a pulsar is the Doppler shift. Since the ground detectors

are carried by the spinning Earth orbiting the Sun, a monochromatic signal in the source

reference will be Doppler modulated by the motion of the detectors. The frequency

modulation due to the Earth's orbital and rotational motion are expected to be seen in

the pulsar signals. These frequency characteristics can be utilized to extract pulsar signals

from the data obtained using GW detectors. Moreover, the changing orientation of the

detectors with respect to the pulsar sources due to the motion of the Earth will change the

'beam pattern' mentioned in Section 1.2, imposing amplitude and phase modulation on

the GW signals. The phase and frequency modulation of the signals broaden the spectral

lines in the power spectrum and spread the power into the frequency bins around the

signal frequencies. This just attenuates the amplitude of the main signal components and

thus imposed an additional demand for increasing the integration time. A solution to this

problem is to build a new time coordinate so that frequency of the signals remains fixed.

Unfortunately, the construction of the new time coordinates means more data analysis

work. The situation becomes extremely worse in the case of searching for sources whose

position and frequencies are not well known since the all-sky and all-frequency search will

be needed.

Hence searching periodic signals is believed to be far more complicated than that for

chirp signals.










1.3.3 Burst Signals

GW bursts originate from mergers of stellar mass binaries, supernovae, the gamma-ray

burst engines and other unknown sources.

A supernova is an explosion of a star which creates an extremely luminous object.

This results in a burst of radiation in the electromagnetic domain which emits as much

energy as the Sun would emit over about 10 billion years. Although only a small fraction

of energy is converted into GWs, it could be big enough to make supernovae important

sources for ground based interferometric detectors. Supernovae is believed to be tl i- 1. I1

in two r- .--4. In the first case, a white dwarf star in a binary system may accumulate

sufficient material from a stellar companion. As the white dwarf approaches the

('I! I.I~I-l,.ekhar limit of roughly 1.4 times the mass of the Sun, it will collapse, tri .111w--i~'~?~

a stellar explosion. In the second case, the iron core of a massive star ceases to create

nuclear fusion energy to resist the gravity. The resulting sudden collapse of the star will

generate a NS or BH, releasing gravitational potential energy that heats and expels the

star's outer l os. r-s. The collapse must he non-axial to produce the change in the quadruple

momentum in order to generate GWs. The waveform of supernovae signals are not

clearly understood. People currently believed that the current sensitivity of ground based

interferometric detectors can only allow us to detect GWs from galactic supernovae and

the signals from the extra-galactic sources would be too weak.

Gamma-ray bursts (GR Bs) are intense flashes of gamma-rays which last from

milliseconds to many minutes, coming from random locations in the sky. GR Bs can he

classified based on their duration as two types: The 'short' GRBs have a life time less than

2 seconds while the 'longf' ones last longer than 2 seconds. The present consensus is that

GRB emission is associated with BH formation processes such as hypernovae, collapsars

and compact binary mergers. Since collapsars and compact binary mergers are both good

GW sources detectable by ground based detectors, we have good reasons to take GRBs as

important targets of the ground based detectors as well.










Since cosmic gamma-ray bursts were first discovered in the late 1960s by the US Vela

nuclear test detection satellites [12], large number of GRB events have been observed. The

detectable event rate could be as high as once per d .v. This is very useful in data analysis

using statistical methods.

The waveforms of gravitational waves from burst sources are poorly known. The

cross-correlation analysis using data from more than one GW detectors is needed to

extract signals from instrumental noise, assuming there are burst signals with enough

strength in the sensitive band of detectors. And benefiting from the non-GW observations

of GRBs, we can use externally tr~i ;r methods to reduce the detection threshold for

GRB signals.

1.3.4 Stochastic Signals

The stochastic background of GWs are expected to be either generated from the

very early universe or a result of the superposition of signals from many unresolved

..I-r ndlli--; 1 11 sources. If stochastic signals can be detected using GW observatories, we

might be able to extend our view of the early universe to a time as early as 10-22 SeCOnds

after the big bang. In comparison, the measurement of the cosmic microwave background

radiation can only provide us the information of the universe about 105 years after the big

bang.

If the stochastic GW background is isotropic, stationary and Gaussian, its spectrum

can then be well characterized by a quantity Row(f) which is defined as

1 dpoW
RGw ( f )= ,(132)
pc d In f

where pc is the critical energy density to close the universe, pow is the GW energy density.

Like what we did to acquire burst signals from the outputs of more than one

detectors, the cross-correlation 2.1, l1i--;-; is also demanded to search for stochastic signals.

The basic idea is to multiply together the outputs from two detectors and integrate them.

Although the waveforms of stochastic signals are not clear, their event rate might be much










higher than that for GW signals of other types. Hence the data analysis for a stochastic

background could possibly be done with much shorter integration time.

1.4 Noise In The Ground based Interferometric Detectors

The sensitivity of an interferonietric detector is limited by several noise sources.

Principal noise sources can he divided into two types : the displacement noise and

the sensing noise. Displacement noise could prevent us from building free-falling test

masses and sensing noise could limit our ability to measure the signature of GWs on the

interferometer, the differential arnt length. Principal displacement noise include seismic

noise, thermal noise, radiation pressure noise, residual gas noise and gravity gradient noise.

Sensing noise include shot noise and optical noise.

Ground based interferonieters will be inevitably affected hv seisnxically driven

vibrations. Suspension isolation systems reduce the seismic noise in the measurement

bandwidth. An ideal isolation system is expected to suppress the seismic noise while

longitudinal motions of mirrors in the GW bandwidth are free of damping. However, the

unavoidable seismic wall at low frequencies prevents current ground based interferonieters

from sensing GW signals from sources such as the mergers of superniassive black holes.

LISA will be a good complenientarity in this regard.

Thermal noise originates through two mechanisms. First, the thermal motion of

the suspension system causes the mirrors to move. Second, the thernially excited

vibration of the mirror defornis the mirror surfaces, i.e., changes the arnt length of the

interferometer. The thermal noise can he characterized through the indirect measurement

of the mechanical loss of the system. The loss can he reduced by improving several aspects

of the system such as providing better coatings for the test mass, utilizing fiber suspension

wires and intplenienting cryogenic cooling devices.

The temperature gradient in the chamber housing the interferometer could produce

differential pressure on the suspended masses via the residual gas molecules. Thus the

thernially introduced motion of the residual gas molecules leads to the displacement of test










masses. Moreover, the residual gas molecules in the beam path of the laser could introduce

optical path length fluctuations in the interferometer [13]. So the residual molecule

noise could also be viewed as a sensing noise. Since the ground based interferometric

GW detectors are all placed in vacuum chambers, the effect due to residual molecules is

considered to be trivial.

The ultimate limit on displacement noise is the gravity gradient noise noise due to

fluctuatingf Newtonian gravitational forces that induce motions in the test masses. An

important gravity gradient noise source is the fluctuating density of the earth near each

of the interferometers test masses. Atmospheric fluctuations could also change the gravity

gradient, but the effect is much weaker than that from earth motions. Gravity gradient

noise due to moving bodies or objects can be more serious than seismic gravity-gradient

noise if such bodies or objects are not kept at an adequate distance from the test

masses. The gravity gradient noise is negligible in the current existing ground based

interferometric detectors. However, advanced interferometers with higher sensitivities

could be limited by gravity gradient noise at frequencies below 10 Hz.

Light inside the interferometer exerts radiation pressure forces on the test masses.

Variations of the light intensity introduces fluctuations in the forces, causing vibrations of

the masses. This is known as the radiation pressure noise. The anti-correlated motion of

the masses can be calibrated in strain sensitivity h as [15]


he, ( ) = ,(1-33)


where Pi, is the optical power in the arm cavity, L is the length of the arm cavity, A is the

wavelength of the laser, and & is the Planck constant. he, increases with the optical power.

Shot noise, also known as the 'photon-counting error', is the fundamental limit to the

measurement of the optical power. It is the statistical fluctuation in the number of the

photons measured by a photo detector causes fluctuations in the amplitude of the signal.

The signal to shot noise ratio increases with the root of the power. The contribution of









shot noise in the output signal, calibrated in the strain sensitivity h, can be calculated as

[15]

hshor ( f) = .~ (1-34)

Radiation pressure noise and shot noise are also called Quantum noise [14]. The total

quantum noise is the quadrature sum of the shot and radiation pressure noise. Minimizing

the quantum noise with respect to the power yields a minimum noise level which is close

to the standard quantum limit (SQL) [16],


Ason (f )= 2(1-35)
(2x rf ) mL2

Here, m is the mass of each mirror, and L is the length of the interferometer arms.

It is noteworthy to mention that the radiation pressure noise we described above is

due to the uncorrelated power fluctuation in two arms that is generated by the quantum

uncertainty. Technical radiation pressure noise which originates from the fluctuation of the

laser power is not related to the SQL.

Amplitude and phase change of the laser field are common noise sources in a perfectly

balanced interferometer. However, when the two arms of the interferometer are not

perfectly identical in terms of having different losses and mirror reflectivities, then the

amplitude and phase changes of the field in each arm are no longer identical. This shows

up as differential displacement noise and can not be distinguished from a GW signal.
1.5 LIGO & Advanced LIGO

1.5.1 Optical Configuration

The laser interferometer gravitational wave observatory (LIGO) operates three

ground based Michelson interferometric detectors. The two interferometers in Hanford,

WA have 2 km and 4 km arms respectively and the one in Livingston, LA has arms of

4 km in length. These interferometers use Fabry-Perot cavities to increase the effective

length of the arms and a power recycling mirror to enhance the optical power circulating

inside the interferometer (see Figure 1-3). The optical configuration of LIGO is plotted





















E TM


made cleaner


TTITM


ILII


E TMI


S~e~7~


Figure 1-3.


Optical configuration of the LIGO detector. BS beam splitter, PBS-
polarizing beam splitter, EOM electro-optic modulator, ITM input test
mass, ETM end test mass, PRM power recycling mirror, AOM acoustic
optical modulator.


P LT









in Figure 1-3. A small fraction of the light from a Nd:YAG laser is split :.l i--i from the

main beam, transmits a polarizing beam splitter (PBS) and passes an acoustic-optic

modulator (AOM) and a A/4 plate. The light is reflected back by a mirror placed after

the A/4 plate, passes through the A/4 plate and the AOM again. The double pass of

light through the A/4 plate rotates the polarization of the light to be orthogonal to the

incoming light's polarization. Hence the light is reflected by the PBS, passes through an

EOM and goes to a reference cavity. A A/4 plate and a PBS are used again to send the

light reflected from the reference cavity to a photodetector. The signal from the photo

detector is demodulated to obtain the error signal that is used to adjust the frequency

of the laser. This is a typical application of using the Pound-Drever-Hall technique [17]

to stabilize the frequency of a laser. The AOM serves as a frequency offset between the

laser frequency and the resonant frequency defined by the reference cavity. This way, the

laser frequency can be tuned/controlled by changing the frequency offset, while the high

frequency stability is ensured. The EOM after the beam splitter (BS) is used to lock the

pre-mode cleaner. The main laser beam is spatially filtered by the pre-mode cleaner first.

The transmitted light is phase modulated by two EOMs placed in series to generate two

pairs of RF sidebands around the carrier field for the length sensing and control of the

mode cleaner and the interferometer. Both carrier and RF sidebands are spatially filtered

again by the mode cleaner before being sent to the interferometer.

The carrier field is kept resonant inside the power recycling cavity formed by the

power recycling mirror (PRM) and the input test masses (IT:\ is). In addition, the carrier

light must be held resonant inside the two Fabry-Perot arm cavities while RF sidebands

are reflected by the ITS1-

To meet the resonance conditions for the cavities and the dark fringe condition

for the Michelson interferometer, the lengths of this optical configuration should be

controlled. It is convenient to describe the two arm cavities by use of the common length,

L+ = L1 + L2, and the differential length, L_ = L1 L2. These two lengths, plus the










length of the power recycling caviityl II =, + ~(lI + 12) and the path length difference in

the interferometer, l_ = 11 12 define the four longitudinal degrees of freedom in a power

recycling interferometer such as LIGO. It is the arm length difference L_ that needs to be

sensed with the most stringent requirement since the variation of L is directly sensed as

the GW signals.

The whole interferometer is controlled via the length sensing and control (LSC)

,thi-h ..The LSC of the interferometer evolves from the Pound-Drever-Hall feed-back

control technique. The modulated laser field reflects at and transmits through the

interferometer. The amplitude and the phase relationship between various frequency

components of the field change as a function of positions of the mirrors that the

interferometer is comprised of. The change is sensed by photo detectors placed at various

locations in the interferometer. The signals from these photo detectors are demodulated at

specified frequencies to create the error signals used in the feed-back control loops.

LIGO has achieved its design goal with peak strain sensitivity of h(f) a 3 x

10-23/2/4 at 150 Hz and Arms a 10-21 in a 100 Hz band. Advanced LIGO is proposed

to have an order of magnitude better strain sensitivity which increases the observational

range by a factor of 10. This means an increment of the event rates by 3 orders of

magnitude since the volume of space that the detector can see grows as the cube of the

distance.

The research work towards an upgrade from LIGO to Advanced LIGO has been

carried out. Besides the development of better materials and suspension systems to reduce

the thermal and seismic noise, the application of a high power laser to increase the signal

to shot noise ratio and the modification of the optical configuration have been proposed.

Compared with the power-recycled scheme in LIGO, the resonant-sideband-extraction,

or RSE topology in Advanced LIGO (see Figure 1-4) improves the performance of the

laser interferometer by adding a signal extraction mirror. The new configuration allows

very high power to build up in its arms with relatively low power at BS. RSE, which was










ETM9~


madtwe cleanerv


RIM Ip lI L2
PS'L -~ BS

is T~M ETM

SRV~


Onljpat madeI ceaner~



Figure 1-4. Optical configuration of the Advanced LIGO detector. PSL laser
pre-stabilization subsystem, IO input optics -Ith-i--rh .. SRM signal
recycling mirror.


first proposed by Mizuno [18], is more practical in Advanced LIGO as compared with

the dual-rel i-l II topology (power recycling and signal recylcing) proposed by Meers

[19], which has the advantage of recycling the signal due to the differential motion of

the arm cavities. As an analog to the way Fabry-Perot cavities optimize the time of

interaction of light with the gravitational wave signal by enlarging the light storage

time, the signal recycling cavity increases the 'signal storage time' by keeping the carrier

resonant inside the cavity formed by the front mirrors of the two arm cavities and the

signal-recycling mirror. Although the signal-recycling configuration was demonstrated

to considerably improve the performance of laser interferometric GW detectors [20], the

storage time for the signal sidebands must be kept short enough to achieve a desired

detection bandwidth. The storage time limit prevents us from using high-finesse arm

cavities in the interferometer. Thus a high power laser incident on the IT?1- becomes

necessary in order to increase the shot noise limited signal to noise ratio. This results in










serious thermal lensing in the BS which is exceedingly difficult to deal with. The signal

extraction mirror is positioned to keep the light anti-resonant inside the signal extraction

cavity. It serves to decrease the storage time for the signal sidebands so that high-finessee

arm cavities can be used without deteriorating the detection bandwidth. The signal

extraction cavity can be detuned to achieve an optimized sensitivity at various frequencies.

This is useful in searching for gravitational wave signals from known narrow band sources.

The introduction of the signal extraction mirror in Advanced LIGO greatly enhances

the complexity of the length sensing and control system. The signal recycling mirror and

two ITS 0; form the signal recycling cavity whose length, i, = i,, + ~(li + 12) has to be

controlled as well as the length in the other five degrees of freedom.

Since the carrier light will not be transferred to the .I-i-inite! r lic port effectively, the

control of the length of the signal recycling cavity requires an additional sideband so that

beat signals between two pairs of sidebands can be used to monitor the variation of the

signal recycling cavity. Length sensing and control in dual-recycling laser interferometer

has been studied through a number of table-top and prototype experiments [21-24].

However, the final control scheme for Advanced LIGO is still to be determined.

It has been proposed to place an output mode cleaner at the .I-i-inite! ( 1;c port of

the interferometer in Advanced LIGO. The output mode cleaner serves to filter out the

light in high order modes which increases the shot noise on the photo detectors does not

contribute to the detection of GW signals.

1.5.2 Readout Scheme

There are two sensing schemes, DC and RF, to extract GW signals encoded in

the optical signal detected by the photo detector at the detection port. A GW signal

with certain frequency f can be seen as a signal sideband around the carrier light. The

amplitude and phase information of the GW signal can be obtained by demodulating beat

notes between the signal sideband and RF sidebands using RF oscillators. RF readout

scheme is currently used in LIGO. The problem associated with this heterodyne detection










scheme is the coupling of the phase noise of RF oscillators to the readout signal. The

coupling will be enhanced when RF sidebands at the detection port of the interferometer

become unbalanced, which is the case in an interferometric gravitational detector with

RSE configuration such as Advanced LIGO.

Advanced LIGO will use the DC sensing scheme, in which the carrier light emerging

from the detection port is used as the local oscillator. As compared with the RF readout

scheme, the DC readout scheme also has several practical advantages: the optical noise

couplings are smaller with a DC readout; the readout electronics will be much simpler

and no RF sidebands emerge from the detection port so that the photo detector will not

he saturated when the laser power is scaled to high levels [25]. The disadvantage of DC

sensing is its subjection to many low frequency noise sources.

1.6 Overview Of The Thesis

As part of the Input Optics group of LIGO at the University of Florida, I has been

involved in instrumentation work for Advanced LIGO. I have mainly worked on two

projects. The first project is about the characterization of a triple pendulum suspension

system. The details of the mechanical analysis of the triple pendulum, which was designed

by the group at Glasgow University, are given in C'!s Ilter 2. Also described in C'!s Ilter 2

are the local control scheme of the pendulum system and some results from the closed-loop

transfer function measurement that was done in the JIF lah at Glasgow University. The

second one is the project of developing a novel RTP < !i--r I1 electro-optic modulator to

meet the requirements of Advanced LIGO. The design and characterization of RTP
based EOl~s are presented in OsI Ilpter 3. The summary and the discussion of the future

work will be given in OsI Ilpter 4.









CHAPTER 2
TRIPLE PENDULUM SUSPENSION SYSTEM

2.1 Seismic Isolation Suspension Systems

The seismic isolation systems in gravitational-wave detectors isolate the test masses

from the seismic vibrations of the ground. This ensures that test masses are as much as

possible in 'free fall'.

The simplest idea of the suspension isolation is to put the test masses on the 'stacks'.

Figure 2-1 shows a simple stack consists of a supporting plate which is held with springs.

The relationship between the ground motion, xo, and the displacement of the mass, xl,

can be expressed as:

mxl = -k (xl xo) (2-1)

where m represents the mass and k is the spring constant. The Laplace transform gives

the transfer function between xl and xo:

xl w,-(2-2)
X0 s2 Lo'2


where s = jw, w is the angular frequency and ao = 2x fo = 2xrl is the resonance

frequency of the spring.

If a viscous damping force is introduced with a damping constant b, the transfer

function can be rewritten as:

( b

xob


The transfer function of the isolation system (see Figure 2-2) is similar to a low-pass filter

so that the seismic noise above the resonance frequency will be effectively filtered out.

The dissipation of the system can be defined by the quality factor Q, which is given by
Lcom
Q = The fall off of the transfer function is proportional to 1/ f2 from fo to Q fo and

becomes 1/ f at higher frequencies.
















I
I I ( Xi
I
I
k
I
I
\\\\\\\\\\\\\\\ \\\\ Xa :


I
I
I


Figure 2-1. A simple spring stack and a pendulum.


x1


102

101

10
10

~j10-2


10-3

10-4

1-s
100


101 102


frequency (Hz)

Figure 2-2. Transfer function of a simple springs stack.










The other way to improve the seismic isolation is to increase the slope of the transfer

function by implementing multiple suspension 11s-c v ;. The isolation will fall off as 1/ f2;z

where n is the number of 111-c v ;. In principle, the attenuation of the seismic noise increases

with the number of 11s-c v ;. However, the complexity of the system prevents us from

increasing the 1.,-c cms without limit. The cross coupling of the horizontal motion to the

vertical motion due to the imperfect symmetry of the mechanical system will become

worse as more 111-c r~s are implemented.

A spring stack can provide good isolations along the vertical direction. Another

example of the suspension isolation is a pendulum (also see Figure 2-1). Pendulum

suspension systems can he used to isolate the test mass from being affected by the

horizontal seismic noise. A simple model for the pendulum is made of a mass which is

suspended on a wire of length 1. Supposing the mass of the wire is negligible and the

damping constant is the same constant b that we mentioned above, the same transfer

function can he derived as that for a simple spring stack except that the resonant

frequency is nlow givenl by wo = 27rT inl the horizontal directions.

2.2 Introduction to the Triple Pendulum Suspension

The suspension system of Advanced LIGO is based on the triple pendulum designed

for GEO600 the German/UK( detector. As shown in Figure 2-3, this suspension system

consists of a two-1 ... r- isolation stack, two cantilever springs which are attached to the

top of the stack, and a triple pendulum. The upper mass of the pendulum is suspended

from the springs with two wires and the double pendulum consisting of the intermediate

mass and the lower mass are suspended from extra four cantilever springs with four

wires. The intermediate mass and the lower mass are also connected with four wires. The

intermediate mass and the lower mass are equal, in order to ensure a monolithic response

to disturbances. The test mass needs to be controlled in order to lock the interferometer.

A reaction pendulum is hence installed parallel to the pendulum which holds the test

mass. Integrated Optical position Sensor/ElectroMagnetic drivers (OSEMs) are mounted


































pendulum pen~dulum

















Figure 2-3. Three dimensional view of the triple pendulum suspension system.
























- upper mass
- canitilever
sp rmig
inate rmne date
mass


test mass


test mass


OSEMW


Figure 2-4. Schematic view of the triple pendulum. The stacks and the damping arm
have been omitted for clarity. This a modified drawing which originates from
[26].


canltilever
sp rinig


X















reactiloni
mass


upper mnass

1,cantilever
spring
inate rmne date!
mlass


photo diode

I magnet

LED










on the pendulum with the flag magnet on the test mass and the coil on the reaction mass.

The motion of the test mass leads to the change in the amount of light which shines

on the photo detector, which introduces the variation of electric current in the coil (see

Figure 2-4). This, in turn, adjusts the feedback force on the magnet.

Details of the triple pendulum suspension system such as the design of the two-1.s-cr~

isolation stack, the design of the cantilever spring, and the active isolation which includes

both the feed-back and the feed-forward control techniques have been given in the

theses of Torrie [26] and Husman [27]. They each built a mechanical model to give a

complete characterization of the mode frequencies and the dynamic response of the single

pendulum, using standard vector analysis and the Lagrangian analysis respectively. The

mechanical analysis is extended to the triple pendulum. However, the details about the

triple pendulum model for all degrees of freedom (DOF) are not completely outlined. In

the following sections, a complete an~ ll-k- of the triple pendula which are installed in the

JIF lah at Glasgow University is carried out. In addition, a 1\atlah model is constructed

with the active control on the top mass being included. This model considers some

differences between the pendulum in the JIF lah and the pendulum installed for GEO 600.

It will be a good supplement to the two triple pendulum models that are mentioned above.

2.3 Mechanical Analysis of the Triple Pendulum

2.3.1 Variables and Parameters

Rigid bodies can move with six DOF in three dimensional space. We can name these

DOF as:

(i) Longitudinal motion :r, along the :i-axis,

(ii) Sid. .--li- motion y, along .0-axis,

(iii) Vertical motion x, along the E-axis

(iv) Roll ~, about the :i-axis,

(v) Pitch 8, about the E-axis,

(vi) Yaw ~, about the 0-axis.





























Figure ~ -- 2-.'-hp' top mass with six magnets attached.


Hence the triple pendulum has 18 degrees of freedom and therefore 18 mode

frequencies. The differential equations of motion of the triple pendulum for all degrees

of freedom are derived here. The control forces on the top mass are applied via six

magnets, which are included in the equations. The global control on the intermediate mass

and the lower test mass has been planned, but is not considered here. The parameters as

specified in Figure 2-7 used are listed below:

(i) Separation of wires in the & direction

at the half separation of the wires on the intermediate mass,

82 the half separation of the low wires on the lower mass.

(ii) Separation of wires in the y direction

no the half separation of the upper wires at the suspension point,

nl the half separation of the upper wires at the suspension point,

n2 the half separation of the upper wires on the upper mass,

n3 the half separation of the intermediate wires on the upper mass,

n4 the half separation of the intermediate wires on the intermediate mass,

as the half separation of the lower wires on the intermediate mass.










































































Figure 2-6. Photograph of the top mass.
45


S"P~FI



i ~1
I rh
gr~ I~









(iii) Height of the suspension points

do height of upper wire break-off (above the center of mass of the upper mass),

dl height of intermediate wire break-off (below the center of mass of the upper mass),

d2 height of intermediate wire break-off (above the center of mass of the intermediate

mass) ,

d3 height of lower wire break-off (below the center of mass of the intermediate

mass) ,

d4 height of lower wire break-off (above the center of mass of the lower mass).

(iv) Separation of magnets on the upper mass

1, the half spacing of magnets (between 3 and 4) acting on pitch,

1, the half spacing of magnets (between 1 and 2) acting on yaw,

le the half spacing of magnets (between the line along 3,4 and 5) acting on roll.

(v) Height of magnets 1, 2 and 6 above the center of mass, i,.

(vi) Length of wires and their projection in the vertical directions

li, 12 and 13 represents the length of upper wires, intermediate wires and lower wires

respectively. Their projection on the vertical directions are let, It2 and lt3, Which are
calculated as

la -( i- o2 (2-4)

It2 83 \i 82 2, (25)

It3 85 \ 84 2. (2-6)

(vi) Momenta of inertia lIg and masses m i = 1, 2, 3 and j = x, y, z. 1, 2, 3 the upper

mass, the intermediate mass and the lower mass and x, y, z directions of the angular
motion.

(vii) Spring constants of the upper cantilever blades, intermediate blades and lower

