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Title: Demonstration of a prototype dual-recycled cavity-enhanced Michelson interferometer for gravitational wave detection
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Title: Demonstration of a prototype dual-recycled cavity-enhanced Michelson interferometer for gravitational wave detection
Physical Description: Book
Language: English
Creator: Delker, Thomas, 1973-
Publisher: University of Florida
Place of Publication: Gainesville Fla
Gainesville, Fla
Publication Date: 2001
Copyright Date: 2001
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Subject: Physics thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Physics -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
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Summary: ABSTRACT: The direct detection of gravitational radiation has long been the goal of a large international collaboration of researchers. The first generation of interferometric gravitational wave detectors is currently being constructed around the world. These detectors will most likely be operating at their designed sensitivities within the next few years. The detectors are expected to give insight into the fundamental nature of the universe as well as operate as observatories for previously unobserved astronomical events. Unfortunately, the event rate that these detectors will be sensitive to is on the order of one per year. A 10-fold increase in sensitivity results in a 1000-fold increase in the event rate. The first generation of the United States' detector, LIGO, is a power-recycled Michelson interferometer with arm cavities. The next upgrade to this detector will come in several forms, one of which is altering the topology. The addition of a signal-recycling mirror to the detection port of the interferometer yields a sensitivity increase of approximately one order of magnitude. The topology alteration also increases the longitudinal degrees of freedom which must be controlled to 5. This dissertation proposes a control scheme for the signal-recycled cavity-enhanced Michelson interferometer with power recycling using frontal modulation. The control scheme is developed through a series of increasingly complex detector topologies. The control of all five degrees of freedom is discussed in detail.
Summary: ABSTRACT (cont.): I present a measure of how independent the control signals are from one another. I also describe the experiment performed to demonstrate the control scheme. The control scheme is simulated in full detail for the tabletop instrument. The results of the simulation and the experiment are compared and show good agreement. A detailed analysis of the operation of the tabletop interferometer is given including the frequency response for two different operating points. The frequency response is shown to be in good agreement with the theory. The results of the experiment are discussed further and some possible future research in this area is proposed.
Summary: KEYWORDS: gravitation, gravity waves, LIGO, physics, gravitational radiation, interferometry
Thesis: Thesis (Ph. D.)--University of Florida, 2001.
Bibliography: Includes bibliographical references (p. 160-162).
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System Details: Mode of access: World Wide Web.
Statement of Responsibility: by Thomas Delker.
General Note: Title from first page of PDF file.
General Note: Document formatted into pages; contains x, 163 p.; also contains graphics.
General Note: Vita.
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
        Page vii
        Page viii
    Abstract
        Page ix
        Page x
    Main
        Page 1
        Page 2
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Full Text











DEMONSTRATION OF A PROTOTYPE DUAL-RECYCLED
CAVITY-ENHANCED MICHELSON INTERFEROMETER FOR
GRAVITATIONAL WAVE DETECTION












By

THOMAS DELKER


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


UNIVERSITY OF FLORIDA


2001















ACKNOWLEDGMENTS


This work would have not be possible without the advice and guidance of my

advisor, Associate Professor David Reitze. I also give many special thanks to Professor

David Tanner, who often acted as a replacement advisor because of the incredible

travel schedule resulting from LIGO involvement.

The most notable piece of advice that I was given during the final years of

completely this study was that a Ph.D. is not necessarily proof that you are intelligent.

Most people show this by the time they complete their first two years of graduate

school. A Ph.D. is instead a badge of stubbornness and persistence. That advice

echoed through my head as more and more challenges appeared along the road to

finishing this dissertation. I believe that I have shown that I am stubborn enough.

The work in this thesis drew on many people's expertise. Dr. Guido Mueller is

largely responsible for the conception of the locking scheme. He also took me down

the first steps in the lab and continually helped with interpreting of results. Dr.

Gerhard Heinzel contributed greatly to this work through his electronic designs and

his electrical circuit simulation program, LISO. Andreas Freise provided an invaluable

tool in FINESSE, which I used on practically a daily basis. Lively discussions at the

LIGO Advance Interferometer Configurations meetings were very informative. For

that I must thank both the participants and the organizer, Dr. Ken Strain. The

thought that other graduate students were also out there working on similar problems

and succeeding (namely Jim Mason and Daniel Shaddock) also was a great help.

The most important component of being able to finish this work was the love

and support of the people in my life: most notable Crystal Lewis, my hiking com-









panion and so much more. I cannot express how important Crystal has been to the

completion of this work, so I won't try. It was my parents who gave me the gift that

allowed me to finish: my stubbornness. I must also thank Kenny for introducing me

to the world of outdoor activities that kept me sane over the past few years. I also

thank Willie for showing me how to succeed in academia.















TABLE OF CONTENTS

ACKNOWLEDGMENTS ............... . . . ii

LIST OF TABLES . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . viii

ABSTRACT . . . . . . . . . . . . . . ix

1 INTRODUCTION . . . . . . . . . . . . . 1
1.1 Gravitational W aves ........................... 1
1.2 Interferometer as a Detector ........... ........... 3
1.2.1 Shot N oise . . . . . . . . . . . . . . 9
1.2.2 Radiation Pressure Noise ................ .. .. . 16
1.2.3 Thermal Noise .................. ....... .. 16
1.3 Sources. .................. ............... .. 17
1.4 Motivation for Future Detectors ................ .. .. 20

2 THEORY . . . . . . . . . . . . . . 23
2.1 Light Fields .................. ............. .. 24
2.1.1 Generating Sidebands .................. ..... 24
2.1.2 Sideband Detection .................. ... .. 26
2.2 Locking M atrix .................. ........... .. 29
2.3 Simple Cavity .................. ............ 33
2.3.1 Light Fields .................. ......... . 33
2.3.2 Error Signals .................. ...... .. .. 40
2.4 M ichelson .................. .............. .. 44
2.4.1 Light Fields .................. ......... .. 44
2.4.2 Error Signals .................. ...... .. .. 46
2.5 Michelson with Arm Cavities .................. ..... 50
2.5.1 Light Fields .................. ......... .. 51
2.5.2 Error Signals .................. . .. 53
2.5.3 Frequency Response .................. .. . 61
2.6 Three-Mirror Coupled Cavity .................. ..... 62
2.6.1 Light Fields .................. ......... .. 62
2.6.2 Error Signals .................. . .. 63









2.7 Power-Recycled Cavity-Enhanced Michelson ..
2.7.1 Light Fields . . . . . . . . .
2.7.2 Error Signals .. ..............
2.8 Dual-Recycled Cavity-Enhanced Michelson .....
2.8.1 Light Fields . . . . . . . . .
2.8.2 Error Signals .. ..............
2.8.3 Frequency Response .. ..........

3 EXPERIMENT . . . ............
3.1 Design ......... . . . . .
3.1.1 Selection of Mirror Parameters . . . ..
3.1.2 Length and Frequency Considerations . .
3.1.3 Physical Layout and Components . . .
3.1.4 Feedback . . . . . . . . . .
3.2 Calculated Locking Matrix . . . . . . ..
3.2.1 Cavity-Enhanced Power-Recycled Michelson
3.2.2 Dual-Recycled Cavity-Enhanced Michelson .
3.3 Measurement of Losses and Resonances . . . .
3.3.1 Arm Cavities . . ............
3.3.2 Simnle Michelson ...............


3.3.3 Power-Recycled Interferometer .
3.3.4 Dual-Recycled Interferometer .
3.4 Measured Locking Matrix . . ...
3.4.1 Power Recycling . . ....
3.4.2 Dual Recycled . . . .....
3.5 Measured Frequency Response . . .
3.5.1 Low Frequency Response . . .
3.5.2 High Frequency Response . .
3.5.3 Detuned Frequency Response .
3.6 Lock Acquisition . . . .......
3.6.1 Initial Lock Acquisition . . .
3.6.2 Repeatable Lock Acquisitions and

4 CONCLUSION . . ..
4.1 Summary of Results . . ......
4.2 Future W ork . . . ..........


Lock


Stability


A GAUSSIAN MODES . .
A.1 Modal Decomposition.
A.2 Propagation of Gaussian


Modes via ABCD Matrices .


B ELECTRONICS . . . . . . . . . . .

REFERENCES . . . . . . . . . . . .


. . 155


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


84
84
84
87
98
102
106
106
110
113
113
114
116
119
122
126
127
129
129
133
135
136
137
139

142
142
144















LIST OF TABLES


Locking matrix for Michelson with arm cavities .. .........
LIGO-like parameters for coupled cavity .. ............
Locking matrix for coupled cavity with in-phase scheme .......
Locking matrix for coupled cavity with quad scheme .........
Locking matrix for LIGO I configuration .. ............

Designed mirror specification........... . . . . .....
Phase shift from arm cavities.. . . . .....
Final length parameters..... . . . ......
PZT tube parameters.. . . . . . .....
Locking matrix for LIGO I configuration . . . . .
Locking matrix for LIGO II configuration . . . . . . ..
Measurement of arm cavity mirrors via FWHM and FSR technique
OSA measurements for LIGO I configuration . . . . . .
OSA measurements for LIGO I configuration . . . . . .
OSA measurements for blocked interferometer . . . . .
OSA measurements for signal-recycling configuration . . . .
OSA measurements for blocked interferometer . . . . .
Buildup of 31 MHz sideband . . . . . . .
Measured locking matrix for power-re.' 11 configuration . . .
Measured locking matrix for dual-recycled configuration . . . .


3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15


86
92
95
101
107
110
114
115
116
116
119
119
122
127
128















LIST OF FIGURES


1.1 A gravitational wave, h+, on a simple Michelson . . . . ... 6
1.2 Example of a delay line to increase detector sensitivity . . ... 10
1.3 Example of a cavity to increase detector sensitivity . . . .... 11
1.4 Power-recycled simple Michelson .................. .. . 12
1.5 Signal-recycled simple Michelson. .................. 13
1.6 Signal recycling with arm cavities ................ . 14
1.7 Shot-noise limited sensitivity ................ .... 15
1.8 Expected noise levels for LIGO II ............. .. . 21

2.1 Carrier with sidebands in frequency space . . . ..... 25
2.2 Feedback loop for a general system .............. .. .. 29
2.3 Fields in a simple cavity .................. ..... 33
2.4 Phase shift of light reflected from an over-coupled cavity ...... .. 36
2.5 Transmission of an impedance matched cavity ............ .. 37
2.6 Light fields with antiresonant sidebands in a cavity . . . .... 41
2.7 In-phase error signal for a cavity in reflection ............. ..43
2.8 Fields in a simple Michelson .................. .. 44
2.9 Transfer function of the Michelson ................ 47
2.10 Fields in Michelson with arm cavities ................. 50
2.11 Frequency response of LIGOI-like instrument ............. .61
2.12 Fields in a three-mirror coupled cavity ................ ..62
2.13 Error signal in reflection for the two degrees of freedom . . ... 66
2.14 Error signal at pick off for the two degrees of freedom . . . ... 66
2.15 Fields in a power-re' ivi1. Michelson ................. 68
2.16 Fields in a signal-recycled Michelson ................. .. 73
2.17 Frequency response of a signal-recycled Michelson with arm cavities 81
2.18 Signal-recycling cavity resonances for recycled and detuned case . 83

3.1 Definition of lengths .................. ...... .. .. 87
3.2 Electric fields in the reflected port ................ 89
3.3 Electric fields in the antisymmetric port ............... ..90
3.4 Phase shift cavities impart to sidebands ................ ..93
3.5 Sidebands in inline cavity .................. .... .. 96
3.6 Sidebands in perpendicular cavity ................ 96
3.7 Sidebands in power-recycling cavity .................. 97
3.8 Sidebands in the Michelson .................. .... .. 97
3.9 Optical components on table .................. .. 99
3.10 31 MHz feedback signals .................. .... 103









3.11 60 MHz feedback signals ............. ... ..... . 103
3.12 31 MHz in-phase signal used to lock ..... . . . . ... 109
3.13 60 MHz quadrature signal used to lock Lpr .............. .109
3.14 31 MHz quadrature signal used to lock 1_ .............. . .110
3.15 60 MHz quadrature signal used to lock Lpr .............. .112
3.16 31 MHz in-phase signal used to lock L.s ............... .112
3.17 OSA in reflection for (a)power-recycled and (b)signal-rc' i, . ... . 123
3.18 OSA at pick off for (a)power-recycled and (b)signal-recycled ..... 123
3.19 OSA at antisym. port for (a)power-recycled and (b)signal-recycled . 124
3.20 Method for measuring locking matrix in-loop . . . . ..... 124
3.21 Power-recycling sensitivity at low frequency ............. ..130
3.22 Signal-recycling sensitivity at low frequency ........... . .130
3.23 Signal-recycling gain at low frequency .... . . . . 131
3.24 Signal-recycling gain on a linear scale .... . . . . 132
3.25 Signal-recycling gain over power-recycling . . . . . 134
3.26 Signal-recycling gain over power-recycling for detuned . . . ... 136
3.27 DC power in the pick-off port .................. ..... 140
3.28 DC power in the reflected port .................. ..... 140
3.29 DC power in the antisymmetric port ................. ..140
3.30 DC power transmitted through the inline cavity . . . ..... 140

A.1 Gaussian mode in a cavity .................. .... 147

B.1 Tunable phase shifter for the local oscillator . . . . 155
B.2 High voltage driver for PZTs ................ ... . 156
B.3 Current driver for the galvometer ................ 157
B.4 PZT feedback loop with f-3/2 roll off ................. ..158
B.5 Feedback loop for the galvometers ................ 159















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 Philosophy

DEMONSTRATION OF A PROTOTYPE DUAL-RECYCLED
CAVITY-ENHANCED MICHELSON INTERFEROMETER FOR
GRAVITATIONAL WAVE DETECTION

By

Thomas Delker

May 2001


C(i iii, i'I: David H. Reitze
Major Department: Physics


The direct detection of gravitational radiation has long been the goal of a large

international collaboration of researchers. The first generation of interferometric grav-

itational wave detectors is currently being constructed around the world. These de-

tectors will most likely be operating at their designed sensitivities within the next few

years. The detectors are expected to give insight into the fundamental nature of the

universe as well as operate as observatories for previously unobserved astronomical

events. Unfortunately, the event rate that these detectors will be sensitive to is on the

order of one per year. A 10-fold increase in sensitivity results in a 1000-fold increase

in the event rate.

The first generation of the United State's detector, LIGO, is a power-recycled

Michelson interferometer with arm cavities. The next upgrade to this detector will

come in several forms, one of which is altering the topology. The addition of a signal-

recycling mirror to the detection port of the interferometer yields a sensitivity increase









of approximately one order of magnitude. The topology alteration also increases the

longitudinal degrees of freedom which must be controlled to 5.

This dissertation proposes a control scheme for the signal-recycled cavity-enhanced

Michelson interferometer with power recycling using frontal modulation. The control

scheme is developed through a series of increasingly complex detector topologies. The

control of all five degrees of freedom is discussed in detail. I present a measure of how

independent the control signals are from one another.

I also describes the experiment performed to demonstrate the control scheme.

The control scheme is simulated in full detail for the tabletop instrument. The results

of the simulation and the experiment are compared and show good agreement. A

detailed analysis of the operation of the tabletop interferometer is given including

the frequency response for two different operating points. The frequency response is

shown to be in good agreement with the theory. The results of the experiment are

discussed further and some possible future research in this area is proposed.















