Laboratory Studies of Arm-Locking Using the Laser Interferometer Space Antenna Simulator at the University of Florida

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Laboratory Studies of Arm-Locking Using the Laser Interferometer Space Antenna Simulator at the University of Florida
THORPE, JAMES IRA ( Author, Primary )
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Subjects / Keywords:
Electrical phases ( jstor )
Error signals ( jstor )
Laser interferometer space antenna ( jstor )
Lasers ( jstor )
Noise measurement ( jstor )
Noise reduction ( jstor )
Propagation delay ( jstor )
Signals ( jstor )
Simulations ( jstor )
Transfer functions ( jstor )

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University of Florida
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Full Text







Copyright 2006


James Ira Thorpe

For my parents, who provided everything and demanded nothing


Many of my colleagues here at UF and elsewhere were truly instrumental in the

production of this work: Shawn Mytrik, my right-hand man with the electronics; Volker

Quetschke, the resident computer wizard; Rodrigo Delgadillo, Michael Hartman, and

Gabriel Boothe, for their hours of faithful service in the lab; Daniel Shaddock, who

provided insight into arm-locking, phasemeters, and other aspects of LISA; Rachel Cruz,

who built the optics side of the LISA simulator; and especially my advisor, Guido Mueller,

for guiding me to this point. Most importantly, I would like to thank my wife, Suzanne,

who put up with nearly three years of separation while I completed this degree.


ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ..................................... 8

LIST OF FIGURES .................................... 9

KEY TO ABBREVIATIONS .. ............................ 13

KEY TO SYMBOLS .................................. 15

A BSTR A CT . . . . . . . . . . 18


1 INTRODUCTION ................................ 20

1.1 Motivation for Gravitational Wave Astronomy ........ ........ 20
1.2 Gravitational Wave Detectors ................... ..... 21
1.3 LISA at the University of Florida ............. ........... 22

2 GRAVITATIONAL WAVES ................... ........ 24

2.1 Overview ....................... ........... 24
2.1.1 Relativity ................... ......... 24
2.1.2 Weak-field GR and Gravitational Waves . . . 27
2.1.3 Properties of Gravitational Waves .. . . . 28
2.1.4 Interaction with Matter ..... . . . 30
2.1.5 Generation of Gravitational Waves .. . . ..... 32
2.1.6 Energy Carried by Gravitational Waves . . . 34
2.2 Sources of Gravitational Waves .................. .... .35
2.3 Detection of Gravitational Waves .................. .. 41
2.3.1 Indirect Detection .................. ....... 41
2.3.2 Direct Detection ............. . . ... 42 Doppler-tracking of spacecraft . . . 43 Pulsar timing ................ .... .. .. 44 Resonant mass detectors ...... . . . 45 Interferometric detectors .. . . . 48


3.1 Introduction ................... . . ... 54
3.2 Sources . .. . . . . .. . . .. 54
3.3 M mission Design ............. . . . ... 55
3.4 The Disturbance Reduction System (DRS) .. . . . 56
3.5 The Interferometric Measurement System (IMS) . . . 57
3.5.1 IM S Overview .................. ....... 58

3.5.2 The Optical Bench ...........................
3.5.3 Time Delay Interferometry ....................... Visualizing TDI ........................ Extensions to TDI .. .................. The zero-signal TDI variable ................. Limitations and noise sources .. .............
3.5.4 A rm -locking . . . . . . . . Closed-loop system dynamics .. ............. Steady-state arm-locking performance .. ......... Transient response .. .................. Alternative arm-locking schemes ............... G W signals . . . . . . . Interaction with pre-stabilization system ..........


4.1 Background ................
4.2 The EPD Concept .. ..........
4.3 Optical Components .. .........
4.3.1 Layout . . . .
4.3.2 Pre-stabilization .. ........
4.4 Electronic Components .. ........
4.5 Phasemeters .. .............
4.5.1 Overview ..............
4.5.2 Phasemeters for LISA-like signals
4.5.3 An IQ phasemeter with a tracking
4.5.4 A Software Phasemeter ...... Design ...........