wires, kyl, k2a, k~3































t










d2

da



S2


I
--- do
*-- ------
I
I

I
I
I
I
I
I
I
I



I
I


d$
~~~~~-~~-~


_JI


Figure 2-7. Parameters of a triple pendulum.


















CI-
d
J


.~t*r


~n~r~r;
i_


Figure 2-8. Photograph of a triple pendulum.


'1










2.3.2 Vertical Response

The vertical motion of the pendulum remains uncoupled from the other five degrees

of freedom motion. The force applied on the lower test mass is -4k3 /3 CO S 0, a/3 is the

extension in the wire from the static equilibrium point, R is the angle of the wires with

respect to the vertical direction, al3 COS R iS the displacement along the z direction. Thus

the equation for the vertical motion of the lower test mass is


m3 3 = -4k3 (3 2 x) (2-7)


where z3, 2a represents the vertical displacement of the lower test mass and the intermediate

mass. Similarly, the equation for the intermediate mass is


m2 2 = -4k2 2x 1i) + 4k3 3x 2a) (2-8)


Considering a control force F, applied vertically on the upper mass via OSEMs, the

equation for the upper mass is


mlxl = -2klx z +4k2 2x 1 Fv) (2-9)


To derive the transfer function, the above equations are rewritten in the Fourier domain,

utilizing the transformation z (t) = eiaz (w). These equations become


-mizz 2 = -2kizi + 4k2 2x 1i v,,

-m2 2 2 = -2k2(x 2 1) + 2k3 (3 2) (2-10)

-m3 3 o2 = -2k~3 3 2 -~)

The relationship between the control force and the vertical motion of the upper mass can

be described in the transfer function given below:

zl 1
Hzv ( ) =2. (2-11)
F, 16k,
miw2 2ki 4k2 _
16k,
m2 o2 4k2~ 4k3 _
m3w2 4k3













S-40
S-60
-80
S-100
-120
10- 100 101



200

S100

0- 0


10- 100 101
Frequency (Hz)

Figure 2-9. Bode plot of H,,.


Figure 2-9 is the Bode plot of the transfer function H,,.

2.3.3 Longitudinal and Pitch Dynamics

The longitudinal motion in the &-direction and the pitch motion are strongly coupled

due to the fact that the break-off position of the wires are either above or below the line

through the center of the mass. The tilt angle 8 and the longitudinal displacement x will

show up together as the variables in the linear differential equations that characterize the

dynamics of the pendulum.

First, the dynamic equations of a single pendulum suspended from two wires of

length I with a spring constant k are derived. Define s as the half separation between two

suspension points and d as the height of the wire breaking off points above the center of

mass. Figure 2-10 shows a single pendulum suspended by two wires. Consider the case

when the mass is tilted by an angle 8 from the horizontal line and displaced by x in the

&-direction. The wires are affected in opposite v-wsi~. Wire 1 is contracted and wire 2 is


































Figure 2-10. Side view of a single pendulum s longitudinal and pitch motion.


stretched as can he seen in Figure 2-11. In Figure 2-11, the unfixed end of wire 1 moves

from N to P'. 1\N is the displacement in the :i-direction.


|ON|I = |NVP| |OP| = :r s, (2-12)


|MO|1 = |RT| = |RO'| |TO'| = s cos 8 d sin 8, (2-13)

|MN|1V = |MO|0 + |ON|I = s cos 8 d sin 0 + r s. (2-14)

For a small angle 8 and a small displacement :r,


| MN|I = :r de. (2-15)


P'Af is the displacement in the @-direction.


|IR|= |OT|= e-s cos 8, (2-16)















la2 P1I5







M IE







Figure 2-11. Longitudinal displacement introduced by the pitch motion.

|P'M| = | P'R| |M~R| (2-17)

A small angle approximation gives,


|P'M| sO.(2-18)

So the length of wire 1 changes from the original length, 1, to

1' = 1-s ) l-s .(2 19)


Hence the change in the length of wire 1 is Al se. We can also give

x de
smn a~ (2-20)

cos a~ 1 (2-21)










Similar geometric analyses shows us that wire 2 is stretched by Al and

x de
sin (2-22)


cos pm 1. (2-23)

In an equilibrium state, the tension in each wire is


Ti = T2 nmg. (2-24)


Wire 1 is contracted such that in the first order approximation,


Ti = ,mg kAl. (2-25)


And the tension in wire 2 becomes


T2 = ,mg + kAl. (2-26)


Hence the force on the mass along the x-direction is


F = Ti sin a~ + T2 Sin p. (2-27)


Substituting Equation 2-25 and 2-26 into Equation 2-27 gives


x de
F = mg (2-28)

Finally, the equation for the longitudinal motion of a single pendulum suspended with two


x de
mx = -F M -mg (2-29)

The net torque which tilt the mass can be calculated as


Q = Ti cos a~ (s de) + Ti sin a~(d + se) T2 COS S (s d) 2 T~Sin P(d se). (2-30)


Substituting Equation 2-20, 2-21, 2-22, 2-23, 2-25 and 2-26 into 2-30 gives
































Figure 2-12. Components of restoringf forces which act to tilt the mass.


d d2
Qm mg-x 2kS2 + mgd + my 0 (2-31)

Hence the equation of motion for pitch is


mgd d2
IO~x 2k~S2 + my dr + 0 (2-32)
1 1

The equations for the longitudinal/pitch motion of a triple pendulum can be derived

based on the mechanical on~ ll-k- of the single pendulum that is described above, i.e., the

equations for each test mass can be given by simply modifying Equation 2-30 and 2-32.

Os and xi are used to define the tilt angle and the linear displacement of each mass of

the triple pendulum (i = 1, 2, 3 upper mass, intermediate mass and lower mass). The

variables x and 8 are replaced with the relative displacement of each mass with respect

to the suspension points and the tilt angles between two .Il11 Il:ent masses. Hence for the










lower mass, the equation for the longitudinal motion is


Z3 Y (3 2ra 4g3 C3 2)] ,
It3


(2-33)


The equation for the pitch motion is


m -id (X t X)- j32/ d4 3 3 2
I~~y 3 3 2) 4k3 3 2 / I 3+.
It3 lt3


(2-34)


It is noteworthy to mention here that the length of the pendulum is the vertical distance

between the suspension points of two stages. The intermediate mass will be stretched

by both the lower wires and the intermediate wires. The intermediate wires stretch the

intermediate mass the same way that the lower wires do the lower mass while the lower

wires affects the intermediate mass the opposite way according to Newton's third law.

Now the tension along intermediate wires have to balance the gravity force due to total

mass of the intermediate mass and the lower mass. The dynamic equations are

(m2 m3) m 3g
m2x 2(x 2 1i 2 82e dle,1 + (3 -- 2~ -d4H3 d3H2) ,
It2 lt3
(2-35)
(m2 m3) */J_-,,,,,,, i, d82 2 1 1l
1,2y 2x 2 1l) 4ks28 2 2 3*/_
It2 1\22t
m -id. m3g l B d4H3 d3H2
(3- 2 +4k38 \--v 2 1J.
It3 lt3
(2-36)

There are only two wires above the upper mass. The tension in the wire is


1
T =(mi + m2 + 3)g.
2


(2-37)


Both its vertical and horizontal components act to tilt the upper mass and the net torque

1S


d2 I (m m 3 d081) (m 1m 2 m3) */Jun1~.


(2-38)










II ,


Ir








Figure 2-13. Side view of the top mass suspended with two wires.


The differential equations for the upper mass have to include the external control force

and the control torque. The force along the x-direction FL and the torque that controls the

pitch motion, Q, are applied via the OSEMs. In this case the equations become



(ml + m2 R3 s 0d (m2 R3 ms 1d
II,81 =(xi doe1) (mi + m2 03 *0) [I -~ (2

-x ) + 4k2s /a2 a da1 02 dl3 )d 2
(2-39)
(ml + m2 R3 m 9 (m2 R3 ms
mlx = (xl doe,) + [(x2 1 2l (,2 1 1 1-F~

(2-40)

The Fourier transformation of Equation 2-33, 2-34, 2-35, 2-36, 2-39 and 2-40 gives:


Z2- 2t3 3 d82d4H3 = 0, (2-41)























































(2-47)


m -ide~ m -ide~ m
Z2 y 3 k8 623 4k38 *a
It3 lt3 2 3,] [ m ll
(s d ) 03 0,


m ./
lt3


(2-42)
(m2 m3),, 903g
1t 1 t


12 jl3L v; g
m3g
2 d4H3 = 0,
lt3


(m2 m3 9

d2 +

(m2 m3) 9 ~

- xl +


did2 1, 602 2 4k


d~:], [r.:t It3


[lnlJ2 It2 3>
(m2 m3)g dl 81


/] :,


m39
X3
t3


(2-43)
m2 03 */.7_ 0II */ .I m */J02 3
+ x2 X 3 + 42 +

S(m2 + 3~ a m3g
2a1d m M2 m3 */J_ 4k3sa




(2-44)
1m a+m~ x2 03 m 02 m3) (m m1 ms2 03
xxa d2
1i It2 1
2m 3 ms9
d282 Fl,


(2-45)


and


(mi m2 + 03Sd 0 m 02 m3 gdl 2m 3 ms dl 2-aI,-4ksa d


(ml + m2 03 1s 02 (3 +d a + ms1 Sd2 03@
It2m>~: ,+(~ s d
+4ks28 ( Ia3 ,,1 2 d2 p

(2-46)


These equations can be written in a matrix form:


Fz

0

0P

q,

0


Z3

a


02










Here ,

~11 mw1

It2lt
(m ~ (m 2 3 m2 R3

014 o 0 6 1,
Ila lt2
(m2 R3 ms
~15 6 2,

(m2 R3 ms


It2 t


23g

It2
(m2 3 9 R39



m -i


SIt3

033 2 Lo3i

035 = -3,

036 = d4,

(mi m2 03 ms) 0d 02 ( 3 96 1s
041=

(m2 03 m) 1d,
~42


044 = o2 ly-42 qjOS 1o M2 (3 900 ms d1-
Ile lt2 i


(m~ (m2) R3 9

~51=









(m2 m3) */.7_- m */J
052 =+
It2 lt3
m .~l


(2-67)


(2-68)


(2-69)


(2-70)

(2-71)


(2-72)


~54= j~S2 (m2 m3 9
It2

11ss = W2 2y 4k282] m28 0 /_- k Y 3d ,
It2 2 3

r56: =~S 4k3 3I~ 4 77 */J.,
I t3
m -ide~
062 =
lt3


~65 = 4k38~ r~4 3,
It3


and


F66 2~ 3y-43 -0*J _0 (2-75)
a It3

Thus the response of the triple pendulum to the external force applied in the longitudinal

direction and the torque that tilts the top mass can be calculated as

Zl 1 FI

x2 0

Z3 0
T ~(2-76)
81 Op

02 0



where T = 0-1. The response of the top mass to F, and Q, can be analytically calculated


X1

Her=

Hap =


(2-77)

(2-78)














a,-40 '
S-60

2 -80C :yi:
-100C ":
10- 100 101



200t :r: r :
100 -



100 10 10 11:: 1: 1:



Frequency (Hz)

Figure 2-14. Bode plot of HI.


Het T41, (2-79)

and

Hep T44. (2-80)

These transfer functions are plotted in Figure 2-14, Figure 2-15, Figure 2-16, and

Figure 2-17 respectively.

2.3.4 Yaw Motion

First, we consider the case of a mass suspended by four wires of length 1, which are

all at the same angle a with respect to the vertical direction as is shown in Figure 2-18.

The tension T in one wire can he projected along two directions. The vertical component

T, balances the gravity force while the horizontal component Tiz along the direction BD is

balanced hv the horizontal force introduced front another wire. However, when the mass

rotates through an angle ~, the net torque by the horizontal components of the tension



















10- 100 101


10- 100 101
Frequency (Hz)


Figure 2-15.


Bode plot of Hrp*


10- 100 101


10- 100 101
Frequency (Hz)


Figure 2-16. Bode plot of Hoo.











S40
a, 20

S-20
-40

10- 100 101



200

v100

a 0


10- 100 101
Frequency (Hz)

Figure 2-17. Bode plot of Hep.


force of four wires will not keep the mass in a balanced state any longer. In Figure 2-19

we can clearly see that it is the force F that acts to rotate the mass. F is the projection

of Th in the direction perpendicular to the line connecting the suspension point and the

center of mass,

F = Th cos y (2-81)

where
1 |BD|
Th = T, tan De = mg (2-82)
4 |CD|
and
|AB|2 + BD|2 AD|2
cos y = (2-83)
2 |AB| |BD|

|AB| and |AD| are determined by the parameters given by the dimensions of the mass:


|AB| = (S2 +a ) (2-84)


































-LIC
'-r
'-


i
L-
--r
il
L~r_


'i
LI
Cc


Figure 2-18. Yaw motion of a single pendulum. The upper part is the view from above.

And the low part is the geometric plot of the effect on one wire when the

mass is rotated through an angle 4.


I
I
I
Iff- I
I li




























B
F D is.O

Ag~ I.

If-







Figure 2-19. Projection of the tension onto .i-- plane which produces the restoring torque.
The upper plot and the lower plot are associated with two different effects on
wires when the mass rotates.









(2-85)


|BD| is calculated from |AB| and |AD|,


|BD| |A~l"l~nLB|2 |D|2- |B| AD co ZAD (2 -86)

Here ZBAD can be 900 + 0 or 900 8, depending on whether a wire stretches or contracts

when the mass rotates. In Figure 2-18, wires 1 and 3 stretch while wires 2 and 4 contract

when the mass rotates in a clockwise direction. The stretch and the contraction changes

|BD| such that


|B~Da| = (S' ,2) 2 () i2 + 2\" (n5 -l ni) sinlc Of ni -i a sd (2-87i)

Therefore, we can write Th and cos y as

(S2 2) 22 2 2) 2 82i 8 4 ) SEn (11 Ry2
cos ye = E ic n r3~J2SifilOj
~2~i2 82 2n Sin) si

82 8 nj) ~fi Sin 8Q 2~i 2 8 2f Sin HQ

( /2n2 ) s 82 ni Sin
82 8 COS2 Sin 8 iCS8 Sin 8,

(2-88)
1 ni nj a
Thf = m -mg (2-89)
4 212 ( lj2

Now the total torque from four wires is

-17~ ni nj + s@ /i n?
Q =2Th cOS7 8~2 @SS + 2Th_ cOS y_ 8~S2 ~ n

1 ni aj -- / af 1l~ mg (n+8
+2-mg Y=

(2-90)
Hence the equation of motion for a four wire suspension is

mg (n? + 82)
I~ = (2-91)


|AD| = ni nj.










Similar to what is outlined in the longitudinal and pith case, the differential equation can

be extended to describe the yaw motion of a triple pendulum in terms of the rotation

angle of each mass with respect to the suspension points of each stage. The tension in the

wires of each stage is proportional to the total weight of the masses below the wires. The

equations of the yaw motion for the lower mass and the intermediate mass are therefore


I -= ),(2-92)


and


I .= (Q-Ia 2a 1 ). ( 3)
It2 it3

The line determined by the two upper suspension points on the top mass passes through

the center of mass. So the horizontal components of tensions in two upper suspension wire

can be written in a relatively simpler form

1 nl no
Th = (mi +m2 m3) g (2-94)
2 Inl

This produces a torque on the top mass when the mass is rotated by a small angle ~,


Q = (mi + m2 03 ms 2 -. (2-95)


Therefore the differential equation which describes the yaw motion of the upper mass

becomes


Iz #1 = # + m)g m 3gn (02 dl 1 ry, (2-967)


where Q, is the external feedback torque used to stabilize the yaw motion of the top mass.

The Fourier transform of Equation 2-92, 2-93 and 2-96 are listed below,


L0'21 3 2) (2-97)





























10- 100 101
Frequency (Hz)

Figure 2-20. Bode plot of H4,.


2(m2 + 3s 9 (n + 2
WIt2

(ml + m2 03 ms S
L02 1z1 1
lay


lt3
(m2 + 3 9 (n 8
It2


(2-98)

(2-99)


From Equation 2.3.4, 2-98 and 2-99, the transfer function that relates the external torque

Q, to the induced rotation of the top mass I1 is derived as


Qw (mi +m2t m3> n a

t39 3
r,~yyItr3 S /
(m2 039 39 (n + ~
It2 lt3


(2 23
I Sa

Ss) W2 ft3 3z
39('T + S)
m39 2n 92 3 W134


The transfer function H4, is plotted in Figure 2-20.


_2 fJ
(2-100)



















I fl


I I"



IH. 1

~O





Figure 2-21. Face on view of the -;I1 .1.li- and the roll motion of a single pendulum.

2.3.5 Sideways and Roll Motion
The coupling between the side i- .1-< and the roll motion is similar to the coupling
between the longitudinal and the pitch motion. However, -;II1 1- li <; and roll coupling
is more complicated. The wires that are used to suspend the masses are angled in the
y-direction. The side i- .1-< motion of the mass will create a torque that can excite the roll
motion while the longitudinal displacement does not tilt the mass. The details will be
discussed below.












I


I DLI
\G H I~


C~ f


cl


P )IO


Figure 2-22. Expanded to show various lengths.

Consider a single pendulum suspended with two wires as is plotted in Figure 2-21.

The two wires have the same length I in the equilibrium state. And the tensions in both

wires are,
1 1
T, = T2 g(211
2"Y It01
After the mass is displaced by a small distance y in the y-direction, the center of mass

moves from O to P. This is followed by a mass rolling by a small angle cp. The suspension

points now move from G and J to E and F. Thus the new lengths of the two wires can be

derived. Fr-om Figure 2-21 and Figure 2-22,


|AE|2=|K |+1,


(2-102)


|AE| = (ny I + y' dy)2 2 2 1 + 2j (n y
1~ 1


d~p) + If 2njltc
nid
'P
(2-103)


Hence wire 1 is stretched by


.1, + njd- nd
1


nj -u n
al = y
1


(2-104)









And for wire 2,


|BF|2 = |LI|2 |+|lt FL||$, (2-105)


ny -~ ll nial d -nidi

(2-106)

Wire 2 is then contracted by Al. Now the tensions in wire 1 and wire 2 become


1 1
Ti = 2mgl + kAl,


(2-107)


(2-108)




cP, (2-109)


and


it (nj us) It
cos R1 = 1 13


(2-110)


cos622 It (nj us) ItV In it (1 .1, + d nid) 1
cos 023 3 .

The equation of motion for the displacement of the center of mass, y7, is


(2-112)


my = -Ti sin 62 T2 Sin 62.


(2-113)


Substituting for TI, T2 from Equation 2-107 and 2-108, for 01 and S22 from Equation

2-109, 2-110, 2-111, and 2-112 and using the first order approximation for small


1 1
T2 mgl kAl.

The angle of wire 1 with respect to the vertical direction a1 satisfies


si x= 1 1 13 13


nj it (1 .1, a d nid) ]
13 p.


The sloping angle of wire 2 satisfies


sin S22 3 ,(211


and

































Figure 2-23. The components of the tension in two wires act to roll the mass.


displacement y and small angle cp gives

my -2k(" 1i2 2

1 2 I t I e ) ( 2 1 1 4 )
2(nj ni) (njle + njd nid) mg (n- )(gl+ d-n)
2k +t d-i (n i nl .nd-nd1

With reference to Figure 2-23, we can see that the component of force which rolls the

mass for wire 1 is Tisina~ and for wire 2 is T2COSp Where


a~ = 900 (a + cp) + 01, (2-115)

and

S= a cp + 02. (2-116)

The angle a is defined by the parameters of the pendulum. Using Equation 2-109, 2-110,

2-111 and 2-112, sin a~ and cos P are calculated to be











1( [


(nyi us)2
13


(nj n) (njl ad
13


- nd)


njle + d (nj us) 1
smn a~ + -
1ln~ + d2 In 2
21 + anjn It (njlt + njd-


" 3


n,
S1


(n i)2
13


agle + d (nj n )
cosp /7
1 n + d2


it (n lt + nad

3j n

ny"~" n, 1


d


1- nd)



us-


(2-117)

3j 2 (n n3 it)


d
\;da 2


(nj n ) (njle + jd
13


ad)


(2-118)


Now the torque for the roll motion is


j~ i)


2 n +d2 1 Sin 0

agl (8; rbi) +d (n;


121

Sn iit


+ my d
-nj i


- T2 COS = 098 i 2


It 12

ad) +d(- ) .1 +a -nd


(2-119)


And the equation of roll motion around the center of mass is


a) (n lt + jd
121


(2-120)


ni le
td1~


S ny a
n" + d2 I

d le
n~ + d2 1


S 1 (nj us)2 i
Iz~( = I mg It 121 2 gj~~ + 2k

mguy[It1 nj (njle + njd nid)1 d~ol (nj n
n, ne agle + d (ny n )nl a a
S It + k1 1J


_ (n i2


12













































Figure 2-24. Relative motion between two .Il11 Il:ent masses.


The extension to a triple pendulum is outlined below. Although the lower mass and the

intermediate mass are both suspended with four wires, the way the wires introduce the

sider-i--li- and the roll motion is not different from the case where they are suspended

with two wires in the manner described above. However, since what we need to consider

here is the relative motion between two .Il11 Il:ent masses, the equations of motion become

complicated. In Figure 2-24, the length of the suspension wires can be calculated by

substituting y with ym y,, d~p with dlc m dwc,, and njcp with v .;. nicp in Equation









Hence Al becomes


S-nicp>[ n

(nj -- us) It
13 m


2-103 and 2-106.

n -


The angle of wire

sin R1
ny







1 +


I1
( m- do n) .


i


ni It (nj ni)
S(Ym Ur)- 7. i) -(dlc do ,) (2-121)
1 1

1 with respect to the vertical direction 01 now satisfies

nj ni + ym Ur> dlcp m docp a 8 ni
- ni ( .;. ni Pn) It (n ni) (dlc m down,) 1
1 1 1
1(Y 2n Um 2i Pn)2
ni I2 1 12
nj us (1 (ny 2 Y n )2 n -n)i
1 13 1m93 -in

1 mdcp dupn> ,


(2-122)

n ) (dlcp down,) 1


i ni~, 2 1 mc
ic, n t (nj ni)
3 1 mcp


(0 1.;. n)It (a

1i i
(Ym Yn 2
(12 _- J2 (c m
- Yn 3


(2-123)


The sin and cos value of the sloping angle of wire 2 now becorne

n, ni ym + yrs + dlc m do a
sin 622
nj ni ( ; ni n)> It (nj ni) (dlp dcp,)
1 (ym Urs)
1 1
yes y di m + do a nj ni It (8;
1Y 1n + U niPn
ny a12 1

(dc,- ~p) nj ne n -n)
(d -d n 1 13 1mMs3 *


I(nj n )2


nj ni
1

- i)
12

- ni n)>


(2-124)









It + .7 ni( n le
nys us( ;.-as Iti~il (ny ni) (dlc m down,)1
1-l. ~ (ym Urs) +
1 1 1

1 + It + 2 m ~7 2 niPn) 2 (dl mp
It (nj us) It 12 -l2I n i
-dua) 1 3 m ~7 3 nic, n 3 dl mp



(2-125)

The force on the lower mass in Figure 2-24 becomes


F = 1 mg-, (sin R1 + sin R2) kAl (sin R1 sin R2) ~ mg t (.i-") 2k

12 It It 1221



1t (ncp us)2

(2-126)

And the equation of motion for the displacement of the lower mass mm is


mmym, = Rm (ym yrs) + (Sounj Rmdl) mp (Some Rmd,) cp. (2-127)


Here ,
1 (ni u)2 _
Rm = -l T. It, 1 + 2kmt (2-128)

(nj us) nj ni it
Sm= -1.*/ 2 2km (2-129)
Im2 Im Im

where ii T, represents the total mass suspended on the wire. If there is no other mass

suspended below mm, ill. = mm.

The angles atl, P, 61, and ,8, as specified in Figure 2-24 can be given using simple

geometric analysis.

atl = 900 (ams + mp) + R1, (2-130)

,6 =em m 2, (2-131)

as, = 900 aral + np R1, (2-132)









,4, = ant + np R2.


(2-133)


Therefore, the torque which rolls the lower mass is

Q =n + d (Ti sin atl T2 cos 61) =nj + d2 ], / Sil tlr CO

kmAl(sin at+cos6) m( -9 Bm ( .;. nicp) + C,ws(dlc m




where


S, 1) -


(2-134)


(ny(n n 2 i) njltm dl nj
12 1 12 +2m1
em m m


ni) nj -- ni
1m


(2-135)
njlem n3 di (nj us) Itm njlem + di (nj us)
Bm ~ i = m1.* 2km (2-136)
Im 1m Im Imim

ny(n -n) (ny- u)2 lRi) ".itm dl (nj ni)
C~~~ws~~ = itf i .el 2 km
li llm im Imim
(2-137)


And the torque which rolls the upper mass is


ni pn) + G, (dlc m do n,) + if */ nj -- n


Q, = Dn (yms Yrs) + E, (11.;


where


1





ne (ny -


_j i)
li


nilt t du (nj i) nj ni
2km
Im Im


(2-139)


0)


us) Itm nil m + d, (nj us)
+2km (2-14
im im

_t 1i itmd j i


(2-141)

Equation 2-127, 2-134, 2-138 can be used as general equations to describe the roll and
sideni--ws- motion of all three masses of the pendulum. The fumetions R, S, V A, B, C, D,


1
A m = i T T e .I


id n,
(2-138)


(nyi s) 2



n, d, (nj -
km Im

n ) (nyi us2)2
+ -T T. .;.7










E and G with variables ii f. us, nj dr, d,, Im and Itm need to be specified for each test

mass. And i f. = C m with i = 1, 2, 3.

For the lower mass,


m3ij3 3 RS(3 Y2) + (S3n5 3 4) (3 (S3n4 R3dS 32, (2-142)


I73m 3 = -A3y2 + A3S3 (B3n4 3d3) (2 [385 C3d4 +3 855 8 _d4 3

(2-143)

where
,1 (us n4 2] n5 4 2244
R3 = -0 / +-43 214
It3 /S3t /3

(us n4) (5 n4) l3
S3 3 i 4k3 2' (2-145)
(n 43 m ,~ 43
1(n (u 42 5 4) 5 3 d4 (5 4) n5 4
A3 = m */1~1 -0*2- k
It3 1a3t 1 3 /3
(2-146)
nalt3 n5 d4 (n5 n4) lt3 n5 3 d4 (5 n4
B3 R113g 2 39 R39 2a 4k3<5 (2147)
13 t3 3~ 3 /3

us (us n4) (5 4 2 (5 4) n5 3 d4 (5 n4)
C3 = m39 m *0I [~ k
13 1 l3 lt3 3 /3
(2-148)

The equations for the intermediate mass are


m292 R 2Y 2 Y1) $_i (S. Rad2) 2p (S282 2ad1 1p 3 RS(3 Y2 -

(S -. R3 4) (3 + (S3n4 R3dS 3P 2 291 2 (R 3 Y 2 R3Y3- (2-149)

(S282 2al 1p 1 is, Rad2 + S3n4 R3d3) 2P (S3n5 R3d4) (3,































+4k2


72m 2y 2 =l A2 B2 (1 8 83 na2l 1 2~ (d2 2 d1 1l

2~, 3 n 3 83 n2 2 _!a 22 -ari D2 3 ) -iu~ E2 5 4 ) 2 4 3m

( 5 n4
It3

(dC~ vi 5 n4
It3~


(2-150)



(2-151)


(2-152)



(2-153)




(2-154)


Here


1 (n3 na2 23 gn(3 -2)
A2 02 (3 *J -3 02 3 93 2
It2 l2t 2
s3t 2 2a (3 2 n ~ 3 2
12 2~
n3 2 3H 2~ 3 2 n~
B2 2 (m 3 + 2~ 02 (m 3 + 2~ 03 (m 2 m
2~ lt2 2
s3t 2 2a (3 na2


lt2
4k2
2


1)3 (13 82) (3 1282 2
C2 ~ ~ M2 1,3 I 2 3 4k2
s3t 2 d2 (n3 na2 3n 2 a

12s n42 -
1~n (u 84 2 5 4 d3 3 5 n4) 85
D2 = 0 */1.- /- k
It3 1213 1 3


n4 3 n85 3S (5 4) l3 n4 3 3 dS5 n4)
E2 = m */~ m3g- +3 2s 4k3
is lt3 32 3 /3

n4 (n5 n4) n 5 n4 2 5 n 4 n4 n3t d3 (5
G2 = m39 m */.. +4k
l i 1 Ilt3 lt3 q, 3 /3


(2-155)


- n4
3.
(2-156)

(2-157)

- n4)


(2-158)


Ra ~ 1 (n3 2 na2lln 83 n22
It2 l 2t 2

(n3 2) (3 2) l2
S2 02 (3 + 23) + 4k2 2
12 2;

























































lt1 (n3 n82 2 t
(m2 + 3s gdl


And for the top mass, the equations for the -;1I i.- li <; and roll motion are


mliyl = R yl + (Sln R do) cp R2 y2 91 28 (~ 3 2 2~~ 2p 2 S

n2 2ai 1p 1 (1 2 Ra 1 2 92~ + (S ni R do + S2n 2 2ad1 1p-

(S283 2 2~~ 2p~


(2-159)


-do) cp Dr (y
n3 n2lt






with

R = (


no nilti + do (ni no)


(2-163)
nilti ni do (ni no)
B = (ml m2 m3Y 2 m m1 M2 R3 (m 1 m1M 3 9
ltl niltl + do (nl noti1
-2ki
11 11


ni (nl no) (n1 no)2
(ml ma 2 ms3 (m 2 m1 02 m3 Sdo
1 1 lt
-no) niltl + do (nl no)


(2-165)
82t 2 d1 (n3 n82
4k2
2~


(3 2)
m2 m3) gn2 2
12


(2-166)


= A 1 y +l Bin Pa + Cpido +1(a~ -dc, (m a + m2 0 9811 8


nIn


2 ~ ~ r ( 1 E(32 21 GI (2 2 l no1' 1 02 3 9


do I- 3l + d2G + (2g R3 9 83 1 2l1
Ilt




(nl no) (ni no) lti
S1 = (mi + m2 03 ms9 2 2ki 2, (2-162)


1 ly

1 (nl -- no)2 81 ~+ 3 n n n0)
(ml + m2 m3) S M 3 o + 2kl
Ile li1t1 ly


(2-164)

l 2ki











n8l2 n2 3 1 83 82 t2 na2t 2 1 (n3 na2
El = (m2 m3) (m 2 m32 R3 (m M2 m3 9 + 4k2
12" l2 2, 2 ~ 2
(2-167)
and


n2 (n3 n2) (3 2 n2
G1 = (m2 03 m9 2m ms2 03 dl
12 1,2 t
(n3 n2)


(2-168)
first Fourier-transformed


The Equation 2-142, 2-149, 2

and then rewritten in matrix form


-159, 2-143, 2-150, 2-160 are


01


A26

A36
0

A56

A66


(2-169)


Here ,


170)



172)



174)

175)

176)


Air = R1 R2 01 2,

A12 -2,

A14 = Sini R do + S282 2ad1,

Als = -S, .+ R2 2,


A22 2 3a+R m~022,

A23 -3,


82 1 3-8
+r 4k2~~1111~~~~111









A24 = -S282 2ad1,


A26 = ~n -S Rno+ ~ R3d4,





A33 = 3 + 3Lo'2,

A35 == -S R3d3,

-36 = 3 5 3 4,

A41 1 A 1,

A42 -1,




A45 = 3El + d2G1 + (m2 -03 ( 83


(2-177)

(2-178)

(2-179)

(2-180)

(2-181)

(2-182)

(2-183)

(2-184)

(2-185)

(2-186)

(2-187)

(2-188)

(2-189)

(2-190)

(2-191)


(2-192)


iiO la "


Asi== 42,

A52 = A2 2~,

A53 -2,


-54 2 nB 2 lC72,

853 -n4

It3 t2 o'2
A56 5 nE2 4G2,

A62 = -A3,

A63 = A3,

-65 3n4 3d3,


(2) ii39


-193)

-194)

-195)

-196)










A66 3ns5 C3 4 + 3g (' 5 4 3 4z02. (2-197)

Hence the sideri-wsi motion induced by F, and the roll motion induced by Q, can be

described as


Y2 0

Y3 I I0
= E (2-198)
(1 r2

2P 0

\P3 0
where E = A-1. The transfer functions that characterize the response of the top mass to

F, and Q, are

H~ = = Ext, (2-199)

Hr = 14, (2-200)

H,, = 41, (2-201)

Hr = = E44. (2-202)
Q,

The Bode plots for the above transfer functions are shown in Figure 2-25, Figure 2-26,

Figure 2-27, and Figure 2-28 respectively.

2.4 Local Control of the Triple Pendulum

Figure 2-29 shows the control loop of the triple pendulum system. Here, C represents

the control channels while S represents the sensing channels. The control signals, which

is the change of the current in coils, introduce forces on the magnets attached on the top

mass. Ti is the actuator matrix that describes the conversion from the current in the coils

to the forces and torques on the top mass,



















10- 100 101


10- 100 101
Frequency (Hz)


Figure 2-25.


Bode plot of Hys.


10- 100 101


10- 100 101
Frequency (Hz)


Figure 2-26. Bode plot of Hr.



















10- 100 101


10- 100 101
Frequency (Hz)


Figure 2-27.


Bode plot of H s.


10- 100 101


10- 100 101
Frequency (Hz)


Figure 2-28. Bode plot of Hr.

























Figure 2-29. Feedback control block diagram of the triple pendulum.


(2-203)


Q,


C6


Ti can be represented by a matrix,


G











1 1 00 0 0

1s i 1, -1, O 0

0 0 11 1 0
T,= (2-204)
ly -ly 0 0 0 0
0 0 00 0 1

0 0 le le -1, i,

The transfer functions from the forces and torques to the motions are included in H. The

resulting motions of the top mass in six DOF are sensed. T2 is the sensor matrix that

describes the conversion from these motions to the change of the positions of magnets,






= T2 ,(2-205)
S4



S6 '

where
11s,0ly,0 0

1 1, O 1, O 0

72 0 1, 1 0 0-1(206
0 1, 1 0 0 le

00100-1

000011,

The induced current variations are fed back as the error signals to adjust the current in

coils. G is the feedback controller.

Only the local control on the top mass has been considered in this model for the triple

pendulum system so far, but the model can be extended to include the global control on































Figure 2-30. Close loop transfer function measurement for the triple suspension system.

all three masses. The model can be used to figure out the mode frequencies of the triple

pendulum. It is also going to be a straightforward practice to add the damping force

terms in the model. Understanding the dynamics of the triple pendulum is essential to,

for example, study the cross coupling effect which is the most serious problem involving

a multistage isolation system, and model the effect of damping the pendulum mode

frequencies.
2.5 Measurement Result

To confirm the mechanical an~ ll--- described above and test the modeling for the

system including the local control part, measurements of transfer functions between

various control channels on the upper mass were carried out. The measurement for the

system in Figure 2-29 is shown in Figure 2-30. White noise input signals generated by an

SR785 model signal analyzer were injected into specified channels. By choosing different

input points, motions of the pendulum of different degrees of freedom were excited while


spectrum analyzer










the feed-back control forces brought the pendulum back to the balanced position. Signals

which represented motions of top mass were splitted out from the control loops and

measured by the signal analyzer. Part of the measurement results are described below.

Table 2-1 lists three sets of measurements with specified input and output points. And the

transfer functions are plotted and compared with the modeling results.

Figure 2-:31 shows the close loop transfer function with channel 1 and channel 2 as

the input points and channel 1 as the output point. This transfer function was used to

confirm the mode frequencies of the longitudinal motion. There is a reasonably good

match between the measurement result (the red curve) and the modeling result (the blue

curve). We can excited longitudinal mode at 6 Hz, 1.3 Hz and 2.3 Hz, which has been

shown in the pendulum transfer function plot (Figure 2-14). The extra dips at around 1

Hz and 2.1 Hz shown in the measurement result are believed to be associated with the

yaw motion that was excited due to the difference between the electrical gain for two coils.

These two mode frequencies are conformed in Figure 2-20. The small dips located between

7 Hz and 8 Hz in the measurement curve represent motions of the supporting stack which

are used to hold the suspension pendulum.