CHAPTER 1
INTRODUCTION

1.1 Gravitational Waves

In order to comprehend the universe on a fundamental level, the intricate details

of mass interaction must be understood. Newton's theory of gravitation is an excel-

lent approximation of masses reacting to other masses at slow speeds, provided the

gravitational potential is not too great. However, some anomalous problems existed,

such as the fact that Maxwell's equations were not invariant under Galilean's princi-

ples of relativity. Newton's Universal Theory of Gravitation agreed with physicists'

everyday encounters with the universe, and thus was thought to be a fundamental

law. As more and more anomalies were encountered, such as the Michelson-Morley

experiment in 1887, a new theory was developed, and special relativity was born. It

was investigated thoroughly by the interaction of electromagnetic waves with matter.

These waves also give us the only available view of the distant universe.

Just as Einstein's Theory of Special Relativity expanded on how mass travels

through space, in 1916 he also expanded on how mass affects space [1]. In his Theory

of General Relativity he laid the groundwork for a new explanation of gravity that

went far beyond the prevailing theory of gravity.

The success of experiments that test the predictions of General Relativity gives

us confidence in the theory's validity. Three classical tests demonstrated the short-

comings of Newtonian gravitation and firmly established General Relativity. In the

first, a finite shift of the perihelion precession of Mercury, predicted to be identically

zero in Newtonian gravity, was explained [2, p.282]. Second, Eddington's famous

experiment during the solar eclipse of 1919 detected a small deflection in the path







2

of a light ray from a distant star as it passed through the sun's gravitational field

[3]. Finally, the redshift as photons climb out of gravitational fields predicted by

General Relativity was observed by Pound and Rebka in 1960 [4]. Many other ex-

periments further verified General Relativity as the appropriate theory of gravity on

cosmological scales.

General Relativity has the additional prediction of the existence of gravitational

radiation. This gravitational radiation is the result of large masses interacting with

each other. Given a sensitive enough instrument, this radiation should be detectable.

Nonetheless, the detection of gravitational radiation has proven to be a difficult goal

and remains unseen almost ninety years after the formulation of General Relativ-

ity. Due to the very weak nature of gravity, gravitational waves continue to elude

physicists.

Some of our strongest evidence for the existence of gravitational radiation comes

from Taylor and Hulse, who made precise observations of the orbital motion of the

binary system PR 1913+16 [5, 6, 7]. One of the stars was a pulsar, providing a very

precise natural clock for the system. Over twenty years they mapped the decay of

the stars' orbit and showed that it very precisely matched what general relativity

predicted if the energy lost by the pair was radiated away by the emission of grav-

itational waves [8]. Their observations provided indirect evidence that gravitational

radiation existed and they were award the Nobel Prize for their work in 1993. There

is additional evidence from other areas of physics that further support the existence

of gravitational radiation [9]. Although this evidence is a massive triumph for general

relativity, it is no replacement for direct detection of gravitational radiation.

The Theory of General Relativity leads to very complex equations, many of

which are immensely difficult, if not impossible to solve. Assumptions are necessary to

make headway into the physics. This leads to a strong desire for confirmation that the

theory is correct, and that the approximations being made are valid. Just as quantum









mechanics was proven with and improved upon by observation of electromagnetic

radiation, detection of gravitational radiation would vastly improve our understanding

of general relativity.

More important however, a gravitational wave detector is more than simply

a confirmation of a theory. It will also be an observatory of the universe. When

astronomers first looked at the universe outside of the visible spectrum they discovered

objects never viewed before. Quasars and Pulsars were discovered. A-i in i .. ii, was

revolutionized, and with it, astrophysics. When the universe is viewed with an entirely

different medium, a revolution of understanding is a very strong possibility. The fact

that gravitational radiation is so hard to detect is also an advantage. Since it interacts

so weakly with matter it will allow observation of events that are otherwise obscured.

1.2 Interferometer as a Detector

In order to detect gravitational radiation, we must first understand how it in-

teracts with the world around us. This can be achieved by the following derivation

of how a gravitational wave affects light traveling to a distant free mass mirror and

back again.

Within the context of General Relativity, a space-time interval can be written

in general as

ds2 g~vdx'dxV, (1.1)


where g,, is the metric tensor that describes the curvature of space. Assuming weak

gravitational fields, the metric is essentially flat (\i 1il:, wski) space with a small added

perturbation, defined as


9gV,, = lpV + h1V )


(1.2)









where


-1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1
0100

0010

0001


(1.3)


and


I h,, I <1


(1.4)


The metric tensor g,, must satisfy Einstein's Field Equation, which in the weak

field approximation reduces to [2, p. 214]


( 2


(1.5)


1 02
c2 t2 O.


Since Einstein's Field Equations are gauge invariant, we can choose a transverse

traceless coordinate system as the gauge. For this gauge, waves propagating in the

z-direction provide a solution to Equation (1.5) when they have the form


0

0
h, =
0

0


0 0 0

hil h12 0

h12 -hll 0

0 0 0


(1.6)


where we define


hll = h+(z ct)


(1.7)


(1.8)


h12 =h (Z ct).






5

where h+ and h, represent two orthogonal polarization states of the wave and the

wave is traveling along the z-axis.

It is easy to see how light can be used to detect this bending of space. Light

alv--v- follows a geodesic, which means ds2 = 0, and thus Equation (1.1) becomes


0 -(cdt)2 + (1 + h+)(dx)2 + (1 h+)(dy)2 + (dz)2. (1.9)


For light traveling along the x-axis, Equation (1.9) becomes


dx c
(1.10)
dt t/1+h +


Assuming that light travels a length I and back again, we can calculate the amount

of time it takes the light to return in the presence of slightly curved space by the

following


21 dx dx= dt' (1.11)


t 1
c i htt') + O(h)}dt'. (1.12)


Rearranging the equation for the amount of time that passes before the light returns

gives
21 1 t
t, -+ h+(t')dt', (1.13)
c 2 2t_21

where we have made the assumption that t, can be replaced by the unperturbed

round trip time, 2, in the integration limits.

Assuming that the gravitational radiation has a simple sinusoidal variation such

that h+(t') = ho cos (cgt'), and writing it as a phase change in the light, where the









light frequency is wo, Equation (1.13) becomes


60(t) ho cos ( )dt' ( 1.14)
2 J_21
ho osin (,t) -sin Lg t ]- } (1.15)

wo sin / /
ho cos t (1.16)




Based on Equation (1.9), it is not difficult to see that if we perform an equivalent

calculation for light that travels along the y-axis, we find that the phase shift is equal

in magnitude but opposite in sign to that found in Equation (1.16). This -ii--.- -1-

that a Michelson interferometer should be a suitable instrument for detecting the

phase shift that accompanies the passage of a gravitational wave since a Michelson

interferometer measures the phase difference of the light in one arm relative to the

other arm. As the gravitational wave propagates down the z-axis perpendicular to the

plane of the interferometer, the phase shift imparted by the wave will modulate the

light paths, hence the intensity of the light hitting the detector. Figure 1.1 shows the

effects of a gravitational wave on a simple Michelson as it propagates perpendicular

to the plane of the detector.








0 7T/2 7t 3T7/2
Figure 1.1: A gravitational wave, h+, on a simple Michelson

From Equation (1.16) several things can be learned. The first point to notice

is that in the limit 91 < 1, the signal scales with the length of the arms. A second
c,









point to note is that the higher the frequency of the light, the larger the phase shift.

A third point is that there is a maximum length, or storage time for light, in the arms

which is

< -. (1.17)
c 2

This becomes extremely important in more advanced designs. For a typical detection

frequency of 300 Hz, the length of the arms would have to exceed 250 km before the

signal would start to degrade. This is equivalent to the gravitational wave interacting

with the light path for a quarter period. Should the length of the arms be so long

that the light interacts with the gravitational wave for a half period, then light phase

would be shifted back to zero, thus showing no apparent effect from the gravitational

wave.

A very important conceptual notion needs to be explored here. The mirror in

the above example was taken to be free of all forces, and space was taken to be pliable.

This was inherent in our choice of a transverse-traceless gauge. In the laboratory, this

is counter-intuitive. The more familiar gauge is one in which all points of space are

taken to be fixed. After all, what object is ever truly free of all forces? But the results

of the previous calculation can't depend on the gauge that is chosen. This would be

fundamentally contradictory to physics. The world does not depend on how we view

it. How do we reconcile the apparent contradiction?

In a gauge where distances are taken to be separated by infinitely rigid rods,

then the free mass in the above example must move in that coordinate system in

order to achieve the same results. This means that the gravitational wave can be

described as a force. The force on a freely falling object in this coordinate system is


1 a2 h11
F = m2 = -ml (1.18)
2 Ot2









If we assume the sinusoidal nature of the gravitational wave, Equation (1.18) becomes


1
F = -mlhou0 cos (wt) (1.19)


The mirror will be moving with a sinusoidal motion that depends on the path length

that the light traveled. This becomes important when we talk about how to generate

synthetic gravitational waves in the experiment.

The description of this gauge is even less intuitive than the previous explana-

tion, since it is hard to understand how the position of the mirror relative to the

measurement point can affect the force that the mirror experiences. However, both

explanations give completely consistent results that are expected from the experiment.

The very precise nature of interferometers makes them excellent choices for mea-

suring small length changes. In fact interferometers can measure length differences

in two arms that are less than the diameter of an atom. Conceptually this can be

difficult to understand. The atom itself is a moving object, and the mirror is made

up of all these objects with uncertain positions. However, the nature of coherent

light sampling all these moving objects together causes this effect to average out, and

high precision interferometry can be achieved. Phase sensitivities comparable to what

needs to be accomplished to detect gravity waves have already been demonstrated at

higher detection frequencies [10].

Unfortunately, interferometry, like any measurement technique, is plagued by

noise. Noise comes in two different forms. The first is fundamental noise associated

with the nature of light. No amount of improvement of materials, isolation, or other

technological advances will reduce this noise. Only by manipulating the light can

advances be made in driving this noise down. The two noise terms here are shot

noise and radiation pressure noise. The second noise term is associated with the

environment in which the interferometer operates. These noise terms are thermal

noise, seismic noise, and many more not mentioned here.









1.2.1 Shot Noise

Shot noise is the result of the random nature of the universe. Heisenberg's

uncertainty relation guarantees that not all the photons that arrive at the beam

splitter and go toward the antisymmetric port, defined by the port that the laser

light does not enter, are ones that were created by the gravitational wave. Some were

shifted due to the uncertain phase information intrinsic in the photon. The quantum

mechanical result for this error is stated as

1
AnA > -, (1.20)
2


where n is the number of photons and Q is the phase of the photons.

An equivalent derivation of the noise uses the classical description of shot noise.

In this case we can assume Poisson statistics for the photons. A detector at the

antisymmetric port would record the number of photons incident at a given time

interval, r. For many measurements the distribution of these measurements would

be centered at n with a variance (An)2 = n or


An (1.21)


For this simple calculation we assume that all the photons detected are the result

of the signal. The power detected is


P =h0on, (1.22)


where h is Planck's constant divided by 27. We can write down the relative error


AP 1
(1.23)
P









where P is the average power in the signal and n is the average number of photons

detected in a time interval r. This immediately shows that the more photons in the

signal, the larger the signal to shot noise ratio. Often we will discuss increasing the

signal at the antisymmetric port in order to increase sensitivity. This is equivalent to

stating that we are lowering the sensitivity to shot noise.

Higher sensitivity can be achieved in several v--=i-. Equation (1.16) tells us that

the signal is proportional to the length of the arms. There is a physical limit on how

long a straight path can be made on earth. A simple way to increase this length is

to fold the arms. This can be accomplished by one of two v---.










Figure 1.2: Example of a d,1 iv line to increase detector sensitivity



1.2.1.1 Delay Lines

The first way is to separate the beams on each reflection. This is shown schemat-

ically in Figure 1.2. In order to maximize the folding, the input beam is injected in

such a way that at each point where the beam reflects off the mirror the light creates a

circle around the mirror with spots at discreet intervals. This way of folding the arms

has the problem that light scattered out of the beam path can enter another beam

path leading to an additional noise term. This scattering becomes critical when the

beams start overlapping, so mirrors have to be big enough to accommodate folding

equal to about 250 km without beam overlap. This is very difficult to achieve [11,

p.86-90].










r2, t2 Einb r, t E
Ewlans 1' 1 r1

I inb *E
1 EL
+ E +
-mo

Figure 1.3: Example of a cavity to increase detector sensitivity

1.2.1.2 Fabry-Perot Cavities

A second method for beam folding is to use cavities in the arms. Figure 1.3 shows

the fields in a cavity. This is such a fundamental building block of a gravitational wave

detectors that we look deeply at the detailed behavior of this object in C'! Ipter 2. For

now the most relevant thing to mention is that the light in the cavity has an average

storage time that is equivalent to an increased arm length. That length increase is

related to the reflectivities of the input and end mirrors. An additional problem is

created by overlapping the beams. The arm cavities must be controlled such that

the light builds up power in the cavity. Although this may seem like an unnecessary

complication, the additional shot noise sensitivity that this achieves is well worth

it. In fact, controlling a single cavity is a simple affair using the Pound-Drever-Hall

technique [12].

1.2.1.3 Power Recycling

Another way to increase the amount of signal is to increase the number of photons

that are present to be phase shifted. This can be accomplished by increasing the

input power. There is a technological limit to the amount of power that can be

delivered to the input of an interferometer which will be quiet enough not to affect

the gravitational wave signal. That limit currently seems to be about 10 watts, and

is hoped to be around 100 watts for the next generation of detectors.

However there is another way to increase the power that is incident on the beam

splitter. An interferometer that is operating on a dark fringe in the antisymmetric









r,, tp





i, ti rpr, tp



rbs, tbs Eo


antisym





Figure 1.4: Power-recycled simple Michelson

port, where no light exits out the antisymmetric port, will reflect all its power back

toward the laser. If a mirror is added at the input port, that power can build up

inside the cavity formed by the end mirrors and the power-recycling mirror [13]. The

fundamental limit here is the amount of losses in the cavity.

We were very careful to call an increase of power in the arms an effective path

lengthening, not a power increase. A fundamental difference is determined by the

nature of the gravity wave. Since the arm cavities are essentially equivalent to in-

creasing the length of the simple Michelson arms, there is a frequency response of the

arm cavities to the gravity-wave signal that will roll off as the signal frequency ap-

proaches the storage time of the arm cavity. The roll off is shown in Figure 1.7 by the

curve with no signal recycling. There is no frequency response associated with power

recycling. The reason that we use arm cavities at all, and do not just remove them in

favor of more power, is because of the complication of the beam splitter. High power

at the beam splitter leads to thermal lensing and high absorption, which significantly








affects the performance of the interferometer, and limits the power recycling factor
that can be achieved.

r,, t





ri, ti
; Erefl


S rbs tbs Eo



rsr, tsr
E +
antisym




Figure 1.5: Signal-recycled simple Michelson


1.2.1.4 Signal Recycling

There is one last cavity that can be formed in order to increase the signal. If a
mirror is introduced to the dark port any signal that is sent out of that port will then
reflect off that mirror and build up in the cavity formed by it and the end mirrors of
the arms. This configuration is known as Signal Recycling and was first proposed by
Meers [14] and demonstrated by Strain [15].
When the signal-recycling mirror is placed in a position so that it builds up the
carrier in the cavity formed by it and the Michelson end mirrors, then the signal
is built up in such a way that shot noise sensitivity is reduced at the expense of








bandwidth. The signal-recycling mirror increases the storage time of the gravitational
wave signal, and when that storage time exceeds a half cycle of the gravitational wave,
then the signal degrades.

r2p, t2p


r1p, tip1b


r2i, t2i li tli


[ 1 Erefl
rbs, tbs E


r, s, t
r tsr5 sr


Figure 1.6: Signal recycling with arm cavities


1.2.1.5 Resonant Sideband Extraction

There is an alternative operating point for the signal-recycling mirror when one
includes arm cavities. When the mirror is placed in a position such that the light is
antiresonant in the cavity formed by it and the arm cavities, then the signal actually
gains bandwidth at the expense of shot noise compared with the equivalent system
on resonance for the carrier. If the arm cavity storage time is then increased, the shot
noise level increases, and the detector has a higher sensitivity at the desired detection
frequency than an equivalent signal recycling instrument. This is known as Resonant










Sideband Extraction (RSE) [16]. The signal on the light created by the gravitational

wave sees an effective lower reflectivity in the end mirrors of the arm cavities than the

carrier sees, and it is extracted by the signal extraction mirror. Without arm cavities

this effect does not occur.