L Results

4.5.5 A Real-

4.6 EPD

time Hardware Phasemeter
Front-end design .....
Back-end design ......
Single-signal PM test with Single-signal PM test with Entangled-phase PM test w Entangled-phase PM test w Performance limitations
U nit . . . . .
Second-generation EPD unit ..
Third-generation EPD unit .....

O . . .

a VCO ......
optical signals ..
ith VCO ....
ith optical signals


5.1 Introduction . . . . . . . . .
5.2 Electronic M odel . . . . . . . .
5.2.1 Method .................. .............

. . .

5.2.2 R results . . . . . . . . .
5.2.3 D discussion . . . . . . . .
5.3 Initial O ptical M odel . . . . . . . .
5.3.1 M ethod ................. ...............
5.3.2 R results . . . . . . . . . Frequency counter measurements .............. Phasem eter data .. ................... Error-point noise . . . . . .
5.3.3 D discussion . . . . . . . .
5.4 Improved Optical M odel .. .......................
5.4.1 System Characterization .. ...................
5.4.2 Filter Design ..... .. .. .. ... .. .. .. .. .. ......
5.4.3 R results . . . . . . . . .

6 CO N CLU SIO N . . . . . . . . .

6.1 Phasemeters and EPD Units ..........................
6.2 A rm -Locking . . . . . . . . .


A DIGITAL SIGNAL PROCESSING .. ....................




Introduction . . . . . . .
Sampling ..........................
A .2.1 A liasing . . . . . .
A.2.2 Upconversion .. ................
D igital Signals . . . . . .
A.3.1 Binary Fractions ...................
A.3.2 Multiplication and Other Operations .. .....
A.3.3 Floating-point Representations ..........
A.3.4 Digitization Noise ..................
D igital Filtering . . . . . .
A.4.1 Time-domain Response ...............
A.4.2 Frequency Response ................
A.4.3 Design M ethods .. ................
A.4.3.1 FIR Filters windowed impulse response
A.4.3.2 IIR Filters bilinear transform method .
A.4.4 Realization and Practicalities ............
A.4.4.1 Filter structures .............
A .4.4.2 Latency . . . . .
A .4.5 CIC Filters . . . . . .
A.4.6 Fractional-Delay Filters ...............


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

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

Table page

2-1 Suggested frequency bands for GWs .................. ...... 40

2-2 Operational GW bar detectors .................. .. ........ .. 47

2-3 Major ground-based GW interferometers ................... ..51

4-1 Major LISA IMS components/signals and their EPD equivalents . ... 90

4-2 Reconstruction algorithms for the hardware PM ... . . ..... 114

4-3 Beat note frequencies and amplitudes for optical entangled-phase measurement 123

4-4 Progression of EPD units ................... ... . 130

A-1 IEEE standard floating point representations ................ ..171


2-1 Tidal distortion of an initially-circular ring of freely-falling test particles















A hypothetical laboratory generator of GWs .....

A binary star system as a generator of GWs .....

Observed shift of periastron for PSR 1913+16 .

Concept for Doppler-tracking detection of GWs .

Sensitivity of ALLEGRO bar detector ...

A Michelson interferometer as a detector of GWs .

Aerial photograph of the LIGO GW detector .....

Sensitivity curves for the LIGO observatory .....

Sources in the LISA observational window ......

Orbital configuration of the LISA constellation .

Diagram of the LISA IMS .. ............

Thiar~m nf na T TA nntir-l hpnrh

b V . . . . .

3-5 "rabbit-ear" diagram for the first-generation TDI X combination .........

3-6 Diagram of a closed-loop SISO system with negative feedback .. ........

3-7 G eneric Nyquist plot . . . . . . . . .

3-8 Nyquist plots for single-arm arm-locking .. ..................

3-9 Bode plot of T,,,(f) with TRT = 33s .. ...................

3-10 Bode plot for a generic arm-locking controller .. ...............

3-11 Closed-loop noise suppression for a generic arm-locking loop .. .........

3-12 Nyquist plot for common arm-locking .. ....................

3-13 Magnitude of square-bracketed term in (3-38) .. ...............

3-14 Combining pre-stabilization and arm-locking with a tuneable cavity .......



Combining pre-stabilization and arm-locking using a sideband cavity lock .