Vertical modes can he seen in the transfer functions plotted in Figure 2-:32.

Mismatches between the mode frequencies shown in the measurement result and the

calculated ones are around 0.4 Hz. This is possibly due to the inaccuracy of the spring

constants of the wires used in the model.

The transfer functions shown in Figure 2-:33 were obtained with the input and output

points connected in channel 6. Both the measurement and modeling results show sideway

Table 2-1. Input and output points for different transfer function measurements.

Motion Input point output point
longitudinal CH1 + CH2 CH1
vertical CH:$ + CH4 CH5
sideway CH6 CH6
























IV










I -30-


-40-- ---------- --


"10 "100 "101






.0 -30 ---- -
S-4 0 -- -- -

-50-


101 "100 "1 0
Frequency (Hz)


Figure 2-31. Close loop transfer function with the input signal being injected to coil 1 and
coil 2 and the output signal from channel 1.















89


























-0 -


(Ir-40 -

H -60 ----:---- .






S-20 C- -------- ----~- ----- -- ---- --- 1-


5 -40-



'10 '100 101
Freqluency (Hz)

Figure 2-:32. Close loop transfer function with the input signal being injected to coils :3, 4
and 5 and the output signal from channel :3.

































- ---- -







Frequency (Hz)


E -


---~- --~---~ ---~-- i -~--~- i


-20~


-40


0


~-20
a,
n
3
-r
~-40
I
-00


Figure 2-33.


Close loop transfer function with the input signal being injected to coil 6 and
the output signal from channel 6.


modes around 0.59 Hz and 1.3 Hz. There is a 2.1 Hz dip in the measurement curve while

the modeling result shows a 2.2 Hz dip.









CHAPTER :3
ELECTRO-OPTIC MODULATOR (EOM)

3.1 Input Optics Subsystem (IO) of Advanced LIGO

The IO works as the bridge between the pre-stabilize laser (PSL) and the main

interferometer (IFO). Figure :3.1 shows the modified schematic view of the IO system

designed for the Advanced LIGO (the original figure is taken from Ref. [28]). The light is

conditioned in the IO to meet the primary scientific requirements for Advanced LIGO.

The laser beam from the PSL is first modulated using radio frequency (RF)

electro-optic modulators (EOhis) to generate the frequency components (sidebands)

for controlling the interferometer. The working mechanism of an EOM will be discussed

in Section :3.2. The mode matching telescope (11 l T) after the EO hs modifies the mode

of the light to match the mode defined by the mode cleaner. The power control part is

based on the conventional design of using a polarizer and a half wave plate. A motorized

rotational stepper stage will be used to rotate the half wave plate. The optical power

incremental step needs to be small and the power changing rate needs to be slow to ensure

that the length-control system of the mode cleaner will be able to track the disturbance on

the mirrors due to the change of the radiation pressure.

The input mode cleaner stabilizes the frequency and suppresses spatial fluctuations

of the laser beam. The length of the mode cleaner cavity determines that the modulation

frequencies need to be integer multiples of the free spectral range (FSR) of the mode
cleaner. That is

S= N-, (:31)
2L'

where R is the modulation frequency, N is an integer, c is the speed of light and L is

the length of the mode cleaner cavity. In this way, both carrier light and sidebands will

resonate inside the mode cleaner and will be transmitted.

The Faraday isolator (FI) performs the conventional role of rejecting the back-reflected

light from propagating towards the input port. The FI designed for the Advanced LIGO























IFO Control
to ISC


PSL
adayr Intensity:
lator Stabilization


MC Length and
Alignment Sensing


ISC


MC ASC
Steering \ctuation Far
Mirrors *Io

MC
From M Vode Matching
PSL Telescope


:RF Power
.Modulation Control Mode
Cleaner









MC Length
Actuation

Figure 3-1. Overall IO schematic.


STo
.COC









uses a birefringence-compensated Faraday rotator (consisting of two TGG ( ni -l .II and a

quartz rotator) and a thermal lens compensation material so that it is suitable for high

power laser applications.

The MMT after the FI matches the beam to the proper mode to satisfy the resonant

condition of the main interferometer. Currently, there are two conceptual designs for the

MMT, according to which two possible designs of the power recycling cavity (PRC) will

be used: either a marginally stable power recycling cavity (j11sPRC) or a stable power

recycling cavity (SPRC) [29]. MSPRC will allow many spatial modes of RF sidebands to

be resonant inside the PRC and introduce losses to the TE,1~,,, mode of the sidebands.

SPRC can reduce the loss in the fundamental mode, but has the potential drawback of

increasing the complexity of the alignment sensing and control (ASC) system and might

contribute to the parametric instabilities [30].

The IO subsystem of Advanced LIGO requires higher quality optical components

in comparison with LIGO. The electro-optic phase modulator (EOM), one of the key

components in the IO, is of interest in this thesis.

3.2 Application of EOMs

When light traverses through an EOM, the phase of the light field which accumulated

along the optical path in the EOM crystal is proportional to the refractive index.

A variation of the refractive index is introduced by the applied electric field via the

electro-optic effect. Since the phase of a laser field is proportional to the light path, the

phase modulation (PM) of the light can be realized by applying a time-dependent voltage

signal across the crystal inside an EOM. In the frequency domain, a phase-modulated

laser field can be viewed as being split into a series of frequency components. The laser

field after the EOM can be described as a mixture of the carrier field, a pair of sidebands

separated from the central frequency component (the carrier) by the modulation frequency

R, and high order sidebands with angular frequencies NVx R away from the central










osomaltor~
lellectr~o de n






elec~trode


Figure 3-2. Configuration of an EOM.


frequency. The light field can be written as

GA = Eoei(wt+msin at) = Jo (m) Eoeibt + Jr (m) Eoei("+")t Jr (m) EoEoei("-")t + (3-2)


Here Eo is the field amplitude, to is the angular frequency of the original laser field, R is

the modulation frequency, and m is the modulation index. The higher order harmonics are

omitted in the equation.

All laser field components, including carrier and sidebands, are delivered to the

interferometer. The reflected, transmitted, and internal pickoff fields which are sensed by

interferometer control photodetectors, which are placed at various detection ports, can be

generally described by frequency dependent transfer functions,


Eout = Eo( Jo (mn) Toe""t + Jr (m) T e~iw+n)t- J1 (m) T_e"(w-n)t} -t (3-3)


To, T+, T_ represent the transfer functions for the carrier, the upper sideband and the

lower sideband to the specific photodetectors. These transfer functions are trivially

dependent on the location of the photodetectors. They are also functions of the relative

positions of the recycling mirrors, beam splitter and test masses. The motions of these

optical components affect the laser field at the photodetector as in different regions of the

interferometer different field components are in different resonance conditions. Figure 3-3

shows the possible locations of the photodetectors used for the length sensing and control










ETMi


L1




II L 2


10[


ITM ~


ETM


Figure 3-3. Possible locations of photodetectors in Advanced LIGO.

in Advanced LIGO. As has been briefly mentioned in (I Ilpter 1, demodulating a specified

beat signal between two frequency components of the laser field will yield an error signal

that reveals a change in certain longitudinal degrees of freedom of certain mirrors.

The proposed control schemes [31, 32] for Advanced LIGO both employ one pair of

sidebands in order to sense the length of the mode cleaner and two pairs of sidebands

to sense the longitudinal degrees of freedom of the interferometer. However, when two

phase modulators are used in series, the sidebands created by the first EOM will be

phase modulated by the following one. This generates 'sidebands on sidebands' which

will beat with the carrier field. The frequencies of the beat signals will be identical to the

frequencies of the beat signals between the sidebands. Beat notes between sidebands will

be used to generate the length sensing signals of Advanced LIGO. Hence the existence of

'sidebands on sidebands' will produce offsets in the real length sensing signals and degrade

the diagfonalization of the locking matrix.


Ougnalf na~tude

























Figure :3-4. The optical configuration of using a Mach-Zehnder interferometer to eliminate
the sidebands on sidebands problem.


There are two techniques proposed as the solution to the sidebands on sidebands

problem. One solution is to split the incoming beam using a Mach-Zehnder interferometer

and place an EOM in each arnt [:33]. In this Mach-Zehnder interferometer, the 'sidebands

on sidebands' do not exist anymore because the sidebands generated by one EOM are

not phase modulated by the other EOM. Another solution, termed complex modulation,

involves the use of one amplitude modulator and one phase modulator to apply specified

(non-sinusoidal) amplitude and phase modulation, resulting in an electro-optically

modulated field which is comprised of desired frequency components without mixing terms

[:34]. The target modulation state is reached by adjusting only the electrical fields which

drive both modulators. In general, the relationship between the original light field E and

the amplitude and the phase modulated light field E' can he described as


E' = Ee 'i~(), (:34)


where ,4 (t) is the amplitude modulation function, and 4 (t) is the phase modulation

function. Given E and a desired modulated field E', these two functions can he calculated:


4 (t) = n ,(:35)


photolp











S(t) = tan-lI(e E'E (3-6)
E'

Detailed discussion of the length sensing and control of the dual-recycling interferometer

is beyond the scope of this thesis. What we focus on are the technical issues related to the

application of EOhis. In general, the rules of thumb are:

i) The implementation of the EOhis should not affect the function of other units of

the interferometer;

ii) The EOhis should not produce noise that could deteriorate the sensitivity of
Advanced LIGO.

3.3 RTP Crystal EOMs

EOhis which are currently used in LIGO are commercially available EOhis made

of LiNhO3
absorption coefficient at 1064 nm, the wavelength of the laser used in LIGO. The proposed

application of 180 W laser power in Advanced LIGO will cause unacceptable thermal

lensingf problems if these EOhis are used. In order to meet the requirements of Advanced

LIGO, we have developed EOhis based on RhTiO2 04 (RTP) < !i--r .IIs instead of LiNbO:3.

3.3.1 Physical Properties of RTP Crystals

Our RTP
absorption coefficient of RTP crystals at 1064 nm wavelength is as low as 50 ppm/cm.

RTP
Table 3-1 lists some physical properties of RTP crystals.

RTP crystals have an orthorhombic < ti -r I1 structure and their EO-coefficient matrix

(linear electro-optic tensor) is:









Table 3-1. Physical properties of RTP crystals.


properties (units) x y z
dn/dT (x10-6/K() a 2.79 0.24
is(W/K/m) b
e (x10-s/K() c 1.2733
n 1064 nm 2.15 2.38 2.27
laser damage threshold (ilW/cm2, 10 ns e 1064nm )e 600
" The dn/dT value for RTP was obtained from [35].
b The thermal conductivity data was obtained from [36].
c The thermal expansion coefficient was obtained from [35].
SThe refractive indices data come from [37].
e The laser damage threshold data was provided by Raicol.



O 0 12.5

0 0 17.1

0 0 39.6
F = pm/V1 (3-7)
0 T42 0

r 0 0

00 0

3.3.2 EOM Configuration

The largest EO-coefficient of the RTP < !i--r I1 iS T33. Therefore, the optimal

configuration is to apply voltage along the 3 direction and let the z-polarized light

propagate along the y direction. If the incoming light's polarization direction is not

perfectly parallel to the 3 direction, the light beam is separated into two beams when it

transmits the front surface of the wedged < t i--r I1 (see Figure 3-5). The difference between

the refractive index of the p-polarized light and that of the s-ligfht results in two different

refraction angles according to Snell's law. Both the front and the end surfaces of the

crystal have the same wedge angle. In Figure 3-5, the light path of the p-polarized light



1 r42 and r51 are currently unknown.































Figure 3-5. RTP

parallels the electrodes so that the deflection angle of the p-polarized light, out~p), equalS

the incident angle, ;, The s-light and p-polarized light no longer overlap each other after

transmitting the wedged-crystal.

The modulation index of the modulator for the p-polarized light is

xrL 3 Vz
my = T7331it zj8

and that for the s-ligfht is
xrL 3 Vz
ms = XTI13R (3 9)

where L is the length of the crystal, d is the thickness in the z-direction, A is the

wavelength of the light, nz and n, are the indices of refraction of the < !i--r I1 along the

z-axis and the x-axis, and Vz is the applied voltage. Our non-wedged RTP
dimension of 4 mm x 4 mm x 15 mm, thus the half-wave voltage V, for the p-polarized

light is 612.5 V.










50 01
input loss IIL


(= Crystal






Figure 3-6. Circuit diagram for the EOM.


Because electro-optic ( i--0 II5 perform like a capacitor in electronic circuits, the

simplest way to build up the voltage across the crystal is to connect it in series with an

inductor to form an L-C resonant circuit. The impedance of the resonant circuit needs to

be matched to the output impedance of the EOM driver (usually 50 R) at the resonant

frequency to reduce the power loss due to standing waves. This is simply realized by

connecting an adjunct capacitor in parallel with the L and Cervsm~. The circuit diagram is

shown in Figure 3-6 2 The inductor L and the capacitor C can be tuned to find a balance

between the consideration for both the voltage built up and the power loss. As compared

with the widely adopted technique of using a resonant transformer to drive an EOM, this

design gained an advantage that the input impedance at low frequencies is very high.
3.4 Technical Features

Technical concerns associated with the application of EOMs in the laser interferometric

interferometer include piezo-resonances, thermal 1& 1,01 residual amplitude modulation,

and potential amplitude and phase noise imposed on the light.



2 Heinzel discussed this design in his thesis [38].










3.4.1 Piezo-resonances

3.4.1.1 Theory

As we mentioned earlier, the electro-optic effect induces the change of the refractive

index of the crystal inside an EOM. Besides this primary effect, the refractive index can

also be changed via a secondary effect due to piezoelectricity and photo-elasticity of the

crystal. An electric field will cause a strain inside the (ni- I1 via the converse piezoelectric

effect, and this in turn will create a change in refractive index through the photo-elastic

effect. Coupling between the electric field applied across the crystal and its mechanical

strain is maximized at the piezoelectric resonance frequencies. Piezo-resonances of

the crystal near the modulation frequency interfere with the modulation and alter the

amplitude and the phase of the sidebands. This becomes a practical limitation for the

application of EOMs especially in cases when the stability of the phase modulation is

demanded.

In general, the piezoelectric resonance frequencies are determined by the velocity of

the surface acoustic wave (SAW), v,,,, and the dimensions of the crystal Le, L, and Lz. It

can be calculated as
v,,
f = "(3-10)
NVLi2'

where i = x, y, z and NV = 1, 2, 3 A SAW is an acoustic wave traveling along the

surface of a material having some elasticity, with an amplitude that typically decays away

from the surface. The physical properties of the SAW and its crystal orientation determine

the SAW velocity. v,,, can be calculated as,


Vswa=. (3-11)


where p is the mass density of the material, c is the effective elastic constant which

depends on the elastic and piezoelectric tensors and their temperature coefficients. Further

discussion of the effective elastic constant can be found in Ref. [39] and examples for

calculating c are given in Ref. [40].













detecto


spect~rum analyzer
V (t
polarizer









polarizerr





Figure 3-7. Experimental arrangement for the piezo resonance measurement of an RTP
crystal.


3.4.1.2 Measurement

Figure 3-7 shows the experimental set up to measure the piezo-resonance frequencies

of the RTP crystal. We shined a 1064nm Nd:YAG laser through two cross polarizers and

align the laser beam on a photodetector (Thorlabs PDA255). The RTP crystal was placed

between the polarizers. The crystal with both ends anti-reflection coated was mounted

between two electrodes which were connected to the source of a spectrum analyzer. The

signal, in swept sine mode from the spectrum analyzer, was applied across the electrodes

to modulate the crystal.

The polarization direction of the incoming light, which was determined by the

first polarizer, was misaligned with the optical axis of the crystal. Applying an electric

field to the < s i--r I1 induced a change in the indices of refraction (hoth ordinary and

extraordinary) giving rise to an electric field dependent birefringence. The < si--r I1 acted as

a variable wave plate with the change of the phase retardation linearly dependent on the














40-





10-

0- "






Frequency (MHz)


Figure :3-8. RTP crystal piezo-resonances.


applied electric field. The second polarizer converted the phase change into an amplitude

modulated signal. The way that this kind of set up converts the phase retardation into

an amplitude modulation signal which was detected by the photodetector is going to

be described in Subsection :3.4.3. The amplitude modulation signal was maximized by

rotating the polarization direction of the polarizers. At the piezo-resonance frequencies

of the (o .1-- I the response of the light to the driving signal was greatly enhanced and

appears as resonant peaks in the spectrum measured via the spectrum analyzer.

Fr-om the transfer function plotted in Figure :3-8, the largest piezo resonance is at 680

kHz. The resonance with the highest frequencies are seen around 6 MHz. Although the

modulation scheme of Advanced LIGO has not been finalized, there is no proposal of using

a resonant modulator below 9 MHz. Consequently, piezo-resonances will not he a technical

problem with the application of the new RTP crystal modulator in Advanced LIGO.










3.4.2 Thermal Lensing

3.4.2.1 Thermal effects in crystals

Thermal lensing in the EOM could affect the spatial mode of the laser field. The

mechanism occurs under the following way. The EOM crystal's optical absorption of the

laser beam traveling through it will create a temperature gradient AT in the crystal. If

the laser beam propagates through the center of the crystal along the S direction and

the heat conducts from the center to the edge of the < s i--r I1 along radial direction r, the

temperature gradient can he found by solving the thermal diffusion equation

4 (r, ~)
v2T =r ) 0, (3-12)


where T (r, x) is the spatial temperature distribution inside the < si--r I1 ,4 (r, x) accounts

for the deposition of heat due to the optical absorption of the laser beam. a is the thermal

conductivity. The solution can he obtained analytically or numerically once the boundary

conditions for Equation 3-12 are given. The boundary conditions are determined by the

shape of the crystal and the way the heat is extracted from the crystal. In our EOhis, the

ground electrode below the crystal which directly contacts the aluminum packaging case

pl1 li-< the role of a heat sink. Without giving an explicit expression for the temperature

profile T(r, x), we know that the center of the crystal is at a higher temperature than the

edges due to the Gaussian distribution of the light intensity. The temperature gradient

AT(r, x) changes the refractive index of the < s i--r I1 along the axes perpendicular to the

propagation axis according to the size of dn/dT. So the laser beam traversing through

the crystal experiences a change in the optical path length due to the position dependent

refractive index. Thermo-elastic deformation and thermally dependent elasto-optic effects

will change the optical path length. In our case, these two effects are smaller than the

dn/dT effect. So we can roughly estimate the spatial dependence of the differential optical















EOMW


W
o Crystal







Figure :3-9. Experimental arrangement for thermal lensingf measurement.


path length as:

AOPL (r) = T(r )d.(:313)

This additional optical path length put an additional spatial phase on the fundamental

TEM1,,,, mode of the beam transmitting an EOM. This effect is known as 'thermal lensing'

because it alters the niodal properties of a beam in a way similar to an ordinary lens. The

thernially induced niodal distortion was characterized hv monitoring the change of the

beam divergence due to the 'thermal lens' created in the EOM crystal.

3.4.2.2 Experiment

Figure :3-9 shows the experimental arrangement used to characterize the thermal

lensing effect. We used a single mode CW laser to probe the 'thermal lens' formed inside a

RTP ( ni- I1 The output power of the laser was 100 W. First, the laser beam was focused

by a lens and the beam divergence was measured using a beam profiler. The radius of

the beam waist after the lens was about 0.4 nin. Then the RTP crystal was placed in the

beam path about 0.25 ni away front the waist where the beam radius was about 600 pm.

This location was within the Rayleigh range of the beam. The beam divergence after the
















E -1 -
E :0.65mm

.~ 0.8 -------e stl
location+
S0.6
600nmm
0.4 -

0.2 __ without RTP
with RTP

0 200 400 600 800 1000
2, distance (mm)






w=0.6mm





Figure :3-10. The measured laser beam divergence with and without the RTP crystal in
the beam path.


crystal was measured again. The change in the beam divergence is plotted in Figure :3-10.

The beam divergence can he fitted into the profiles with a Af2 value of about 1.75, which

is due to the laser modal distortion created hv the other optical components heated by the

high power laser. Using simple geometric optics, the focal length f of the thermal lens can

he derived from

=tan a~ tan /3. (:314)


Here, w is the beam radius at the crystal, and the angles a~ and /3 represent the far-field

divergence of the laser beam with and without the crystal in the beam path. Hence,

we calculated a lower limit on the 'thermal lens' of about 9 m. This 9 m 'thermal lens'

corresponded to a change in the laser beam's wavefront sagittal depth, 6.s, of about 10 nm














Ir




Figure 3-11. C'!I. lily of the wave front at the beam waist.


(see Figure 3-11). This was estimated from:


6e a (3-15)
4f

The thermal lensing effect is basically determined by the the optical absorption coefficient

a~, the thermal conductivity n, the temperature dependence of the refractive index dn/dT,

the length of the < s i--r I1 L and the laser power P. The sag change of the wavefront be can

be evaluated using the equation from [41]:

a~LP du
be a 0.105~- (3-16)
SdT

From the measurement result as described above, the optical absorption coefficient a~ of

the RTP
In Advanced LIGO, the laser power will be 180 W. So the thermal lensingf magnitude

in terms of 6e can be estimated to be 1.8 times the be introduced by a 100 W laser, which

is 18 nm. Supposing that all light can be coupled into the arm cavities when the input

laser's power is low, the change of the wave front when the laser power is increased to 180

W will cause mode mismatch. The fraction of light that will not be coupled to the arm

cavities due to the change of the wave front can be estimated to be [42]


6PP (~I)I(3-17)

So the maximum power that could be lost due to the thermal lensing in an RTP
EOM is 0.25' assuming that we don't compensate this in the mode-matching telescope.











imt+m IsHR Sfpdlimtrinctit+mzini+
cote +ingn~




Fiur 3-2 eiulapiuemdlto det h iaineto nEM







cuveinsie e <12 Rs -ida amlud Hencaio e the 'termallens'descrbed hre i an ida perica




lens. In principle, the change of the mode matching due to an additional spherical lens

in the beam path can be corrected by adjusting the mode-matching telescope. In reality,

the non-spherical part of the 'thermal lens' is unavoidable, which will create high order

modes that cannot be coupled to arm cavities. However, the power loss due to the thermal

lensing inside RTP crystal EOMs in Advanced LIGO will be much less than 0.25' and the

losses become negfligfible.

3.4.3 Residual Amplitude Modulation (RAM)

3.4.3.1 Generation mechanism

Misalignment of the crystal axis with respect to the polarization direction of the

incoming light causes residual amplitude modulation when an EOM is used to modulate

the phase of the light field. This is illustrated in Figure 3-12 which shows the incoming

light as a linearly polarized light with a wave vector normal to the front surface of the

crystal. The angle between the polarization direction of the incoming light and the e axis

of the < !i--r I1 is p. The light field can be seen as consisting of two orthogonal components

along the principal axes of the crystal. The optical path lengths for the two components

are different due to the birefringfence of the < s i--r I1 The resulting phase retardation causes









the light after the EOM to become elliptically polarized. The light is then 'filtered' by

subsequent polarization sensitive optical components, e.g., a polarizer. Should there be a

misalignment between the ( i-- I1 axis and the direction of the polarizer, the output light

field will become both phase- and amplitude-modulated. Given an incoming light field


fo = Eoeiwt, (3-18)

the output field can be written as


2 = Eo (cos p cos yeist+m," sin a2t + sin p sin ye"""+m,' sin ot+"' (3-19)

Here y is the angle between the crystal axis and the direction of the polarizer. R is

the modulation frequency. mi and m2 arT tWO modulation indices along two principal

crystallographic axes, which are plotted as the e-axis and the o-axis in Figure 3-12 and

lay in the plane vertical to the beam propagation direction. A4 is the phase retardation

between the light field along the e-axis and the light field along the o-axis. This is due to

the difference between the refractive indices, no and ne, along these two axes such that,


ag = (no n1e) WL. (3-20)


Equation 3-19 can be expanded as

A ~ (c~os 3 c~os _Jo (m1I) + sinl P sin ye"^*Jo (b2j) e'" Eo + (c~os P c~os yJ1 (mi) +t

sin 4 sin ye"^* J (mn2)) e"("'"!"Eo (cos # cos qJ (mi~) + sinl sin ye'^* JI (m2.~) e'"iwn)iEo
(3-21)
Thus the complex amplitude of the carrier field is


Ec = (cos P .os TyJo (mi)j + sin P sin yje""OJo (m2)) Eo. (3-22)

Since p and y can be minimized to be very small through alignment, Ec can be approximated

as (Jo (mi7) + ?jehmelo (m27~)) Eo. Hence the carrier field canl be seen as consisting of two
parts: the original field Jo (ml) Eoeibt and a residual component pyeinsdo (m2) Eoeiwt



















A ~F~ EC




Fiur -1.C~ is-ofte rena ion ofy prncpa axswt epett h ighil
whe avotae s ppledacos te !i-i.

SimilarlyEJ anR ieadas ossso rniplcmoetJ m)Ee~~~ n






caewihi hw nFigure 3-12. C'!I!~ f h rettonsie mo re geneal cases wht sentt the icmn light il

an~~~we elcria field bein applied across the
toiaccratly, cluate thsebn ams ounits of lgthat prnis amlitd c mpodulated byl Eo "th EOM

know tiong the misalignment angles. In Figure 3-13, is the light fied it a wavlife vcor





Before applying voltage across the crystal, the principal crystallographic axes are along the

X, Y and Z axes, respectively. And the orientation and relative magnitude of refractive
indices of the crystal can be described by the index ellipsoid equation,

x 2 22
a+ a+ = 1. (3-23)
n2 2 g2










n,, n, and nz are refractive indices along three axes and L is the distance that the

light travels inside the < ti--r I1 When an electrical field is built up across the < ti--r I1

the principal axes will rotate and the magnitude of the principal refractive indices will

change. Consider the specific example of RTP < !i--r I1- The index ellipsoid equation in the

presence of an electric field E can be written as


+ T1Ez x2 (d 2r3Ez)~ 92 733Ez 52 2r42Egyz + 2r51ExzX = 1, (3-24)


where E,, Ew and Ez are components of the applied electric field along the X, Y and Z

directions. This new ellipsoid reduces to the original one (defined by Equation 3-23) when

EF= 0.

A new set of principal axes (X', Y', Z' axes in Figure 3-13) can be found by a

principal-axis transformation of the quadratic form

+ T13Ez 0 T51Ez

S = 0 2 + 13Ez T51Ey (3-25)

T51Ez T51Ey T 33Ez

The S matrix can be transformed to a diagonal matrix S'. The components of S' are

eigenvalues of S. The magnitudes of the principal axes of the index ellipsoid (i.e., the

refractive indices along new principal axes) are


ni = -, (3-26)


where i = X', Y', Z'. They change as the electric field EF changes. In general, the

orientation of the new principal axes also change with E~ But in some cases, the new