There is of course a broad range of intermediate states, where the signal-recycling

mirror is neither in the recycled case or the RSE case, but rather in between. This is

known as detuned signal recycling.



-- Broadband Signal Recycling
100000- Detuned 10 Degrees From Broadband
-....... Detuned 87.4 Degrees From Broadband (LIGO II)
----- Resonant Sideband Extraction
10000 No Signal Recycling

S1000

100

10 .-----.-.---.... .. ---...-.-
o 10


0.1-
0.01 0.1 1 10 100 1000 10000
Frequency (Hz)

Figure 1.7: Shot-noise limited sensitivity


Figure (1.7) shows the shot noise limit for different positions of the signal-

recycling mirror. The parameters of the interferometer are those of the initial design

for the United States dual-re' i, .1 gravitational wave detector, LIGO II. The effects

of tuning are very clear. As the signal-recycling mirror is tuned from broadband to

RSE, the signal has a peak sensitivity at a specific frequency at the cost of bandwidth.

The RSE configuration has the highest bandwidth with the lowest sensitivity for fixed

arm cavities.

The detector was designed to operate at a specific tuning. If it was desirable

to operate in RSE mode, for example, then the interferometer would be redesigned









for peak performance. In this case, the arm cavity finesses would be increased to

increase the sensitivity and to lower the bandwidth. It is important to realize that

signal recycling will work with a simple Michelson, but RSE needs cavities in the

arms. Without arm cavities the signal simply doesn't build up in the signal-recycling

cavity, the shot noise becomes larger, and no advantage is gained.

1.2.2 Radiation Pressure Noise

Another noise that is quantum mechanical in nature is radiation pressure noise

[17]. This noise term is created by the discreet arrival of photons at the mirror surface.

This noise term is proportional to the square root of the power in the arms over the

mirror mass. This doesn't bode well for increasing the power on the end mirror, but

there is an additional component to this noise that has been ignored. The frequency

response of the radiation pressure noise falls off at one over the frequency squared.

At low frequencies this dominates the fundamental noise spectrum, but at higher

frequencies shot noise is dominant.

1.2.3 Thermal Noise

Thermal noise [18] mostly comes in the problematic form of thermal motion.

That is, any object that is used to hold the mirror, and the mirror itself, has some

thermal energy that causes the mirror to move, creating noise in the detection signal.

There are several tricks to pl i, here.

The first trick is to suspend the mirror from a wire [19]. This concentrates all

the thermal motion into very distinct frequencies that are defined by the pendulum

mode and the "violin" modes of the wire-mass system, with a spectral distribution

controlled by the Q value, or quality factor. The Q value depends on the material of

the wire, material of the mirror, and the manner in which the wire is attached to the

mirror and the supporting structure. These parameters can be fine-tuned so that the









resonances are far enough away from frequencies of interest, and so that the Q value

is as high as possible.

The thermal vibration in the mirror substrate itself is slightly more difficult to

manipulate. Once cylindrical mirrors have been chosen as the obvious shape, the

free parameters are the height-to-diameter ratio and the material of the substrate.

Picking a material with a good Q is extremely important, but nature has been kind

here, and fused silica works well for the first generation of detectors [18].

Another form of thermal noise is Braginsky Noise, or thermoelastic noise. This is

created by the fact that the mirror substrate absorbs quanta of energy from the light.

This leads to a nonuniform thermal distribution in the surface of the mirror, creating

small scattering centers for the light. This noise level is expected to be below other

noise levels for the first generation of detectors, but causes serious problems when the

power levels increase inside the interferometer in advanced detectors [20].

1.3 Sources

The most common source of gravitational radiation is from two objects orbiting

around each other. In order to achieve some sense of how large the gravitational wave

signal is, it is useful to calculate the strain for two objects orbiting each other in the

far field limit.

Making direct analogies with electromagnetic radiation we arrive at an equation

for the gravitational radiation. There will be no monopole radiation since it doesn't

exist in electromagnetic radiation. This is the result of the conservation of energy.

There will be no dipole radiation because there is only one "(
The quadrupole moment [11, p.30] can be written as


/1, = dV (, -t 6r2) p (r) (1.24)









Writing the gravitational equivalent of quadrupole radiation gives


2G
h Rc = R4 /. (1.25)


Equation (1.25) can actually be derived more rigorously from Equation (1.5). [2,

p.226-233]

For this simple example, we assume each object has a mass M, that they are

separated by 2ro, and that they have a constant angular velocity w. If the z direction

is perpendicular to the plane of orbit, the quadrupole moments are


I, = 2Mr cos2 (t) (1.26)


I 2Mr2 cos2 (Lt)- (1.27)


and


Iy = Iy, = 2Mr2 cos (uw) sin (wt) (1.28)


Assuming the observation happens directly along the z-axis at a distance R then

Equation (1.25) becomes


32x22G
h, = -hy, = Mr cos (2wt) (1.29)
Rc4 0


and


-322G 2 sin (2) (.30)
he,= Ae = R MrYf2 sin (2wt). (1.30)
RC4









Using classical gravity to give the orbital angular velocity as


GM
2 GM (1.31)
4r3'


and defining the Schwarzschild radius for a point mass as


rs = 2 (1.32)


we can write the amplitude of the gravitational wave as


Ihl rss (1.33)
rR


Assume that the two orbiting objects are a binary neutron star system. The

mass of neutron stars is 1.4 times our sun, or 3 x 1030 kg. If the stars are almost

,uii' lir,.- then ro = 20 km, and w ; 400 Hz. Putting the binary stars in the Virgo

System means R w 4.5 x 1023 m. This gives


Ihl I 1 x 10-21. (1.34)


This is a very impressively low strain, but unfortunately does not tell the whole story.

An important question to ask is how many events like this are detectable by a

realistic earthbound detector. Binary coalescence actually will come in three forms,

Neutron Star-Neutron Star, Neutron Star-Black Hole, and Black Hole-Black Hole.

The coalescence will have to be in its final stages in order to emit the strongest

waves possible, since that is the point at which the masses are closest. Trying to

calculate the number of such entities that will be detectable by the first generation of

gravitational wave detectors is extremely difficult, and has huge errors, ranging over

several decades [21].









Using observations of how many neutron star binaries are in our galaxy, the best

guess at this time is that an event will be detectable every 10-6 to 10-7 times per

year that originates in our galaxy [22, 23, 24]. Making assumptions that they are

distributed evenly over all space, then a detector with 10-23 strain sensitivity will be

able to detect 1 event every 1000 to 10000 years. However, if the strain sensitivity

is improved by one order of magnitude, the number of events goes up by 103, since

Equation (1.33) is inversely proportional to distance from earth, but the number of

events goes up as the distance from earth to the third power. That means about

one event per year will be detectable. The lowest estimate is around one event every

10 years. Any improvement to the detector significantly increases the detection rate

[21].

Fortunately neutron star binaries are not the only source of gravitational waves.

They are the best understood, and most predictable. Other possible sources of grav-

itational waves are non-axial symmetric pulsars, non-axial symmetric supernovas,

coalescing black holes, and a stochastic background that is still present from the Big

Bang. There is also the important fact that there are many things about strong field

gravity that is not fully understood. This leaves open the possibility of strong sources

that we haven't yet recognized. One thing is very clear though. In order to detect

gravitational waves every possible avenue for reducing noise and increasing strain

sensitivity must be explored.

1.4 Motivation for Future Detectors

There is currently a huge effort in the experimental gravity wave world commu-

nity to build detectors that will detect gravitational waves. There are six detectors

that will come on line in the next 10 years. An array of detectors is important in

order to correlate the data since the signal to noise ratio will not be very large. The

United States project, LIGO, will be the one most frequently discussed in this work.

It is a power-recycled Michelson with 4-km long arm cavities [25].










Sapphire test masses
1 0 - :::: ::: : ::::: :::: :: :: :: : ::: : : ::::: : -: :::: : '': -:

......... .. .... : ,... .i i .. .. I . th e rm a l ........ ....... ... ..
....... Shot noise
Int. thermal
Susp. thermal
S. . -- Radiation pressure
10-22 ::::, :- Total noise



..-23




10-24




101 102 103
flHz

Figure 1.8: Expected noise levels for LIGO II


The first generation of LIGO has been designed to give a strain sensitivity of

S10-22/Hz1/2. The more favorable estimates of event rates that this detector will

be able to sense is not better than a few per year, and most likely no events will be

detectable. In order to guarantee that the detector will see gravitational radiation

at a rate of better than one per year the sensitivity must be increased by at least a

factor of 10. In order to do that all the noise sources mentioned previously must be

improved upon.

Figure (1.8) shows the expected noise in LIGO II for sapphire test masses [26].

Every noise source contributes significantly to the total noise curve. The most critical

are internal thermal noise of the test masses themselves and the shot noise of the

detector.

The next generation of gravitational wave detectors will require all the techniques

to reduce shot noise discussed in Section 1.2.1. In order for the detector to work a

scheme for locking all the mirrors so that the light is resonant in the appropriate

places needs to be devised. That is the main thrust of this work.









Chapter 2 will analyze the sensitivity of different interferometer configurations.

A detailed transfer function for different configurations will be derived. Control

schemes for the different configurations will be discussed and when possible, calcu-

lated. A method for comparing different control schemes for the same configuration

will be presented. A detailed explanation for how the experimental dual-recycled

tabletop instrument was controlled will be presented. The frequency response of a

signal-recycled instrument will be discussed in more detail.

Chapter 3 will discuss the prototype dual-recycled cavity-enhanced interferom-

eter that was constructed on a tabletop. It will explain in detail how the parameters

were decided on. It will give a detailed explanation for the error signals expected

from the instrument. The instrument will be characterized through measurements

of losses and the amount of power buildup in cavities. The performance of the error

signals will be assessed with a measurement of the locking matrix. The low frequency

sensitivity of the instrument will be shown. The measurement of the high frequency

response of the dual-recycled and slightly detuned instrument will be presented and

compared to the predictions.

Chapter 4 will summarize the work done. It will draw some final conclusions.

Possible future work will be discussed.















CHAPTER 2
THEORY

The analysis and control of a complex interferometer requires some general the-

ory along with detailed calculations. This chapter will discuss the input light field,

the generation of additional frequency components on that light field and detection of

the fields. It will then explain how those detected light fields can be used to control

the interferometer. The next step will be a careful analysis of the different topologies

for the instrument, including a section on the control signals for each topology.

There are two regimes in which an optical configuration can be analyzed. The

first is its response to time dependent perturbations. This analysis ignores the storage

times of complex cavities, but it is very useful in developing a locking scheme for the

interferometer. It will also explain the slow time response of the interferometer to

gravitational radiation, where the signal frequency is much lower than the storage

time of any of the cavities. The second regime is the response to time dependent

perturbations. This frequency analysis is fundamental to how the interferometer will

respond to the gravity wave in general.

In order to understand how signals are generated in complicated multi-cavity

interferometers, it is useful to build up a model for the static case using subsections.

This will allow us eventually to express a full transfer function of the complete inter-

ferometer, which will depend on the frequency of the light. Although there are pro-

grams which allow modeling of different interferometer configurations [27], one needs

to go through this exercise in order to develop and understand a locking scheme.

The frequency analysis will be accomplished largely through the modeling program

Finesse.









2.1 Light Fields

In most cases the light field going into a complex interferometer has some fre-

quency shifted components. The advantages of this will be explored more thoroughly

when we discuss locking the cavities and the Michelson interferometer to the center

frequency, or carrier. The frequency shifted fields come in two flavors, either single

sideband or paired sideband. Since our experiment uses only paired sidebands, that

is the situation that we consider in detail.

2.1.1 Generating Sidebands

In order to generate a pair of sidebands the carrier passes through an electro-

optic modulator (EOM) [28]. The modulator is driven with a sine wave typically in

the RF frequency region which modulates the index of refraction through the electro-

optic effect. This results in an effective path length change that oscillates at the drive

frequency.

The output field is then


Epr = Eoe-i(wot+msin nt), (2.1)


where Wo is the light frequency, and Q is the frequency at which the EOM is driven.

m is related to the amplitude of the driving frequency. If we assume m is much less

than one, as would normally be the case, then Equation (2.1) becomes


Epm Eoe-iot (1 im sin Qt) (2.2)

SE e-wo + -i(wo+Q)t m i(wo-n)t (2.3)


Figure (2.1) shows how the electric field looks in a frequency domain. There is

a phase relationship between the carrier and the sidebands that changes with time.









Eo





E,



E-1



EB,

Figure 2.1: Carrier with sidebands in frequency space

The sidebands rotate in the complex plane in opposite directions with respect to the

carrier at frequency f. For a good discussion of this, see Mizuno's thesis [29, p.20-28].
At this point it is important to note that light reflecting off a mirror oscillating
in the direction of the incident light z at a given frequency has the exact same effect

on the light field. It takes some portion off the carrier and frequency shifts it, such
that it creates a pair of sidebands. This is the same effect as doppler shifting. As the
wavefronts reflect off a mirror moving opposite the direction of incidence, the reflected
wave fronts are closer together than the original wavefronts, shifting the frequency
up. As the light reflects off a mirror moving .1-.v li, from the incident light it shifts the
reflected light's frequency down. For a mirror moving sinusoidally the reflected light
has the form of Equation (2.1)
It follows that when a gravity wave interacts with an arm of the interferometer,

given Equation (1.19), the effect the gravitational wave has on the light field in the
instrument can be described as a pair of frequency shifted sidebands. This realization

becomes very useful in synthesizing signals. Rather than trying to move a mirror to
simulate a gravitational wave, light can be added to the system through the end








mirror of one of the arms. If the added light is offset from the carrier by a frequency

ug then it simulates one half of the pair of sidebands that would be created by the
interaction of a gravitational wave of frequency ug with the detector.

2.1.2 Sideband Detection

Once this field has traveled through an optical device, be it either a cavity or
complex interferometer, the sideband amplitudes and phases have changed due to
the frequency dependent transfer function, T (k). We can write an electric field with
paired sidebands after it has interacted with an arbitrary transfer function as


E = [ (ko) + T (k +) e- T(k_) e] Eoe-iw (2.4)
2 2

where k = ko kn, k = ", and Eo is the amplitude of the carrier before interacting
with the system.

A photodetector will only detect the power of the total field, or


P EE*, (2.5)


which leads to


EE* =D.C. + -{T (ko) T* (k+) + T* (ko) T (+) (2.6)
2
T (ko) T* (k_) T* (ko)T (k_)} IEo 2 cos Qt

+ {T (ko) T* (k) T* (ko)T (k)
2T(o (_)- ()(_)} t (2.7)
+ T (ko) T* (k-) T* (ko) T (k-) I} I Eo 2 sin Qt (2.7)









Realizing that a + a* = 2R {a} and a a* = 2S {a} allows us to write


P =D.C. + mR {T (ko) [T (k+) T (k_)]*} IE 2 cos Qt (2.8)

ms {T (ko) [T (k+) + T (k_)]*} Eo 12 sin t (2.9)


Taking the photocurrent and using heterodyne detection [30, p.885-902], the

signal is beat against a sine wave with the appropriate demodulation phase, Q, such

that

2,
s= P cos (Qt + ) dt. (2.10)


Choosing the appropriate phase, Q, picks up a factor of 1 from the integration

and yields the final results of


Sinpase 72 {T (ko) [T (k+) + T (k_)]*} IEo 2 (2.11)
2
Squadrature {T (ko) [T (k+) T (ko)]*} IEol2 (2.12)
2

These two signals, the in-phase and the quadrature, can be generated by two

different processes. The in-phase component will be generated when there is a phase

shift between the sidebands and the carrier. Assuming that the sideband phase does

not change as the system departs from its operating point, but that the carrier receives

some phase shift, then it is the beat of the carrier with the sidebands acting as a local

oscillator that would create a signal which gives information about how the system

phase shifted the carrier. This signal could be used to control the system so that the

phase shift imparted to the carrier does not change with time. This signal is known

as an error signal. The in-phase signal will be the sine of the phase difference between

the paired sidebands and the carrier, which is a good error signal since it is zero when









the phases are the same, and has opposite signs depending on whether the phase shift

is positive or negative.

A quadrature signal will be created when there is an imbalance in the paired

sideband amplitudes. Assuming that the carrier and sideband phases do not change

as the system departs from its operating point, but that the amplitude of the two

sidebands changes antisymmetricly, then an error signal is created in this component.

Again, the error signal is zero at the operating point and changes sign depending on

which direction it has departed.

There are other v--,v in which the error signals can be created. For example, a

change in the phase of the sidebands will show up in the quadrature signal. Often the

transfer function of the system acting on the input light field will create a signal that

appears in both quadratures. If, for example, the transfer function depends on two

degrees of freedom, it is desirable to have signals to sense these degrees of freedom

that are independent from each other. The input light field and the free parameters in

the transfer function are usually chosen such that the degree of freedom that creates

the signal is as pure in one component as possible. This leaves the other component

of the signal to be used for the other degree of freedom, and the two processes then

create error signals that are independent from one another.

There may be some confusion here as to why the in-phase signal is obtained

when mixing down with sin Qt and the quadrature signal is obtained when mixing

down with cos Qt. Typically the situation would be the reverse, since the sine is the

imaginary part and the cosine the real part of the complex optical field. This should

not be a concern since it depends on the phase used to demodulate the signal. If

one started the derivation with sidebands being generated with a cosine modulation

rather than the sine that was chosen here, then the final demodulation phases would

have been shifted by 7 and you would have the standard concept of in-phase and









quadrature. The important thing to realize is that there are two signals that are

orthogonal to each other.

2.2 Locking Matrix

Ultimately the goal of the interferometer is to obtain the highest sensitivity

to the gravitational wave. To accomplish this, complex optical configurations are

used with many degrees of freedom. These degrees of freedom must be sufficiently

controlled such that they don't introduce noise into the gravity-wave channel.

Optimally a change of one of the degrees of freedom from the operating point

would not affect the other degrees of freedom. However, that is usually not the case

in complex optical configurations. Take the example of a power-recycled Michelson

with arm cavities. A change in the length of the arm cavities results in a phase shift of

the carrier coming from the arms. This phase shift causes the power-recycling cavity

to no longer have a buildup inside it, since the incoming light is no longer perfectly

constructively interfering with the light in the cavity. The power recycling factor is

reduced, thus reducing the sensitivity to the gravitational wave signal.


AL- M S












Figure 2.2 Feedback loop for a general system
Figure 2.2: Feedback loop for a general system









There is a general way to discuss how independent the signals used to control

the degrees of freedom are and how to decide on a control scheme. Figure 2.2 shows

a general system with feedback. The matrix M is the optical system, and in general

not diagonal. The matrix G is the gain matrix, and for simplicity sake is usually

diagonal. The vector S are the error signals that we use to feed back to the degrees

of freedom. The vector AL is the disturbance of each of the degrees of freedom. We

can now write down the error signals in general as


S MAL + GM S. (2.13)


Solving this equation for the signal vector we get


M
S M AL. (2.14)
1- GM


The situation that requires analysis is two degrees of freedom, where AL =

{AL1, AL2} and M is a 2 by 2 matrix. We will use this as a basic building block for

much more complicated situations. In this case


G1 0
G (2.15)
0 G2


The locking matrix is created from the equations that govern how S1 and S2

depend on the degrees of freedom. Writing down the error signals as


as1 as1
= AL1 + AL2 (2.16)
aL1 aL2
as2 aS2
2 = AL + AL2 (2.17)
aL1 aL2









The locking matrix is the Jacobian, defined as


as1 as1
tL1 tL2
M (2.18)
aS2 aS2
9L1 tL2

It is now useful to solve for the two signals in order to see how one signal depends

on the two degrees of freedom. After some linear algebra the results are


(-G2det M + AL + a AL2
Sa (2.19)
1 -&, aSG2- GG2detM

-G1 det M + AL2 + JAL1
S2 (2.20)
1 1 G 1G2 -GG2det M


There are a few cases to consider. The first case is when the two degrees of

freedom truly are independent from each other. That is equivalent to assuming that

M is diagonal. We can also assume G1,2 > 1 for the steady state solution, since

we want enough feedback to hold the resonant conditions for the light. Then these

equations become

9S1 ALl
S = AL L 1 (2.21)


as ALI
aCLl


2 aL2 AL2 --2 (2.22)
1 9Sz rAL
1 aG2 G2

The second case to consider is when the off-diagonal terms in M start to get

large compared to the diagonal terms. In this case the signals start to get mixed.

Without taking special steps with the gain distributions, changes in one degree of

freedom appear in the error signal for the other degree of freedom causing the system









to try to compensate for this signal. This is a bad situation, and in general we want

Si to be dependent only on ALi

Taking Equation (2.20), we realize that for S2 to be dominated by AL2 then the

condition
as2 aS2
G1 det M 2 > (2.23)
OL2 OLi

must be true.

Investigating the case where the degree of freedom L1 dominates signal S2, then
a2- >> 2. We can now write a simple expression for G1.


aS2
GI det (2.24)
det M

G1 > os, s (2.25)
0L1 &S2 8%2
OL1 OL1

OS2 OS1
If L2= IL2 corresponding to the det M = 0, the signals are linearly dependent.
OL1 OL1
We cannot lock this system since there is no unique locking point.

As long as the system is not very linearly dependent, the gain does not need to

be abnormally high. As the system becomes more and more linearly dependent, then

det M 0 and in order to achieve lock Gi oo.

There is an absolute measure for how orthogonal the signals are to one another,

given by the cross product of the two signals. The angle between the two signals is

defined as
Six S
sina -- (2.26)
S1 S2









Looking at equations (2.16) and (2.17) we can see that the cross product of the two

signals is

aS1 aS2 as1 aS2
sina 2 (2.27)
as, ++ a S s2 2s2
Ay 1 aL2 aL1 aL2

which we recognize as


det M
sin a (2.28)
sSi S as1 as2 I S2
y aL1 L2 yL1 a iL2

The smaller a is, the more linearly dependent the two signals are. This gives us an

absolute measure with which to compare locking schemes.

2.3 Simple Cavity

The simple cavity is the most fundamental improvement for an interferometric

gravitational wave detector. It is very useful to explore the properties of a simple

cavity completely. Since a suspended cavity cannot detect gravitational radiation the

frequency analysis will be saved for when it is applicable in a detector.



r1'
Eansr2, t2 Einb r, t]

+" L + E"

Eino

Figure 2.3: Fields in a simple cavity


2.3.1 Light Fields

Figure (2.3) shows the notation used. We adopt a few conventions here that

are not necessarily universal. Light reflected off the coated side of a mirror gets

a minus sign. Light reflected off the substrate side of the mirror gets a plus sign.









All transmitted light is real and positive and the substrates have zero thickness.

As for notation, all fields traveling away from a mirror have the subscript o and

all fields traveling to the mirror have the subscript b. The mirror reflectivities and

transmissivities, ri and ti, are for field amplitudes. For power quantities they are

related by Ri =|ri2 and T = It 2.

Writing all the equations for the fields gives


Eino = t1E0 r1Einb (2.29)

Einb -r2-2ikLEEno (2.30)

Etrans = t2-ikLEino (2.31)

refl t= rIE + tlEinb. (2.32)


Continuing on in painful detail, we substitute Equation (2.29) into Equation (2.30)

and solve for Einb


Enb = -2-2ikL (tiE rTEib) (2.33)

Eib (1 r1r2e-2ikL) -r2 -2ikLE (2.34)
r2t1e-2ikL
Einb 2= Eo (2.35)
1 rlr2e-2ikLe


Solving for the other fields gives :


Eino --2ikL E (2.36)
1 rr2e2
tlt26-ikL
Etrans = r -2ikL E0 (2.37)
1 rr2e 2kf
ri (2 + 2) e-2ikL
Ef -2kL Eo. (2.38)
1 rir26









It is useful to define a complex reflectivity of the cavity such that


Erefl = -rFPEo. (2.39)


Assuming that there are no losses in the coating (generally a good assumption), so

that r2 + t 1, we can write


-r1 + r26-2ikL
rFpp 1 22ikL (2.40)
1 rlr2C-2ikL


There is a lot to be learned from these equations. If we assume that the cavities

are resonant for the carrier, such that L ', we can define the buildup inside the

cavity as
Eio 1
io > 1. (2.41)
tlEo 1 r1r2

This number is ahv-l- greater than or equal to 1 since the limit where r2 -- 0 implies

that the field inside is simply the light transmitted through the first mirror. As r2

gets larger than 0 the amplitude of electric field in the cavity increases. This changes

if the light is not on resonance.

The light reflected from the cavity is


Erefl r1 72
(2.42)
Eo 1 r1r2


The numerator is negative if r2 > r1, and reflected light is shifted by 180 degrees in

phase from the incoming light. This condition is known as an over-coupled cavity. If

r2 < r7 then the reflected light has the same phase as the incoming light, which is

known as an under-coupled cavity. If r2 r1, then there is no light reflected from

the cavity, and it is impedance matched, or critically coupled. If there are no losses,

then the transmitted power of the impedance matched cavity is equal to the input

power. Losses in the cavity can often be combined to be included into the r2 term,










such that r2 2 r2 1 sf and t2 is unchanged. This approximation concentrates

all the losses in the second mirror, and does not affect any of the equations used thus

far.

It is also useful to calculate the phase shift that the light receives from a cavity.

From Equation (2.42) we calculate


ri (1 + R2)- 2 + R1) cos (2kL)
tan r2 sin(2L)
T2 (1 1) Sin (2kL)


(2.43)


Figure 2.4 shows the phase shift that the reflected light receives from an over-

coupled cavity. When the light is resonant with the cavity the reflected light is shifted

by 180 degrees. When the light is antiresonant the phase shift is zero or 27. This will

be a useful fact for deciding where to place sidebands with respect to a cavity's free

spectral range in order to get the most useful locking signals.


2






Ca
0)1
Ca


04--
-FSR/2


0
Frequency (MHz)


FSR/2


Figure 2.4: Phase shift of light reflected from an over-coupled cavity


There are two quantities that are often used to define a cavity. They are related

to the length of the cavity and the reflectivities of the mirrors. The free spectral












FSR


-c/2L 0 c/2L
Input Light Frequency


Figure 2.5: Transmission of an impedance matched cavity


range of a cavity is defined as


FSR


(2.44)


('!I ,:1i5; the cavity's length on the order of wavelengths causes the cavity to go

through resonances every free spectral range. The resonances occur at a length change

equal to a half wavelength of the input light. If there are sidebands on the light, and

they are at a frequency that equals the FSR then they will also be resonant in the

cavity when the carrier is resonant.

The finesse of the cavity is defined as


F= -
1 rr2


(2.45)


This quantity is a measure of how much light is built up in the cavity.

Figure (2.5) shows the transmitted field of a cavity as it is scanned over a wave-

length. The physical parameters that can be measured are the FSR and the full









width half maximum (FWHM). The FWHM is related to the Finesse and the FSR

[31, p.408-436] by


FSR
FWHM F
F


(2.46)


One can determine the reflectivities of the mirrors of a cavity by measuring the

FWHM, the FSR, and the losses of the cavity. By measuring the FWHM and the

FSR the finesse can be calculated from Equation (2.46). Solving Equation (2.45) for

the square root of the reflectivities gives


A r-I 2


-Tr /r2 + 4F2
2F


(2.47)


(2.48)


A2
F1


Defining the amplitude loss as


IErefl Sloss IEol ,


we can write Equation (2.42) as


losefl
Sloss P0
V =P


and solving for r1 gives


loss (1 A2) V lo)2 (1 A2)2 + 4A2
2


and


(2.49)


(2.50)


(2.51)









Note that s/2 is a direct measurement of the total power loss in the cavity. This

is accomplished by measuring the power reflected from the cavity on resonance and

dividing it by the power reflected when the cavity is off-resonant.

Losses in a cavity are rarely negligible. Assuming 1/8th of a percent losses on

each bounce off a mirror, which seems to be pretty realistic for the mirrors in this

experiment, a cavity with a finesse of 60 has total losses of almost 10' ,

If the light isn't on resonance then the analysis needs to be redone. This will be

useful to calculate the length of a cavity when the RF sideband is sitting somewhere

near the FWHM of the cavity. We can calculate the length of the detuning from the

measurement of the power buildup inside the cavity.


1E22 2
I -i--12 2- B (2.52)
T1 Eol2 r1r2eL-2ikL


gives


1 + RIR2 1
cos (2L) = 2 (2.53)
2rir2

Here we've used the familiar r = Ri. In order to calculate the effective reflectivity of

the end mirror we need to use the reflected field. We'll divide the incoming field by

the reflectivity of the first mirror, since normally the fields measured from a cavity

are the reflected field when the cavity is locked, and the reflected field when a beam

block as been inserted into the cavity. Solving these equations


|Erep|2 1 1 26-2ikL 2
I E2 1 1 rr- 2e-2kL C, (2.54)
Eo12 1 rlr2e-2 ikL '
RB R
C 1 r26-2ikL 2 (2.55)
B R,









gives

1 RI( 1) 1
R2 + R B B (2.56)
1 R1


It will be useful to calculate the reflectivity of a cavity for small changes in

length. This will not only be used for calculating how the gravity-wave signal shows

up in the light fields of the cavity, but also will be used to generate error signals for

the cavity.

For a change in length of the cavity L -- L 61, Equation (2.40) can be written

as


-rl + r26-2ik(Ll)
rFp 1 rlr2e-2ik(Ll) (2.57)


Assuming that the arm cavity is on resonance for the light, Equation (2.57) becomes


rFp = _r 2ik(- ) (2.58)



2.3.2 Error Signals

There are two common processes that generate an error signal [12]. We showed

that signals can be generated either by the change of the phase of the carrier with

respect to the sidebands, resulting in an in-phase signal, or by an amplitude difference

in the paired sidebands, resulting in a quadrature signal.

An in-phase signal is created when the carrier receives a phase shift relative to

the sidebands from a slight change in length of the cavity, but the amplitudes of the

sidebands stay that same. The obvious place to put the sidebands would be at the

antiresonant points of the cavity, which is half the free spectral range. At that point

the sidebands are very insensitive to length changes of the cavity. Figure 2.6 shows













1.0


0.8


U5 0.6
Eo

0.4-
9Lq

0.2



-FSR 0 FSR
Frequency


Figure 2.6: Light fields with antiresonant sidebands in a cavity


where the electric fields lie with respect to the cavity resonances in frequency space.