orientation of the ellipsoid could coincide with the unperturbed axes. For example, when E

is applied along Z-axis of an RTP crystal, the quadratic form becomes










/1
2 1

S = 0


The orientation of the principal

are






n',=


\


1
2 2


(3-27)


0 0 2~ T33Ez

axes remains unchanged while the new refractive indices


1,
Sny -n T13Ez,
2



Sn, 28 T23Ez,


(3-28)




(3-29)




(3-30)


1
n z = n1


The orientation and magfnitudes of the new principal axes determine the polarization of

the light field E' after the EOM. E' can be written as


A' = (Exe"'PZ + E~Je"' j + E~e'ei4) is",


(3-31)


niL
where ft, #2 and 43, Which satisfy ~ = 2xr i = x, y, z, are phases accumulated by three

orthogonal field components E', E, and E' as the light travels through the crystal. This

elliptically polarized light field can be converted into an amplitude modulated, linearly

polarized light by a polarizer. In Figure 3-13, plane ABCD is the incident plane defined by

the polarizer. The field of the transmitted p-polarized light is given by


E' =' Ee" cos a~ + E' ei"" cos + E' ,,,3CO) iwt


(3-32)


where a~, p and y are the angles between X', Y', Z' axes and plane ABCD.

RAM can also be introduced via the Fabry-Perot cavity effect if an un-wedged crystal

is used inside an EOM. Back-reflections between the end surfaces of the ( ni- I1 will alter









the harmonic content of the modulated optical beam and form a measurable amplitude

modulation component onto the light field. In Figure 3-14, the relationship between the

incident field E and the transmitted field E' can be written as

tit26-i
E' = E--, (3-33)
1 rir26

where rl, 1 are the amplitude reflectivity and transmissivity of the front surface, T2, 2a arT

those of the end surface, 4 is the phase determined by the optical path length

2xrLn
= ~(3-34)


where A is the wavelength of the light field, L is the length of the ( i-- I1 and n is the

refractive index along the the 3 direction. When a sinusoidal voltage signal Vsinal is

applied across the crystal along the 3 direction, the refractive index becomes

1,V
n = no nr.?33 (3-35)
2 d

according to Equation 3-30, where no is the refractive index before applying the driving

voltage. Hence the phase 4 can be written as


# = 00 + m sin Rt, (3-36)

where
2xrLno
~o =(3-37)

xrL ,V
m = rof .(3 -38)

Hence Equation 3-33 becomes

1 26e-i(#o+msin 1t) 1 12 (ei(#~sn I""t"") T1 T2
E' = E = ,E. (3-39)
1 rir26-"i(#ommsin"t) 1 2rlr2 COS 0~ m Sin Rt) T

Equation 3-39 implies that the field E' becomes an amplitude-modulated field. The

magnitude of the amplitude modulation increases with the reflectivity rl and T2-















E E,

L


Figure 3-14. Fabry-Perot cavity effect due to the back reflection between the front and end
surfaces of the ( i i-- 1













Figure 3-15. The existence of RAM changes phase relationship between the leaked carrier
light and the RF sidebands at the dark port of LIGO.

3.4.3.2 RAM in Advanced LIGO

As mentioned in Section 1.5, changes in the amplitude and phase relationship between

the carrier field and the sidebands reflect the relative positions of the mirrors in an

interferometric GW detector. However, RAM also changes the amplitudes and phases of

the light fields. These changes are indistinguishable from the signature due to the motion

of the mirrors. RAM results in unwanted offsets in the length sensing and control signals.

This is intuitively understandable with a phasor-diagram (see Figure 3-15). The left part

of Figure 3-15 shows us the light fields at the dark port of LIGO for an ideal case that

RAM is completely eliminated. C represents the carrier field and S represents an RF

sideband. When the interferometer is locked, the phase difference between the leaked

carrier light and the RF sidebands at the dark port will be exactly 900. The deviation of










the interferometer from its operation point will cause the phase difference to be away from

900. The heat notes between the carrier and the RF sidebands will be demodulated to

generate error signals which are used to adjust the mirrors of the interferometer back to

operation positions via the length sensing and control system.

The right phasor-diagram describes how R AM creates a heat note between the

carrier light and the RF sidebands when the lengths of the interferometer satisfy the

resonant condition. Here C4 and SA are residual components added on the carrier and

an RF sideband respectively. The resulting carrier field C' and RF sideband S' are not

orthogonal to each other so that the heat note will lead to the generation of feed-back

control forces which act on the mirrors and push the interferometer away from the ideal

resonant condition. The amplitude of R AM must he kept below a certain level to ensure

that the static length-errors of the interferometer are within the design tolerances. Errors

signals due to R AM can also be balanced out by adding electrical voltage offsets in the

feed-back control loops.

Possible offsets generated in various control loops depends on the length sensing and

control scheme of the interferometric detector. In LIGO, the fractional RAM is required to

be less than 10-~ to ensure that the static error of the differential arm length is less than

the residual deviation requirement of 10-13 [4:3]. The length sensing and control scheme for

Advanced LIGO is not finalized, but will be much more complicated. However, because of

the intrinsic .I-i-iss!!~ i n. -I in the detuned Advanced LIGO interferometer and DC sensing,

the requirement of R AM will probably not he tightened.

3.4.3.3 Characterization

The EOM crystal will expand or contract when its temperature fluctuates. The

refractive indices n,, and n, are also temperature-dependent From Equations :320

and :319, we can see that R AM will vary when the temperature drifts. This could

affect the operation of an interferometric detector since the adjustment of laser power

will potentially lead to a temperature drift which in turn changes the R AM. So a











Specma-lln ma~ller


N:YLF


Oscillosope

PBS' Polariter AC C
L3 t DC
L4" FP2 O Phtoto detetor
TFTFP



FI AlZ Polarite L1 L2

Figure 3-16. Experimental setup to characterize the residual amplitude modulation
created by an EOM heated by a 1053nm Nd:YLF laser.


pre-installation test of the power dependent RAM imposed by an EOM is essential for

high power laser interferometers, such as Advanced LIGO.

Figure 3-17 is the optical layout of the experiment that is used to measure the RAM

generated by an EOM being heated by a laser at different power levels. The pump-probe

scheme experiment used a 500 mw non-planar ring oscillator (NPRO) Nd:YAG laser as

the probe laser to sense the amplitude modulation signal and a 50 W Nd:YLF laser as the

heating source. A Faradayl~i Isolator (FI) was used to protect the probe laser from being

affected by the back-reflected light or light from the pump laser. A half wave plate and a

polarizing beam splitter were used to control the laser power. The polarizing beam splitter

was aligned so that the transmitted light was s-polarized with respect to the optical table.

Its divergence was adjusted by a MMT consisting of lenses L1 and L2. The beam was then

reflected by a thin film polarizer TFP1. The brewster angle of the thin film polarizer is

about 560. The EOM was aligned as precisely as possible to match the crystal's principal

axis to the vertical direction. The beam diameter of the probe light at the EOM was

about 1 mm. The probe light is then reflected by another thin film polarizer TFP2. Part

of the light was then split off and sent towards an optical spectrum analyzer (OSA) which





('. YiJ ,I~ Y"~,


Residual Amplitude Modulation
I I


?r. rr
i h
n'lc' h~lli
I


x 10-5
8

6-


20 40 60 80 100 '120 140 160 180
time (minutes)
Heating Power


1_

r"xl--S' '


20
0


20 40 60 80 100 120
time (minutes)


140 160 180 200


Figure 3-17. Correlation between the heating power and RAM.


was used to measure the modulation index of the EOM. The rest transmitted another

cube polarizer which was aligned in the same way as the first one. The transmitted laser

beam was focused on a photodetector. The photodetector has separated DC and AC

output ports. The amplitude of the DC signal from the photodetector is proportional

to the total intensity of the light detected while the AC signal was used to monitor the

amplitude modulation of the laser field. The heating beam from the YLF laser propagated

in the opposite direction, with the beam divergence being adjusted by a MMT consisting

of lenses L3 and L4. Rotating the half wave plate in front of TFP2 changed the intensity

of the p-polarized light that was delivered to the EOM. The output power of the YLF

laser was fixed. So we knew the power of the heating beam that transmits the EOM by

monitoring the laser power reflected at TFP 2. The diameter of the heating beam was

controlled to be about 2 mm at the EOM to avoid clipping. The extinction ration of

the thin film polarizer is greater than 1000:1 (Tp:Ts) so that most of the heating beam









transmitted TFP1. A small amount of YFL light that was delivered to the probe laser was

rejected by the FI.

The amplitude of the AC signal was measured using a spectrum analyzer and the

DC signal was measured using an oscilloscope. These signals were continuously recorded.

We started with tracking the drift of the residual amplitude modulation for more than 20

minutes. Then we exposed the EOM to a 5 W heating laser and kept on measuring the

temporal variation of the RAM. The heating power was then increased to 20 W and finally

to 40 W in the following two steps and the time series measurements were repeated.

The modulation index is maintained to be about 0.2 during the measurement. From

the measurement result plotted in Figure 3-17, we clearly saw the correlation between the

heating power and the RAM created by the EOM. We also noticed that realigning the

EOM can bring the RAM back to the original low levels. RAM created by the EOM that

we used can be minimized to be below 10-5 of total light intensity. We believed this is due

to the Fabry-Perot cavity effect imposed by the unwedged crystal inside the EOM, since

the reflectivity of the AR-coated end surfaces of the ( ni- I1 was measured to be about

0."' .~ which is high enough to create a 10-s amplitude modulation.

3.4.4 Laser Amplitude and Phase Noise Produced by EOMs

3.4.4.1 Generation mechanism

We have seen that both the amplitude and phase of a laser will be modulated as it

traverses through EOMs. Instabilities of the modulation will create amplitude and phase

noise on the carrier field and sidebands. Amplitude noise imposed by EOMs, along with

the intrinsic amplitude noise of the laser will be coupled to the readout channel such that

the strain sensitivity of GW interferometric detectors can be deteriorated. Variation of

the phase between the RF sidebands and the carrier field will be converted into laser

frequency noise via the length sensing and control loops. Laser noise coupling in Advanced

LIGO has been studied by Somiya [44]. The final requirement on the laser noise level is










to be determined, but is expected to be very stringent. The amplitude and phase noise

imposed by EOhis on the laser should not exceed these requirements.

The laser noise associated with an EOM is generated through several mechanisms.

Noise sources are due to either the instability of the modulation signal, or unstable

physical properties of the EOM crystal. Two noise generation mechanisms are addressed

here.

First, variant modulations in the index causes amplitude noise of both the carrier

field and sidebands. The change of the amplitude of the carrier field as a function of the

modulation index can he written as

8 Jo (WI)
E, = .Io(mo + 6m) Eo Io (mo) Eo + o=,z, GmEo = *Io (mo) Eo *I (mo) GmEo.

(:340)

The relative amplitude noise of the carrier due to the variation of the modulation index is

6E,. .I (mo) 6m
RAI,N,= (:341)
E,. .o (m o)

The amplitude of an RF sideband changes with the modulation index

8.71 (mo) 6m
Es = .I (mo + 6m) Eo I (mo) Eo +m In |m=,,z Eo (*I (mo) + )Eo. (:342)
Bum 2

The relative amplitude noise of the sideband is


RAI,4N=. (:34:3)
2.1((mo)

Second, variations of R AM also contribute to the amplitude and the phase noise of

the light field. Equation :321 describes R AM as a function of the modulation index

ni, misalignment angles a~ and /3 and the phase retardation A4. Fluctuations of these

parameters at frequencies in the GW bandwidth will lead to additional laser noise that

could interfere with the signal.

We can see from Equations :38 and :39 that the stability of the modulation index

of an EOM depends on the stability of the driving signal, refractive indices, electro-optic



















Figure 3-18. Deformation of end surfaces of a crystal when an acoustic wave passes by.


coefficients and dimensions of the crystal. The amplitude and phase stability of the

driving voltage is determined by the quality of the oscillator used to drive the EOM.

Ultra-low noise oscillators are demanded. Refractive indices and electro-optic coefficients

are temperature dependent. Temperature fluctuations will also lead to the expansion or

contraction of the < s i--r I1 Moreover, the thermal-elasto effect could create strain inside

the crystal, which in turn changes the index ellipsoid via the elasto-optic effect. In sum,

it is oscillator noise and temperature variations that affect the modulation index. These

two noise sources also cause fluctuations of RAM. First, fluctuations of the modulation

index induced by these two noise sources will lead to fluctuations of RAM. Thermally

induced variations of the refractive indices and dimensional changes of the crystal result

in fluctuations of Af. Variations of the principal axes of the crystal cause variations of

the misaligfnment angles a~ and P. In sum, we believe two principal noise sources that

perturb the amplitude and the phase of the laser via an EOM are oscillator noise and

temperature variations.

Another potential noise source is acoustic noise. Acoustic waves could change the

dimensions of the < !i--r I1 via the elasto-acoustic coupling as they pass through the crystal,

perturb the optical path length which a light field experiences inside the EOM. This in

turn leads to the variation of the modulation index and RAM, generating residual laser

noise. A straightforward case is shown in Figure 3-18. The deformation depths of two end

surfaces of the crystal are expected to be different due to the energy dissipation of the











NPROl e


Phase meter


PLL controller g,

S~ynlchr~onization
Clock


Figure 3-19. Experimental setup to characterize the laser amplitude and noise imposed by
an EOM.


acoustic wave. Uncorrelated vibrations of two surfaces result in fluctuations of the optical

path length.

3.4.4.2 Characterization

Noise associated with amplitude and phase variations of an optical signal is most

easily studied in the frequency domain. Unfortunately, direct measurements of amplitude

or phase noise of a laser field at Advanced LIGO required levels is very difficult because

of limitations on the dynamic range and the intrinsic noise of optical spectrum analyzers.

We developed a heterodyne technique to measure the residual optical noise imposed by an

EOM when it was used to phase modulate a laser beam. The light field to be measured

was mixed with another light field to create an optical beat signal. By locking two light

fields with a constant frequency offset via a PLL, both the amplitude and the phase could

be precisely tracked using a novel phase meter [45]. Figure 3-19 shows the electro-optical

layout of the experiment. The two lasers were non-planar ring oscillator (NPRO) Nd:YAG

lasers with a 1064 nm wavelength. The laser field of NPRO 1 propagates through a

half-wave plate and a polarizer first, allowing the power to be controlled by rotaing the

half-wave plate. The laser field was then modulated via an EOM which was driven










hy a sine wave signal with a frequency R. The divergence of the transmitted light

was adjusted by a mode-matching telescope consisting of lenses L1 and L2. The laser

beam from NPRO 2 also propagates through a half-wave plate and a polarizer, with

the polarization direction adjusted to be parallel to the polarizer in front of NPRO 1.

Another mode-matching telescope consisting of lenses L:3 and L4 is used to control the

divergence of this laser beam. Two laser beams were superimposed at a power beam

splitter (BS). The mode matching between the two lasers was optimized by adjusting two

mode-matching telescopes.

The laser field from NPR O 2 heated with the carrier field and RF sidebands of NPR O

1 at the two photodetectors (PD 1 and PD 2). Demodulating the heat notes as measured

by PD 1 with frequency Do using a function generator and a double balanced mixer

created an error signal. The demodulated signal was filtered by the PLL controller and

used to phase-lock NPRO 2 with respect to NPRO 1. The constant phase offset between

two lasers was maintained by tuning the frequency of NPRO 2. The frequency/phase

conversion added a 6 dB/octave slope to the loop transfer function of the phase locking

loop (PLL), which was a part of the proportional-integral (PI) control designed for the

system. Figure :3-20 shows the loop transfer function of the PLL measured using a network

spectrum analyzer. We could see that the unity gain frequency of the PLL was around :30

K(Hz.

If the field of NPR O 1 is represented by El and the field of NPR O 2 is represented by

E2, the photocurrent of each photodetector could be written as


S) = ,+E2E E 2 (nz) Ef + E + 2Jo (nz) E1E2 o ('OS + a

c-c (:344)
+ 2.1 (nz) E1E2 ('OS [( 02) t + a ] -211 (nz) E1E2 o [(RO + 00t+
C-S

Figure :3-21 shows the frequency intervals between fields of two lasers.

The signal from PD 2 was analyzed by the phase meter developed by the LISA group

at the University of Florida. The phase meter is a digital signal processing device that
















40-

-20


10o 101 102 103 104 105




S180


i 160



10o 101 102 103 104 105
frequency (Hz)


Figure 3-20. Loop transfer function of PLL.


can be used for lock-in detection of RF signals. It had four synchronized input channels

and the analog-to-digital converter (ADC) of each channel digitizes the incoming signal

with a 100 MHz sampling rate. The digital filters inside the phase meter can decimate

the digital signal by a factor of 210. The phase meter provided the amplitude and the

phase information of the input signals via the in-phase/quadrature (I/Q) demodulation

technique. The schematic of the I/Q demodulation of the digitized signal is shown in

Figure 3-22. In order to overcome the dynamic range limitation of 900 or 1800 for phase

measurement, the local oscillator (LO), a numerically controlled oscillator (NCO), of the

I/Q demodulator was phase locked to the RF signal under test by tuning the frequency of

the LO signal around the frequency of the input signal.

The function generators used to drive the EOM and to provide the reference signal

for phase-locking, and the phase meter were synchronized to a synthesized clock generator










NPRO 1

Carrier


Sideband


L ~ -n o .I


Figure :3-21. The fields of two phase-locked lasers.


fequ~ency #1trer


siin 27Tfft

i ~ -- Q atan(UQ)-~


A destruction

'-----






-->p .0," apped


Figure :3-22. Block diagram of a I/Q measurement system inside a phase meter.


(Stanford research system, model CG6:35) in order to avoid the phase noise due to the

time jitter.

In our experiment, we used three channels of the phase meter to measure the

amplitude and the phase of the carrier-carrier (C-C) beat note of two lasers, of the

heat note between the carrier of NPRO 2 and one RF sideband of NPRO 1 (C-S), and of

the modulation signal front the EOM driver simultaneously. The angular frequencies of the

NCOs for :3 channels were tuned to be Ro, R S2o and R, respectively. These amplitude

and phase data were recorded in the form of time series. The linear spectral density (LSD)

of the amplitude or the phase could be calculated via the Fourier transformation.

We tried our first test on an unwedged crystal EOM (manufactured by New Focus).

The EOM was driven hv a 27 MHz sinusoidal signal front a function generator (Fluke

model 6062A). The reference signal in the PLL was a 10 MHz sinusoidal signal front

another function generator (Stanford research system model DS:345). The amplitude noise


Sideband











: C-C
C-S
1 0 ..- ) c-sic-c
laser intensity
V()
04







~106

100 101 102 103
freuqency (Hz)

Figure 3-23. Amplitude noise spectra of the laser beat notes and the modulation signal
from the EOM driver as measured by the phase meter,


and phase noise spectra of the 10 MHz C-C beat note and 17 MHz C-S beat note are

shown in Figure 3-23 and Figure 3-24, respectively.

As far as we know, the amplitude noise of beat notes come from several sources

besides the amplitude noise imposed by the EOM: intrinsic laser amplitude noise, variation

of the overlapping (beam jitter) between two laser beams, and electronic noise associated

with the phase meter (e.g. digitization noise). Noticing that the first two effects will

impose noise on the C-C and the C-S beat notes with the same relative amplitude, such

noise, being labeled as 'common-mode' noise, can be eliminated via a common-mode

rejection analysis. Specifically, if the relative magnitude of the 'common-mode' amplitude

noise is represented by 6Ae while 6A1 and 6A2 represent the relative amplitude noise of the

C-C and C-S beat notes in 'differential-mode', we can calculate the amplitude ratio of the

C-S beat notes to the C-C beat note

Ac-s (1 6Ae 6A2) Ac-s (1 6Ac) (1 +6A2) Ac-s
F = (1 + 6A2 6A1) (3-45)
Ac-c (1 +6Ae 6A1) Ac-c (1 6Ac) (1 +6A1) Ac-c










Here Ac-c, Ac-s are amplitudes of the C-C and C-S beat notes. According to Equation

3-45, the 'common-mode' noise should not show up in the noise spectrum of F. The LSD

of F, labeled as 'c-s/c-c' (the green curve in Figure 3-23) represents only the amplitude

noise in 'differential-mode.' As can be seen, the 'common-mode' noise below 40 Hz is

successfully reduced via this common-mode rejection an~ lli--- We believe that the laser

amplitude noise which has been eliminated from the data is largely due to the beam

jitter that dominates the low frequency region, since the measured laser intensity noise

(represented by the brown curve in Figure 3-23) is more than an order magnitude lower

than the measured amplitude noise.

As has been discussed in Section 3.4.4, it is oscillator noise and temperature

fluctuations that cause an EOM to produce additional amplitude and phase noise on

the laser beam. Our result shows that the oscillator noise (labeled as V(R) in Figure 3-23)

only counts for a small part of the amplitude noise floor. The thermally induced noise can

be estimated once we know the thermal expansion coefficient, refractive indices and dn/dT

values. Thermally induced changes in electro-optic coefficients and the thermal-elasto

effect are not considered here. From Equations 3-21 and 3-33, the carrier field after the

EOM can be approximated as

0o'
-i--L

E~w 1 26C 0 OS 8 COS ( 01)+ Sin in 76i"''e"b 0 02)) Eo. (3-46)
-2i--L
1 rir26 C

The modulation indices are
wo Vz
mi = Lr33n 37
c d)

m2~ = WLr.Sil3V 38
cd

And

L = Lo + RAT, (3-49)

d = do + Ez AT, (3-50)















-i: -4
1~04





S10" ---noise floor


10
10o 101 102 109
frequency (Hz)

Figure 3-24. Linear spectral density of phase noise in laser beat notes and the modulation
signal from the EOM driver.


n, = neo + AT, (3-51)
dT

nz = nzo + AT. (3-52)
dT

The calculation using Equations from 3-46 to 3-52 shows that temperature fluctuations at

a level of 0.1 mKl/H could only introduce a 10-9 TelatiVe amplitude noise in the carrier

field. And the electronic noise floor that the phase could introduce in the measurement

is not fully understood by the time this thesis is finished, it is highly possible that we are

still limited by the measurement sensitivity to see the amplitude noise imposed by the

EOM. The amplitude noise curve in Figure 3-23 can be seen as the upper limit of the

amplitude noise that is produced by the EOM without stabilizing the temperature of the

crystal and applying acoustic isolation methods.

If the electronic noise associated with the phase meter can be omitted, the measured

difference between the phase of the C-S beat notes #c-s and that of the C-C beat note

e-c, is the residual phase imposed by the EOM. The LSD of #c-s Oc-c (green curve in

Figure 3-24) is above 10-5 ; (1. I4~l/ This overlaps with the LSD of the phase of the



















S10o 10 101
frqec (Hz):
Figure~~~~~~~~~~~~~~~- 3-5 ierseta estyo hs os nlsrbetntsadtemdlto







expect o ispparone 0 oe blo tisnoise floor. hswscnimdb nte



measureme25 Lntwere theFlkfuctr dnityon geertoase repeilaed y bettnter on te (Sultanord



Research Syste DS. 34)henwmesrmetrsuti shown in Figure 3-25. i h S t rsda ae Ivn bot





measurements, the phase noise due to temperature fluctuations or acoustic perturbations

should be below the level set by the noise floor.









CHAPTER 4
CONCLUSION

A complete characterization of each mechanical, electrical or optical component

that has been planned to be used in Advanced LIGO is an important part of the

instrumentation work. The performance of each component must meet the system

requirements with consideration given to the improvement of the strain sensitivity and the

operational feasibility.

Both the triple pendulum suspension system and the RTP crystal EOM have been

planned to be used in the IO subsystem of Advanced LIGO. The triple pendulum

described in this thesis is the one that has been installed and tested in the JIF lah at

Glasgow University. however, the triple pendulum to be used in Advanced LIGO will be

different from this prototype pendulum. The mechanical model that has been developed

by the author will be a useful reference model for the design and test of the final version of

the triple pendulum suspension system.

The RTP (ni-- I EOM is going to be installed in the 40-m prototype interferometer

(at California Institute of Technology) and Enhanced LIGO (a mid-step between LIGO

and Advanced LIGO). The characterizations of the EOM has been done at ITF is

essentially a pre-installation test. The measurement results provide useful information

for the final decision of its application in Advanced LIGO.

The modeling of a triple pendulum suspension and the characterization of the new

EOM in terms of its practical limitations have been presented in this thesis. The results

are summarized below.

4.1 Triple Pendulum Suspension Model

The dynamic model of the triple pendulum can he used to investigate the mode

frequencies or transfer function for all degrees of freedom. The model can he used to

simulate the response of the pendulum to the external excitation applied through one or

multiple local control channels. Based on the mechanical analysis that has been completed

in this thesis, the expansion of this model to a model that includes a complete global










control on all three test masses will be a straight forward practice. The cross coupling

effect between control channels for motions of different degrees of freedom can also

predicted using this model. As compared with other existing models, it is more complete

in details. For example, two OSEMs that are used to adjust the longitudinal and vaw

motion of the pendulum are located above the center of the upper mass. A pitch motion

will be introduced when the control force is applied via these two OSEMs. The sensing

and control matrix described in ChI Ilpter 2 has taken this into consideration. The mode

frequencies of the vertical, longitudinal and sideway motions calculated by this model have

been confirmed by the measurement results. The errors of theoretical values are less than

0.2 Hz in most part. The N----- -1 disagreement (about 0.4 Hz) shows up in a 5 Hz vertical

mode. More characterization measurements are needed to confirm mode frequencies of the

pitch, yaw and roll motions predicted by the model.

The damping constants for the motions of three test masses need to be verified and

incorporated in this model.

4.2 EOM

The RTP ( i--r I1 EOM has been shown to be a quality EOM candidate for Advanced

LIGO. It has excellent optical properties and a less than 1000 ppm/cm absorption

coefficient at 1064nm wavelength. The residual amplitude modulation can he maintained

below a 10-4 level without taking any efforts to stabilize the temperature of the crystal.

The experimental studies of the amplitude and phase noise imposed by the EOM on

the laser field have demonstrated that the relative amplitude noise is below a level of

2 x 10-5/2/& and the phase noise is below a level of 10-5 cyclesl/Hx in the LIGO

bandwidth. The phase noise floor which is not limited by the EOM driver is shown to be

around 3 x 10-6 cyclesl/Hx.

The heterodyne technique which has been developed to measure the EOM noise can

he improved and used for ultra-sensitive measurement of both the amplitude and phase










of optical signals. This kind of technique is essential for the testing of future electro-opic

components to be used in Advanced LIGO.









APPENDIX A
MATLAB MODEL FOR THE TRIPLE PENDULUM

A.1 Close Loop Transfer Function (clp.m)

The file clp.m is used to calculate and plot the close loop transfer functions


T ( f )= uc (A-1)
1 + Ti HG T2l

with G, H, Ti and T2 defined in Figure 2-29.

*II I *** *** *** ***** ** ** ***** ** ** **** ** ** ** ****

clear all;

param; .parameter file

matrixT1T2; control matrix

In = [1, 1, 0, 0, 0, 0]; .~ input channels

Out = [1, 0, 0, 0, 0, 0]; .~ output channels

In = transpose(In); fmin = 1.0e-1;

fmax = 12;

bmin = log10(fmin);

bmax = log10(fmax);

N =1000;

for n = 1:1:N;

freq = 10^(bmin + (bmax bmin)*(n-1)/(N-1)); .~ log span

w = 2*pi~freq;

s = i* w;

model; the mechanical model

localCtr; .local control servo

H = T2 M T1;

OL = H ;

compel = Out *LC* OL In;

CL = 1 /(1 + compel);









f(n) = freq;

TF(n) = CL;
end

mag = abs(TF);

mag = 20 log10 (mag);

phi = unwrap(angle(TF)) 180/pi;

subplot(2,1,1)

semilogx(f, mag)

axis([fmin fmax min(mag) max(mag)])

set(gca, 'FontSize', 14);

xlabel('Frequency (Hz) )

ylabel(':\i I .g .1ude (dB) )

grid

subplot(2,1,2)

semilogx(f, phi)

axis([fmin fmax min(phi)

max (phi)])

set(gca, 'FontSize', 14);

xlabel('Frequency (Hz) )

ylabel('Phase (deg.) )

grid
1 ** ** ** ** ** ** ** ** *** ** ** ** ** ** ** ** ** **

A.2 Parameters (parameter.m)

The file (parameter.m) represents the input parameters for the triple pendulum

installed in JIF lah. The parameters are defined in ChI Ilpter 2.

1 ***********************************************

g = 9 81;










.X direction separation

su = 0.00; .~ 1/2 separation of upper wires

si = 0.03; .~ !/2 separation of intermediate wires

sl = 0.005; .~ 1/2 separation of lower wires

s1 = si; s2 = sl;

.arms

Lp = 0.04; half spacing of coils acting on pitch

ly = 0.08; .half spacing of coils acting on yaw

Ir = 0.1:3; .half spacing of coils acting on roll

.masses

ml = 2.95+0.1; m2 = 2.71; m:3 = 2.71;

m2:3 = m2 + m:3; ml:$ = ml + m2 + m:3;

.moments of inertia

Ilx = 0.0189; I2x = 0.0066; I;3x = 0.0066;

Ily = 0.00:35; I2y =0.005:30; I;3y = 0.005:3;

Ilz = 0.0189; I2z =0.00524; I;3z = 0.00524;

.spring constants

k1 = :3.6066e+002;

k2 = 6.9:301e+002/2;

k:3 = 9.7506e+003/2;

ir = 0.06:35; ..!~!!. Inini a~ of intermediate mass (cylinder)

tr = 0.06:35; ..!~!!. Ininia~ of lower mass (cylinder)

dO = 0.001; .~! I .I!,!! of upper wire break-off (above c.of m. upper mass)

dl = 0.001; H. u !,!! of intermediate wire break-off (below c.of m. upper mass)

d2 = 0.001; H. !lit, of intermediate wire break-off (above c.of m. of int. mass)

d:3 = -0.001; H. u !,!! of lower wire break-off (below c.of m. intermediate mass)

d4 = 0.001; H. u !,!! of lower wire break-off (above c.of m.test mass)










nO = 0.06; .~ 1/2 separation of upper wires at suspension point

n1 = 0.1; .~ 1/2 separation of upper wires at upper mass

n2 = 0.03; .~ 1/2 separation of intermediate wires at upper mass

n:3 = ir-0.005+0.01; .~ 1/2 separation of intermediate wires at intermediate mass

n4 = ir-0.005+0.005; .~ 1/2 separation of lower wires at intermediate mass

n5 = tr-0.005+0.005; .~ 1/2 separation of lower wires at test

It1 = sqrt(11^\2 (nO-n1)^\2);

It2 = sqrt(12^\2 (n2-n:3)^\2);

It:$ = sqrt(13^\2 (n4-n5)^\2);

15 = 0.1; height of coils 1, 2, and 6 above the center of mass

ains of six channels

gl =1;gR2 =1;g3 =1;gR4 =1;gR5 =1;g6= 1;

G= [g100000

0 g20000

00 g3000

000 g400

0000 g50

00000 g6]i;

*II ************************************************

A.3 Model (model.m)

The file model.m provides the transfer function front the external forces and torques

to the induced motion of the upper mass in six degrees of freedom.

1 ***********************************************

.vertical

Hzv = 1/(2*k1 + 4*k2 + nil s^\2 (nt: s^\2 + 4*k:3)*16*k2^2 /( ( 4*k2 + 4*k:3

+ n2*s^2)*( n13*s^2 + 4*k:3 )- 16*k:3^2 ) );

************************************************









.longitudinal / pitch

T(1,1) =-ml~s^2- (m2+m3)*g/1t2 (ml+m2+m3)*g/1tl;

T(1,2) = (m2+m3)*g/1t2; T(1,4) = (ml+m2+m3)*g~d0/1t1 + (m2+m3)*g~dl/1t2;

T(1,5) =- (m2+m3)*g~d2/1t2;
T(2,1) = (m2+m3)*g/1t2;

T(2,2) = -m2*s^2 (m2+m3)*~g/1t2-m3*g/1t3;

T(2,3) = m3*g/1t3;

T(2,4) = -(m2+m3)*g~dl/1t2;

T(2,5) = (m2+m3)*g~d2/1t2+m3*g~d3/1t3;
T(2,6) = -m3*g~d4/1t3;

T(3,2) = 1; T(3,3) = -lt3*s^2/g 1;

T(3,5) = -d3; T(3,6) = d4;

T(4,1) = (ml+m2+m3)*g~d0/1t1 + (m2+m3)*g~dl/1t2;

T(4,2) = -(m2+m3)*g~dl/1t2;
T(4,4) = -s^2*Ily + 4*k2*sl^2-(ml +m2+m3)*g~d0^2/1t1 (ml+m2+m3)*g~d0 -

(m2+m3)*~g~dl^2/1t2;

T(4,5) =-4*k2*sl^2 + (m2+m3)*g~d1 + (m2+m3)*~g~dl~d2/1t2;

T(5,1) = -(m2+m3)*g~d2/1t2;

T(5,2) = (m2+m3)*g~d2/1t2 + m3*g~d3/1t3;

T(5,3) = -m3*g~d3/1t3;
T(5,4) = 4*k2*sl^2 + (m2+m3)*~g~dl~d2/1t2;

T(5,5) = -s^2*I2y 4*k2*sl^2 (m2+m3)*g~d2^2/1t2 (m2+m3)*g~d2 4*k3*s2^2 -

m3*g~d3^2/1t3 + m3*g~d3;

T(5,6) = 4*k3*s2^2 + m3*g~d3*d4/1t3 ;

T(6,2) = -m3*g~d4/1t3;
T(6,3) = m3*g~d4/1t3;

T(6,5) = 4*k3*s2^2 + m3*g~d4*d3/1t3;









T(6,6) = -s^2*I3y 4*k3*s2^2 m3*g~d4 m3*g~d4^2/1t3;

Gamma = inv(T);


.yaw
Hxl = Gamma(1,1) ;

Hxp = Gamma(1,4);
Hal = Gamma(4,1);

Hap = Gamma(4,4);
P1 =(ml+m2+m3)*~g~nl~nO/tl;

P2 = (m2+m3)*g*(n2^2 + sl^\2)/1t2;

P3 = (m2+m3)*g*(n3^2 + sl^\2)/1t2;

P4 = m3*g*(n4^2 + s2^\2)/1t3;

P5 = m3*g*(n5^2 + s2^\2)/1t3;

Hby = 1/( P1 + P2*( 1 P3/( P3 + P4* ( 1 P5/( P5 + s^2*I3z ) ) + s^2*I2z ) ) +

s^2*Ilz );




Al = (ml+m2+m3)*g~d0*( 1/1t1 -( n1 n )^2/11^\2/1t1 )- (ml+m2+m3)*g~nl*(

n1-n0 )/11^\2 + 2*kl* ( n1*1t1 + d0*(n1-nO) )*(n1-nO)/11^2;

B1 =(ml+m2+m3)*~g~nl*tl/11\2 (ml+m2+m3)*g~nl/1t1 + (ml+m2+m3)*g~d0*

(n1-nO)/11^\2 2*kl*tl*( n1*1t1 + d0*(n1-nO))/11^2;
C1 = (ml+m2+m3)*g~nl* (n1-nO)/11^\2 + (ml+m2+m3)*g~d0*( (n1-nO)^\2/11^\2/1t 1

- 1/1t1 ) 2*kl*( n1 n0 )*( nl*1t1 + d0*(n1-nO) )/11^\2;
D1 =(m2+m3)*g~dl*( 1/1t2 (n3 -n2 )^\2/12^\2/1t2 ) -(m2+m3)*g~n2*( n3-n2

)/12^\2 4*k2* ( n2*1t2 + dl*(n3-n2) )*(n3-n2)/12^2;
E = (m2+m3)*~g~n2*1t2/12^2 -(m2+m3)*g~n3/1t2 -(m2+m3)*g~dl* (n3-n2)/12^\2+

4*k2*1t2/12*( n2*1t2 + dl*(n3-n2) )/12;









G1 = (m2+m3)*g~n2* (n3-n2)/12^\2 + (m2+m3)*g~dl*( (n3-n2)^\2/12^\2/1t2 1/1t2 )

+ 4*k2* ( n2*1t2 + dl*(n3-n2) )*(n3-n2)/12^2;
R1 =-( ml+m2+m3 )*g*( 1/1t1 (n1- n0 )^2/11^\2/1t1 ) +2*kl*(n1-nO)^\2/11^\2;

S1 =-(ml+m2+m3)*g* (n1-nO)/11^\2- 2*kl*tl*(n1-nO) /11^\2;
A2 = (m2+m3)*g~d2*( 1/1t2 ( n3 n2 )^\2/12^\2/1t2 ) (ml+m2+m3)*g~n3*( n3-n2

)/12^\2 + 4*k2* ( n3*1t2 + d2*(n3-n2) )*(n3-n2)/12^2;
B2 = (m2+m3)*~g~n3*1t2/12^2 (m2+m3)*g~n3/1t2 + (m2+m3)*g~d2* (n3-n2)/12^\2 -

4*k2*1t2*( n3*1t2 + d2*(n3-n2) )/12^\2;
C2 = (m2+m3)*g~n3*(n3-n2)/12^\2 + (m2+m3)*g~d2*( (n3-n2)^\2/12^\2/1t2 1/1t2 )-

4*k2*( n3 n2 )*( n3*1t2 + d2*(n3-n2) )/12^\2;
D2 = m3*g~d3*( 1/1t3 ( n5 n4 )^\2/13^\2/1t3 ) m3*g~n4*( n5-n4 )/13^\2 4*k3* (

n4*1t2 + d3*(n5-n4) )*(n5-n4)/13^2;
E2 = m3*g~n4*1t3/13^2 -m3*g~n5/1t3 -m3*g~d3* (n5-n4)/13^\2 + 4*k3*1t3/13*( n4*1t3

+ dl*(n5-n4) )/13;
G2 = m3*g~n4* (n5-n4)/13^\2 + m3*g~d3*( (n5-n4)^\2/13^\2/1t3 1/1t3 ) + 4*k3* (

n4*1t3 + d3*(n5-n4) )*(n5-n4)/13^2;
R2 = -( m2 + m3 )*g*( 1/1t2 ( n3 n2 )^\2/12^\2/1t2 ) + 4*k2*( n3 n2)^\2/12^\2;

S2 = -( m2 + m3 )*g*( n3 n2)/12^\2 4*k2*1t2*( n3 n2)/12^\2;
A3 = m3*g~d4*( 1/1t3 ( n5 n4 )^\2/13^\2/1t3 ) m3*g~n5*( n5-n4 )/13^\2 + 4*k3* (

n5*1t3 + d4*(n5-n4) )*(n5-n4)/13^2;
B3 = m3*g~n5*1t3/13^2 m3*g~n5/1t3 + m3*g~d4*(n5-n4) /13^\2 4*k3*1t3*( n5*1t3

+d4*(n5-n4) )/13^\2;
C3 = m3*g~n5* (n5-n4)/13^\2 + m3*g~d4*( (n5-n4)^\2/13^\2/1t3 1/1t3 ) 4*k3*( n5 -

n4 )*( n5*1t3 + d4*(n5-n4) )/13^\2;
R3 = -m3 *g*( 1/1t3 ( n5 n4 )^\2/13^\2/1t3 ) + 4*k3*( n5 n4)^\2/13^\2;
S3 = -m3 *g*( n5 n4)/13^\2 4*k3*1t3*( n5 n4)/13^\2;

Q(1,1) = R1 + R2 ml~s^2;









Q(1,2) =-R2;

Q(1,4) = S1*n1- R1*d0 + S2*n2 R2*dl;

Q(1,5) = -S2*n3 + R2*d2;

Q (2, 1) =-R2;

Q(2,2) = R2 + R3 m2*s^2;

Q(2,3) = -R3;

Q(2,4) = -S2*n2 + R2*dl;

Q(2,5) = S2*n3 R2*d2 + S3*n4 R3*d3;

Q(2,6) = -S3*n5 + R3*d4;

Q(3,2) = -R3;

Q(3,3) = R3 m3*s^2;

Q(3,5) = -S3*n4 + R3*d3;

Q(3,6) = S3*n5 R3*d4;

Q(4,1) = Al + D1;

Q(4,2) = -D1;

Q(4,4) = Bl~n1 + n2*E1 + dl*G1 + (ml + m2 + m3)*g*(nl*(n1 nO)/1t1 dO) -
Ilx~s^2;

Q(4,5) = -n3*E1 d2*G1 (m2 + m3)*g*(n3* (n3 n2)/1t2 dl);

Q (5, 1) = A2;

Q(5,2) = A2 + D2;

Q(5,3) = -D2;

Q(5,4) = -n2*B2 dl*C2;

Q(5,5) = n3*B2 + d2*C2 + n4*E2 + d3*G2 + (m2 + m3)*g*(n3*(n3 n2)/1t2 d2)-

m3*g*(n4*(n5 n4)/1t3 d3)- I2x~s^2;

Q(5,6) = -n5*E2 d4*G2;

Q(6,2) = -A3;

Q(6,3) = A3;









Q (6,5) = -B:3*n4 C:3*d3;

Q(6,6) = B:3*n5 + C:3*d4 + n13*g* (n5* (n5 n4)/1t:3 d4)- I;3x~s^2;

Qt = inv(Q);

Hyt = Qt(1,1);

Hyr = Qt(1,4);

Hct = Qt(4,1);

Her = Qt(4,4);



Al = [ Hxl Hxp 0 0 0 0 .~ x

Hal Hap 0 0 0 0 theta = a

00OHzv00 z

00 OHhy0 ~phi =

0 0 0 0 Hyt Hyr y

0 0 0 0 Hct Her ]; .~ psi = c

A.4 Transformation IVatrix (matrixT1T2.m)

The file matrixT1T2.m represents the conversion front the input signals to the six

coils to the output forces and torques applied on the top mass.



.transformation front 6 coil voltages to 6 forces

12:3456

T1= [110000 FI

1s 15 Ip -lp 0 0 Alp

001110 F

ly -ly 0 0 0 0 My
000001 Ft



.transformation front 6 degrees of freedom to 6 magnet positions










.x theta z phi y psi

T2 = [11Is 01y 00 .~ 1

11Is 0-ly00 2

Olp 1001r .3

0 -lp 1 00 Ir 4

0 01 0 0- -I.5

000011s]; .~ 6

CII ************************************************

A.5 Local Control Servo (LocalCtr.m)

The file LocalCtr.m gives the transfer function of the local control servo. The poles

and zeros are calculated using the matlab model written by Torrie [26].

1 ** ** ** ** ** ** ** ** *** ** ** ** ** ** ** ** ** **

.the local control servo

num = (s+b6 1;:1n+00)* (s+2.4669e-14)* (s+2. 1991e+00);

den = ((s+5.6549e+01)*(s+4.3982e+00))^\2;

kf = 2.0e5;

servo = kf num/den;

LC = G servo;

*II *** *** *** *** ***** ** ** ******** ** ** ** ** *******









APPENDIX B
ELECTRONICS


Figure B-1. Circuit schematic of the phase lockingf servo.









REFERENCES

[1] C. Cutler and K(. S. Thorne in Proc. of the 16th International Conference on General
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[2] J. Weber, Phys. Rev. D, 117, 306 (1960).

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[5] Massimo Tinto, Phys. Rev. D, 53, 5354 (1996).

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[7] G. E. Moss, L. R. Miller, and R. L. Forward, Applied Optics, 10, 2495 (1971).

[8] R. Weiss, "Electromagnetically Coupled Broadband Gravitational Antenna," Res.
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[9] R. W. P. Drever, J. Hough, W. A. Edelstein, J. R. Pugh, and W. Martin, in Proc.
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Dei Lincei, Rome, Italy, 1979), p. 365.

[10] H. Billing, K(. Maischberger, A. Rdiger, R. Schilling, L. Schnupp, W. Winkler, J.
Phys. E, 12, 1043 (1977).

[ 11] W. M. Folkner and et. al., in S3rd AIAA Aerospace Sciences M~eeting and Ex~hibit,
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[13] S. E. Whitcomb, "Optical pathlength fluctuations in an interferometer due to
residual gas," Technical Report (Caltech, 1984).

[14] Carlton M. Caves, "Quantum-mechanical noise in an interferometer," Physical
Review D, 23, pl693 (1981).

[15] P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors
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and H. Ward, Applied Physics B: Lasers and Optics, 30, 97 (1983).

[18] J. Mizuno, K(. A. Strain, P. G. Nelson, J. M. C!. is, R. Schilling, A. Ruediger,
W. Winkler, K(. Danzmann Phys. Lett. A, 175, 273 (1993).









[19] B. J. Meers, Phys. Rev. D, 38, 2317 (1988).

[20] K. A. Strain and B. J. Meers, Phys. Rev. Lett, 66, 1391, 1991.

[21] K(. A. Strain, G. Mueller, T. Delker, D. H. Reitze, D. B. Tanner, J. E. Mason, P. A.
Willems, D. A. Shaddock, M. B. Gray, C. M. Lowry, D. E. McClelland, Appl. Opt,
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[22] G. Mueller, T. Delker, D. B. Tanner, D. H. Reitze, Appl. Opt, 42, 1257 (2003).

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[24] K(. Somiya, P. Beyersdorf, K(. Arai, S. Sato, S. K~awamura, O. Miyakawa,
F. K~awazoe, S. Sakata, A. Sekido, N. Mio, Appl. Opt, 44, 3179 (2005).

[25] R. Adhikari, "DC readout," Technical Pl. m:i,;, Session of the LSC M~eeting,
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[26] C. I. Torrie, "Development of suspension for the GEO600 gravitational wave
detector ," PhD thesis, Glasgow University, Glasgow (2001).

[27] M. E. Husman, "Suspension and control for interferometric gravitational wave
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[28] M. Adrain, A. Lucianetti, R. Martin, G. Mueller, V. Quetschke, D. H. Reitze,
D. B. Tanner, L. Williams, W. Wu, "Input optics subsystem preliminary design
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[29] V. Quetschke, G. Mueller, "Design of Stable Power-Recycling Cavities," LIGO
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configuration to eliminate sidebands of sidebands," LIGO Document Number
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[36] V. G. Dimitriev, G. G. Gurzadyan, and D. N. Nikogco i- Ilr Handbook of Nonlinear
Cr; lerl (Springer Series in Optical Science, Berlin, 1991).









[37] http://www.coretech.com.cn/English/Upload/06211128pf product
information page from Coretech Crystal Division of Shanda Luneng Information
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London, 1961).

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wave applications," IEEE transactions on ultrasonics. ferroelectrics. and frequency
control, 41, 53 (1994).

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S. Yoshida, D. H. Reitze, Applied Optics, 40, 366 (2001).

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(1991).

[43] P. Fr-itsel, G. Gonzalez, A. Marin, N. Mavalavala, D. Quimette, L. Sievers, D. Sigg,
M. Zucker, "Length sensing and control subsystem preliminary design," LIGO
Document Number T970122-00-D (Caltech/ \! 1', 1997).

[44] K(. Somiya and Y. Chen and S. K~awamura and N. Mio, Phys. Rev. D, 73, 122005
(2006).

[45] J. I. Thorpe, "Laboratory- studies of arm-locking using the laser interferometry space
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Gainesville (2006).









BIOGRAPHICAL SKETCH

I was born in Chan!~ I on September 13th, 1979. I became a student when I was 5 years

old and have kept this profession ever since. I enrolled at the University of Science and

Technology of CluI, I in 1996 and got my Bachelor of Science in Materials Science and

Engineering in 2001. Since I found that physics classes are more interesting than other

classes that I took at college, I decided to go to graduate school to learn more. I became

a graduate student at the University of Florida in the fall of 2001 and joined the LIGO

group the year after. I got a chance to go to Glasgow University and worked on triple

suspension system in the JIF lab. After a 7-month-long interesting and also troublesome

experience in Scotland, I went back UF and continued to do some experiments testing

the optical components which are designed for the Advanced LIGO Input Optics. I am

supposed to end my career as a student at UF.





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ThanksforallthehelpIhavereceivedduringmygraduatestudies.MyadvisorDr.Tannerandco-advisorDr.ReitzeoeredmetheopportunitytojointheLIGOgroupatUFandsupportedmewithfullassistantshipuntiltheendofmygraduatestudy.Withouttheirencouragementandhelp,thisthesiswouldneverhavebeencompleted.IamalsoindebtedtoDr.Mueller,whooftenactedasareplacementadvisorandkeptonhelpingmewithdetailedinstructions.Withouthisguidance,Iwouldhavebeenlostinthedarknessalotoftimes.IhavetoappreciatethetutoringIreceivedfromRupalAminwhenIcametotheLIGOlabforthersttime.Hehadgreatpatienceinshowingmethefundamentallabskillsevenifmyresponseisalwaysdumb.Hishard-workingstylescaredmeawayfrombeinglazy.IalsolearnedalotfromMalikRakhmanov.HeisthebestphysicstutorIhaveevermet.Thephysicsthathetaughtmecoveredawidescopefromthefundamentalmechanicstotheadvancedoptics.Hisdetail-orientedteachingstylemadethelearningprocessreallyenjoyable.ManythankstoVolkerQuetschkeforhishelponelectronics.Heisacomputergeniusandanexperiencedcontrolengineer.ThanksalsogoestoKateDooley,NaveenMargankunte,RodrigoDelgadillo,andHsin-jungLinfortheirkindhelpofcorrectingthegrammaticalerrorsinmythesis.RodisalsomylunchtimepartneralongwithMikeHartmanandDanielArenas.TheyjustkeptoncheeringmeupwhenIwasfrustratedbythetediousthesiswritingwork. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1GravitationalWaves .............................. 15 1.1.1Introduction ............................... 15 1.1.2GravitationalWaveSources ...................... 18 1.1.3DetectionofGravitationalWaves ................... 18 1.2GravitationalWaveDetectionUsingMichelsonInterferometricDetectors 19 1.3SignalsForTheGroundBasedInterferometricDetectors .......... 24 1.3.1ChirpSignals .............................. 24 1.3.2PeriodicSignals ............................. 24 1.3.3BurstSignals .............................. 27 1.3.4StochasticSignals ............................ 28 1.4NoiseInTheGroundbasedInterferometricDetectors ............ 29 1.5LIGO&AdvancedLIGO ............................ 31 1.5.1OpticalConguration .......................... 31 1.5.2ReadoutScheme ............................. 36 1.6OverviewOfTheThesis ............................ 37 2TRIPLEPENDULUMSUSPENSIONSYSTEM .................. 38 2.1SeismicIsolationSuspensionSystems ..................... 38 2.2IntroductiontotheTriplePendulumSuspension ............... 40 2.3MechanicalAnalysisoftheTriplePendulum ................. 43 2.3.1VariablesandParameters ........................ 43 2.3.2VerticalResponse ............................ 49 2.3.3LongitudinalandPitchDynamics ................... 50 2.3.4YawMotion ............................... 60 2.3.5SidewaysandRollMotion ....................... 68 2.4LocalControloftheTriplePendulum ..................... 82 2.5MeasurementResult .............................. 87 3ELECTRO-OPTICMODULATOR(EOM) .................... 92 3.1InputOpticsSubsystem(IO)ofAdvancedLIGO .............. 92 3.2ApplicationofEOMs .............................. 94 5

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............................... 98 3.3.1PhysicalPropertiesofRTPCrystals .................. 98 3.3.2EOMConguration ........................... 99 3.4TechnicalFeatures ............................... 101 3.4.1Piezo-resonances ............................. 102 3.4.1.1Theory ............................. 102 3.4.1.2Measurement ......................... 103 3.4.2ThermalLensing ............................ 105 3.4.2.1Thermaleectsincrystals .................. 105 3.4.2.2Experiment .......................... 106 3.4.3ResidualAmplitudeModulation(RAM) ................ 109 3.4.3.1Generationmechanism .................... 109 3.4.3.2RAMinAdvancedLIGO ................... 115 3.4.3.3Characterization ....................... 116 3.4.4LaserAmplitudeandPhaseNoiseProducedbyEOMs ........ 119 3.4.4.1Generationmechanism .................... 119 3.4.4.2Characterization ....................... 122 4CONCLUSION .................................... 130 4.1TriplePendulumSuspensionModel ...................... 130 4.2EOM ....................................... 131 APPENDIX AMATLABMODELF0RTHETRIPLEPENDULUM ............... 133 A.1CloseLoopTransferFunction(clp.m) ..................... 133 A.2Parameters(parameter.m) ........................... 134 A.3Model(model.m) ................................ 136 A.4TransformationMatrix(matrixT1T2.m) ................... 141 A.5LocalControlServo(LocalCtr.m) ....................... 142 BELECTRONICS ................................... 143 REFERENCES ....................................... 144 BIOGRAPHICALSKETCH ................................ 147 6

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Table page 2-1Inputandoutputpointsfordierenttransferfunctionmeasurements. ...... 88 3-1PhysicalpropertiesofRTPcrystals. ......................... 99 7

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Figure page 1-1EectoftheincidentGWswithtwopolarizationsontestparticlesarrangedinacircle.TheGWspropagateinadirectionperpendiculartothepage.GWreferstothephaseoftheGWs. ........................... 17 1-2AMichelsoninterferometerwithtwoorthogonalarmslyingontheplane.ThepropagationdirectionofGWsintwopolarizationdirectionisverticaltotheplane.BS-beamsplitter,M-mirror,PD-photodetector. ........... 20 1-3OpticalcongurationoftheLIGOdetector.BS-beamsplitter,PBS-polarizingbeamsplitter,EOM-electro-opticmodulator,ITM-inputtestmass,ETM-endtestmass,PRM-powerrecyclingmirror,AOM-acousticopticalmodulator. 32 1-4OpticalcongurationoftheAdvancedLIGOdetector.PSL-laserpre-stabilizationsubsystem,IO-inputopticssubsystem,SRM-signalrecyclingmirror. ..... 35 2-1Asimplespringstackandapendulum. ....................... 39 2-2Transferfunctionofasimplespringstack. ..................... 39 2-3Threedimensionalviewofthetriplependulumsuspensionsystem. ........ 41 2-4Schematicviewofthetriplependulum.Thestacksandthedampingarmhavebeenomittedforclarity.Thisamodieddrawingwhichoriginatesfrom[ 26 ]. .. 42 2-5`T-shape'topmasswithsixmagnetsattached. ................... 44 2-6Photographofthetopmass. ............................. 45 2-7Parametersofatriplependulum. .......................... 47 2-8Photographofatriplependulum. .......................... 48 2-9BodeplotofHzv. ................................... 50 2-10Sideviewofasinglependulum'slongitudinalandpitchmotion. ......... 51 2-11Longitudinaldisplacementintroducedbythepitchmotion. ............ 52 2-12Componentsofrestoringforceswhichacttotiltthemass. ............ 54 2-13Sideviewofthetopmasssuspendedwithtwowires. ............... 56 2-14BodeplotofHxl. ................................... 60 2-15BodeplotofHxp. ................................... 61 2-16BodeplotofHl. ................................... 61 2-17BodeplotofHp. ................................... 62 8

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............................. 63 2-19Projectionofthetensiononto^x-^yplanewhichproducestherestoringtorque.Theupperplotandthelowerplotareassociatedwithtwodierenteectsonwireswhenthemassrotates. ............................. 64 2-20BodeplotofHy. ................................... 67 2-21Faceonviewofthesidewaysandtherollmotionofasinglependulum. ..... 68 2-22Expandedtoshowvariouslengths. ......................... 69 2-23Thecomponentsofthetensionintwowiresacttorollthemass. ......... 71 2-24Relativemotionbetweentwoadjacentmasses. ................... 73 2-25BodeplotofHys. ................................... 83 2-26BodeplotofHyr. ................................... 83 2-27BodeplotofH's. ................................... 84 2-28BodeplotofH'r. ................................... 84 2-29Feedbackcontrolblockdiagramofthetriplependulum. .............. 85 2-30Closelooptransferfunctionmeasurementforthetriplesuspensionsystem. ... 87 2-31Closelooptransferfunctionwiththeinputsignalbeinginjectedtocoil1andcoil2andtheoutputsignalfromchannel1. .................... 89 2-32Closelooptransferfunctionwiththeinputsignalbeinginjectedtocoils3,4and5andtheoutputsignalfromchannel3. .................... 90 2-33Closelooptransferfunctionwiththeinputsignalbeinginjectedtocoil6andtheoutputsignalfromchannel6. .......................... 91 3-1OverallIOschematic. ................................. 93 3-2CongurationofanEOM. .............................. 95 3-3PossiblelocationsofphotodetectorsinAdvancedLIGO. ............. 96 3-4TheopticalcongurationofusingaMach-Zehnderinterferometertoeliminatethesidebandsonsidebandsproblem. ........................ 97 3-5RTPcrystals(wedgedandnon-wedged)mountedbetweentwoelectrodes. .... 100 3-6CircuitdiagramfortheEOM. ............................ 101 9

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103 3-8RTPcrystalpiezo-resonances. ............................ 104 3-9Experimentalarrangementforthermallensingmeasurement. ........... 106 3-10ThemeasuredlaserbeamdivergencewithandwithouttheRTPcrystalinthebeampath. ...................................... 107 3-11Changeofthewavefrontatthebeamwaist. .................... 108 3-12ResidualamplitudemodulationduetothemisalignmentofanEOM. ...... 109 3-13Changeoftheorientationofprincipalaxeswithrespecttothelighteldwhenavoltageisappliedacrossthecrystal. ....................... 111 3-14Fabry-Perotcavityeectduetothebackreectionbetweenthefrontandendsurfacesofthecrystal. ................................ 115 3-15TheexistenceofRAMchangesphaserelationshipbetweentheleakedcarrierlightandtheRFsidebandsatthedarkportofLIGO. ............... 115 3-16ExperimentalsetuptocharacterizetheresidualamplitudemodulationcreatedbyanEOMheatedbya1053nmNd:YLFlaser. .................. 117 3-17CorrelationbetweentheheatingpowerandRAM. ................. 118 3-18Deformationofendsurfacesofacrystalwhenanacousticwavepassesby. .... 121 3-19ExperimentalsetuptocharacterizethelaseramplitudeandnoiseimposedbyanEOM. ....................................... 122 3-20LooptransferfunctionofPLL. ........................... 124 3-21Theeldsoftwophase-lockedlasers. ........................ 125 3-22BlockdiagramofaI/Qmeasurementsysteminsideaphasemeter. ....... 125 3-23AmplitudenoisespectraofthelaserbeatnotesandthemodulationsignalfromtheEOMdriverasmeasuredbythephasemeter. ................. 126 3-24LinearspectraldensityofphasenoiseinlaserbeatnotesandthemodulationsignalfromtheEOMdriver. ............................. 128 3-25LinearspectraldensityofphasenoiseinlaserbeatnotesandthemodulationsignalfromtheEOMdriver. ............................. 129 B-1Circuitschematicofthephaselockingservo. .................... 143 10

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TheLaserInterferometerGravitationalWaveObservatory(LIGO)isnowoperatingatitsdesignsensitivity.Toextendtheabilitytosensegravitationalwavesignalsinabroaderrangewithreasonablerateofdetections,theAdvancedLIGOhasbeenplannedandR&Dforitisunderway.AdvancedLIGOisdesignedtohaveanorderofmagnitudebettersensitivitythenthecurrentdetector.Thiswillbeachievedbyimprovingseveraltechnicalfeatures,includingtheuseoftriplependulumsuspensionsystemsandanorderofmagnitudehigherlaserpower. GoingfromLIGOtoAdvancedLIGOrequiresnewdesignsforsomekeycomponentsorsubsystems.AspartoftheLIGOgroupattheUniversityofFlorida,IwasinvolvedintheresearchworkrelatedtotwocomponentsofAdvancedLIGO.TheseincludethetriplependulumsuspensionsystemwhichhasbeendesignedandinstalledintheJIFlabatGlasgowUniversity.Myresearchprovidesacompletemechanicalanalysisforthetriplependulumanddescribethelocalcontrolsystemwithfeedbackforcesappliedonthetopmassviasixelectro-magneticactuators.Alsopresentedisthecharacterizationmeasurementofthetransferfunctionofthissuspensionsystem. AnotherresearchprojectIhaveworkedonfocusesonthedevelopmentandtestofanovelelectro-opticphasemodulator(EOM),whichisakeycomponentoftheinputopticssubsystem.Theexperimentsincludemeasurementsofthephysicalpropertiesof 13

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1.1.1Introduction 2Rg=8G c4T(1{1) Thecurvedgeometry,beinginterpretedasthegravitationaleldofthemattersource,ischaracterizedbytheRiccitensorR,theRicciscalarRandthemetrictensorgintheleftpartoftheequation.Tisthestress-energytensor,andtheconstantisgivenintermsofc(thespeedoflight)andG(thegravitationalconstant).TheEFEisusedtodeterminethemetrictensorg. Thegeneraldenitionofthespace-timeintervalinGeneralRelativityis: Becausegravitationalwaves(GWs)arefromastronomicalobjectsveryfarawayfromusandthusveryweak,theycanbeseenassmalldisturbancestotheatmetric(Minkowskispace),denedas ^g=^+^h;(1{3) where ^=0BBBBBBB@10000100001000011CCCCCCCA:(1{4) Thesmalldisturbancehcanbesimpliedintoausefulandsimpleformwhenthetransverse-traceless(TT)gaugeisappliedtosolveforGWsinvacuum, 15

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Generalrelativity(GR)predictsthatGWstravelwiththespeedoflightcandcanbetreatedasplanewaves,beingseentobenon-dispersiveinthetheoreticaltreatment.ThequantityhisthewaveformwhichcanbeinterpretedasthesuperpositionoftwoorthogonalpolarizationstatesforGWs, ^h=0BBBBBBB@00000h+0000h+000001CCCCCCCA+0BBBBBBB@000000h00h0000001CCCCCCCA:(1{6) whereh+=hxx=hyyandh=hxy=hyxarethestrainamplitudeofGWswithtwodistinctpolarizations.ApassingGWwillimposeacompressingorstretchingeecton`freefalling'testmassesdependingonthephaseandpolarizationofthewave.Forexample,theeectontestparticlesarrangedinacircularringbyGWsoftwopolarizationscomingintotheplaneisshowninFigure 1-1 .IfListheoriginaldiameteroftheringandtwoGWshaveequalstrainamplitudeh+=h=h,thechangeofthediameterasindicatedinFigure 1-1 canbegivenas L=hL:(1{7) Inotherword,thestrainamplitudeofaGWcanbecharacterizedas L:(1{8)Intuitivelyspeaking,GWsareemittedwhenthemassdistributionofaobjectorsystemchangesinanoscillatoryway.Thisissimilartothewaythatachangeincharge 16

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EectoftheincidentGWswithtwopolarizationsontestparticlesarrangedinacircle.TheGWspropagateinadirectionperpendiculartothepage.GWreferstothephaseoftheGWs. distributioncreatesanelectricaldipolemomentandthusradiateselectro-magneticwaves.However,thelowestorderradiationmultipolethatthemassdistributionvariationcanproduceisquadrupoleradiation.GravitationalmonopolesexistandappearasNewtoniangravity,butoscillatorychangesinthemonopolemomentofmassdistributionareforbiddenduetomassconservation.Andchangesinthegravitationaldipolemomentareforbiddenduetoconservationofmomentum.Specically,GWsareemittedbyphysicalobjectswithachangingquadrupolemoment.Thestrainofthequadrupoleradiationhcanbeevaluatedas R1 HereRisthedistancetothesourceand^Qisthereducedquadrupolemomenttensorofthesourcemass,denedas 3r2dV;(1{10) where(r)isthemassdensity,theristhedistancefromthecenterofthemass,andtheintegralisoverthevolumeofthesource.Theamplitudeofthestrainisverysmall,whichmakesthedirectdetectionofGWsaverychallengingtask.Takeatypicalcoalescing 17