From figure 2.4 we can see that the sidebands do not receive any phase shift when

they are placed at half the free spectral range.

Referring to Equation (2.38) and using the fact that the sidebands are antires-

onant, we can write the reflected sideband fields as


rl + r2
E-1 refl = 12 E- input (2.59)
1 + rir2


The electric fields reflected off the cavity can now be expressed as



E = rl r2-2iko + 72 -it 1 +r2 Ci Eoe-2iwot (2.60)
1 r7r2e-2iko06 2 1 + r1r2 2 1 + rr2


Comparing this field to equations (2.11) and (2.12), we can calculate the error

signal. Since the sideband transfer function amplitudes are equal, as expected, then









the quadrature component is zero. The sideband transfer functions are also real as

written, and so we just need to calculate the imaginary part of the reflected carrier.




S{Eo re{(rl- (2.61)
1 + RIR2 2r1r2 cos (2ko61)
r2 ( R1) sin (2ko61)
1 + RIR2 2r1r2 cos (2ko61)

Putting this into Equation (2.11) gives


r + r2 r2( R1) sin 2kol6
Sin chase = -m (2.63)
h 1 + r1r2 1 + R1R2 2r1r2 cos 2ko61

In hindsight, the general shape of Equation (2.63) would have been easy to

guess. The carrier picks up an extra phase shift from movement of the back mirror

of the cavity that the sideband doesn't see since it is reflected directly off the front

mirror. The phase shift would be 2ko6l. Equation (2.11) shows that the overall error

signal would go like the sine of the phase shift. Since that equation also -o- that the

amplitude of the error signal corresponds to the amplitude of the carrier, we would

also predict the denominator. The amplitudes of the sidebands are also what would

be expected.

Figure (2.7) shows what the in-phase error signal looks like when the sidebands

have been placed at the antiresonances of the cavity. The slope at the center of the

error signal is related to the finesse of the cavity. The features at half the free spectral

range is the signal created by the sidebands becoming resonant in the cavity. The

locking range of the signal is defined as when the signal changes sign. In this case the

locking range is A/4. For more complicated systems the locking range can become

small, and it is then a concern since the system must be very close to the operating

point in order to generate the correct locking signal.












1.0



0.5



I 0.0-



0 -0.5



-1.0
-/4 0 1/4
Scanning Cavity Length

Figure 2.7: In-phase error signal for a cavity in reflection


The quadrature error signal can not be produced without also producing the

in-phase signal. There is no frequency for the sidebands such that a change in length

of the cavity results in a change of amplitude of the sidebands but not the phase of

the sidebands relative to the carrier.

An important point here is the concept of a local oscillator. For the two error

signal quadratures, the carrier and the sidebands shift in their role of local oscillator.

In the case where a signal is produced by a phase shift of the carrier, the sidebands

are the local oscillator, providing a constant phase reference. There must be sideband

present in order to detect an error signal. In the case where the error signal is created

by an imbalance in the amplitudes of the sidebands the carrier is 1p giving the role of

the local oscillator. This means that if a cavity is impedance matched for the carrier,

then there is an in-phase locking signal in reflection, but there is no quadrature signal.

If one can take the signal from inside the cavity, then both error signals are present.








2.4 Michelson
The simple Michelson is the fundamental building block for a gravitational wave
detector. The Michelson's control is important to investigate in order to understand
how to accomplish that control in a more complicated configuration. The frequency
response is a simple cosine function as would be expected in any Michelson, and does
not deserve any special attention.

+






r, t Epot Epb


Eib refl



EE0
+ E B +t
rbs, tbs
1 i

antisym
Figure 2.8: Fields in a simple Michelson


2.4.1 Light Fields
The sensitivity of a simple Michelson to a differential change in the arm lengths
is a relatively easy calculation. A few conventions need to be chosen. As in the case
of the simple cavity all reflectivities and transmissivities of simple mirrors are real
and positive. Light reflected off the coated side of the optic in air will get a minus









sign, light reflected off the coated side in the substrate will have a plus sign. This

applies to the beam splitter as well.

Figure (2.8) shows the notation that will be used. The fields are all drawn as

they interact with the beam splitter. A quick explanation of the subscripts is useful.

Fields and optics in the arm that are in line with the incoming beam get a subscript i.

Fields and optics that are perpendicular get a subscript p. Fields traveling away from

the optic get a subscript o and fields traveling back toward the optic get a subscript

b.


tbsEO


re- 2ikliEio


(2.64)

(2.65)


ritbse-2ikliEo


rbsEo


rpe-2ikl Epo
r p e p


These equations give us

antisymmetric port:


Eantisym

T (k)antisym


(2.66)

(2.67)


rp bs -2iklpEo


a solution for the transfer function of the light to the


rbsEib + tbsEpb

rbstbs (rb p-2ikl ie- 2ikl')
M Ip+li k je- Pi
rbtbse -2ik pe -2ik

tMI (k)


(2.68)

(2.69)

(2.70)

(2.71)


rieZiz ) "









If we assume that the lengths of the two arms are unchanged except for a small

differential movement such that


li liJ + 6l,


Ip l= 61


(2.72)


and that the end mirrors have the same reflectivity, r2 then (2.70) becomes


T (k)antisym = 2irbstbs2 sin r2k (1- 1


247"
6)] g2"(1


(2.73)


We can see that the choice of how we define reflections and transmissions at surfaces

has created an antisymmetric port that is dark for equal length arms. It is also useful

to write down the transfer function for the reflected port


S s2 -2ik T rit2 2ikC) -2ik'

-rMI (k).


(2.74)

(2.75)


2.4.2 Error Signals

We have already derived the transfer functions of the Michelson interferometer.
It makes sense to lock the Michelson at the antisymmetric port using an in-phase

signal. In order to do this we need to set the sideband at


SRESMI
R=


RESMI -
2 (lp li)


(2.76)


(2.77)


This ensures that the sidebands are bright in the antisymmetric port.


T (k),,f


where












1.0

0

0.5



0.0



S-0.5- E E
E b sb



-1.0
-4 0 1/4
Scanning Cavity Length

Figure 2.9: Transfer function of the Michelson


Figure (2.9) shows where the fields lie with respect to the carrier. The two

sidebands in fact do have the same sign at the antisymmetric port. This is the

result of having opposite signs at the input of the Michelson, and the fact that the

transfer function of the Michelson to the antisymmetric port flips the sign of the

upper sideband, but not the lower sideband, as is clear from the figure. The carrier

has a sign opposite that of the sidebands.

It is useful to solve this situation analytically. Writing down the field for the

carrier at the antisymmetric port using a 50/50 beam splitter and full reflector end

mirrors we get

Eo ,tis,, i sin (2kol)e--'" Eo (2.78)


The sidebands take a little more work to get. From Equation (2.70) we get


E = -ie-2k sin 2kP i (2.79)
It~~~~~~~ isueu)osleti iuto nltial.Wiigdw h il o h









Substituting lp = lp 61, li = li + 61 and realizing that setting the carrier to be dark

at this port, and the sidebands to be bright gives the conditions 2k-O2- n7 and

2k-P2 (n + 1) yields


E = Ie-2ik Ip 2k6
E ie-2ik 2 cos 2k161.


(2.80)


Writing


cos 2 (ko kQ) 61 = cos (2ko61) cos (2kQ1l) F sin (2ko61) sin (2kQl) ,


(2.81)


and taking 61 << ka, which is true since typically we are locking to a fraction of the

carrier wavelength, and the sidebands are RF, gives


cos 2 (ko ka) 61 cos 2ko61


(2.82)


Finally we can write the sideband fields at the antisymmetric port as


e-2ikplico2ko
E =-Fie- 2 cos 2ko61


(2.83)


The electric field at the antisymmetric port is


E
E0 input


D.C. + e-'


.m 2ik
sin (2kol) i-e cos (2ko6) eit
2
I M-2ik- 1+ cos (2ko6l) et.
2


(2.84)


When calculating the error signal the overall phase shift of the carrier is lost, and we

get


EE* 2m sin (2ko61) cos (2ko,6) cos 2ko + f) Eoincident2 (2.85)









Demodulating this signal with cos 2kA + Qt) gives us the locking signal


Squad = -msin (2ko6l) cos (2;o6l) |Eo incident 2 (2.86)


The demodulation phase 2ko-^- can be further simplified. Using the fact that

Q = gives

demod P + I (2.87)
2 1P Ii

We have called this signal a quadrature signal because the signal came from

Equation (2.12). However, it is created by an in-phase process. The signal arises

from a phase shift in the carrier with respect to the sidebands. As the Michelson

drifts from the locking point the sideband amplitudes stay the same relative to each

other. It is the phase of the carrier that changes with respect to the sidebands. The

sidebands themselves are transmitted with opposite signs from their original sign,

resulting in a signal that is created by an in-phase process but showing up in the

quadrature component. The demodulation phase is picked to optimize the size of the

signal. We will hold off plotting this error signal until we discuss the Michelson with

arm cavities, since the error signals are essentially the same.

As was the case for the cavity error signal, it would have been relatively easy to

predict Equation (2.86). The sine dependence is a direct result of the transfer function

of the Michelson on the carrier. The cosine dependence could also be guessed at. Since

the sideband is bright in the antisymmetric port, and we can expect length changes

on the order of the carrier wavelength to change the amount of sideband present,

the cosine function is the obvious result. The demodulation phase is the phase the

sidebands experience that the carrier does not.

Since there is a very clear symmetry between the antisymmetric port and the

reflected port, there is a second possible port for the locking signal. Since Equation

(2.86) does not have any dependence of the sidebands, it is easy to argue that the








result is the same with a minus sign from the fact that the sign of the sidebands are
different.
The common movement of the arms has been completely ignored. In fact, com-
mon movement of the end mirrors in a simple Michelson will not change the locking
signal nor the sensitivity. Since the only quantity that depends on the common mode
of the Michelson is the demodulation phase which is determined experimentally, the
common mode of the Michelson can be ignored.

r2p, 2p' t2



Lp


rip, tip +




T2i, t2i rli, ti Epo Epb
T Eib Ere

_10
EO
+ + Eio + -

Sbs tbs
antisym

Figure 2.10: Fields in Michelson with arm cavities


2.5 Michelson with Arm Cavities

This improvement allows the physical length from the beam splitter to the end
mirrors in the arms to be shortened while increasing the sensitivity. The control









becomes more complicated, but can still be calculated analytically. The frequency

response is more complicated due to the nature of the arm cavities.

2.5.1 Light Fields

The transfer function for a Michelson with arm cavities follows very easily from

previous results. Including arm cavities into the Michelson means substituting the

reflectivity of the Fabry-Perot cavities for the end mirrors into Equation (2.70) and

gives

., I+l, / Ipl-i ., lv-Ipi
T (k)antisym = bstbs -2ik (rFPp-2ik _2 i r 2e ) (2.88)


For the carrier we can simplify this equation. Using Equation (2.58) for the

reflectivities of the arm cavities, and assuming that the carrier is dark at the an-

tisymmetric port, and that the cavities in both arms are the same, a differential

movement of the end mirrors of the cavity gives


21rbs bs2 (1- 1) sin (2k61) __,,
E0 antisym =1 + R 2 2 cos e o. (2.89)
1 + RIR2 2rlr2 cos 2ko61

The difference between Equation (2.73) and Equation (2.89) is the term


EO antisym w/ arms 1 R1
(2.90)
Eo antisym w/o arms 1 + RIR2 2rlr2 cos 2ko61

Assuming the length change in the arm cavities is much less than the wavelength of

the light, then Equation (2.90) becomes


Eo antisym w/ arms 1 rf
)2 (2.91)
Eo antisym w/o arms (1 rlr2)2

Assuming the cavity finesse has already been optimized for frequency response,

then we can derive how rl and r2 should be related. From Equation (2.45) it is









evident that a fixed finesse value is equivalent to the product rlr2 being fixed. We

immediately recognize the denominator in Equation (2.41) as the power buildup in

the arms, a fixed value. This then leaves the numerator as the only free parameter.

It is very clear that ri should be as small as possible to give the largest response to

the gravity wave. It then follows that r2 needs to be as large as possible. Without

optical gain (which is not practical) we are limited to r2 being less than or equal to

1.

If the end mirrors are highly reflective then Equation (2.91) becomes


EO antisym w/ arms 1 + rl (2
(2.92)
Eo antisym w/o arms 1 r1


For a high Finesse over-coupled cavity the numerator is approximately 2 and the

denominator is the buildup of light in the cavity.

For the case where the cavity is impedance matched r2 rl, Equation (2.91)

becomes
Eo antisym w/ arms 1 (2
." (2.93)
E0 antisym w/o arms 1 r"

This is essentially half of the response to the over-coupled cavity. The impedance

matched cavity transmits half of the signal through the cavity, increasing the shot

noise.

Since there is a maximum amount of time the light can spend in the arm before

the gravity-wave signal starts degrading, and using the result that r2 should be as

large as possible, this sets a limit on rl such that


w9L rl 7r
< -. (2.94)
c 1- r 2









For the error signals we will need the light reflected from the Michelson. From

Equation (2.75) we can write


/ ),, I -i Ip + Ii IW li
T (k) = e-2i +FP t 2ik )-2i (2.95)



2.5.2 Error Signals

Locking this configuration requires a combination of techniques from the previ-

ous examples. The combination of Michelson and arm cavities do add a significant

complication.

It's useful to define the degrees of freedom that we want to lock. There are three

degrees of freedom in this system: the two cavities and the Michelson. The places

to get error signals are either the reflected port or the antisymmetric port. The

placement of detectors affects our choice of the degrees of freedom. A common length

increase in the arm cavities appears in the reflected port. Differential length changes

of the arm cavities will appear in the antisymmetric port. Common movement of the

short Michelson arms is unimportant. Differential movement of the short Michelson

arms shows up in both the reflected and antisymmetric port.

If we use the same locking scheme as was used for the short Michelson, the

sidebands are completely bright in the antisymmetric port. The differential Fabry-

Perot locking signal shows up strongly. The differential Michelson locking signal is

also present there, and is in fact in the same quadrature as the differential Fabry-

Perot error signal since they are both in-phase signals, relying on the phase shift of

the carrier with respect to the sidebands to carry all the information necessary to

determine the offset. The error signal for the differential Michelson is much weaker

than the cavities.

In the reflected port there are no locking signals at all when the sideband is bright

in the antisymmetric port. This is caused by the fact that there are no sidebands









present in the reflected port. The sidebands act as a local oscillator for the carrier,

just as a reference sine wave is a local oscillator used to mix down the photodiode

signal in order to retrieve the error signal. The sideband needs to be present in order

to act as a local oscillator to detect the phase shift between the sidebands and the

carrier.

There are two solutions to this dilemma. The first is to use a second pair of

sidebands that are bright in the reflected port. Although this solution is easy to

comprehend, it is not very elegant.

Another solution manipulates where the sideband lies with respect to the free

spectral range of the Michelson. To understand what is happening here it is useful

to write down the fields in the reflected port.