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1 ].Thersttypeofsourcesemitwavesinthe`extremelylow'frequencyband(1015to1018Hz),anditisbelievedthatinitialdensityperturbationsassociatedwithinationoftheuniverseintroducegravitational-eldperturbationinthisfrequencyband.Thesecondtypeofsourcesare`verylow'frequencyones(107to109Hz).Theexpectedsourcesincludesuper-massivebinaryblackhole(BH)systems.Thethirdtypesources,the`low'frequencyones(106to1Hz),includetheinspiralofthestellar-massbinaryobjects,shortperiodstellar-massbinaries,themergerofsuper-massivebinaries,thegravitationalcollapeofsuper-massivestars,andacosmicgravitationalwavebackground.Andsourcesofthefourthtype,the`high'frequencysources(1Hzto10KHz),consistofthemergerofstellar-massbinaries,supernovae,astochasticbackgroundandnon-axisymmetricspinningneutronstars(NSs). Ingeneral,heaviersourcescorrespondtolowerfrequencies.AndthesamephysicalobjectsmaygenerateGWradiationsatdierentfrequenciesastheyevolve.Forexample,theinspiralofastellar-massbinaryatearlystageradiatesinthe`low'frequencyband.Theradiationperioddecreasesastwocomponentsofthebinaryapproacheseachother.HencetheGWradiationfrequencyincreasesgradually.Inthelastfewsecondsofthebinary'sinspiral,thefrequencyhasalreadybeeninthe`high'frequencyband.ThefrequencyoftheGWradiationkeepsonincreasingasthebinarymergesandringsdown. 2 ].In1974,RussellHulseandJosephTaylorprovide 18

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3 4 ].However,thisindirectdetectionoftheGWsgeneratedbyabinarypulsardidnotprovideustheinformationcarriedbythewavesthemselves.HencethedirectmeasurementoftheGWstrainh(t)isneeded.DirectdetectionofGWswillopenupawayforexaminingthetheoreticalpredictionofh(t),testingtheastrophysicalmodels,andcharacterizingtheastronomicalsystemsthatgenerateGWs. SeveraltechniquesformeasuringGWsdirectly,besidestheresonantbardetectors,havebeenproposedorimplemented.ThesetechniquesincludespacecraftDoppler-tracking[ 5 ],pulsartimingsearches[ 6 ],ground-basedMichelsonlaserinterferometricdetectors[ 7 { 10 ],andthelaserinterferometerspaceantenna(LISA)[ 11 ].Theground-basedlaserinterferometricdetectors,asthemostpromisingGWdetectorsthatarecurrentlyinoperation,aregoingtobediscussedinSection 1.2 1-2 .Anideallaserinterferometricgravitationalwavedetectorisbuiltupbymirrorswhicharefreeofforces.Thelaserbeamissplitintotwobeamsatthebeamsplitter(BS).ThedistancebetweentheBSandtheendmirrors(M),thearmlength,canbealteredbyapassingGW.ThelaserinterferometerdenesitsowncoordinateswhicharedierentfromtheTTcoordinates.Inthiscoordinatesystem,thexandyaxesaredeterminedbythedirectionsofthetwoarmswiththezaxispointingperpendicularlytothem(seeFigure 1-2 ). Itisthelightthatisusedastherulertomeasurethechangeofthearmlength,whichisthesignaturebythepassingGWsontheinterferometer.Weknowthattheproper

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AMichelsoninterferometerwithtwoorthogonalarmslyingontheplane.ThepropagationdirectionofGWsintwopolarizationdirectionisverticaltotheplane.BS-beamsplitter,M-mirror,PD-photodetector. lengthbetweentwospace-timeeventslinkedbyalightiszero,whichcanbeexpressedas g0dxdx=0;(1{11) whereg0isthemetrictensordenedinthecoordinatesassociatedwiththedetector.Forlightinoneofthearms(e.g.thearmalongthex-axis), whereh0xxisthestrainalongthex-armoftheinterferometer.ThedistancethatlighttravelsfromtheBStotheendofthex-armandbackcanbecalculatedas 2Lx=ZLx0dxZ0Lxdx=Zttxc 2h0xx(t)dt0;(1{13) 20

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where!gisthefrequencyofaGWsignal,Equation 1{13 canthenbewrittenas 2Lx 2h0cos(!gt)dt0=xh0sin(!gt)sin(!g(tx)) 2!g:(1{15) Equation 1{15 canberewrittenas 2!g=2Lx 2!gx 2!gx:(1{16) Therefore,thephasethatlightaccumulatesduringaroundtripinthex-armis 2!gx 2!gx:(1{17) Asimilarexpressioncanbewrittenforthelighttravelsinthey-arm.Thisisgivenas 2!gy 2!gy:(1{18) ThemotionoftheendmirrorsineacharmsintroducedbyGWswillbeinadierentialmode.Thisintroducesvariationsofthephasedierence=xybetweenthetwobeamswhentheyarerecombinedattheBS,changingtheinterferencepatternatthedarkport.Thischangecanbepreciselysensedbyaphotodetector(PD).ThustheinformationofGWsisencodedinthereadoutsignalfromthePD. WhentheroundtriptimeisshortcomparedtotheGWperiodsuchthat!gx1and!gy1,thephasedierencebecomes 2!0LxLy 2!0h0(x+y)cos!gt2!0 21

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wherei=x;y,xwilldecreaseasLxincreaseswhileywillincreasewithLy,ascanbeseenfromEquation 1{17 andEquation 1{18 .Hencethesignalwilldegrade.Sothereisalimitonthearmlength,whichis Oncethelengthofthearmsbecomelongerthanthelimit,theGWstrainstartstooscillateasthelightpropagates,shiftingtheroundtripphasebacktowardszero.ThisdeterioratestheresponseoftheinterferometertoGWs. ThecoordinatedenedbytheinterferometerandtheTTcoordinatecanberelatedbyarotationaltransformation, where~xTrepresentsthecoordinatesassociatedwiththeTTgaugeand~xrepresentsthecoordinatesassociatedwiththedetector.Therotationaltransformationinducesthetransformationofthemetric.Thestraintensorinthedetectorframeh0,canbefoundfromtheTTtensorhbymeansoftheinducedtransformation, Euleranglesareusedtospecifytheorientationofthegravitational-waveframewithrespecttothedetectorframe, where',,aretheanglesdenedasshowninFigure 1-2 .Therotationsaroundthex,y,andzaxesarethusgivenby 22

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Theserotationsarethetransformationswithin3-dimensionalspace.Thestraintensorh0canthusberepresentedbya33matrix.Hencewecandescribeasthesumoftwopartseachoriginatingfromanindependentpolarization, =F+h++Fh:(1{28) Theinformationofthestrainamplitudeh+,histhusencodedin. 2(1+cos2)cos2cos2'cossin2sin2';(1{29) 2(1+cos2)cos2sin2'+cossin2cos2':(1{30)'isthepolarizationangleofthewave.andkeeponchangingastheearthorbitsaroundthesunandrotatesdaily.ThismotionalsogeneratesaDopplereectofthegravitationalwavesignals.Thebeampatternsarenearlyomnidirectionalexceptforcertainorientations.Hencethegravitationalwaveinterferometersaresensitivetoaverylargeareaoftheskyatalltimes. 23

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1{9 Thewaveformsofchirpsignalsfromcoalescingbinarieshavebeenknowntoahighdegreeofprecision.Hencetheclassicalmatchedlteringtechniquecanbeappliedontheoutputdataofthedetectorstoenableareliabledetection.However,onlyafewCoalescingNS/NSsystemshavebeenobservedwhilenoNS/BHorBH/BHhaveyetbeendiscovered,whichimpliesthatthedistributiondensityofcoalescingbinariesintheuniversecouldbesmall.HencethedetectionrateofthosesignalsusingGWdetectorsisveryuncertain. 24

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Thecharacteristicamplitudeofgravitationalwavesfrompulsarscanbegivenas hc=162Gf2 whereIisthemomentofinertiaofthepulsararoundtherotationaxis,fistheGWfrequency,",theequatorialellipticity,isameasureoftheasymmetryofthepulsar,andristhedistancebetweentheobserverandthesource.Forexample,ThecharacteristicfrequencyofgravitationalwavegeneratedbytheCrabpulsaris59.6Hz,anupperlimitofstrainamplitudeoftheGWsignalfromtheCrabpulsariscalculatedtobeh1024witha"7104andr=1:8kpc.WeakperiodicsignalslikethiswillbeburiedinthenoiseoftheoutputsignalsoftheGWdetectors.However,thesignaltonoiseratiocanbeincreasedbyintegratingtheoutputsignaloveralongobservationtime. ThedetectionofperiodicGWsignalswillberealizedbyapplyingfastFouriertransform(FFT)analysisonthedetectoroutputandconstructthepowerspectrum.IfwecouldkeeponsearchingtheskywithlongenoughobservationtimeT,thesignalswithnearlyxedfrequenciespeakswillshowupinthepowerspectrumsincethesignaltonoiserationgrowsasT.ThestatisticallyimportantpeaksinthespectrumcouldbeassociatedwiththeseperiodicGWsignals.However,suchdataanalysisalgorithmjustimposesabigcomputationalburdenonthesignalprocessingprocedure.ThisisbecausethattheintegrationrequiredtimeinordertoachieveareasonablesignaltonoiseratiowilllikelybesolongthatahugenumberofdatapointsneedtobehandledintheFFT 25

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Pulsarscanloserotationalenergybyelectromagneticradiation,theemissionofparticles,andgravitationalwavedissipation.Thatiswhytherotationalfrequencyvariesoveratimescaleintheorderofthepulsarage,whichiscalledthe`spindown'.ThisleadstothefrequencydriftoftheGWsignals.TheothermechanismthatcausesthefrequencychangeoftheobservedGWsfromapulsaristheDopplershift.SincethegrounddetectorsarecarriedbythespinningEarthorbitingtheSun,amonochromaticsignalinthesourcereferencewillbeDopplermodulatedbythemotionofthedetectors.ThefrequencymodulationduetotheEarth'sorbitalandrotationalmotionareexpectedtobeseeninthepulsarsignals.ThesefrequencycharacteristicscanbeutilizedtoextractpulsarsignalsfromthedataobtainedusingGWdetectors.Moreover,thechangingorientationofthedetectorswithrespecttothepulsarsourcesduetothemotionoftheEarthwillchangethe`beampattern'mentionedinSection 1.2 ,imposingamplitudeandphasemodulationontheGWsignals.Thephaseandfrequencymodulationofthesignalsbroadenthespectrallinesinthepowerspectrumandspreadthepowerintothefrequencybinsaroundthesignalfrequencies.Thisjustattenuatestheamplitudeofthemainsignalcomponentsandthusimposedanadditionaldemandforincreasingtheintegrationtime.Asolutiontothisproblemistobuildanewtimecoordinatesothatfrequencyofthesignalsremainsxed.Unfortunately,theconstructionofthenewtimecoordinatesmeansmoredataanalysiswork.Thesituationbecomesextremelyworseinthecaseofsearchingforsourceswhosepositionandfrequenciesarenotwellknownsincetheall-skyandall-frequencysearchwillbeneeded. Hencesearchingperiodicsignalsisbelievedtobefarmorecomplicatedthanthatforchirpsignals. 26

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Asupernovaisanexplosionofastarwhichcreatesanextremelyluminousobject.ThisresultsinaburstofradiationintheelectromagneticdomainwhichemitsasmuchenergyastheSunwouldemitoverabout10billionyears.AlthoughonlyasmallfractionofenergyisconvertedintoGWs,itcouldbebigenoughtomakesupernovaeimportantsourcesforgroundbasedinterferometricdetectors.Supernovaeisbelievedtobetriggeredintwoways.Intherstcase,awhitedwarfstarinabinarysystemmayaccumulatesucientmaterialfromastellarcompanion.AsthewhitedwarfapproachestheChandrasekharlimitofroughly1.4timesthemassoftheSun,itwillcollapse,triggeringastellarexplosion.Inthesecondcase,theironcoreofamassivestarceasestocreatenuclearfusionenergytoresistthegravity.TheresultingsuddencollapseofthestarwillgenerateaNSorBH,releasinggravitationalpotentialenergythatheatsandexpelsthestar'souterlayers.Thecollapsemustbenon-axialtoproducethechangeinthequadruplemomentuminordertogenerateGWs.Thewaveformofsupernovaesignalsarenotclearlyunderstood.PeoplecurrentlybelievedthatthecurrentsensitivityofgroundbasedinterferometricdetectorscanonlyallowustodetectGWsfromgalacticsupernovaeandthesignalsfromtheextra-galacticsourceswouldbetooweak. Gamma-raybursts(GRBs)areintenseashesofgamma-rayswhichlastfrommillisecondstomanyminutes,comingfromrandomlocationsinthesky.GRBscanbeclassiedbasedontheirdurationastwotypes:The`short'GRBshavealifetimelessthan2secondswhilethe`long'oneslastlongerthan2seconds.ThepresentconsensusisthatGRBemissionisassociatedwithBHformationprocessessuchashypernovae,collapsarsandcompactbinarymergers.SincecollapsarsandcompactbinarymergersarebothgoodGWsourcesdetectablebygroundbaseddetectors,wehavegoodreasonstotakeGRBsasimportanttargetsofthegroundbaseddetectorsaswell. 27

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12 ],largenumberofGRBeventshavebeenobserved.Thedetectableeventratecouldbeashighasonceperday.Thisisveryusefulindataanalysisusingstatisticalmethods. Thewaveformsofgravitationalwavesfromburstsourcesarepoorlyknown.Thecross-correlationanalysisusingdatafrommorethanoneGWdetectorsisneededtoextractsignalsfrominstrumentalnoise,assumingthereareburstsignalswithenoughstrengthinthesensitivebandofdetectors.Andbenetingfromthenon-GWobservationsofGRBs,wecanuseexternallytriggermethodstoreducethedetectionthreshholdforGRBsignals. IfthestochasticGWbackgroundisisotropic,stationaryandGaussian,itsspectrumcanthenbewellcharacterizedbyaquantityGW(f)whichisdenedas GW(f)=1 wherecisthecriticalenergydensitytoclosetheuniverse,GWistheGWenergydensity. Likewhatwedidtoacquireburstsignalsfromtheoutputsofmorethanonedetectors,thecross-correlationanalysisisalsodemandedtosearchforstochasticsignals.Thebasicideaistomultiplytogethertheoutputsfromtwodetectorsandintegratethem.Althoughthewaveformsofstochasticsignalsarenotclear,theireventratemightbemuch 28

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Groundbasedinterferometerswillbeinevitablyaectedbyseismicallydrivenvibrations.Suspensionisolationsystemsreducetheseismicnoiseinthemeasurementbandwidth.AnidealisolationsystemisexpectedtosuppresstheseismicnoisewhilelongitudinalmotionsofmirrorsintheGWbandwidtharefreeofdamping.However,theunavoidableseismicwallatlowfrequenciespreventscurrentgroundbasedinterferometersfromsensingGWsignalsfromsourcessuchasthemergersofsupermassiveblackholes.LISAwillbeagoodcomplementarityinthisregard. Thermalnoiseoriginatesthroughtwomechanisms.First,thethermalmotionofthesuspensionsystemcausesthemirrorstomove.Second,thethermallyexcitedvibrationofthemirrordeformsthemirrorsurfaces,i.e.,changesthearmlengthoftheinterferometer.Thethermalnoisecanbecharacterizedthroughtheindirectmeasurementofthemechanicallossofthesystem.Thelosscanbereducedbyimprovingseveralaspectsofthesystemsuchasprovidingbettercoatingsforthetestmass,utilizingbersuspensionwiresandimplementingcryogeniccoolingdevices. Thetemperaturegradientinthechamberhousingtheinterferometercouldproducedierentialpressureonthesuspendedmassesviatheresidualgasmolecules.Thusthethermallyintroducedmotionoftheresidualgasmoleculesleadstothedisplacementoftest 29

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13 ].Sotheresidualmoleculenoisecouldalsobeviewedasasensingnoise.SincethegroundbasedinterferometricGWdetectorsareallplacedinvacuumchambers,theeectduetoresidualmoleculesisconsideredtobetrivial. Theultimatelimitondisplacementnoiseisthegravitygradientnoise-noiseduetouctuatingNewtoniangravitationalforcesthatinducemotionsinthetestmasses.Animportantgravitygradientnoisesourceistheuctuatingdensityoftheearthneareachoftheinterferometerstestmasses.Atmosphericuctuationscouldalsochangethegravitygradient,buttheeectismuchweakerthanthatfromearthmotions.Gravitygradientnoiseduetomovingbodiesorobjectscanbemoreseriousthanseismicgravity-gradientnoiseifsuchbodiesorobjectsarenotkeptatanadequatedistancefromthetestmasses.Thegravitygradientnoiseisnegligibleinthecurrentexistinggroundbasedinterferometricdetectors.However,advancedinterferometerswithhighersensitivitiescouldbelimitedbygravitygradientnoiseatfrequenciesbelow10Hz. Lightinsidetheinterferometerexertsradiationpressureforcesonthetestmasses.Variationsofthelightintensityintroducesuctuationsintheforces,causingvibrationsofthemasses.Thisisknownastheradiationpressurenoise.Theanti-correlatedmotionofthemassescanbecalibratedinstrainsensitivityhas[ 15 ] wherePinistheopticalpowerinthearmcavity,Listhelengthofthearmcavity,isthewavelengthofthelaser,and~isthePlanckconstant.hrpincreaseswiththeopticalpower. Shotnoise,alsoknownasthe`photon-countingerror',isthefundamentallimittothemeasurementoftheopticalpower.Itisthestatisticaluctuationinthenumberofthephotonsmeasuredbyaphotodetectorcausesuctuationsintheamplitudeofthesignal.Thesignaltoshotnoiseratioincreaseswiththerootofthepower.Thecontributionof 30

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15 ] RadiationpressurenoiseandshotnoisearealsocalledQuantumnoise[ 14 ].Thetotalquantumnoiseisthequadraturesumoftheshotandradiationpressurenoise.Minimizingthequantumnoisewithrespecttothepoweryieldsaminimumnoiselevelwhichisclosetothestandardquantumlimit(SQL)[ 16 ], Here,misthemassofeachmirror,andListhelengthoftheinterferometerarms. Itisnoteworthytomentionthattheradiationpressurenoisewedescribedaboveisduetotheuncorrelatedpoweructuationintwoarmsthatisgeneratedbythequantumuncertainty.TechnicalradiationpressurenoisewhichoriginatesfromtheuctuationofthelaserpowerisnotrelatedtotheSQL. Amplitudeandphasechangeofthelasereldarecommonnoisesourcesinaperfectlybalancedinterferometer.However,whenthetwoarmsoftheinterferometerarenotperfectlyidenticalintermsofhavingdierentlossesandmirrorreectivities,thentheamplitudeandphasechangesoftheeldineacharmarenolongeridentical.ThisshowsupasdierentialdisplacementnoiseandcannotbedistinguishedfromaGWsignal. 1.5.1OpticalConguration 1-3 ).TheopticalcongurationofLIGOisplotted 31

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OpticalcongurationoftheLIGOdetector.BS-beamsplitter,PBS-polarizingbeamsplitter,EOM-electro-opticmodulator,ITM-inputtestmass,ETM-endtestmass,PRM-powerrecyclingmirror,AOM-acousticopticalmodulator. 32

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1-3 .AsmallfractionofthelightfromaNd:YAGlaserissplitawayfromthemainbeam,transmitsapolarizingbeamsplitter(PBS)andpassesanacoustic-opticmodulator(AOM)anda=4plate.Thelightisreectedbackbyamirrorplacedafterthe=4plate,passesthroughthe=4plateandtheAOMagain.Thedoublepassoflightthroughthe=4platerotatesthepolarizationofthelighttobeorthogonaltotheincominglight'spolarization.HencethelightisreectedbythePBS,passesthroughanEOMandgoestoareferencecavity.A=4plateandaPBSareusedagaintosendthelightreectedfromthereferencecavitytoaphotodetector.Thesignalfromthephotodetectorisdemodulatedtoobtaintheerrorsignalthatisusedtoadjustthefrequencyofthelaser.ThisisatypicalapplicationofusingthePound-Drever-Halltechnique[ 17 ]tostabilizethefrequencyofalaser.TheAOMservesasafrequencyosetbetweenthelaserfrequencyandtheresonantfrequencydenedbythereferencecavity.Thisway,thelaserfrequencycanbetuned/controlledbychangingthefrequencyoset,whilethehighfrequencystabilityisensured.TheEOMafterthebeamsplitter(BS)isusedtolockthepre-modecleaner.Themainlaserbeamisspatiallylteredbythepre-modecleanerrst.ThetransmittedlightisphasemodulatedbytwoEOMsplacedinseriestogeneratetwopairsofRFsidebandsaroundthecarriereldforthelengthsensingandcontrolofthemodecleanerandtheinterferometer.BothcarrierandRFsidebandsarespatiallylteredagainbythemodecleanerbeforebeingsenttotheinterferometer. Thecarriereldiskeptresonantinsidethepowerrecyclingcavityformedbythepowerrecyclingmirror(PRM)andtheinputtestmasses(ITMs).Inaddition,thecarrierlightmustbeheldresonantinsidethetwoFabry-PerotarmcavitieswhileRFsidebandsarereectedbytheITMs. TomeettheresonanceconditionsforthecavitiesandthedarkfringeconditionfortheMichelsoninterferometer,thelengthsofthisopticalcongurationshouldbecontrolled.Itisconvenienttodescribethetwoarmcavitiesbyuseofthecommonlength,L+=L1+L2,andthedierentiallength,L=L1L2.Thesetwolengths,plusthe 33

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2(l1+l2)andthepathlengthdierenceintheinterferometer,l=l1l2denethefourlongitudinaldegreesoffreedominapowerrecyclinginterferometersuchasLIGO.ItisthearmlengthdierenceLthatneedstobesensedwiththemoststringentrequirementsincethevariationofLisdirectlysensedastheGWsignals. Thewholeinterferometeriscontrolledviathelengthsensingandcontrol(LSC)subsystem.TheLSCoftheinterferometerevolvesfromthePound-Drever-Hallfeed-backcontroltechnique.Themodulatedlasereldreectsatandtransmitsthroughtheinterferometer.Theamplitudeandthephaserelationshipbetweenvariousfrequencycomponentsoftheeldchangeasafunctionofpositionsofthemirrorsthattheinterferometeriscomprisedof.Thechangeissensedbyphotodetectorsplacedatvariouslocationsintheinterferometer.Thesignalsfromthesephotodetectorsaredemodulatedatspeciedfrequenciestocreatetheerrorsignalsusedinthefeed-backcontrolloops. LIGOhasachieveditsdesigngoalwithpeakstrainsensitivityofh(f)31023=p TheresearchworktowardsanupgradefromLIGOtoAdvancedLIGOhasbeencarriedout.Besidesthedevelopmentofbettermaterialsandsuspensionsystemstoreducethethermalandseismicnoise,theapplicationofahighpowerlasertoincreasethesignaltoshotnoiseratioandthemodicationoftheopticalcongurationhavebeenproposed. Comparedwiththepower-recycledschemeinLIGO,theresonant-sideband-extraction,orRSEtopologyinAdvancedLIGO(seeFigure 1-4 )improvestheperformanceofthelaserinterferometerbyaddingasignalextractionmirror.ThenewcongurationallowsveryhighpowertobuildupinitsarmswithrelativelylowpoweratBS.RSE,whichwas 34

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OpticalcongurationoftheAdvancedLIGOdetector.PSL-laserpre-stabilizationsubsystem,IO-inputopticssubsystem,SRM-signalrecyclingmirror. rstproposedbyMizuno[ 18 ],ismorepracticalinAdvancedLIGOascomparedwiththedual-recycledtopology(powerrecyclingandsignalrecylcing)proposedbyMeers[ 19 ],whichhastheadvantageofrecyclingthesignalduetothedierentialmotionofthearmcavities.AsananalogtothewayFabry-Perotcavitiesoptimizethetimeofinteractionoflightwiththegravitationalwavesignalbyenlargingthelightstoragetime,thesignalrecyclingcavityincreasesthe`signalstoragetime'bykeepingthecarrierresonantinsidethecavityformedbythefrontmirrorsofthetwoarmcavitiesandthesignal-recyclingmirror.Althoughthesignal-recyclingcongurationwasdemonstratedtoconsiderablyimprovetheperformanceoflaserinterferometricGWdetectors[ 20 ],thestoragetimeforthesignalsidebandsmustbekeptshortenoughtoachieveadesireddetectionbandwidth.Thestoragetimelimitpreventsusfromusinghigh-nessearmcavitiesintheinterferometer.ThusahighpowerlaserincidentontheITMsbecomesnecessaryinordertoincreasetheshotnoiselimitedsignaltonoiseratio.Thisresultsin 35

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TheintroductionofthesignalextractionmirrorinAdvancedLIGOgreatlyenhancesthecomplexityofthelengthsensingandcontrolsystem.ThesignalrecyclingmirrorandtwoITMsformthesignalrecyclingcavitywhoselength,ls=lsr+1 2(l1+l2),hastobecontrolledaswellasthelengthintheothervedegreesoffreedom. Sincethecarrierlightwillnotbetransferredtotheasymmetricporteectively,thecontrolofthelengthofthesignalrecyclingcavityrequiresanadditionalsidebandsothatbeatsignalsbetweentwopairsofsidebandscanbeusedtomonitorthevariationofthesignalrecyclingcavity.Lengthsensingandcontrolindual-recyclinglaserinterferometerhasbeenstudiedthroughanumberoftable-topandprototypeexperiments[ 21 { 24 ].However,thenalcontrolschemeforAdvancedLIGOisstilltobedetermined. IthasbeenproposedtoplaceanoutputmodecleanerattheasymmetricportoftheinterferometerinAdvancedLIGO.TheoutputmodecleanerservestolteroutthelightinhighordermodeswhichincreasestheshotnoiseonthephotodetectorsdoesnotcontributetothedetectionofGWsignals. 36

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AdvancedLIGOwillusetheDCsensingscheme,inwhichthecarrierlightemergingfromthedetectionportisusedasthelocaloscillator.AscomparedwiththeRFreadoutscheme,theDCreadoutschemealsohasseveralpracticaladvantages:theopticalnoisecouplingsaresmallerwithaDCreadout;thereadoutelectronicswillbemuchsimplerandnoRFsidebandsemergefromthedetectionportsothatthephotodetectorwillnotbesaturatedwhenthelaserpowerisscaledtohighlevels[ 25 ].ThedisadvantageofDCsensingisitssubjectiontomanylowfrequencynoisesources. 2 .AlsodescribedinChapter 2 arethelocalcontrolschemeofthependulumsystemandsomeresultsfromtheclosed-looptransferfunctionmeasurementthatwasdoneintheJIFlabatGlasgowUniversity.ThesecondoneistheprojectofdevelopinganovelRTPcrystalelectro-opticmodulatortomeettherequirementsofAdvancedLIGO.ThedesignandcharacterizationofRTPcrystalbasedEOMsarepresentedinChapter 3 .ThesummaryandthediscussionofthefutureworkwillbegiveninChapter 4 37

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Thesimplestideaofthesuspensionisolationistoputthetestmassesonthe`stacks'.Figure 2-1 showsasimplestackconsistsofasupportingplatewhichisheldwithsprings.Therelationshipbetweenthegroundmotion,x0,andthedisplacementofthemass,x1,canbeexpressedas: wheremrepresentsthemassandkisthespringconstant.TheLaplacetransformgivesthetransferfunctionbetweenx1andx0: wheres=j!,!istheangularfrequencyand!0=2f0=2r mistheresonancefrequencyofthespring. Ifaviscousdampingforceisintroducedwithadampingconstantb,thetransferfunctioncanberewrittenas: ms !20+s2+b ms:(2{3) Thetransferfunctionoftheisolationsystem(seeFigure 2-2 )issimilartoalow-passltersothattheseismicnoiseabovetheresonancefrequencywillbeeectivelylteredout.ThedissipationofthesystemcanbedenedbythequalityfactorQ,whichisgivenbyQ=!0m b.Thefalloofthetransferfunctionisproportionalto1/f2fromf0toQf0andbecomes1/fathigherfrequencies. 38

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Asimplespringstackandapendulum. Figure2-2. Transferfunctionofasimplespringstack. 39

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Aspringstackcanprovidegoodisolationsalongtheverticaldirection.Anotherexampleofthesuspensionisolationisapendulum(alsoseeFigure 2-1 ).Pendulumsuspensionsystemscanbeusedtoisolatethetestmassfrombeingaectedbythehorizontalseismicnoise.Asimplemodelforthependulumismadeofamasswhichissuspendedonawireoflengthl.Supposingthemassofthewireisnegligibleandthedampingconstantisthesameconstantbthatwementionedabove,thesametransferfunctioncanbederivedasthatforasimplespringstackexceptthattheresonantfrequencyisnowgivenby!0=2p 2-3 ,thissuspensionsystemconsistsofatwo-layerisolationstack,twocantileverspringswhichareattachedtothetopofthestack,andatriplependulum.Theuppermassofthependulumissuspendedfromthespringswithtwowiresandthedoublependulumconsistingoftheintermediatemassandthelowermassaresuspendedfromextrafourcantileverspringswithfourwires.Theintermediatemassandthelowermassarealsoconnectedwithfourwires.Theintermediatemassandthelowermassareequal,inordertoensureamonolithicresponsetodisturbances.Thetestmassneedstobecontrolledinordertolocktheinterferometer.Areactionpendulumishenceinstalledparalleltothependulumwhichholdsthetestmass.IntegratedOpticalpositionSensor/ElectroMagneticdrivers(OSEMs)aremounted 40

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Threedimensionalviewofthetriplependulumsuspensionsystem. 41

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Schematicviewofthetriplependulum.Thestacksandthedampingarmhavebeenomittedforclarity.Thisamodieddrawingwhichoriginatesfrom[ 26 ]. 42

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2-4 ).This,inturn,adjuststhefeedbackforceonthemagnet. Detailsofthetriplependulumsuspensionsystemsuchasthedesignofthetwo-layerisolationstack,thedesignofthecantileverspring,andtheactiveisolationwhichincludesboththefeed-backandthefeed-forwardcontroltechniqueshavebeengiveninthethesesofTorrie[ 26 ]andHusman[ 27 ].Theyeachbuiltamechanicalmodeltogiveacompletecharacterizationofthemodefrequenciesandthedynamicresponseofthesinglependulum,usingstandardvectoranalysisandtheLagrangiananalysisrespectively.Themechanicalanalysisisextendedtothetriplependulum.However,thedetailsaboutthetriplependulummodelforalldegreesoffreedom(DOF)arenotcompletelyoutlined.Inthefollowingsections,acompleteanalysisofthetriplependulawhichareinstalledintheJIFlabatGlasgowUniversityiscarriedout.Inaddition,aMatlabmodelisconstructedwiththeactivecontrolonthetopmassbeingincluded.ThismodelconsiderssomedierencesbetweenthependulumintheJIFlabandthependuluminstalledforGEO600.Itwillbeagoodsupplementtothetwotriplependulummodelsthatarementionedabove. 2.3.1VariablesandParameters (i)Longitudinalmotionx,alongthe^x-axis, (ii)Sidewaysmotiony,along^y-axis, (iii)Verticalmotionz,alongthe^z-axis (iv)Roll,aboutthe^x-axis, (v)Pitch,aboutthe^z-axis, (vi)Yaw,aboutthe^y-axis. 43

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`T-shape'topmasswithsixmagnetsattached. Hencethetriplependulumhas18degreesoffreedomandtherefore18modefrequencies.Thedierentialequationsofmotionofthetriplependulumforalldegreesoffreedomarederivedhere.Thecontrolforcesonthetopmassareappliedviasixmagnets,whichareincludedintheequations.Theglobalcontrolontheintermediatemassandthelowertestmasshasbeenplanned,butisnotconsideredhere.TheparametersasspeciedinFigure 2-7 usedarelistedbelow: (i)Separationofwiresinthe^xdirection (ii)Separationofwiresinthe^ydirection 44

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Photographofthetopmass. 45

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(iv)Separationofmagnetsontheuppermass (v)Heightofmagnets1,2and6abovethecenterofmass,ls. (vi)Lengthofwiresandtheirprojectionintheverticaldirections (vi)MomentaofinertiaIi;jandmassesmi,i=1,2,3andj=x,y,z.1,2,3-theuppermass,theintermediatemassandthelowermassandx,y,z-directionsoftheangularmotion. (vii)Springconstantsoftheuppercantileverblades,intermediatebladesandlowerwires,k1,k2,k3. 46

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Parametersofatriplependulum. 47

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Photographofatriplependulum. 48

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wherez3,z2representstheverticaldisplacementofthelowertestmassandtheintermediatemass.Similarly,theequationfortheintermediatemassis ConsideringacontrolforceFvappliedverticallyontheuppermassviaOSEMs,theequationfortheuppermassis Toderivethetransferfunction,theaboveequationsarerewrittenintheFourierdomain,utilizingthetransformationz(t)=ei!tz(!).Theseequationsbecome Therelationshipbetweenthecontrolforceandtheverticalmotionoftheuppermasscanbedescribedinthetransferfunctiongivenbelow: 49

PAGE 50

BodeplotofHzv. Figure 2-9 istheBodeplotofthetransferfunctionHzv. First,thedynamicequationsofasinglependulumsuspendedfromtwowiresoflengthlwithaspringconstantkarederived.Denesasthehalfseparationbetweentwosuspensionpointsanddastheheightofthewirebreakingopointsabovethecenterofmass.Figure 2-10 showsasinglependulumsuspendedbytwowires.Considerthecasewhenthemassistiltedbyananglefromthehorizontallineanddisplacedbyxinthe^x-direction.Thewiresareaectedinoppositeways.Wire1iscontractedandwire2is 50

PAGE 51

Sideviewofasinglependulum'slongitudinalandpitchmotion. stretchedascanbeseeninFigure 2-11 .InFigure 2-11 ,theunxedendofwire1movesfromNtoP0.MNisthedisplacementinthe^x-direction. Forasmallangleandasmalldisplacementx, 51

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Longitudinaldisplacementintroducedbythepitchmotion. Asmallangleapproximationgives, Sothelengthofwire1changesfromtheoriginallength,l,to Hencethechangeinthelengthofwire1isls.Wecanalsogive sinxd l;(2{20) cos1:(2{21) 52

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sinxd l;(2{22) cos1:(2{23) Inanequilibriumstate,thetensionineachwireis 2mg:(2{24) Wire1iscontractedsuchthatintherstorderapproximation, 2mgkl:(2{25) Andthetensioninwire2becomes 2mg+kl:(2{26) Hencetheforceonthemassalongthe^x-directionis SubstitutingEquation 2{25 and 2{26 intoEquation 2{27 gives l:(2{28) Finally,theequationforthelongitudinalmotionofasinglependulumsuspendedwithtwowiresis l:(2{29)Thenettorquewhichtiltthemasscanbecalculatedas SubstitutingEquation 2{20 2{21 2{22 2{23 2{25 and 2{26 into 2{30 gives 53

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Componentsofrestoringforceswhichacttotiltthemass. lx2ks2+mgd+mgd2 Hencetheequationofmotionforpitchis lx2ks2+mgd+d2 Theequationsforthelongitudinal/pitchmotionofatriplependulumcanbederivedbasedonthemechanicalanalysisofthesinglependulumthatisdescribedabove,i.e.,theequationsforeachtestmasscanbegivenbysimplymodifyingEquation 2{30 and 2{32 .iandxiareusedtodenethetiltangleandthelineardisplacementofeachmassofthetriplependulum(i=1,2,3-uppermass,intermediatemassandlowermass).Thevariablesxandarereplacedwiththerelativedisplacementofeachmasswithrespecttothesuspensionpointsandthetiltanglesbetweentwoadjacentmasses.Henceforthe 54

PAGE 55

x3=g lt3[(x3x2)(d43d32)];(2{33) Theequationforthepitchmotionis Itisnoteworthytomentionherethatthelengthofthependulumistheverticaldistancebetweenthesuspensionpointsoftwostages.Theintermediatemasswillbestretchedbyboththelowerwiresandtheintermediatewires.TheintermediatewiresstretchtheintermediatemassthesamewaythatthelowerwiresdothelowermasswhilethelowerwiresaectstheintermediatemasstheoppositewayaccordingtoNewton'sthirdlaw.Nowthetensionalongintermediatewireshavetobalancethegravityforceduetototalmassoftheintermediatemassandthelowermass.Thedynamicequationsare lt2[(x2x1)(d22d11)]+m3g lt3[(x3x2)(d43d32)];(2{35) lt3s22(32)+m3gd33+d43d32 2(m1+m2+m3)g:(2{37) Bothitsverticalandhorizontalcomponentsacttotilttheuppermassandthenettorqueis 55

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Sideviewofthetopmasssuspendedwithtwowires. Thedierentialequationsfortheuppermasshavetoincludetheexternalcontrolforceandthecontroltorque.Theforcealongthe^x-directionFlandthetorquethatcontrolsthepitchmotion,QpareappliedviatheOSEMs.Inthiscasetheequationsbecome lt1(x1d01)+(m2+m3)g lt2[(x2x1)(d22d11)]+Fl:(2{40) TheFouriertransformationofEquation 2{33 2{34 2{35 2{36 2{39 and 2{40 gives: 56

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lt3d4d32+!2I3y4k3s22m3gd4m3g lt3(s22+d24)3=0;(2{42) (m2+m3)g lt2x1+m2!2(m2+m3)g lt2m3g lt3x2+m3g lt3x3(m2+m3)g lt2d11+m3g lt3d3+(m2+m3)g lt2d22m3g lt3d43=0;(2{43) lt2d1d21+"!2I2y4k2s21(m2+m3)g lt2d22(m2+m3)gd24k3s22m3g lt3d23#2+4k3s22+m3g lt3d3d4+m3gd33=0;(2{44) lt2(m1+m2+m3)g lt1x1+(m2+m3)g lt2x2+(m1+m2+m3)g lt1d0+(m2+m3)g lt2d11(m2+m3)g lt2d22=Fl;(2{45) and lt2d211+(m2+m3)gd1+4k2s21+(m2+m3)g lt2d1d22=Qp:(2{46) Theseequationscanbewritteninamatrixform: 57

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11=m1!2(m2+m3)g lt2(m1+m2+m3)g lt1;(2{48) 12=(m2+m3)g lt2;(2{49) 14=(m1+m2+m3)g lt1d0+(m2+m3)g lt2d1;(2{50) 15=(m2+m3)g lt2d2;(2{51) 21=(m2+m3)g lt2;(2{52) 22=m2!2(m2+m3)g lt2m3g lt3;(2{53) 23=m3g lt3;(2{54) 24=(m2+m3)g lt2d1;(2{55) 25=(m2+m3)g lt2d2+m3g lt3d3;(2{56) 26=m3g lt3d4;(2{57) 32=1;(2{58) 33=!2lt3 35=d3;(2{60) 36=d4;(2{61) 41=(m1+m2+m3)gd0 42=(m2+m3)gd1; lt2(2{63) 44=!2I1y4k2s21(m1+m2+m3)gd0 lt2d21;(2{64) 45=4k2s21+(m2+m3)gd1+(m2+m3)g lt2d1d2;(2{65) 51=(m2+m3)gd2 58

PAGE 59

53=m3gd3 54=4k2s21+(m2+m3)g lt2d1d2;(2{69) 55=!2I2y4k2s21(m2+m3)g lt2d22(m2+m3)gd24k3s22m3g lt3d23;(2{70) 56=4k3s22+m3g lt3d3d4+m3gd3;(2{71) 62=m3gd4 63=m3gd4 65=4k3s22+m3g lt3d4d3;(2{74) and 66=!2I3y4k3s22m3gd4m3g lt3d24:(2{75) Thustheresponseofthetriplependulumtotheexternalforceappliedinthelongitudinaldirectionandthetorquethattiltsthetopmasscanbecalculatedas whereT=1.TheresponseofthetopmasstoFlandQpcanbeanalyticallycalculatedas 59

PAGE 60

BodeplotofHxl. and ThesetransferfunctionsareplottedinFigure 2-14 ,Figure 2-15 ,Figure 2-16 ,andFigure 2-17 respectively. 2-18 .ThetensionTinonewirecanbeprojectedalongtwodirections.TheverticalcomponentTvbalancesthegravityforcewhilethehorizontalcomponentThalongthedirection!BDisbalancedbythehorizontalforceintroducedfromanotherwire.However,whenthemassrotatesthroughanangle,thenettorquebythehorizontalcomponentsofthetension 60

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BodeplotofHxp. Figure2-16. BodeplotofHl. 61

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BodeplotofHp. forceoffourwireswillnotkeepthemassinabalancedstateanylonger.InFigure 2-19 wecanclearlyseethatitistheforceFthatactstorotatethemass.FistheprojectionofThinthedirectionperpendiculartothelineconnectingthesuspensionpointandthecenterofmass, where 4mgjBDj jCDj(2{82) and cos=jABj2+jBDj2jADj2 62

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Yawmotionofasinglependulum.Theupperpartistheviewfromabove.Andthelowpartisthegeometricplotoftheeectononewirewhenthemassisrotatedthroughanangle. 63

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Projectionofthetensiononto^x-^yplanewhichproducestherestoringtorque.Theupperplotandthelowerplotareassociatedwithtwodierenteectsonwireswhenthemassrotates. 64

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Here\BADcanbe90+or90,dependingonwhetherawirestretchesorcontractswhenthemassrotates.InFigure 2-18 ,wires1and3stretchwhilewires2and4contractwhenthemassrotatesinaclockwisedirection.ThestretchandthecontractionchangesjBDjsuchthat Therefore,wecanwriteThandcosas cos=(s2+n2i)2+(s2+n2i)2+(ninj)22p ninj!p ninjsin!1p ninj!p ninjsin=nicos ninjsin;(2{88) 4mgninjs Nowthetotaltorquefromfourwiresis 2mgninj+s 2mgninjs Hencetheequationofmotionforafourwiresuspensionis 65

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and Thelinedeterminedbythetwouppersuspensionpointsonthetopmasspassesthroughthecenterofmass.Sothehorizontalcomponentsoftensionsintwouppersuspensionwirecanbewritteninarelativelysimplerform 2(m1+m2+m3)gn1n0 Thisproducesatorqueonthetopmasswhenthemassisrotatedbyasmallangle, Thereforethedierentialequationwhichdescribestheyawmotionoftheuppermassbecomes whereQyistheexternalfeedbacktorqueusedtostabilizetheyawmotionofthetopmass.TheFouriertransformofEquation 2{92 2{93 and 2{96 arelistedbelow, 66

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BodeplotofHy. FromEquation 2.3.4 2{98 and 2{99 ,thetransferfunctionthatrelatestheexternaltorqueQytotheinducedrotationofthetopmass1isderivedas ThetransferfunctionHyisplottedinFigure 2-20 67

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Faceonviewofthesidewaysandtherollmotionofasinglependulum. 68

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Expandedtoshowvariouslengths. ConsiderasinglependulumsuspendedwithtwowiresasisplottedinFigure 2-21 .Thetwowireshavethesamelengthlintheequilibriumstate.Andthetensionsinbothwiresare, 2mgl lt(2{101) Afterthemassisdisplacedbyasmalldistanceyinthe^y-direction,thecenterofmassmovesfromOtoP.Thisisfollowedbyamassrollingbyasmallangle'.ThesuspensionpointsnowmovefromGandJtoEandF.Thusthenewlengthsofthetwowirescanbederived.FromFigure 2-21 andFigure 2-22 l':(2{103) Hencewire1isstretchedby l=njni l':(2{104) 69

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l':(2{106) Wire2isthencontractedbyl.Nowthetensionsinwire1andwire2become 2mgl lt+kl;(2{107) 2mgl ltkl:(2{108) Theangleofwire1withrespecttotheverticaldirection1satises sin1=njni l(njni)(njlt+njdnid) and cos1=lt Theslopingangleofwire2satises sin2=njni l(njni)(njlt+njdnid) and cos2=lt Theequationofmotionforthedisplacementofthecenterofmass,y,is SubstitutingforT1,T2fromEquation 2{107 and 2{108 ,for1and2fromEquation 2{109 2{110 2{111 ,and 2{112 andusingtherstorderapproximationforsmall 70

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Thecomponentsofthetensionintwowiresacttorollthemass. displacementyandsmallangle'gives lt"1(njni)2 ltd(njni)(njlt+njdnid) 2-23 ,wecanseethatthecomponentofforcewhichrollsthemassforwire1isT1sinandforwire2isT2coswhere and Theangleisdenedbytheparametersofthependulum.UsingEquation 2{109 2{110 2{111 and 2{112 ,sinandcosarecalculatedtobe 71

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l(njni)(njlt+njdnid) cosnjlt+d(njni) l(njni)(njlt+njdnid) Nowthetorquefortherollmotionis lt(njni)(njlt+njdnid) l+mgdnjnjni Andtheequationofrollmotionaroundthecenterofmassis lt(njni)(njlt+njdnid) l':(2{120) 72

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Relativemotionbetweentwoadjacentmasses. Theextensiontoatriplependulumisoutlinedbelow.Althoughthelowermassandtheintermediatemassarebothsuspendedwithfourwires,thewaythewiresintroducethesidewaysandtherollmotionisnotdierentfromthecasewheretheyaresuspendedwithtwowiresinthemannerdescribedabove.However,sincewhatweneedtoconsiderhereistherelativemotionbetweentwoadjacentmasses,theequationsofmotionbecomecomplicated.InFigure 2-24 ,thelengthofthesuspensionwirescanbecalculatedbysubstitutingywithymyn,d'withdl'mdu'n,andnj'withnj'mni'ninEquation 73

PAGE 74

and 2{106 .Hencelbecomes l=njni Theangleofwire1withrespecttotheverticaldirection1nowsatises sin1njni+ymyndl'm+du'n cos1ltnj'm+ni'n Thesinandcosvalueoftheslopingangleofwire2nowbecome sin2njniym+yn+dl'mdu'n 74

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TheforceonthelowermassinFigure 2-24 becomes 2mgl lt(sin1+sin2)kl(sin1sin2)(mg"1 Andtheequationofmotionforthedisplacementofthelowermassmmis Here, whereMmrepresentsthetotalmasssuspendedonthewire.Ifthereisnoothermasssuspendedbelowmm,Mm=mm. Theanglesl,l,uanduasspeciedinFigure 2-24 canbegivenusingsimplegeometricanalysis. 75

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Therefore,thetorquewhichrollsthelowermassis 2Mmglm where Andthetorquewhichrollstheuppermassis where Equation 2{127 2{134 2{138 canbeusedasgeneralequationstodescribetherollandsidewaysmotionofallthreemassesofthependulum.ThefunctionsR,S,VA,B,C,D, 76

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Forthelowermass, where Theequationsfortheintermediatemassare 77

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Here 78

PAGE 79

with 79

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and TheEquation 2{142 2{149 2{159 2{143 2{150 2{160 arerstFourier-transformedandthenrewritteninmatrixform Here, 11=R1+R2+m1!2;(2{170) 12=R2;(2{171) 14=S1n1R1d0+S2n2R2d1;(2{172) 15=S2n3+R2d2;(2{173) 21=R2;(2{174) 22=R2+R3+m2!2;(2{175) 23=R3;(2{176) 80

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25=S2n3R2d2+S3n4R3d3;(2{178) 26=S3n5+R3d4;(2{179) 32=R3;(2{180) 33=R3+m3!2;(2{181) 35=S3n4+R3d3;(2{182) 36=S3n5R3d4;(2{183) 41=A1+D1;(2{184) 42=D1;(2{185) 44=B1n1+n2E1+d1G1+(m1+m2+m3)gn1n1n0 45=n3E1+d2G1+(m2+m3)gn3n3n2 51=A2;(2{188) 52=A2+D2;(2{189) 53=D2;(2{190) 54=n2B2d1C2;(2{191) 55=n3B2+d2C2+n4E2+d3G2+(m2+m3)gn3n3n2 56=n5E2d4G2;(2{193) 62=A3;(2{194) 63=A3;(2{195) 65=B3n4C3d3;(2{196) 81

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HencethesidewaysmotioninducedbyFsandtherollmotioninducedbyQrcanbedescribedas where=1.ThetransferfunctionsthatcharacterizetheresponseofthetopmasstoFsandQrare TheBodeplotsfortheabovetransferfunctionsareshowninFigure 2-25 ,Figure 2-26 ,Figure 2-27 ,andFigure 2-28 respectively. 2-29 showsthecontrolloopofthetriplependulumsystem.Here,CrepresentsthecontrolchannelswhileSrepresentsthesensingchannels.Thecontrolsignals,whichisthechangeofthecurrentincoils,introduceforcesonthemagnetsattachedonthetopmass.T1istheactuatormatrixthatdescribestheconversionfromthecurrentinthecoilstotheforcesandtorquesonthetopmass, 82

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BodeplotofHys. Figure2-26. BodeplotofHyr. 83

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BodeplotofH's. Figure2-28. BodeplotofH'r. 84

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Feedbackcontrolblockdiagramofthetriplependulum. 85

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ThetransferfunctionsfromtheforcesandtorquestothemotionsareincludedinH.TheresultingmotionsofthetopmassinsixDOFaresensed.T2isthesensormatrixthatdescribestheconversionfromthesemotionstothechangeofthepositionsofmagnets, where T2=0BBBBBBBBBBBBBB@1ls0ly001ls0ly000lp100lr0lp100lr00100lr00001ls1CCCCCCCCCCCCCCA:(2{206) Theinducedcurrentvariationsarefedbackastheerrorsignalstoadjustthecurrentincoils.Gisthefeedbackcontroller. Onlythelocalcontrolonthetopmasshasbeenconsideredinthismodelforthetriplependulumsystemsofar,butthemodelcanbeextendedtoincludetheglobalcontrolon 86

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Closelooptransferfunctionmeasurementforthetriplesuspensionsystem. allthreemasses.Themodelcanbeusedtogureoutthemodefrequenciesofthetriplependulum.Itisalsogoingtobeastraightforwardpracticetoaddthedampingforcetermsinthemodel.Understandingthedynamicsofthetriplependulumisessentialto,forexample,studythecrosscouplingeectwhichisthemostseriousprobleminvolvingamultistageisolationsystem,andmodeltheeectofdampingthependulummodefrequencies. 2-29 isshowninFigure 2-30 .WhitenoiseinputsignalsgeneratedbyanSR785modelsignalanalyzerwereinjectedintospeciedchannels.Bychoosingdierentinputpoints,motionsofthependulumofdierentdegreesoffreedomwereexcitedwhile 87

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2-1 liststhreesetsofmeasurementswithspeciedinputandoutputpoints.Andthetransferfunctionsareplottedandcomparedwiththemodelingresults. Figure 2-31 showsthecloselooptransferfunctionwithchannel1andchannel2astheinputpointsandchannel1astheoutputpoint.Thistransferfunctionwasusedtoconrmthemodefrequenciesofthelongitudinalmotion.Thereisareasonablygoodmatchbetweenthemeasurementresult(theredcurve)andthemodelingresult(thebluecurve).Wecanexcitedlongitudinalmodeat6Hz,1.3Hzand2.3Hz,whichhasbeenshowninthependulumtransferfunctionplot(Figure 2-14 ).Theextradipsataround1Hzand2.1Hzshowninthemeasurementresultarebelievedtobeassociatedwiththeyawmotionthatwasexcitedduetothedierencebetweentheelectricalgainfortwocoils.ThesetwomodefrequenciesareconformedinFigure 2-20 .Thesmalldipslocatedbetween7Hzand8Hzinthemeasurementcurverepresentmotionsofthesupportingstackwhichareusedtoholdthesuspensionpendulum. VerticalmodescanbeseeninthetransferfunctionsplottedinFigure 2-32 .Mismatchesbetweenthemodefrequenciesshowninthemeasurementresultandthecalculatedonesarearound0.4Hz.Thisispossiblyduetotheinaccuracyofthespringconstantsofthewiresusedinthemodel. ThetransferfunctionsshowninFigure 2-33 wereobtainedwiththeinputandoutputpointsconnectedinchannel6.Boththemeasurementandmodelingresultsshowsideway Table2-1. Inputandoutputpointsfordierenttransferfunctionmeasurements. MotionInputpointoutputpoint longitudinalCH1+CH2CH1verticalCH3+CH4CH5sidewayCH6CH6 88

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Closelooptransferfunctionwiththeinputsignalbeinginjectedtocoil1andcoil2andtheoutputsignalfromchannel1. 89

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Closelooptransferfunctionwiththeinputsignalbeinginjectedtocoils3,4and5andtheoutputsignalfromchannel3. 90

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Closelooptransferfunctionwiththeinputsignalbeinginjectedtocoil6andtheoutputsignalfromchannel6. modesaround0.59Hzand1.3Hz.Thereisa2.1Hzdipinthemeasurementcurvewhilethemodelingresultshowsa2.2Hzdip. 91

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3.1 showsthemodiedschematicviewoftheIOsystemdesignedfortheAdvancedLIGO(theoriginalgureistakenfromRef.[ 28 ]).ThelightisconditionedintheIOtomeettheprimaryscienticrequirementsforAdvancedLIGO. ThelaserbeamfromthePSLisrstmodulatedusingradiofrequency(RF)electro-opticmodulators(EOMs)togeneratethefrequencycomponents(sidebands)forcontrollingtheinterferometer.TheworkingmechanismofanEOMwillbediscussedinSection 3.2 .Themodematchingtelescope(MMT)aftertheEOMsmodiesthemodeofthelighttomatchthemodedenedbythemodecleaner.Thepowercontrolpartisbasedontheconventionaldesignofusingapolarizerandahalfwaveplate.Amotorizedrotationalstepperstagewillbeusedtorotatethehalfwaveplate.Theopticalpowerincrementalstepneedstobesmallandthepowerchangingrateneedstobeslowtoensurethatthelength-controlsystemofthemodecleanerwillbeabletotrackthedisturbanceonthemirrorsduetothechangeoftheradiationpressure. Theinputmodecleanerstabilizesthefrequencyandsuppressesspatialuctuationsofthelaserbeam.Thelengthofthemodecleanercavitydeterminesthatthemodulationfrequenciesneedtobeintegermultiplesofthefreespectralrange(FSR)ofthemodecleaner.Thatis =Nc whereisthemodulationfrequency,Nisaninteger,cisthespeedoflightandListhelengthofthemodecleanercavity.Inthisway,bothcarrierlightandsidebandswillresonateinsidethemodecleanerandwillbetransmitted. TheFaradayisolator(FI)performstheconventionalroleofrejectingtheback-reectedlightfrompropagatingtowardstheinputport.TheFIdesignedfortheAdvancedLIGO 92

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OverallIOschematic. 93

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TheMMTaftertheFImatchesthebeamtothepropermodetosatisfytheresonantconditionofthemaininterferometer.Currently,therearetwoconceptualdesignsfortheMMT,accordingtowhichtwopossibledesignsofthepowerrecyclingcavity(PRC)willbeused:eitheramarginallystablepowerrecyclingcavity(MSPRC)orastablepowerrecyclingcavity(SPRC)[ 29 ].MSPRCwillallowmanyspatialmodesofRFsidebandstoberesonantinsidethePRCandintroducelossestotheTEM00modeofthesidebands.SPRCcanreducethelossinthefundamentalmode,buthasthepotentialdrawbackofincreasingthecomplexityofthealignmentsensingandcontrol(ASC)systemandmightcontributetotheparametricinstabilities[ 30 ]. TheIOsubsystemofAdvancedLIGOrequireshigherqualityopticalcomponentsincomparisonwithLIGO.Theelectro-opticphasemodulator(EOM),oneofthekeycomponentsintheIO,isofinterestinthisthesis. 94

PAGE 95

CongurationofanEOM. frequency.Thelighteldcanbewrittenas HereE0istheeldamplitude,!istheangularfrequencyoftheoriginallasereld,isthemodulationfrequency,andmisthemodulationindex.Thehigherorderharmonicsareomittedintheequation. Alllasereldcomponents,includingcarrierandsidebands,aredeliveredtotheinterferometer.Thereected,transmitted,andinternalpickoeldswhicharesensedbyinterferometercontrolphotodetectors,whichareplacedatvariousdetectionports,canbegenerallydescribedbyfrequencydependenttransferfunctions, 3-3 showsthepossiblelocationsofthephotodetectorsusedforthelengthsensingandcontrol 95

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PossiblelocationsofphotodetectorsinAdvancedLIGO. inAdvancedLIGO.AshasbeenbrieymentionedinChapter 1 ,demodulatingaspeciedbeatsignalbetweentwofrequencycomponentsofthelasereldwillyieldanerrorsignalthatrevealsachangeincertainlongitudinaldegreesoffreedomofcertainmirrors. Theproposedcontrolschemes[ 31 32 ]forAdvancedLIGObothemployonepairofsidebandsinordertosensethelengthofthemodecleanerandtwopairsofsidebandstosensethelongitudinaldegreesoffreedomoftheinterferometer.However,whentwophasemodulatorsareusedinseries,thesidebandscreatedbytherstEOMwillbephasemodulatedbythefollowingone.Thisgenerates`sidebandsonsidebands'whichwillbeatwiththecarriereld.Thefrequenciesofthebeatsignalswillbeidenticaltothefrequenciesofthebeatsignalsbetweenthesidebands.BeatnotesbetweensidebandswillbeusedtogeneratethelengthsensingsignalsofAdvancedLIGO.Hencetheexistenceof`sidebandsonsidebands'willproduceosetsinthereallengthsensingsignalsanddegradethediagonalizationofthelockingmatrix. 96

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TheopticalcongurationofusingaMach-Zehnderinterferometertoeliminatethesidebandsonsidebandsproblem. Therearetwotechniquesproposedasthesolutiontothesidebandsonsidebandsproblem.OnesolutionistosplittheincomingbeamusingaMach-ZehnderinterferometerandplaceanEOMineacharm[ 33 ].InthisMach-Zehnderinterferometer,the`sidebandsonsidebands'donotexistanymorebecausethesidebandsgeneratedbyoneEOMarenotphasemodulatedbytheotherEOM.Anothersolution,termedcomplexmodulation,involvestheuseofoneamplitudemodulatorandonephasemodulatortoapplyspecied(non-sinusoidal)amplitudeandphasemodulation,resultinginanelectro-opticallymodulatedeldwhichiscomprisedofdesiredfrequencycomponentswithoutmixingterms[ 34 ].Thetargetmodulationstateisreachedbyadjustingonlytheelectricaleldswhichdrivebothmodulators.Ingeneral,therelationshipbetweentheoriginallighteldEandtheamplitudeandthephasemodulatedlighteldE0canbedescribedas whereA(t)istheamplitudemodulationfunction,and(t)isthephasemodulationfunction.GivenEandadesiredmodulatedeldE0,thesetwofunctionscanbecalculated: 97

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Detaileddiscussionofthelengthsensingandcontrolofthedual-recyclinginterferometerisbeyondthescopeofthisthesis.WhatwefocusonarethetechnicalissuesrelatedtotheapplicationofEOMs.Ingeneral,therulesofthumbare: i)TheimplementationoftheEOMsshouldnotaectthefunctionofotherunitsoftheinterferometer; ii)TheEOMsshouldnotproducenoisethatcoulddeterioratethesensitivityofAdvancedLIGO. 3-1 listssomephysicalpropertiesofRTPcrystals. RTPcrystalshaveanorthorhombiccrystalstructureandtheirEO-coecientmatrix(linearelectro-optictensor)is: 98

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PhysicalpropertiesofRTPcrystals. properties(units)xyz 35 ]. 36 ]. 35 ]. 37 ]. 3-5 ).Thedierencebetweentherefractiveindexofthep-polarizedlightandthatofthes-lightresultsintwodierentrefractionanglesaccordingtoSnell'slaw.Boththefrontandtheendsurfacesofthecrystalhavethesamewedgeangle.InFigure 3-5 ,thelightpathofthep-polarizedlight 99

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RTPcrystals(wedgedandnon-wedged)mountedbetweentwoelectrodes. parallelstheelectrodessothatthedeectionangleofthep-polarizedlight,'out(p),equalstheincidentangle,'in.Thes-lightandp-polarizedlightnolongeroverlapeachotheraftertransmittingthewedged-crystal. Themodulationindexofthemodulatorforthep-polarizedlightis r33n3zVz andthatforthes-lightis r13n3xVz whereListhelengthofthecrystal,disthethicknessinthe^z-direction,isthewavelengthofthelight,nzandnxaretheindicesofrefractionofthecrystalalongthez-axisandthex-axis,andVzistheappliedvoltage.Ournon-wedgedRTPcrystalshaveadimensionof4mm4mm15mm,thusthehalf-wavevoltageVforthep-polarizedlightis612.5V. 100

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CircuitdiagramfortheEOM. Becauseelectro-opticcrystalsperformlikeacapacitorinelectroniccircuits,thesimplestwaytobuildupthevoltageacrossthecrystalistoconnectitinserieswithaninductortoformanL-Cresonantcircuit.TheimpedanceoftheresonantcircuitneedstobematchedtotheoutputimpedanceoftheEOMdriver(usually50)attheresonantfrequencytoreducethepowerlossduetostandingwaves.ThisissimplyrealizedbyconnectinganadjunctcapacitorinparallelwiththeLandCcrystal.ThecircuitdiagramisshowninFigure 3-6 38 ]. 101

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3.4.1.1Theory Ingeneral,thepiezoelectricresonancefrequenciesaredeterminedbythevelocityofthesurfaceacousticwave(SAW),vsaw,andthedimensionsofthecrystalLx,LyandLz.Itcanbecalculatedas wherei=x;y;zandN=1;2;3.ASAWisanacousticwavetravelingalongthesurfaceofamaterialhavingsomeelasticity,withanamplitudethattypicallydecaysawayfromthesurface.ThephysicalpropertiesoftheSAWanditscrystalorientationdeterminetheSAWvelocity.vsawcanbecalculatedas, :(3{11) whereisthemassdensityofthematerial,cistheeectiveelasticconstantwhichdependsontheelasticandpiezoelectrictensorsandtheirtemperaturecoecients.FurtherdiscussionoftheeectiveelasticconstantcanbefoundinRef.[ 39 ]andexamplesforcalculatingcaregiveninRef.[ 40 ]. 102

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ExperimentalarrangementforthepiezoresonancemeasurementofanRTPcrystal. 3-7 showstheexperimentalsetuptomeasurethepiezo-resonancefrequenciesoftheRTPcrystal.Weshineda1064nmNd:YAGlaserthroughtwocrosspolarizersandalignthelaserbeamonaphotodetector(ThorlabsPDA255).TheRTPcrystalwasplacedbetweenthepolarizers.Thecrystalwithbothendsanti-reectioncoatedwasmountedbetweentwoelectrodeswhichwereconnectedtothesourceofaspectrumanalyzer.Thesignal,insweptsinemodefromthespectrumanalyzer,wasappliedacrosstheelectrodestomodulatethecrystal. Thepolarizationdirectionoftheincominglight,whichwasdeterminedbytherstpolarizer,wasmisalignedwiththeopticalaxisofthecrystal.Applyinganelectriceldtothecrystalinducedachangeintheindicesofrefraction(bothordinaryandextraordinary)givingrisetoanelectricelddependentbirefringence.Thecrystalactedasavariablewaveplatewiththechangeofthephaseretardationlinearlydependentonthe 103

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RTPcrystalpiezo-resonances. appliedelectriceld.Thesecondpolarizerconvertedthephasechangeintoanamplitudemodulatedsignal.ThewaythatthiskindofsetupconvertsthephaseretardationintoanamplitudemodulationsignalwhichwasdetectedbythephotodetectorisgoingtobedescribedinSubsection 3.4.3 .Theamplitudemodulationsignalwasmaximizedbyrotatingthepolarizationdirectionofthepolarizers.Atthepiezo-resonancefrequenciesofthecrystal,theresponseofthelighttothedrivingsignalwasgreatlyenhancedandappearsasresonantpeaksinthespectrummeasuredviathespectrumanalyzer. FromthetransferfunctionplottedinFigure 3-8 ,thelargestpiezoresonanceisat680kHz.Theresonancewiththehighestfrequenciesareseenaround6MHz.AlthoughthemodulationschemeofAdvancedLIGOhasnotbeennalized,thereisnoproposalofusingaresonantmodulatorbelow9MHz.Consequently,piezo-resonanceswillnotbeatechnicalproblemwiththeapplicationofthenewRTPcrystalmodulatorinAdvancedLIGO. 104

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3.4.2.1Thermaleectsincrystals whereT(r;z)isthespatialtemperaturedistributioninsidethecrystal.A(r;z)accountsforthedepositionofheatduetotheopticalabsorptionofthelaserbeam.isthethermalconductivity.ThesolutioncanbeobtainedanalyticallyornumericallyoncetheboundaryconditionsforEquation 3{12 aregiven.Theboundaryconditionsaredeterminedbytheshapeofthecrystalandthewaytheheatisextractedfromthecrystal.InourEOMs,thegroundelectrodebelowthecrystalwhichdirectlycontactsthealuminumpackagingcaseplaystheroleofaheatsink.WithoutgivinganexplicitexpressionforthetemperatureproleT(r;z),weknowthatthecenterofthecrystalisatahighertemperaturethantheedgesduetotheGaussiandistributionofthelightintensity.ThetemperaturegradientT(r;z)changestherefractiveindexofthecrystalalongtheaxesperpendiculartothepropagationaxisaccordingtothesizeofdn=dT.Sothelaserbeamtraversingthroughthecrystalexperiencesachangeintheopticalpathlengthduetothepositiondependentrefractiveindex.Thermo-elasticdeformationandthermallydependentelasto-opticeectswillchangetheopticalpathlength.Inourcase,thesetwoeectsaresmallerthanthedn=dTeect.Sowecanroughlyestimatethespatialdependenceofthedierentialoptical 105

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Experimentalarrangementforthermallensingmeasurement. pathlengthas: OPL(r)=dn dTLZ0T(r;z)dz:(3{13) ThisadditionalopticalpathlengthputanadditionalspatialphaseonthefundamentalTEM00modeofthebeamtransmittinganEOM.Thiseectisknownas`thermallensing'becauseitaltersthemodalpropertiesofabeaminawaysimilartoanordinarylens.Thethermallyinducedmodaldistortionwascharacterizedbymonitoringthechangeofthebeamdivergenceduetothe`thermallens'createdintheEOMcrystal. 3-9 showstheexperimentalarrangementusedtocharacterizethethermallensingeect.WeusedasinglemodeCWlasertoprobethe`thermallens'formedinsideaRTPcrystal.Theoutputpowerofthelaserwas100W.First,thelaserbeamwasfocusedbyalensandthebeamdivergencewasmeasuredusingabeamproler.Theradiusofthebeamwaistafterthelenswasabout0.4mm.ThentheRTPcrystalwasplacedinthebeampathabout0.25mawayfromthewaistwherethebeamradiuswasabout600m.ThislocationwaswithintheRayleighrangeofthebeam.Thebeamdivergenceafterthe 106

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ThemeasuredlaserbeamdivergencewithandwithouttheRTPcrystalinthebeampath. crystalwasmeasuredagain.ThechangeinthebeamdivergenceisplottedinFigure 3-10 .ThebeamdivergencecanbettedintotheproleswithaM2valueofabout1.75,whichisduetothelasermodaldistortioncreatedbytheotheropticalcomponentsheatedbythehighpowerlaser.Usingsimplegeometricoptics,thefocallengthfofthethermallenscanbederivedfrom f=tantan:(3{14) Here,wisthebeamradiusatthecrystal,andtheanglesandrepresentthefar-elddivergenceofthelaserbeamwithandwithoutthecrystalinthebeampath.Hence,wecalculatedalowerlimitonthe`thermallens'ofabout9m.This9m`thermallens'correspondedtoachangeinthelaserbeam'swavefrontsagittaldepth,s,ofabout10nm 107

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Changeofthewavefrontatthebeamwaist. (seeFigure 3-11 ).Thiswasestimatedfrom: 41 ]: dn dT:(3{16) Fromthemeasurementresultasdescribedabove,theopticalabsorptioncoecientoftheRTPcrystalcanbeestimatedtobeabout250ppm/cmusingEquation 3{16 InAdvancedLIGO,thelaserpowerwillbe180W.Sothethermallensingmagnitudeintermsofscanbeestimatedtobe1.8timesthesintroducedbya100Wlaser,whichis18nm.Supposingthatalllightcanbecoupledintothearmcavitieswhentheinputlaser'spowerislow,thechangeofthewavefrontwhenthelaserpowerisincreasedto180Wwillcausemodemismatch.Thefractionoflightthatwillnotbecoupledtothearmcavitiesduetothechangeofthewavefrontcanbeestimatedtobe[ 42 ] Ps 2:(3{17) SothemaximumpowerthatcouldbelostduetothethermallensinginanRTPcrystalEOMis0.25%assumingthatwedon'tcompensatethisinthemode-matchingtelescope. 108

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ResidualamplitudemodulationduetothemisalignmentofanEOM. Itisnoteworthytomentionthatthe0.25%modemismatchisderivedbasedonanassumptionthatthethermallyinducedopticalpathlengthgradientfollowsaparaboliccurveinsidethecrystal.Hencethe`thermallens'describedhereisanidealsphericallens.Inprinciple,thechangeofthemodematchingduetoanadditionalsphericallensinthebeampathcanbecorrectedbyadjustingthemode-matchingtelescope.Inreality,thenon-sphericalpartofthe`thermallens'isunavoidable,whichwillcreatehighordermodesthatcannotbecoupledtoarmcavities.However,thepowerlossduetothethermallensinginsideRTPcrystalEOMsinAdvancedLIGOwillbemuchlessthan0.25%andthelossesbecomenegligible. 3.4.3.1Generationmechanism 3-12 whichshowstheincominglightasalinearlypolarizedlightwithawavevectornormaltothefrontsurfaceofthecrystal.Theanglebetweenthepolarizationdirectionoftheincominglightandtheeaxisofthecrystalis.Thelighteldcanbeseenasconsistingoftwoorthogonalcomponentsalongtheprincipalaxesofthecrystal.Theopticalpathlengthsforthetwocomponentsaredierentduetothebirefringenceofthecrystal.Theresultingphaseretardationcauses 109