2.5.2.1 Common Mode Fabry-Perot

Since it is the common mode that we are interested in we substitute Lp

L + 61 and Li -- L + 61, where we have assumed the arm cavities to have the same

macroscopic length L, into Equation (2.95) and we get

r1 r2e-2ik(L+61) ( / 2ik+~1
T (k) = cr2ik(L+) cos 2k e-2) (2.96)
1- rr2e-2ik(L+l) 2 /

Specifically writing down the transfer function for the sidebands, putting them exactly

at the antiresonances of the cavities, and assuming that r2 t 1 gives


T (k) = cos (2kL e-2i ) (2.97)


Continuing, we expand out the cosine term to get


cos (2kL P -i cos (2koP cos (2klP T

sin (2ko0 2 I) sin (2k (2.98)









Once again using the fact that the carrier is completely reflected, that is r2 = 1, then

2ko0 = nr and the transfer function for the sidebands becomes


T(Vk) cos (2k1 e-2i ). (2.99)


Writing down the electric field in the reflected port we get

Er-fl C 2ik, 1 r2-2-iko6l IP _i(i 2k,, +QiLt)
E e o --{ + cos 2ka 2
EO t rer26-2ikol + c m 22

2 2
cos (2k 2 i ) j2ks i 2 Q)}


We can now use equations (2.11) and (2.12) to get the error signal if we demod-

ulate the signal with the correct phase. It is immediately evident that the sideband

amplitudes do not change as the arm cavities common mode length is dithered. This

is as expected, and there is no quadrature signal. The sideband transfer function is

completely real. All that is needed is the imaginary part of the carrier transfer func-

tion, which was calculated in Equation (2.62). We can now write down the common

mode Fabry-Perot error signal.


r2 (1 R) sin (2ko61) os I 1
Sinphase = -m cos 2ka (2.101
1 + R1R2 2rlr2 cos (2ko61) 2


where the demodulation phase is


Pdemod 2k +. (2.102)
2

The common mode error signal is exactly the same as the error signal reflected

directly from a cavity. This is as expected, since the Michelson should be essentially

invisible to the common mode of the cavities. At this point it is important to realize

that this exercise has not been in vain. This calculation must be done in order to









understand how the signal is coupled out of the interferometer. This will become even

more evident for the differential Fabry-Perot mode, where the error signal actually

changes which quadrature it is in.

There is an additional term in the error signal which comes from the amplitude

of the sidebands. If the Michelson is made completely bright in the symmetric port,

then this term goes to one. This would be an undesirable operating point, since there

would be no local oscillator in the antisymmetric port for the differential mode of the

arm cavities. Another difference in the equations is the result of assuming r2 is close

to unity. This simplified the equations significantly, and for a gravity-wave detector

this is the situation that is desired. However, the final formula can easily be modified

by including the term calculated from the simple cavity.

2.5.2.2 Differential Michelson

For the differential Michelson it is most useful to take the signal in the reflected

port. It will turn out that it will be orthogonal to the common mode Fabry-Perot

signal in this port. Writing down the transfer function again in the reflected port,

and using Equation (2.95), substituting lp lp 61 and 1i 1i + 61, and make the

two arm cavities the same we get


T (k) -rF (k) cos 2k 2 61 e (2.103)


Assuming that r2 m 1 and r, < 1 then for the carrier


rFp (ko) 1 (2.104)


and for the sideband


rFp = -1.


(2.105)









The transfer function for the carrier is then


T (ko) = cos (2ko6l) e-2 (2.106)


using the fact that the carrier is dark in the antisymmetric port.

For the sidebands we can show that


cos 2k ( 61 cos (2ko61) cos (2kQl1)
2 ) (2.107)
+ sin (2ko6l) sin (2kl6) .

This lets us write down the electric field.




Ef = e-2io cos (2ko6) + m {A + B} -2
2, (2.108)
m {A B }ei21 +Q
2

where A is the product of the sine terms in Equation (2.107) and B is the product of

the cosine terms.

Calculating the error signal, it is immediately evident that the in-phase signal is

zero because both the sideband and carrier transfer functions are real. The quadrature

signal is then


Squad = -mcos (2ko6l) sin (2ko6l) sin 2kQ (2.109)


where



Pdemrod 2ka l (2.110)
2









This is the same result as for the simple Michelson in the antisymmetric port

except for the additional phase shift and the slight change in the sideband amplitude.

Note that this is in the quadrature phase and the Fabry-Perot common mode was

in the in-phase, with the same demodulation phase. This puts these two signals

exactly ninety degrees out of phase, and thus completely orthogonal. This signal

is a pure quadrature signal because all of the in-phase signal is transmitted to the

antisymmetric port because of the nature of a Michelson.

2.5.2.3 Fabry-Perot Differential Mode

The transfer function of a differential length change of the two arm cavities to

the antisymmetric port for the carrier is


ir2 (1 R1) sin (2ko61) ,
T (ko) +RR2 r12 0 (2.111)
1 -+ R R2 2rlr2 COS 2ko61

For the sidebands it is


T (k) isin (2k e-2i2 (2.112)


Expanding the sine

/ /- li I /I/ /I I.
sin 2k = sin 2ko cos 2kQ + cos 2ko sin 2kQ (2.113)
2 2 2 2 2
sin 2kn (2.114)
2

using the fact that the carrier is dark in the antisymmetric port. This shows the

Michelson flipping the sign of one of the sidebands but not the other, as was seen in

the case of the simple Michelson.









Writing down the electric field in the antisymmetric port


S. _,- r2(1 R) sin(2ko61)
1 + RiR2 2rlr2 cos 2ko61

sin (2k e-i 2k) (2.115)

m ( l 2 ) i(2k,1P +Qt)
-- sm 2kei)
2 2

The transfer function of the sidebands is real and has opposite signs. This results

in the in-phase being zero. The quadrature error signal is


r2 (1- R) sin (2ko61) IP 1(2.
Squad = -m sin 2kan (2.116)
1 + R1R2 2rlr2 cos 2ko61 2

This is the same result as the common mode but it shows up in the quadrature

signal. The differential Michelson will also show up in this quadrature, and thus the

differential Fabry-Perot error signal is not orthogonal to the differential Michelson.

2.5.2.4 Locking Matrix

Our optical configuration is now complicated enough to introduce the locking

matrix. Although for this example the locking matrix is fairly well diagonalized it is

useful to use as an illustration of what a good locking matrix is. Again, the locking

matrix is a measure of how linearly independent our locking signals are to one another.

It is an extremely important item because it will decide how to design the locking

loops of the interferometer.

The locking matrix is defined as


Mn, D.O.F. (2.117)
OD.O.F.


The degrees of freedom are defined by the physical system. In the case of the Michel-

son with arm cavities the degrees of freedom are the common mode of the arm cav-









ities, L+, the differential mode of the arm cavities L_, and the differential mode of

the Michelson, l_.

S, is the signal that is being used. This signal depends on several variables. In

the current case

S, = S, {L_, L+, 1, Q, vdemod} (2.118)

The subscript n is the number of ports that have photodetectors multiplied by two,

the in-phase and quadrature component in each of those ports.

We are now prepared to write down the locking matrix for a Michelson with

arm cavities using the locking scheme previously discussed. Although there are four

locking signals, since there are only three degrees of freedoms to lock, one of the

locking signals will not be used. The three signals are picked in such a way that they

give us the most diagonal locking matrix.

The only signal that is not orthogonal is the in-phase antisymmetric port. As-

suming that the cavity end mirrors are highly reflective, we calculate


OSantisym I 1- R P -- li
m -m sin 2kQ (2.119)
BL (1- ri)2 2


and


-satm sin 2kn- (2.120)
01_ 2


Writing the locking matrix where each signal has been normalized by the degree

of freedom that is has been designed to detect yields Table 2.1.

The off-diagonal term in the antisymmetric port is the differential Michelson

contribution. We don't expect this to be a problem since the error signal for the

Fabry-Perot is much sharper then for the differential Michelson. In fact, as R1 1










Table 2.1: Locking matrix for Michelson with arm cavities


then this term goes to zero. Even at an intensity reflectivity for R1 of 91iI' this term

is almost a 100 times less than the differential Fabry-Perot signal.





L = 4000 m,F= 625
1 -------- L = 4000 m, F= 206


-- - -- - -- -------------- - --._ . _ _\
C)


0.1






0.01
1 10 100 1000
Frequency (Hz)


Figure 2.11: Frequency response of LIGOI-like instrument



2.5.3 Frequency Response

The frequency response of a Michelson with arm cavities is the first configuration

that we have encountered that has some structure to it. The response is governed

by the arm cavities. Since the cavity is locked to the wavelength of the carrier and

the gravity wave can be interpreted as frequency shifted light, then the amount of


L_ l_ L+
Antisym Q 1 (1"), 0
Refl Q 0 1 0
Refl I 0 0 1









gravity-wave signal that is built up in the cavity is dependent on the finesse and free

spectral range of the cavity.

Figure 2.11 shows the frequency response of a Michelson with arm cavities for two

different finesses, and with the same free spectral range. The low frequency response

is larger for the higher finesse, but the 3 dB point is lower. This is commensurate with

idea that as the gravity-wave signal frequency becomes higher than the linewidth of

the cavity the amount of gravitational wave signal is reduced.

r3t3 r2, t2 Erl, t1
E b E ref



+ + Em + Eo

Figure 2.12: Fields in a three-mirror coupled cavity



2.6 Three-Mirror Coupled Cavity

2.6.1 Light Fields

The next step to improve the sensitivity of the interferometer is to introduce a

coupled cavity. Before we include this in our instrument, it is useful to look at it as

a pure cavity. Figure 2.12 shows the fields. The front cavity is formed by M1 and

i.[. and the back cavity is formed by i_ and I[.. We can reduce the second cavity

into a single mirror with a complex reflectivity, rFp. We can write down the transfer

functions for the light fields reflected and inside the cavity


-T ( 2 73e-2ikL
r (k) 3r 2ik (2.121)
1 T3e26

1 rlrpe-2ikl
Ti, (k) 1 2ikl (2.123)
1 rrpe-i
(2.124)









2.6.2 Error Signals

Locking the three mirror coupled cavity is a much more challenging task than

previously undertaken. There are two philosophies for locking this type of system.

The first is to use the largest locking signal for the cavities, the in-phase signal.

Both cavities will create an in-phase signal, but it is possible to create a locking

matrix that is linearly independent out of the signals from the reflected port and

from a pick-off inside the first cavity.

Developing a locking scheme for the in-phase approach is relatively straight

forward. The sideband needs to be antiresonant in the back cavity, and resonant in

the front cavity. Resonance in the second cavity is at half the free spectral range

because the sidebands get a 180 degree phase shift upon reflection from the second

cavity when the carrier is on resonance. For a LIGO-like coupled cavity, we can use

the following parameters.


Table 2.2: LIGO-like parameters for coupled cavity


Parameter Value
R1 95'.
R2 97'
R3 1
L 4000m
I 6m


The FWHM, or line width, of the second cavity is very small. The free spectral

range is 37.5 kHz and the FWHM is 182 Hz. As long as we don't pick a frequency

that lies within the line width, it is effectively antiresonant. For the first cavity, a

12.5 MHz sideband is exactly resonant in it, and is sufficiently antiresonant in the

second cavity.

The locking matrix that we get in this case is shown in table 2.3. The error

signals have been normalized such that they are a vector row with length unity. That










Table 2.3: Locking matrix for coupled cavity with in-phase scheme


L I
Refl I 1 3.1 x 10-4
P.O. I 1 -1.0 x 10-2


allows us to immediately see what the determinant of it is. We can see that this

matrix is not linearly dependent, but also not very orthogonal. Calculating the angle

between the two vectors yields


a 10 mrads (2.125)


so they are fairly parallel. We will need a locking hierarchy such that G1 ~ 100G2.

This is very possible, and similar to a previous experiment [32].

A second way to lock this configuration is the use of the quadrature signal. By

using the signal created by an imbalance in the amplitudes of the sidebands, we get

a locking signal for one cavity that will be relatively independent of the other cavity.

It is still desirable to have the sidebands antiresonant in the second cavity. The

in-phase signal that this creates will be used to lock this degree of freedom. In order to

create an imbalance in the two sideband amplitudes, the sideband needs to resonate

to some degree in the cavity that it will lock. A sideband with a frequency of 12.614

MHz satisfies these requirements for the parameters in table 2.3.

The cost of using the quadrature signal is that the demodulation phase must be

picked to minimize the in-phase component from the second cavity in the signal that

locks the first cavity. The accuracy of the demodulation phase, and the noise related

to phase jitter, is something that must be analyzed, and unfortunately will only be

touched on in this work.

As we can see, using the quadrature signal to lock the first cavity has almost

completely decoupled the degrees of freedom. They are now almost completely or-









Table 2.4: Locking matrix for coupled cavity with quad scheme


L I
Refl I 1 5.2 x 10-3
P.O. Q 0 1


thogonal, and in fact if the demodulation phase is picked with perfect accuracy both

off-diagonal terms can be driven to zero. The amount that they can be decoupled

is related to the phase noise in the signal being used to demodulate the signal. In

this scheme the in-phase signal starts to become of the order of the quadrature signal

when the phase noise exceeds 1.7 mrads.

Figures 2.13 and 2.14 shows the actually error signals in each of the two ports.

As we can see, the signals are very orthogonal. In the reflected port some of the in-

phase signal created by the front cavity shows up. This coupling is unavoidable since

those two in-phase signals are essentially parallel in phase space. At the pick-off the

in-phase signal from the rear cavity has been essentially zeroed, and all that remains

is the quadrature signal created by the front cavity. Before we can claim this scheme

a complete success we have to realize that in the case of a full interferometer there is

the complication of the differential Michelson.

Another possible problem with using this scheme is the question of how to find

the correct phase. For the in-phase technique the phase is tuned to maximize the

signal. Since the signal has such a broad maximum, this is rather easy. In the case

of the quadrature signal, finding a zero requires being very close to it.

There are two possible v--v to solve this problem. The first is to use the in-

phase locking scheme with the detuned frequency. Once the instrument is locked up

the in-phase component can be zeroed using a second channel, and then switching to

that channel for lock. Lock acquisition after this has been achieved should be easier



















-- Scanning L
0.010- ----- Scanning 1

0

P-4
0.005



0.000- ------ ------- -------------



S-0.005 -



-0.010


-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

Mirror Detuning (Degrees)



Figure 2.13: Error signal in reflection for the two degrees of freedom


0.20-


- Scanning L
-----Scanning 1


0.00 ----------------

-0.05

-0.10-

-0.15-

-0.20
-4 -2 0 2 4

Mirror Detuning (Degrees)


Figure 2.14: Error signal at pick off for the two degrees of freedom









because the error signals are relatively independent of each other, and lock stability

should be better, since noise in one degree of freedom doesn't feed into the other.

A second way to acquire the demodulation phase is to misalign the first mirror,

and lock the second cavity with the in-phase signal. The demodulation phase for

the quadrature detector can then be tuned to zero the in-phase component. This

is not the final demodulation phase, but it does give an absolute measurement for

the demodulation phase. The misaligned case can be simulated, and from that the

final demodulation phase can be calculated. Tuning the quadrature signal to that

phase and realigning the cavity should be enough to get the two degrees of freedom

independent. Experimentally both of these techniques are used, and both work to

varying degrees of success.

Although the complication of tuning the demodulation phase seems large, the

same steps must be taken to separate the common mode signal from the differential

Michelson signal in the reflected port. The only simplification is that those two

signals are exactly ninety degrees out of phase, where as in this case there is no

simple relationship with the demodulation phase used for the pick-off signal and that

used for the reflected signal.

2.7 Power-Recycled Cavity-Enhanced Michelson

The power-recycled cavity-enhanced Michelson is the configuration that LIGO

will be using for its first generation of detectors. The control scheme is much more

complicated and analytical solutions are now more obscure due to the fact that not all

locking configurations involve sidebands either on or off resonance. When sidebands

sit on the edges of cavity resonances then expressions for the fields are nontrivial.

The frequency response is the same as without power recycling since the effect of

the power-recycling mirror can be explained completely as an increase in laser power

which clearly has no effect on the frequency response.









+
rFPP





rFPi o Epbr rp tpr

pr
SE prb E
Ii rbs, tbs E
E Epro


antisym

Figure 2.15: Fields in a power-re, 1.1 Michelson

2.7.1 Light Fields

Figure (2.15) shows the fields defined for a power-recycled interferometer. Using

previous results to reduce the work, if we substitute Equation (2.75) for r2 and r,,, tp

for rl, t1 in the cavity equations (2.35), (2.36), (2.38), this gives the equations for the

fields in the power-recycling cavity and reflected from the power-recycling mirror.


Eo = tpr 2 Eo (2.126)
1 rprrMie-C
--MIpri -2ikl"
Eprb = It 2ik Eo (2.127)
1 rMIrpre -2iklp
Spr- (MI (r + t 2ikl ,
Erefl = r 2r Eo. (2.128)
1 TprTMIC-2iklpr


For the antisymmetric port, we need to multiply Equation (2.70) by Eproe-iklpI

to get the transfer function to the dark port, giving us

T (k) tMItpr _-ikl. (2.129)
S()antisym 1 rprrMIe 2ikl


To include arm cavities is a simple matter of substituting rFP into tMI and rMi.









For a simple analysis on how the power-recycling mirror affects the sensitivity

we can use the simple Michelson. Assuming that the end mirrors are completely

reflective, that the beam splitter is nearly to 50/50, that the arm lengths obey (2.72),

and that lpr + 'P is resonant for the light, then for the carrier


rMI = -2ik cos (2iko61) e-2ik0 (2.130)


and


itpe C
Eantsym sin (2ko61) E. (2.131)
1 rpr


Comparing this to the same equation for the simple Michelson, Equation (2.78), we

have gained the factor


r > 1. (2.132)
1 rpr 1 rpr


This is the buildup of power in the cavity formed by the end mirrors of the Michelson

and the power-recycling mirror.

We already know how the arm cavity mirrors would have their reflectivities

determined by the ,in i, .-;i of the Michelson with arm cavities. R2 is as large as

possible, usually only transmitting around 10 parts per million. R1 is selected so that

the storage time of the arm cavities does not exceeds half the cycle of a gravity wave

at the frequency of interest. For LIGO I this means R1 is 97'.

The power recycling reflectivity is optimal when the power buildup in the cavity

is as large as possible. In order to pick a value for the reflectivity of the power-recycling

mirror we need to express the power build in the cavity in terms of the reflectivity

of the power-recycling mirror Rpr and the total losses in the interferometer for the

carrier, Lo. Then we need to take a derivative of this with respect to Rpr and set it









equal to zero. This yields the result that Rpr = Lo. The carrier is impedance matched

into the interferometer. For LIGO I Rpr is around 97'.

2.7.2 Error Signals

For this optical configuration there are four degrees of freedom. We still have

the same three as in the case without the power-recycling mirror, L_, L, 1_ but we

have picked up control of the power-recycling cavity, Lpr. The physical length of the

power-recycling cavity is lpr + 2

Once again the entire system can be locked with a single pair of sidebands if

the sidebands are picked to lie at the correct frequencies with respect to the different

cavity and Michelson free spectral ranges.

There is now a third port from which a locking signal can be obtained. Besides

the normal antisymmetric port and reflected port, we can now use a signal that is

reflected off the antireflection coated side of the beam splitter. This signal will create

a linearly independent signal from the reflected port.

The sidebands should again be antiresonant for the arm cavities. This will allow

us to essentially use the same locking signals for L_, L+, and 1_. Again, L_ is locked in

the dark port, and depends weakly on the differential Michelson signal. The reflected

port locks L+ and 1_, which are ninety degrees out of phase with each other.

In order to lock the coupled degrees of freedom, L+ and Lpr using the in-phase

technique discussed in Section 2.6, the sideband should be resonant in the power-

recycling cavity. Taking the signal at the pick-off we have a signal that is linearly

independent from the common mode of the Fabry-Perot, but is far from orthogonal.

Table 2.5 shows the locking matrix for this scheme using LIGO-like parameters

published by Martin Regehr [32, p.69]. We can see that the differential signals are lin-

early independent and fairly orthogonal. The coupling of the differential Fabry-Perot

in the signal used to lock the differential Michelson is weak because the differential

Michelson is taken in the reflected port, and very little differential Fabry-Perot shows










Table 2.5: Locking matrix for LIGO I configuration


L_ 1_ L+ Lpr
Antisym Q 1 7.4 x 10-3 0 0
Refl Q 4.5 x 10-3 .58 -4.8 x 10- -.82
Refl I 0 3.2 x 10-4 1 9.9 x 10-4
P.O. I 0 -1.6 x 10-5 1 -2.4 x 10-3


up in that port. The differential Michelson is very weak in the antisymmetric port

compared to the Fabry-Perot.

The reflected port quadrature signal is demodulated with a phase such that

the in-phase common mode signal is zeroed. Ninety degrees from this phase is the

maximum common mode Fabry-Perot signal because they are orthogonal signals. The

power recycling degree of freedom is starting to show up in the differential Michelson

because it is also on this channel. However, because one of these signals is common

mode and the other is a differential mode they will ahiv-- be linearly independent.

We are using an in-phase signal to lock both the common mode Fabry-Perot and

power recycling. We immediately know that they are linearly independent because

the sign of the product of the diagonal terms is different than the sign of the product

of the off-diagonal terms. Using our cross product to show how linearly independent

they are gives us


sin a+ = 3.4 x 10-3 (2.133)


As we can see the angle between the two signals is very small. As long as the angle

is large enough to separate them, then we can still lock the system. The question of

how large is large enough is determined by how much gain is needed to overcome the

coupling.

Since they are not completely orthogonal, a gain hierarchy needs to be estab-

lished in order to eliminate the coupling. From the calculation in Section 2.2 we know









how large the gain in the common mode Fabry-Perot needs to be in order to suppress

that degree of freedom in the signal used to lock power recycling.

We could also have used the quadrature signal to lock the power-recycling cavity

in order to separate the degrees of freedom. Since the analysis is similar to that done

for the full signal-recycled interferometer, we will put off the detailed discussion for

that section. We will note that for this scheme both the differential Michelson and

the power-recycling cavity have similar demodulation phases.

LIGO I has the possibility of using a different locking scheme. Although the

above locking scheme has proven to work well [32], there is not enough signals to

control the angular degrees of freedom of the mirrors. For that reason a second

sideband is used. This second sideband could give us a chance to decouple the common

mode Fabry-Perot and the power-recycling cavity even more. Making it antiresonant

in the power-recycling cavity, we can use it to control the common mode of the arm

cavities. Since it is completely antiresonant it is much more sensitive to L+. Putting

the original sideband on resonance with the power-recycling cavity and having some

leak out the dark port gives us the remaining locking signals.

2.8 Dual-Recycled Cavity-Enhanced Michelson

The final upgrade to an interferometer that this work will analyze is the use

of a signal-recycling mirror. The introduction of the signal-recycling mirror makes

the equations much more complex. The solutions for the light fields are not terrible

difficult to arrive at but, do require some extensive algebra. The frequency response

of such an instrument is rather complicated, and also deserves some attention.

2.8.1 Light Fields

The light fields in this instrument are much more intermixed than before. There

is now a second Michelson that has be introduced to the interferometer consisting

of the fields entering the beam splitter from the antisymmetric port. Any field that












P
p +






Blb E p pr r

li rbs, ts
+ Eio Epro Eo

rSO
BEsror srb

I i r s, t s r

Eantisym


Figure 2.16: Fields in a signal-recycled Michelson

goes towards the antisymmetric port will be reflected off the signal-recycling mirror
and will see the Michelson from the other side. Light fields that are bright in the
antisymmetric port will then enter the second Michelson and be bright in the reflected
port. A second cavity has also been added. For light that is dark in the antisymmetric
port, the cavity is formed by the signal-recycling mirror and the arm cavities. For light
that is bright in the antisymmetric port, the cavity is formed by the signal-recycling
mirror and the power-recycling mirror.
Figure (2.16) shows the fields inside a signal-re 1'. 1 Michelson. In order to
solve for the electric fields we want to solve the equations for the case without power
recycling. It's a simple matter to add power recycling in at the end. We will also call
the reflectivity of the arm cavities rp, ri in order to save some subscripts. We need to
keep in mind that they are the reflectivities of the Fabry-Perot cavities.









Writing down the equations


tbsEo + "bsEsrb

-r_ ie-2iklEi _

-rbsEo + tbsEsrb

_p e-2iklpEpo

tbsEpb + rbsEib

-r se-2ikl, Eo
I~~~~ ygO -


rie-2ikl (tbsEo + rbsEsrb)



-rpe-2ikl ( -rbEo + tbsEsrb)



-rre-2ikl (tbsEpb + rbsEib)


Unfortunately most of the work done in the previous sections will not really help for

this configuration due to the mixing created by the signal-recycling mirror.

In order to make the equations more manageable some abbreviations must be

used. The average length of the common mode Michelson and differential mode

Michelson can be defined as


12k + l
2
S= 2k -

2
c = 2klr.
(Psr = 2klsr-.


(2.140)

(2.141)

(2.142)

(2.143)


The fields reflected back from the beam splitter towards the power-recycling

mirror and in the antisymmetric port are


S { (rrpe- r r+2 ~~ r- [rt4 + 2rrpt r r ++ r} ]-+
rp -% bs' I 2.i1s4 I s pbsbs P I bsl-
1 + rtb2s rir) e-i(2y ++rPs() rPr t r, +( rt ) g(e-(+S+S+
-r (k) (2.144)


(2.134)

(2.135)

(2.136)

(2.137)

(2.138)

(2.139)









and


r(t_ +r i r1)t8 r(1 1 + I-
T 1 + rf rt (1 rpr,) e-i(2,p++sp,r) rs(ter 2 C-, + jei ) e e-++)
t' (k) (2.145)


Now using the normal cavity equations we can write the reflected, transmitted

and intra-power-recycling cavity transfer functions.


Tref (k) = M (k) -2il (2.146)
1 r1rIM (k) e-2iklpr
I t, P (k) p-iklpr
Tati (k) 1 (k) e-ikl (2.147)
1 rpr MI (k) e-2ikl,

Tp.*. (k) = 2il (2.148)
1 rperMI (k) e-Cp2i

We want to now calculate the sensitivity gain of this configuration over one

without signal recycling. For this we can assume that there are no arm cavities, that

the end mirrors are completely reflective, and that there is no power recycling. Then

the transfer function to the antisymmetric port for a differential movement in the

arms reduces to


2itsrrbstbs Soin 2k P 1 9 -2
T (k)antm L (2.149)
1 rsr cos L2k (P2 61 2ik (ls,+


For the carrier we assume that it is dark in the antisymmetric port and that it is

resonant in the signal-recycling cavity formed by the signal-recycling mirror and the

end mirrors of the arms. We also assume that the movement of the end mirrors is

very small compared to the wavelength. Then Equation (2.149) becomes


T (ko)2itsrFbstbs sin (2kol6) e-2ik
T ( ansym r(2.150)
1 rs,.









Comparing this to Equation (2.78) gives us


T (ko)sR tsr2
T (ko)MI rsr

This result is not a surprise and could have be written down immediately without

this detailed analysis. The gain is simply the buildup of the carrier in the cavity

formed by the signal-recycling mirror and the ends of the Michelson mirror. Since

the Michelson is held dark all differential movements go towards the antisymmetric

port, and then encounter the signal-recycling mirror. The signal-recycling mirror is

then controlled so that the differential signal is resonant in that cavity.

2.8.2 Error Signals

There are several possible v--,v- to lock this configuration. The approach that

was taken in this work was that the locking scheme should be relatively easy to

implement. This rules out using single sidebands, since generation of single sidebands

on a carrier is rather complex and involves frequency stabilizing two lasers. This is

not an insurmountable problem, but since there are other efforts ongoing to lock this

configuration that use single sidebands, it would have been a redundant effort [33, 34].

The next criterion was that it would lock the interferometer in a dual-recycled state,

where the carrier is built up in the signal-recycling cavity. The locking scheme was

not designed to operate in a detuned state. However, small detunings should be

possible if the signals are sufficiently independent. The third criterion was that it

would be locked using only frontal modulation. Injecting light from other ports for

locking would not be allowed. The reason for this is mainly the complex noise and

implementation issues for an external modulation locking scheme when applied to a

full scale gravitational wave detector.

There are five degrees of freedom in this configuration. There are several differ-

ent v-,v to express these degrees of freedom. For all cases the common mode and









differential mode of the arm cavities and the differential Michelson are the most con-

venient ones to use because of the properties of the Michelson. The power-recycling

cavity length is also a fairly usual choice for control. For the signal-recycling mir-

ror the most obvious degree of freedom is the power-recycling, signal-recycling cavity

length. This degree of freedom will be controlled by a sideband that enters in through

the power-recycling mirror, must reach the signal-recycling mirror, and must sense

its movement.


2.8.2.1 Common Modes

Control of the common modes is relatively straight forward, and similar to con-

trol schemes already discussed. We want to control these degrees of freedom with one

of pair of sidebands, Q2 (This sideband has the subscript 2 because in the final design

it has the higher frequency). In order to control the common modes, we are going to

have to deal with the coupling of the common mode of the Fabry-Perot cavities and

the power-recycling mirror.

For our scheme we want to use an in-phase signal for the locking of the arm

cavities. The sideband should be antiresonant in this case. To lock the power-

recycling cavity the sideband should sit in the edges of the resonances, somewhere

near the full width, half maximum. This will generate a quadrature signal for the

power-recycling cavity degree of freedom.


2.8.2.2 Signal Recycling

The signal-recycling mirror is controlled by an in-phase signal created by a side-

band that is resonant in the cavity formed by it and the power-recycling mirror. This

cavity is rather complex and samples many optics in the interferometer twice before

it makes a round trip in the cavity.

The key to sensing the signal-recycling mirror is to have the second pair of

sidebands, fi, bright in the antisymmetric port (Since it will turn out that this









pair of sidebands is lower in frequency than the other pair, it has the subscript 1).

When f1 is reflected back into the interferometer from the signal-recycling mirror it

sees the Michelson from the opposite side, and is then bright in the reflected port.

The signal-recycling cavity length is then adjusted so that it forms a cavity with the

power-recycling mirror and signal-recycling mirror.

Ideally it would only sense a cavity formed by the signal-recycling mirror and

the arm cavities. The limitation of frontal modulation that we imposed excludes this

because there is no way for a light field to be bright for the antisymmetric port when

coming from the reflected port, but dark for the reflected port when coming from the

antisymmetric port. The symmetry of the system forbids it. External modulation

would allow injecting a light field in the antisymmetric port that would sample only

the signal-recycling cavity and the arm cavities.

With this resonance condition on fR we get the signal-recycling mirror error

signal. A change in the position of the signal-recycling mirror results in a phase

shift of f with respect to the carrier. Again, the concept of a local oscillator is

important because the carrier acts as a local oscillator for the sidebands to sense this

degree of freedom. That means that there is no locking signal for this sideband in

the antisymmetric port because the carrier is not present there. The locking signal

must be taken either in reflection or at the pick off. Both these ports give the same

signal with slightly different shot noise.

The signal-recycling mirror control is very difficult to obtain. Since the signal-

recycling cavity will have a low finesse, the error signal simply will not be as strong

as the other degrees of freedom. The cavity it forms also samples every other degree

of freedom. This is unlike the other degrees of freedom, except for the differential

Michelson which rivals it in complexity, since other signals in general only have a

strong dependence on one other parameter.