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theoutputeldcanbewrittenas Hereistheanglebetweenthecrystalaxisandthedirectionofthepolarizer.isthemodulationfrequency.m1andm2aretwomodulationindicesalongtwoprincipalcrystallographicaxes,whichareplottedasthee-axisandtheo-axisinFigure 3-12 andlayintheplaneverticaltothebeampropagationdirection.isthephaseretardationbetweenthelighteldalongthee-axisandthelighteldalongtheo-axis.Thisisduetothedierencebetweentherefractiveindices,noandne,alongthesetwoaxessuchthat, =(none)! cL:(3{20) Equation 3{19 canbeexpandedas Thusthecomplexamplitudeofthecarriereldis Sinceandcanbeminimizedtobeverysmallthroughalignment,EccanbeapproximatedasJ0(m1)+eiJ0(m2)E0.Hencethecarriereldcanbeseenasconsistingoftwoparts:theoriginaleldJ0(m1)E0ei!tandaresidualcomponenteiJ0(m2)E0ei!t. 110

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Changeoftheorientationofprincipalaxeswithrespecttothelighteldwhenavoltageisappliedacrossthecrystal. Similarly,anRFsidebandalsoconsistsofaprincipalcomponentJ1(m1)E0ei(!+)tandaresidualtermeiJ1(m2)E0ei(!+)t. Equation 3{19 describesRAMasafunctionofmisalignmentanglesforasimpliedcasewhichisshowninFigure 3-12 .Considermoregeneralcaseswhentheincominglightisnotverticaltoendsurfacesofthecrystal.ThewaythatrefractiveindiceschangewithanelectricaleldbeingappliedacrossthecrystalneedstobespeciedindetailinordertoaccuratelycalculatetheamountoflightthatisamplitudemodulatedbytheEOM,knowingthemisalignmentangles.InFigure 3-13 ,~Eisthelighteldwithawavevector~k.Beforeapplyingvoltageacrossthecrystal,theprincipalcrystallographicaxesarealongtheX,YandZaxes,respectively.Andtheorientationandrelativemagnitudeofrefractiveindicesofthecrystalcanbedescribedbytheindexellipsoidequation, 111

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where"x,"yand"zarecomponentsoftheappliedelectriceldalongtheX,YandZdirections.Thisnewellipsoidreducestotheoriginalone(denedbyEquation 3{23 )when~"=0. Anewsetofprincipalaxes(X0,Y0,Z0axesinFigure 3-13 )canbefoundbyaprincipal-axistransformationofthequadraticform TheSmatrixcanbetransformedtoadiagonalmatrixS0.ThecomponentsofS0areeigenvaluesofS.Themagnitudesoftheprincipalaxesoftheindexellipsoid(i.e.,therefractiveindicesalongnewprincipalaxes)are wherei=X0;Y0;Z0.Theychangeastheelectriceld~"changes.Ingeneral,theorientationofthenewprincipalaxesalsochangewith~".Butinsomecases,theneworientationoftheellipsoidcouldcoincidewiththeunperturbedaxes.Forexample,when~"isappliedalongZ-axisofanRTPcrystal,thequadraticformbecomes 112

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Theorientationoftheprincipalaxesremainsunchangedwhilethenewrefractiveindicesare 1 2n3xr13"z;(3{28) 1 2n3yr23"z;(3{29) 1 2n3zr33"z:(3{30) Theorientationandmagnitudesofthenewprincipalaxesdeterminethepolarizationofthelighteld~E0aftertheEOM.~E0canbewrittenas where1,2and3,whichsatisfyi=2niL ;i=x;y;z,arephasesaccumulatedbythreeorthogonaleldcomponentsE0x,E0yandE0zasthelighttravelsthroughthecrystal.Thisellipticallypolarizedlighteldcanbeconvertedintoanamplitudemodulated,linearlypolarizedlightbyapolarizer.InFigure 3-13 ,planeABCDistheincidentplanedenedbythepolarizer.Theeldofthetransmittedp-polarizedlightisgivenby where,andaretheanglesbetweenX0,Y0,Z0axesandplaneABCD. RAMcanalsobeintroducedviatheFabry-Perotcavityeectifanun-wedgedcrystalisusedinsideanEOM.Back-reectionsbetweentheendsurfacesofthecrystalwillalter 113

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3-14 ,therelationshipbetweentheincidenteldEandthetransmittedeldE0canbewrittenas wherer1,t1aretheamplitudereectivityandtransmissivityofthefrontsurface,r2,t2arethoseoftheendsurface,isthephasedeterminedbytheopticalpathlength :(3{34) whereisthewavelengthofthelighteld,Listhelengthofthecrystal,andnistherefractiveindexalongthethe^zdirection.WhenasinusoidalvoltagesignalVsintisappliedacrossthecrystalalongthe^zdirection,therefractiveindexbecomes 2n30r33V d(3{35) accordingtoEquation 3{30 ,wheren0istherefractiveindexbeforeapplyingthedrivingvoltage.Hencethephasecanbewrittenas where r33n30V d:(3{38)HenceEquation 3{33 becomes Equation 3{39 impliesthattheeldE0becomesanamplitude-modulatedeld.Themagnitudeoftheamplitudemodulationincreaseswiththereectivityr1andr2. 114

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Fabry-Perotcavityeectduetothebackreectionbetweenthefrontandendsurfacesofthecrystal. Figure3-15. TheexistenceofRAMchangesphaserelationshipbetweentheleakedcarrierlightandtheRFsidebandsatthedarkportofLIGO. 1.5 ,changesintheamplitudeandphaserelationshipbetweenthecarriereldandthesidebandsreecttherelativepositionsofthemirrorsinaninterferometricGWdetector.However,RAMalsochangestheamplitudesandphasesofthelightelds.Thesechangesareindistinguishablefromthesignatureduetothemotionofthemirrors.RAMresultsinunwantedosetsinthelengthsensingandcontrolsignals.Thisisintuitivelyunderstandablewithaphasor-diagram(seeFigure 3-15 ).TheleftpartofFigure 3-15 showsusthelighteldsatthedarkportofLIGOforanidealcasethatRAMiscompletelyeliminated.CrepresentsthecarriereldandSrepresentsanRFsideband.Whentheinterferometerislocked,thephasedierencebetweentheleakedcarrierlightandtheRFsidebandsatthedarkportwillbeexactly90.Thedeviationof 115

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Therightphasor-diagramdescribeshowRAMcreatesabeatnotebetweenthecarrierlightandtheRFsidebandswhenthelengthsoftheinterferometersatisfytheresonantcondition.HereCAandSAareresidualcomponentsaddedonthecarrierandanRFsidebandrespectively.TheresultingcarriereldC0andRFsidebandS0arenotorthogonaltoeachothersothatthebeatnotewillleadtothegenerationoffeed-backcontrolforceswhichactonthemirrorsandpushtheinterferometerawayfromtheidealresonantcondition.TheamplitudeofRAMmustbekeptbelowacertainleveltoensurethatthestaticlength-errorsoftheinterferometerarewithinthedesigntolerances.ErrorssignalsduetoRAMcanalsobebalancedoutbyaddingelectricalvoltageosetsinthefeed-backcontrolloops. Possibleosetsgeneratedinvariouscontrolloopsdependsonthelengthsensingandcontrolschemeoftheinterferometricdetector.InLIGO,thefractionalRAMisrequiredtobelessthan103toensurethatthestaticerrorofthedierentialarmlengthislessthantheresidualdeviationrequirementof1013[ 43 ].ThelengthsensingandcontrolschemeforAdvancedLIGOisnotnalized,butwillbemuchmorecomplicated.However,becauseoftheintrinsicasymmetriesinthedetunedAdvancedLIGOinterferometerandDCsensing,therequirementofRAMwillprobablynotbetightened. 3{20 and 3{19 ,wecanseethatRAMwillvarywhenthetemperaturedrifts.ThiscouldaecttheoperationofaninterferometricdetectorsincetheadjustmentoflaserpowerwillpotentiallyleadtoatemperaturedriftwhichinturnchangestheRAM.Soa 116

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ExperimentalsetuptocharacterizetheresidualamplitudemodulationcreatedbyanEOMheatedbya1053nmNd:YLFlaser. pre-installationtestofthepowerdependentRAMimposedbyanEOMisessentialforhighpowerlaserinterferometers,suchasAdvancedLIGO. Figure 3-17 istheopticallayoutoftheexperimentthatisusedtomeasuretheRAMgeneratedbyanEOMbeingheatedbyalaseratdierentpowerlevels.Thepump-probeschemeexperimentuseda500mwnon-planarringoscillator(NPRO)Nd:YAGlaserastheprobelasertosensetheamplitudemodulationsignalanda50WNd:YLFlaserastheheatingsource.AFaradayIsolator(FI)wasusedtoprotecttheprobelaserfrombeingaectedbytheback-reectedlightorlightfromthepumplaser.Ahalfwaveplateandapolarizingbeamsplitterwereusedtocontrolthelaserpower.Thepolarizingbeamsplitterwasalignedsothatthetransmittedlightwass-polarizedwithrespecttotheopticaltable.ItsdivergencewasadjustedbyaMMTconsistingoflensesL1andL2.ThebeamwasthenreectedbyathinlmpolarizerTFP1.Thebrewsterangleofthethinlmpolarizerisabout56.TheEOMwasalignedaspreciselyaspossibletomatchthecrystal'sprincipalaxistotheverticaldirection.ThebeamdiameteroftheprobelightattheEOMwasabout1mm.TheprobelightisthenreectedbyanotherthinlmpolarizerTFP2.Partofthelightwasthensplitoandsenttowardsanopticalspectrumanalyzer(OSA)which 117

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CorrelationbetweentheheatingpowerandRAM. wasusedtomeasurethemodulationindexoftheEOM.Theresttransmittedanothercubepolarizerwhichwasalignedinthesamewayastherstone.Thetransmittedlaserbeamwasfocusedonaphotodetector.ThephotodetectorhasseparatedDCandACoutputports.TheamplitudeoftheDCsignalfromthephotodetectorisproportionaltothetotalintensityofthelightdetectedwhiletheACsignalwasusedtomonitortheamplitudemodulationofthelasereld.TheheatingbeamfromtheYLFlaserpropagatedintheoppositedirection,withthebeamdivergencebeingadjustedbyaMMTconsistingoflensesL3andL4.RotatingthehalfwaveplateinfrontofTFP2changedtheintensityofthep-polarizedlightthatwasdeliveredtotheEOM.TheoutputpoweroftheYLFlaserwasxed.SoweknewthepoweroftheheatingbeamthattransmitstheEOMbymonitoringthelaserpowerreectedatTFP2.Thediameteroftheheatingbeamwascontrolledtobeabout2mmattheEOMtoavoidclipping.Theextinctionrationofthethinlmpolarizerisgreaterthan1000:1(Tp:Ts)sothatmostoftheheatingbeam 118

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TheamplitudeoftheACsignalwasmeasuredusingaspectrumanalyzerandtheDCsignalwasmeasuredusinganoscilloscope.Thesesignalswerecontinuouslyrecorded.Westartedwithtrackingthedriftoftheresidualamplitudemodulationformorethan20minutes.ThenweexposedtheEOMtoa5WheatinglaserandkeptonmeasuringthetemporalvariationoftheRAM.Theheatingpowerwasthenincreasedto20Wandnallyto40Winthefollowingtwostepsandthetimeseriesmeasurementswererepeated. Themodulationindexismaintainedtobeabout0.2duringthemeasurement.FromthemeasurementresultplottedinFigure 3-17 ,weclearlysawthecorrelationbetweentheheatingpowerandtheRAMcreatedbytheEOM.WealsonoticedthatrealigningtheEOMcanbringtheRAMbacktotheoriginallowlevels.RAMcreatedbytheEOMthatweusedcanbeminimizedtobebelow105oftotallightintensity.WebelievedthisisduetotheFabry-PerotcavityeectimposedbytheunwedgedcrystalinsidetheEOM,sincethereectivityoftheAR-coatedendsurfacesofthecrystalwasmeasuredtobeabout0.2%,whichishighenoughtocreatea105amplitudemodulation. 3.4.4.1Generationmechanism 44 ].Thenalrequirementonthelasernoiselevelis 119

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ThelasernoiseassociatedwithanEOMisgeneratedthroughseveralmechanisms.Noisesourcesareduetoeithertheinstabilityofthemodulationsignal,orunstablephysicalpropertiesoftheEOMcrystal.Twonoisegenerationmechanismsareaddressedhere. First,variantmodulationsintheindexcausesamplitudenoiseofboththecarriereldandsidebands.Thechangeoftheamplitudeofthecarriereldasafunctionofthemodulationindexcanbewrittenas Therelativeamplitudenoiseofthecarrierduetothevariationofthemodulationindexis J0(m0):(3{41) TheamplitudeofanRFsidebandchangeswiththemodulationindex Therelativeamplitudenoiseofthesidebandis Second,variationsofRAMalsocontributetotheamplitudeandthephasenoiseofthelighteld.Equation 3{21 describesRAMasafunctionofthemodulationindexm,misalignmentanglesandandthephaseretardation.FluctuationsoftheseparametersatfrequenciesintheGWbandwidthwillleadtoadditionallasernoisethatcouldinterferewiththesignal. WecanseefromEquations 3{8 and 3{9 thatthestabilityofthemodulationindexofanEOMdependsonthestabilityofthedrivingsignal,refractiveindices,electro-optic 120

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Deformationofendsurfacesofacrystalwhenanacousticwavepassesby. coecientsanddimensionsofthecrystal.TheamplitudeandphasestabilityofthedrivingvoltageisdeterminedbythequalityoftheoscillatorusedtodrivetheEOM.Ultra-lownoiseoscillatorsaredemanded.Refractiveindicesandelectro-opticcoecientsaretemperaturedependent.Temperatureuctuationswillalsoleadtotheexpansionorcontractionofthecrystal.Moreover,thethermal-elastoeectcouldcreatestraininsidethecrystal,whichinturnchangestheindexellipsoidviatheelasto-opticeect.Insum,itisoscillatornoiseandtemperaturevariationsthataectthemodulationindex.ThesetwonoisesourcesalsocauseuctuationsofRAM.First,uctuationsofthemodulationindexinducedbythesetwonoisesourceswillleadtouctuationsofRAM.Thermallyinducedvariationsoftherefractiveindicesanddimensionalchangesofthecrystalresultinuctuationsof.Variationsoftheprincipalaxesofthecrystalcausevariationsofthemisalignmentangles-and.Insum,webelievetwoprincipalnoisesourcesthatperturbtheamplitudeandthephaseofthelaserviaanEOMareoscillatornoiseandtemperaturevariations. Anotherpotentialnoisesourceisacousticnoise.Acousticwavescouldchangethedimensionsofthecrystalviatheelasto-acousticcouplingastheypassthroughthecrystal,perturbtheopticalpathlengthwhichalighteldexperiencesinsidetheEOM.ThisinturnleadstothevariationofthemodulationindexandRAM,generatingresiduallasernoise.AstraightforwardcaseisshowninFigure 3-18 .Thedeformationdepthsoftwoendsurfacesofthecrystalareexpectedtobedierentduetotheenergydissipationofthe 121

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ExperimentalsetuptocharacterizethelaseramplitudeandnoiseimposedbyanEOM. acousticwave.Uncorrelatedvibrationsoftwosurfacesresultinuctuationsoftheopticalpathlength. 45 ].Figure 3-19 showstheelectro-opticallayoutoftheexperiment.Thetwolaserswerenon-planarringoscillator(NPRO)Nd:YAGlaserswitha1064nmwavelength.ThelasereldofNPRO1propagatesthroughahalf-waveplateandapolarizerrst,allowingthepowertobecontrolledbyrotaingthehalf-waveplate.ThelasereldwasthenmodulatedviaanEOMwhichwasdriven 122

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ThelasereldfromNPRO2beatedwiththecarriereldandRFsidebandsofNPRO1atthetwophotodetectors(PD1andPD2).DemodulatingthebeatnotesasmeasuredbyPD1withfrequency0usingafunctiongeneratorandadoublebalancedmixercreatedanerrorsignal.ThedemodulatedsignalwaslteredbythePLLcontrollerandusedtophase-lockNPRO2withrespecttoNPRO1.TheconstantphaseosetbetweentwolaserswasmaintainedbytuningthefrequencyofNPRO2.Thefrequency/phaseconversionaddeda6dB/octaveslopetothelooptransferfunctionofthephaselockingloop(PLL),whichwasapartoftheproportional-integral(PI)controldesignedforthesystem.Figure 3-20 showsthelooptransferfunctionofthePLLmeasuredusinganetworkspectrumanalyzer.WecouldseethattheunitygainfrequencyofthePLLwasaround30KHz. IftheeldofNPRO1isrepresentedbyE1andtheeldofNPRO2isrepresentedbyE2,thephotocurrentofeachphotodetectorcouldbewrittenas {z }C-C+2J1(m)E1E2cos[(0)t+]| {z }C-S2J1(m)E1E2cos[(+0)t+](3{44) Figure 3-21 showsthefrequencyintervalsbetweeneldsoftwolasers. ThesignalfromPD2wasanalyzedbythephasemeterdevelopedbytheLISAgroupattheUniversityofFlorida.Thephasemeterisadigitalsignalprocessingdevicethat 123

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LooptransferfunctionofPLL. canbeusedforlock-indetectionofRFsignals.Ithadfoursynchronizedinputchannelsandtheanalog-to-digitalconverter(ADC)ofeachchanneldigitizestheincomingsignalwitha100MHzsamplingrate.Thedigitalltersinsidethephasemetercandecimatethedigitalsignalbyafactorof210.Thephasemeterprovidedtheamplitudeandthephaseinformationoftheinputsignalsviathein-phase/quadrature(I/Q)demodulationtechnique.TheschematicoftheI/QdemodulationofthedigitizedsignalisshowninFigure 3-22 .Inordertoovercomethedynamicrangelimitationof90or180forphasemeasurement,thelocaloscillator(LO),anumericallycontrolledoscillator(NCO),oftheI/QdemodulatorwasphaselockedtotheRFsignalundertestbytuningthefrequencyoftheLOsignalaroundthefrequencyoftheinputsignal. ThefunctiongeneratorsusedtodrivetheEOMandtoprovidethereferencesignalforphase-locking,andthephasemeterweresynchronizedtoasynthesizedclockgenerator 124

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Theeldsoftwophase-lockedlasers. Figure3-22. BlockdiagramofaI/Qmeasurementsysteminsideaphasemeter. (Stanfordresearchsystem,modelCG635)inordertoavoidthephasenoiseduetothetimejitter. Inourexperiment,weusedthreechannelsofthephasemetertomeasuretheamplitudeandthephaseofthecarrier-carrier(C-C)beatnoteoftwolasers,ofthebeatnotebetweenthecarrierofNPRO2andoneRFsidebandofNPRO1(C-S),andofthemodulationsignalfromtheEOMdriversimultaneously.TheangularfrequenciesoftheNCOsfor3channelsweretunedtobe0,0and,respectively.Theseamplitudeandphasedatawererecordedintheformoftimeseries.Thelinearspectraldensity(LSD)oftheamplitudeorthephasecouldbecalculatedviatheFouriertransformation. WetriedourrsttestonanunwedgedcrystalEOM(manufacturedbyNewFocus).TheEOMwasdrivenbya27MHzsinusoidalsignalfromafunctiongenerator(Flukemodel6062A).ThereferencesignalinthePLLwasa10MHzsinusoidalsignalfromanotherfunctiongenerator(StanfordresearchsystemmodelDS345).Theamplitudenoise 125

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AmplitudenoisespectraofthelaserbeatnotesandthemodulationsignalfromtheEOMdriverasmeasuredbythephasemeter. andphasenoisespectraofthe10MHzC-Cbeatnoteand17MHzC-SbeatnoteareshowninFigure 3-23 andFigure 3-24 ,respectively. Asfarasweknow,theamplitudenoiseofbeatnotescomefromseveralsourcesbesidestheamplitudenoiseimposedbytheEOM:intrinsiclaseramplitudenoise,variationoftheoverlapping(beamjitter)betweentwolaserbeams,andelectronicnoiseassociatedwiththephasemeter(e.g.digitizationnoise).NoticingthatthersttwoeectswillimposenoiseontheC-CandtheC-Sbeatnoteswiththesamerelativeamplitude,suchnoise,beinglabeledas`common-mode'noise,canbeeliminatedviaacommon-moderejectionanalysis.Specically,iftherelativemagnitudeofthe`common-mode'amplitudenoiseisrepresentedbyAcwhileA1andA2representtherelativeamplitudenoiseoftheC-CandC-Sbeatnotesin`dierential-mode',wecancalculatetheamplituderatiooftheC-SbeatnotestotheC-Cbeatnote =ACS(1+Ac+A2) 126

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3{45 ,the`common-mode'noiseshouldnotshowupinthenoisespectrumof.TheLSDof,labeledas`c-s/c-c'(thegreencurveinFigure 3-23 )representsonlytheamplitudenoisein`dierential-mode.'Ascanbeseen,the`common-mode'noisebelow40Hzissuccessfullyreducedviathiscommon-moderejectionanalysis.Webelievethatthelaseramplitudenoisewhichhasbeeneliminatedfromthedataislargelyduetothebeamjitterthatdominatesthelowfrequencyregion,sincethemeasuredlaserintensitynoise(representedbythebrowncurveinFigure 3-23 )ismorethananordermagnitudelowerthanthemeasuredamplitudenoise. AshasbeendiscussedinSection 3.4.4 ,itisoscillatornoiseandtemperatureuctuationsthatcauseanEOMtoproduceadditionalamplitudeandphasenoiseonthelaserbeam.Ourresultshowsthattheoscillatornoise(labeledasV()inFigure 3-23 )onlycountsforasmallpartoftheamplitudenoiseoor.Thethermallyinducednoisecanbeestimatedonceweknowthethermalexpansioncoecient,refractiveindicesanddn/dTvalues.Thermallyinducedchangesinelectro-opticcoecientsandthethermal-elastoeectarenotconsideredhere.FromEquations 3{21 and 3{33 ,thecarriereldaftertheEOMcanbeapproximatedas Themodulationindicesare And 127

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LinearspectraldensityofphasenoiseinlaserbeatnotesandthemodulationsignalfromtheEOMdriver. ThecalculationusingEquationsfrom 3{46 to 3{52 showsthattemperatureuctuationsatalevelof0.1mK=p 3-23 canbeseenastheupperlimitoftheamplitudenoisethatisproducedbytheEOMwithoutstabilizingthetemperatureofthecrystalandapplyingacousticisolationmethods. Iftheelectronicnoiseassociatedwiththephasemetercanbeomitted,themeasureddierencebetweenthephaseoftheC-SbeatnotescsandthatoftheC-CbeatnoteccistheresidualphaseimposedbytheEOM.TheLSDofcscc(greencurveinFigure 3-24 )isabove105cycles=p 128

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LinearspectraldensityofphasenoiseinlaserbeatnotesandthemodulationsignalfromtheEOMdriver. modulationsignalfromtheEOMdriver(blackcurveinFigure 3-24 )atfrequenciesbelow1000Hz.AlsoshowninFigure 3-24 istheLSDoftheresidualphasegivenascscc.Thecorrelationbetweenthephasenoiseofcsccandisexpectedtodisappearoncegoesbelowthisnoiseoor.ThiswasconrmedbyanothermeasurementwheretheFlukefunctiongeneratorwasreplacedbyabetterone(StanfordResearchSystemDS345).ThenewmeasurementresultisshowninFigure 3-25 .Inbothmeasurements,thephasenoiseduetotemperatureuctuationsoracousticperturbationsshouldbebelowthelevelsetbythenoiseoor. 129

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Acompletecharacterizationofeachmechanical,electricaloropticalcomponentthathasbeenplannedtobeusedinAdvancedLIGOisanimportantpartoftheinstrumentationwork.Theperformanceofeachcomponentmustmeetthesystemrequirementswithconsiderationgiventotheimprovementofthestrainsensitivityandtheoperationalfeasibility. BoththetriplependulumsuspensionsystemandtheRTPcrystalEOMhavebeenplannedtobeusedintheIOsubsystemofAdvancedLIGO.ThetriplependulumdescribedinthisthesisistheonethathasbeeninstalledandtestedintheJIFlabatGlasgowUniversity.however,thetriplependulumtobeusedinAdvancedLIGOwillbedierentfromthisprototypependulum.Themechanicalmodelthathasbeendevelopedbytheauthorwillbeausefulreferencemodelforthedesignandtestofthenalversionofthetriplependulumsuspensionsystem. TheRTPcrystalEOMisgoingtobeinstalledinthe40-mprototypeinterferometer(atCaliforniaInstituteofTechnology)andEnhancedLIGO(amid-stepbetweenLIGOandAdvancedLIGO).ThecharacterizationsoftheEOMhasbeendoneatUFisessentiallyapre-installationtest.ThemeasurementresultsprovideusefulinformationforthenaldecisionofitsapplicationinAdvancedLIGO. ThemodelingofatriplependulumsuspensionandthecharacterizationofthenewEOMintermsofitspracticallimitationshavebeenpresentedinthisthesis.Theresultsaresummarizedbelow. 130

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2 hastakenthisintoconsideration.Themodefrequenciesofthevertical,longitudinalandsidewaymotionscalculatedbythismodelhavebeenconrmedbythemeasurementresults.Theerrorsoftheoreticalvaluesarelessthan0.2Hzinmostpart.Thebiggestdisagreement(about0.4Hz)showsupina5Hzverticalmode.Morecharacterizationmeasurementsareneededtoconrmmodefrequenciesofthepitch,yawandrollmotionspredictedbythemodel. Thedampingconstantsforthemotionsofthreetestmassesneedtobeveriedandincorporatedinthismodel. TheheterodynetechniquewhichhasbeendevelopedtomeasuretheEOMnoisecanbeimprovedandusedforultra-sensitivemeasurementofboththeamplitudeandphase 131

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132

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1+T1HGT2;(A{1) withG,H,T1andT2denedinFigure 2-29 % param;%parameterle matrixT1T2;%controlmatrix In=[1,1,0,0,0,0];%inputchannels Out=[1,0,0,0,0,0];%outputchannels In=transpose(In);fmin=1.0e-1; fmax=12; bmin=log10(fmin); bmax=log10(fmax); N=1000; forn=1:1:N; freq=10^(bmin+(bmax-bmin)*(n-1)/(N-1));%logspan w=2*pi*freq; s=i*w; model;%themechanicalmodel localCtr;%localcontrolservo H=T2*M*T1; OL=H; compol=Out*LC*OL*In; CL=1/(1+compol); 133

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TF(n)=CL; end mag=abs(TF); mag=20log10(mag); phi=unwrap(angle(TF))*180/pi; subplot(2,1,1) semilogx(f,mag) axis([fminfmaxmin(mag)max(mag)]) set(gca,'FontSize',14); xlabel('Frequency(Hz)') ylabel('Magnitude(dB)') grid subplot(2,1,2) semilogx(f,phi) axis([fminfmaxmin(phi) max(phi)]) set(gca,'FontSize',14); xlabel('Frequency(Hz)') ylabel('Phase(deg.)') grid % 2 % 134

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su=0.00;%1/2separationofupperwires si=0.03;%1/2separationofintermediatewires sl=0.005;%1/2separationoflowerwires s1=si;s2=sl; %arms lp=0.04;%halfspacingofcoilsactingonpitch ly=0.08;%halfspacingofcoilsactingonyaw lr=0.13;%halfspacingofcoilsactingonroll %masses m1=2.95+0.1;m2=2.71;m3=2.71; m23=m2+m3;m13=m1+m2+m3; %momentsofinertia I1x=0.0189;I2x=0.0066;I3x=0.0066; I1y=0.0035;I2y=0.00530;I3y=0.0053; I1z=0.0189;I2z=0.00524;I3z=0.00524; %springconstants k1=3.6066e+002; k2=6.9301e+002/2; k3=9.7506e+003/2; ir=0.0635;%dimensionofintermediatemass(cylinder) tr=0.0635;%dimensionoflowermass(cylinder) d0=0.001;%heightofupperwirebreak-o(abovec.ofm.uppermass) d1=0.001;%heightofintermediatewirebreak-o(belowc.ofm.uppermass) d2=0.001;%heightofintermediatewirebreak-o(abovec.ofm.ofint.mass) d3=-0.001;%heightoflowerwirebreak-o(belowc.ofm.intermediatemass) d4=0.001;%heightoflowerwirebreak-o(abovec.ofm.testmass) 135

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n1=0.1;%1/2separationofupperwiresatuppermass n2=0.03;%1/2separationofintermediatewiresatuppermass n3=ir-0.005+0.01;%1/2separationofintermediatewiresatintermediatemass n4=ir-0.005+0.005;%1/2separationoflowerwiresatintermediatemass n5=tr-0.005+0.005;%1/2separationoflowerwiresattest lt1=sqrt(l1^2-(n0-n1)^2); lt2=sqrt(l2^2-(n2-n3)^2); lt3=sqrt(l3^2-(n4-n5)^2); ls=0.1;%heightofcoils1,2,and6abovethecenterofmass %gainsofsixchannels g1=1;g2=1;g3=1;g4=1;g5=1;g6=1; G=[g100000 0g20000 00g3000 000g400 0000g50 00000g6]; % % Hzv=-1/(2*k1+4*k2+m1*s^2-(m3*s^2+4*k3)*16*k2^2/((4*k2+4*k3+m2*s^2)*(m3*s^2+4*k3)-16*k3^2)); %

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T(1,1)=-m1*s^2-(m2+m3)*g/lt2-(m1+m2+m3)*g/lt1; T(1,2)=(m2+m3)*g/lt2;T(1,4)=(m1+m2+m3)*g*d0/lt1+(m2+m3)*g*d1/lt2;T(1,5)=-(m2+m3)*g*d2/lt2; T(2,1)=(m2+m3)*g/lt2; T(2,2)=-m2*s^2-(m2+m3)*g/lt2-m3*g/lt3; T(2,3)=m3*g/lt3; T(2,4)=-(m2+m3)*g*d1/lt2; T(2,5)=(m2+m3)*g*d2/lt2+m3*g*d3/lt3; T(2,6)=-m3*g*d4/lt3; T(3,2)=1;T(3,3)=-lt3*s^2/g-1; T(3,5)=-d3;T(3,6)=d4; T(4,1)=(m1+m2+m3)*g*d0/lt1+(m2+m3)*g*d1/lt2; T(4,2)=-(m2+m3)*g*d1/lt2; T(4,4)=-s^2*I1y+4*k2*s1^2-(m1+m2+m3)*g*d0^2/lt1-(m1+m2+m3)*g*d0-(m2+m3)*g*d1^2/lt2; T(4,5)=-4*k2*s1^2+(m2+m3)*g*d1+(m2+m3)*g*d1*d2/lt2; T(5,1)=-(m2+m3)*g*d2/lt2; T(5,2)=(m2+m3)*g*d2/lt2+m3*g*d3/lt3; T(5,3)=-m3*g*d3/lt3; T(5,4)=4*k2*s1^2+(m2+m3)*g*d1*d2/lt2; T(5,5)=-s^2*I2y-4*k2*s1^2-(m2+m3)*g*d2^2/lt2-(m2+m3)*g*d2-4*k3*s2^2-m3*g*d3^2/lt3+m3*g*d3; T(5,6)=4*k3*s2^2+m3*g*d3*d4/lt3; T(6,2)=-m3*g*d4/lt3; T(6,3)=m3*g*d4/lt3; T(6,5)=4*k3*s2^2+m3*g*d4*d3/lt3; 137

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Gamma=inv(T); % Hxl=Gamma(1,1); Hxp=Gamma(1,4); Hal=Gamma(4,1); Hap=Gamma(4,4); P1=(m1+m2+m3)*g*n1*n0/lt1; P2=(m2+m3)*g*(n2^2+s1^2)/lt2; P3=(m2+m3)*g*(n3^2+s1^2)/lt2; P4=m3*g*(n4^2+s2^2)/lt3; P5=m3*g*(n5^2+s2^2)/lt3; Hby=1/(P1+P2*(1-P3/(P3+P4*(1-P5/(P5+s^2*I3z))+s^2*I2z))+s^2*I1z); % A1=(m1+m2+m3)*g*d0*(1/lt1-(n1-n0)^2/l1^2/lt1)-(m1+m2+m3)*g*n1*(n1-n0)/l1^2+2*k1*(n1*lt1+d0*(n1-n0))*(n1-n0)/l1^2; B1=(m1+m2+m3)*g*n1*lt1/l1^2-(m1+m2+m3)*g*n1/lt1+(m1+m2+m3)*g*d0* (n1-n0)/l1^2-2*k1*lt1*(n1*lt1+d0*(n1-n0))/l1^2; C1=(m1+m2+m3)*g*n1*(n1-n0)/l1^2+(m1+m2+m3)*g*d0*((n1-n0)^2/l1^2/lt1-1/lt1)-2*k1*(n1-n0)*(n1*lt1+d0*(n1-n0))/l1^2; D1=(m2+m3)*g*d1*(1/lt2-(n3-n2)^2/l2^2/lt2)-(m2+m3)*g*n2*(n3-n2)/l2^2-4*k2*(n2*lt2+d1*(n3-n2))*(n3-n2)/l2^2; E1=(m2+m3)*g*n2*lt2/l2^2-(m2+m3)*g*n3/lt2-(m2+m3)*g*d1*(n3-n2)/l2^2+4*k2*lt2/l2*(n2*lt2+d1*(n3-n2))/l2; 138

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R1=-(m1+m2+m3)*g*(1/lt1-(n1-n0)^2/l1^2/lt1)+2*k1*(n1-n0)^2/l1^2; S1=-(m1+m2+m3)*g*(n1-n0)/l1^2-2*k1*lt1*(n1-n0)/l1^2; A2=(m2+m3)*g*d2*(1/lt2-(n3-n2)^2/l2^2/lt2)-(m1+m2+m3)*g*n3*(n3-n2)/l2^2+4*k2*(n3*lt2+d2*(n3-n2))*(n3-n2)/l2^2; B2=(m2+m3)*g*n3*lt2/l2^2-(m2+m3)*g*n3/lt2+(m2+m3)*g*d2*(n3-n2)/l2^2-4*k2*lt2*(n3*lt2+d2*(n3-n2))/l2^2; C2=(m2+m3)*g*n3*(n3-n2)/l2^2+(m2+m3)*g*d2*((n3-n2)^2/l2^2/lt2-1/lt2)-4*k2*(n3-n2)*(n3*lt2+d2*(n3-n2))/l2^2; D2=m3*g*d3*(1/lt3-(n5-n4)^2/l3^2/lt3)-m3*g*n4*(n5-n4)/l3^2-4*k3*(n4*lt2+d3*(n5-n4))*(n5-n4)/l3^2; E2=m3*g*n4*lt3/l3^2-m3*g*n5/lt3-m3*g*d3*(n5-n4)/l3^2+4*k3*lt3/l3*(n4*lt3+d1*(n5-n4))/l3; G2=m3*g*n4*(n5-n4)/l3^2+m3*g*d3*((n5-n4)^2/l3^2/lt3-1/lt3)+4*k3*(n4*lt3+d3*(n5-n4))*(n5-n4)/l3^2; R2=-(m2+m3)*g*(1/lt2-(n3-n2)^2/l2^2/lt2)+4*k2*(n3-n2)^2/l2^2; S2=-(m2+m3)*g*(n3-n2)/l2^2-4*k2*lt2*(n3-n2)/l2^2; A3=m3*g*d4*(1/lt3-(n5-n4)^2/l3^2/lt3)-m3*g*n5*(n5-n4)/l3^2+4*k3*(n5*lt3+d4*(n5-n4))*(n5-n4)/l3^2; B3=m3*g*n5*lt3/l3^2-m3*g*n5/lt3+m3*g*d4*(n5-n4)/l3^2-4*k3*lt3*(n5*lt3+d4*(n5-n4))/l3^2; C3=m3*g*n5*(n5-n4)/l3^2+m3*g*d4*((n5-n4)^2/l3^2/lt3-1/lt3)-4*k3*(n5-n4)*(n5*lt3+d4*(n5-n4))/l3^2; R3=-m3*g*(1/lt3-(n5-n4)^2/l3^2/lt3)+4*k3*(n5-n4)^2/l3^2; S3=-m3*g*(n5-n4)/l3^2-4*k3*lt3*(n5-n4)/l3^2; Q(1,1)=R1+R2-m1*s^2; 139

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Q(1,4)=S1*n1-R1*d0+S2*n2-R2*d1; Q(1,5)=-S2*n3+R2*d2; Q(2,1)=-R2; Q(2,2)=R2+R3-m2*s^2; Q(2,3)=-R3; Q(2,4)=-S2*n2+R2*d1; Q(2,5)=S2*n3-R2*d2+S3*n4-R3*d3; Q(2,6)=-S3*n5+R3*d4; Q(3,2)=-R3; Q(3,3)=R3-m3*s^2; Q(3,5)=-S3*n4+R3*d3; Q(3,6)=S3*n5-R3*d4; Q(4,1)=A1+D1; Q(4,2)=-D1; Q(4,4)=B1*n1+n2*E1+d1*G1+(m1+m2+m3)*g*(n1*(n1-n0)/lt1-d0)-I1x*s^2; Q(4,5)=-n3*E1-d2*G1-(m2+m3)*g*(n3*(n3-n2)/lt2-d1); Q(5,1)=-A2; Q(5,2)=A2+D2; Q(5,3)=-D2; Q(5,4)=-n2*B2-d1*C2; Q(5,5)=n3*B2+d2*C2+n4*E2+d3*G2+(m2+m3)*g*(n3*(n3-n2)/lt2-d2)-m3*g*(n4*(n5-n4)/lt3-d3)-I2x*s^2; Q(5,6)=-n5*E2-d4*G2; Q(6,2)=-A3; Q(6,3)=A3; 140

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Q(6,6)=B3*n5+C3*d4+m3*g*(n5*(n5-n4)/lt3-d4)-I3x*s^2; Qt=inv(Q); Hyt=Qt(1,1); Hyr=Qt(1,4); Hct=Qt(4,1); Hcr=Qt(4,4); % HalHap0000%theta=a 00Hzv000%z 000Hby00%phi=b 0000HytHyr%y 0000HctHcr];%psi=c % %123456 T1=[110000%Fl lslslp-lp00%Mp 001110%Fv ly-ly0000%My 000001%Ft 00lrlr-lrls];%Mr %transformationfrom6degreesoffreedomto6magnetpositions 141

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T2=[1ls0ly00%1 1ls0-ly00%2 0lp100lr%3 0-lp100lr%4 00100-lr%5 00001ls];%6 % 26 ]. % num=(s+6.2832e+00)*(s+2.4669e-14)*(s+2.1991e+00); den=((s+5.6549e+01)*(s+4.3982e+00))^2; kf=2.0e5; servo=kf*num/den; LC=G*servo; %

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FigureB-1. Circuitschematicofthephaselockingservo. 143

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IwasborninChinaonSeptember13th,1979.IbecameastudentwhenIwas5yearsoldandhavekeptthisprofessioneversince.IenrolledattheUniversityofScienceandTechnologyofChinain1996andgotmyBachelorofScienceinMaterialsScienceandEngineeringin2001.SinceIfoundthatphysicsclassesaremoreinterestingthanotherclassesthatItookatcollege,Idecidedtogotograduateschooltolearnmore.IbecameagraduatestudentattheUniversityofFloridainthefallof2001andjoinedtheLIGOgrouptheyearafter.IgotachancetogotoGlasgowUniversityandworkedontriplesuspensionsystemintheJIFlab.Aftera7-month-longinterestingandalsotroublesomeexperienceinScotland,IwentbackUFandcontinuedtodosomeexperimentstestingtheopticalcomponentswhicharedesignedfortheAdvancedLIGOInputOptics.IamsupposedtoendmycareerasastudentatUF. 147