We will find that the differential Michelson causes the most problems with the

signal-recycling mirror. It is important that we create a signal for the differential

Michelson that has a different phase from signal recycling. The amount that the

differential Michelson degree of freedom appears in the signal recycling error signal

can then be zeroed with the phase of the local oscillator for the mixer.

2.8.2.3 Differential Modes

Control of the differential Fabry-Perot degrees of freedom will again be similar

to previously discussed schemes. In this case the sideband will be fairly bright in the

antisymmetric port. This gives us a strong local oscillator for measuring the in-phase

signal created by the differential Fabry-Perot Cavities.

The difficult degree of freedom is the differential Michelson. For the case of a

simple Michelson with signal recycling, the locking signal could be taken in the dark

port. This gives us a strong in-phase error signal that is created by the differential

Michelson, since the sidebands act as the local oscillator, and the carrier gets the

phase shift. There is no locking signal contribution from the signal-recycling mirror

since there is no carrier in the dark port to act as the local oscillator.

For the case of arm cavities the dark port error signal will ahl--i be dominated

by the differential Fabry-Perot signal. That leaves the reflected port and the pick off

signal. For the differential Michelson and the signal recycling error signals there is

no difference in the pick-off and the reflected port, except for an overall amplitude

change. The locking matrix for these two signals in those two ports is completely

linearly dependent. This would not be the case for the signals in the antisymmetric

port and the pick off signal. The locking matrix for these two ports as a function of

the differential Michelson and signal-recycling mirror would be linearly independent

if error signals were present. Unfortunately there is no locking signal for the signal-

recycling mirror in the dark port. A differential Michelson signal created by an in-









phase mechanism would be completely overwhelmed by the differential Fabry-Perot

in the antisymmetric port.

There is another way to get a locking signal for the differential Michelson that

is linearly independent from the signal recycling signal. A quadrature signal could

be used. By slightly detuning the lengths of the Michelson we can put the sideband

so it is not completely dark in the antisymmetric port. As the differential Michelson

changes the amplitude of the two sidebands will change, and a quadrature signal will

be created.

In order to ensure that the differential Michelson and the signal-recycling mirrors

are linearly independent the sideband has to be completely resonant in the signal-

recycling cavity. There would then be essentially no quadrature signal created by

the signal-recycling mirror. By tuning the phase of the local oscillator the amount

that the signal recycling degree of freedom appears in the differential Michelson error

signal can be completely zeroed.

We can again get the differential Fabry-Perot modes in the antisymmetric port

and the differential Michelson signal will be obtained at the pick-off inside the power-

recycling cavity. This can be the same port that signal recycling is take in. There

is no advantage in taking one in reflection and the other at the pick off. Again,

the reflected port and the pick off are completely linearly dependent for the signal

recycling and differential Michelson error signals.

Note that the error signal to lock the differential Michelson is now created by

a completely different mechanism than what will be used to lock the LIGO I con-

figuration. Simulations using Finesse show that this locking technique does indeed

give linearly independent solutions. The full locking matrix will be discussed in the

experimental section were there are known parameters for calculating the locking

scheme.







81



Broadband Signal Recycling
Detuned 10 Degrees From Broadband
.----... Detuned 87.4 Degrees From Broadband (LIGO II)
------- Resonant Sideband Extraction
10- No Signal Recycling


I 1 -




0.01

S1E-3

o 1E-4

1E-5
0.01 0.1 1 10 100 1000 10000
Frequency (Hz)


Figure 2.17: Frequency response of a signal-recycled Michelson with arm cavities


2.8.3 Frequency Response

The frequency response of a cavity-enhanced signal-re. 1, ,1 Michelson has al-

ready been touched on in the introduction. It requires a little more explanation

though.

Figure 2.17 shows the frequency responses of a signal-re' v 1' 1 interferometer

for various tunings of the signal-recycling mirror. The parameters used are from the

LIGO II conceptual design, so that the frequency behavior is what will be evident

in the real instrument. They have been normalized for the instrument without any

signal-recycling mirror. There are several features in the responses that should be

understood.

One feature in every curve is that there is an overall high frequency cut off. The

source of this cut off is the cavities which the gravity-wave signal sees. The cut off

changes frequency as the tuning of the signal-recycling mirror changes.









For the dual-recycled case the mirror is in the position that the carrier is resonant

in the cavity formed by it and the arm cavities. The -3 dB point is lower than if there

is no signal recycling, because the storage time for the gravitational wave signal is

effectively longer, resulting in a lower frequency at which the phase of the signal

already in the cavity is 180 degrees out of phase with the new signal being generated.

It's at this point that the signal begins to roll off. The frequency response for this

cavity is the same as a Fabry-Perot cavity with a higher finesse.

For the resonant sideband extraction case the mirror is positioned such that the

carrier is exactly antiresonant in the cavity formed by the signal-recycling mirror and

the arm cavities. In this case the -3 dB point is actually higher in frequency than if

there were no signal-recycling mirror. The reason for this is that the signal-recycling

mirror results in a low storage time in the arm cavities. The three-mirror cavity for

the gravity-wave signal formed by the arm cavities and the signal-recycling mirror

can be described as a two-mirror cavity. The gravity-wave sideband enters through

the end mirror and then encounters a complex mirror formed by the signal-recycling

mirror and the input arm cavity mirror. This complex mirror has an overall lower

reflectivity than the input arm cavity mirror, and the signal is coupled out through

the signal-recycling mirror. The response is exactly like a Fabry-Perot cavity with a

lower finesse.

Every case in between these two extremes results in a cavity that has a peak

response for a certain frequency. Figure 2.18 explains pictorial why that is so. The

signal-recycling cavity is detuned in such a way that the frequency that is resonant

is no longer the same as the carrier. This results in a frequency response that is

peaked at a frequency that is a function of the detuning. There is some cost for

this. As is clear in the figure 2.18, only one sideband can be recycled at the detuning

frequency. This is why the peak with 10 degrees detuning is approximately half that

of the broadband case. As the detuning approaches the RSE case the effective finesse













6 -- No Detuning
------ Detuned

5- \


a 4
o I
3-



Lower
1 Gravity Wave Sid and Upper
GravityjWave Sideband
--- -----------


Frequency


Figure 2.18: Signal-recycling cavity resonances for recycled and detuned case


of the cavity is lowered because of the effect discussed in the RSE case. This explains

the lower peak in the case where the detuning is 87.4 degrees with respect to the 10

degree case.

We can also explain the terminology broadband and narrow band instrument.

Although there is no narrowing of the resonance peak, the fact that only one sideband

is recycled makes the detuned case 1v 1i row band". Both the RSE and dual-recycled

case are termed broadband since they recycle both signal sidebands equally.















CHAPTER 3
EXPERIMENT

This chapter will concern itself with the tabletop dual-recycled cavity-enhanced

Michelson interferometer built to demonstrate the locking scheme. Section 3.1 will

discuss the physical parameters and components used in the experiment. Section

3.2 will present the locking matrices resulting from simulations. Section 3.3 will

characterize the losses and resonances for the carrier and sidebands in the tabletop

instrument. Section 3.4 will present the measured locking matrix and compare it

to the simulation. Section 3.5 will show the experimental results of the sensitivity

measurements as a function of frequency and compare them to the theory.

3.1 Design

The first step in designing the tabletop interferometer, detailed in Section 2.8.2,

is creating a locking scheme. The next step is to decide on mirror reflectivities.

Once that is done, the length and sidebands are specified to match the resonance

conditions stipulated in the locking scheme. This section will focus on these two

steps. The design section will also show the physical layout of all the components

and discuss the key components used to control and monitor the interferometer.

3.1.1 Selection of Mirror Parameters

Our locking scheme requires that sidebands are not albv--i- resonant or antireso-

nant in a particular cavity. In order to calculate all the conditions that are necessary

for the sidebands, we have to take into account the phase shifts encountered by the

sidebands from cavities when they are not fully resonant. Also, in order to set the

sideband in the full width, half maximum of the power-recycling cavity (a stipulation









of our locking scheme) we must know the finesse of the cavities. These considerations

dictate that we select mirror reflectivities before sideband frequencies and cavity

lengths are chosen.

We begin by specifying the arm cavity mirrors. The losses that are incurred in

these cavities will constrain the power-recycling mirror reflectivity. The first condi-

tion on these optics is that the carrier should be strongly over-coupled in the entire

interferometer. In order to achieve this, we must estimate the mirror losses that we

will be using. We selected mirrors manufactured by CVI Lasers for cost and avail-

ability reasons. The CVI high reflector (HR) mirrors were found to have losses of

approximately 0.001;:'. from our measurements. This limited the cavity input cou-

pler to 91,' reflectivity in order to stay well within the stipulation that the overall

interferometer be over-coupled. The finesse of the arm cavities is then limited to ap-

proximately 60. An added advantage of going with lower finesse arm cavities is that

the cavities are much more stable since their linewidth is larger. More stable cavities

are definitely an advantage when dealing with such a complex instrument, and the

mirror parameters chosen still provide a solid test of principles.

The power-recycling mirror can now be specified. With the losses in the arm

cavities around I' it seems prudent to choose a power-recycling mirror of 1I' .

reflectivity to ensure over-coupling. In the end, the interferometer to be very close

to impedance matched because of additional losses in the power-recycling cavity.

Without losses the power recycling factor would be about 17.

The signal-recycling mirror should give a reasonable recycling factor. It should

also keep the power-recycling, signal-recycling cavity over-coupled for the sideband

that will be used to lock it. We selected a signal-recycling mirror of I,.'. reflectivity.

If the differential Fabry-Perot signal is taken from behind the signal-recycling mirror,









then we would have a signal gain of


G, -tsr 3. (3.1)
1 rsr


The mirror diameters are chosen according to several different considerations.

For the mirrors that are actuated directly, the mirror should weigh as little as possible

because of resonant frequency considerations. This is the case for the cavity end

mirrors and the signal-recycling mirror. The cavity end mirrors were chosen to be

0.5 inches, and the cavity input mirror were also 0.5 inches in diameter for symmetry

reasons. The signal-recycling mirror is 1 inch, but, as will be described later, this

degree of freedom has the lowest feedback gain, and the resonant frequency is not

a concern. The power-recycling mirror is not directly actuated. A 1 in mirror was

chosen. The beam splitter is at 45 degrees and it is easiest to use a 2 inch optic in

order to avoid any of the beams clipping on an edge.


Table 3.1: Designed mirror specification


Designed Measured Mirror Mirror
Parameter Reflectivity Reflectivity Curvature Diameter
Power Recycling '. 82. !' 2 m 1 in
Arm Cavity Input Coupler 91i' 911' 4 m 0.5 in
Arm Cavity End Mirror HR 99.>' 6 m 0.5 in
signal-recycling mirror 51', 86.1 4 m 1 in
Beam Splitter 50'. 50'. Flat 2 in


The actually values vary from the specified value by a few percent, which is as

close as the manufactures generally quotes them. The only optic that we needed to

make sure is very close to the actual value specified is the beam splitter. A beam

splitter that is not 50/50 would create many problems, and would not be a good test

of the locking scheme.









There is also the matter of picking the correct radius of curvature for the mirrors

so that it matches the gaussian mode. The details of this will not be gone through in
this work, but appendix A will describe how to do the calculation. The cavity lengths
must be known before one can calculate the modes.


L,



SEo.t ERefl


4 E






/ j antisym


Figure 3.1: Definition of lengths


3.1.2 Length and Frequency Considerations

Figure 3.1 shows the layout and length definitions on our tabletop interferometer.

The main difference between this layout and the typical signal recycling topology is

that the signal-recycling cavity has been turned, and the partially reflective optic is

the turning mirror. The turned signal-recycling cavity has the advantage that the

end mirror in the signal-recycling cavity can be blocked, but all the light fields are

still present on the photodetectors. With the turned signal-recycling cavity, blocking

the signal-recycling mirror allows the LIGO I configuration to be locked. Ideally the
interferometer will remained locked when the beam block is removed from in front of

the signal-recycling cavity, resulting in a very simple lock acquisition.









The first condition of the locking scheme which must be satisfied is that one on

the sidebands, Q2 is bright in the reflected port for the Michelson. The other sideband,

Q1, should be almost bright in the antisymmetric port. This has the unfortunately

consequence that they are multiples of each other since 2 is at the free spectral range

of the Michelson and QR is half the free spectral range of the Michelson. We will solve

this problem later.

The determination of the radio frequency (RF) of the sidebands is governed by

physical constraints. The wavelength of the RF sidebands must not be so long that

the cavity lengths can not fit on the table. This puts a lower limit on the frequencies

around 10 meters, or 30 MHz. In general, it is desirable to select the lowest possible

frequency since electronics become exponentially more difficult to find and build as

their frequency increases. The fact that the sidebands are multiples of each other

results in the other sideband being 60 MHz.

The next step is determining the .- i-vi ii. I1 ry in the Michelson arm lengths, i -lp.

We will put the 60 MHz sideband, Q2, bright in the reflected port. That immediately

sets the arm length mismatch. The sidebands will be bright in the reflected port

every free spectral range of the Michelson. In this case


c
FSRMI = ( 1 =60 MHz, (3.2)
2 (- lp)


or



li 1p = 2.5 m. (3.3)


We will find that 1p = .3 m and i = 2.8 m satisfy our requirements well.

The next length that we specify is the power recycling length, i,,. First we

calculate the length of the power-recycling cavity so that the 60 MHz sideband is

resonant in it. We then detune the length of the power-recycling cavity so that the









sideband is sitting somewhere within the full width, half maximum. This will give us

a strong quadrature signal to lock the power-recycling cavity.

In LIGO I, the sideband used to lock the power-recycling cavity is on resonance

for the power-recycling cavity when the frequency is at integer multiples plus a half of

the free spectral length. The reason for this is because the Michelson arm .-i.!liii_. I ry

is small, 0.58m, and the modulation frequency has a long wavelength, 24m. The

Michelson acts as a small amplitude loss mechanism for the sideband, but otherwise

does not disturb it. The power-recycling mirror and the arm cavities behave as a

normal three mirror coupled cavity. The carrier receives a r phase shift from the

cavity. In order for the carrier to be resonant in the power-recycling cavity the cavity

length must be shifted by j and the cavity now has resonances at FSR (n + '). Since

the sidebands do not get this r phase shift they need to be at an integer plus a half

free spectral range in order to be resonant.




1.0-
0

0.5- + 30MHz


H 0.0



S-05- 30MHz


-1.0
-3k/8 0 3R/8
Scanning Differential Michelson Length

Figure 3.2: Electric fields in the reflected port


In contrast we place the sideband used to lock the power-recycling cavity, Q2

completely in the next dark fringe. This gives an additional r phase shift to Q2 that










the carrier does not see. This is shown pictorially in Figure 3.2. The carrier sits in a

bright fringe. The 30 MHz sideband pair is transmitted fully to the dark port. The

60 MHz sideband pair sits at the next bright fringe, which has given each sideband

a sign flip from when it entered the Michelson. Since the carrier does not receive

this sign flip, but does receive a sign flip from the arm cavities, they are both again

resonant at integer multiples of the free spectral range.

We are now ready to specify the length of our power-recycling cavity.


c
= 60 MHz, (3.4)
2 + 'Pj+


or



lp, + = 2.5 m (3.5)
2


For our previous picks of lp and Ii, then lpr = 0.95 m.




1.0-
0.


0.5



0.0
r-e


S-0.5 30MHz + 30MHz


-1.0 .
-3/8 0 3/W8
Scanning Michelson Length


Figure 3.3: Electric fields in the antisymmetric port




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