Combining pre-stabilization and arm-locking using an offset PLL ...




























The EPD technique applied to a single LISA arm . .

Optical layout of the UF LISA interferometry simulator .

Frequency noise in the L1 Lo beat note . .....

Phase noise in the L1 Lo beat note . .......

Overview of the DSP system from Pentek Corporation .

Schematic of a IQ phasemeter with feedback . ...

Laplace domain model of the system in Figure 4-6 . .

Bode plot of G(s) for the software PM . .......

Bode plot of H(s) for the software PM . ......

Expected behavior of the software PM . .......

Observed behavior of the software PM . .......

Schematic of the real-time hardware PM . ......

Schematic of a Direct Digital Synthesizer . .....

CIC decimation filter near the first aliasing band . .

Passband flatness of the CIC filter in the hardware PM .

Feedback filter for hardware PM tracking loop . ..

Packing format for PM data transferred over the VIM interfac

VCO phase noise measured by four channels of the hardware F

Linear spectral density of laser beat note phase . ..

Qualitative amplitude spectrum of interfering beams with sho

Timeseries for entangled phase test using VCOs . .

LSD of an entangled phase test using VCOs . ....

Analog electronics used to prepare beat signals for PM .

Linearly-detrended phase for optical entangled-phase measure

Quadratically-detrended phase for optical entangled-phase measurement

Linear spectral density for optical entangled-phase measurement . .

Noise suppression in optical entangled-phase measurement . .

. 125

. 126

. 127

. 88

. . 9 1

. 92

. 93

. 94

. 99

. . 101

. 104

. . 105

. . 106

. 107

. . 108

. . 109

. . 110

. . 111

. . 112

e . . 113

M . . 117

. 118

t noise . .. 119

. . . 122

. . . 122

. . 124

ent . 124










5-6 Laplace-domain model of the system in Figure 5-5

Bode plot of controller for initial optical arm-locking system . .

Response of interferometer to phase modulation . .

Timeseries of L2 Lo beat note for locked and unlocked cases . .

Detrended timeseries of L2 Lo beat note for locked and unlocked cases

Close-up of locked case in Figure 5-9. . .....

Spectrum of locked and unlocked frequency noise . .

Timeseries of beat note phase for unlocked and locked cases . .

Phase noise spectra for the unlocked and locked cases . . ..

Closed-loop noise suppression for optical arm-locking . .

Error-point noise for locked and unlocked cases . .

Modification of electronics for improved optical arm-locking . .

Raw phase timeseries of arm-locking system characterization data .

Detrended timeseries of arm-locking system characterization data .

Linear spectral densities of arm-locking system characterization data .

Magnitude of transfer function for arm-locking system . .

Timeseries of filtered and unfiltered frequency noise from VCO input

Measured and designed transfer functions of arm-locking control filter

Histogram of residual phase for laser beat note in hardware PM . . .

Schematic of the NCO used in the 3rd-generation EPD unit . . .

Detrended phase of VCO signal in EPD test . . . .

Linear spectral densities in EPD test with VCO signals . . . .

Experimental setup for electronic arm-locking experiment . . . .

Transfer functions for electronic arm-locking experiment . . .

Linear spectral density of arm-locked VCO signal . . .

Closed-loop noise suppression for electronic arm-locking experiment . .

Experimental arrangement for the initial optical arm-locking experiments .



























. . 142

. . 143

. . 144

. . 145

. 146

. . 146

. . 147

. . 148

. . 148

. . 149

. . 150

. . 151

. . 153

. . 154

. . 154

. . 155

. . 157

. . 157

A-1 An overview of the sampling process . ........

A-2 The phenomenon of aliasing . ............

A-3 An overview of the upconversion process . ......

A-4 Assumed PDF for quantization error . ........

A-5 Non-uniform, non-white quantization error . ....

A-6 Comparison of Laplace and z domains . .......

A-7 Windowed impulse response method for designing FIR filters .

A-8 The direct-form I filter structure . ..

A-9 The direct-form II transposed filter structure . ...

A-10 Magnitude response of a generic two-stage CIC filter . .

. . 163

. . 165

. . 167

. . 172

. . 173

. . 177

. . . 179

. . 181

. . 182

. . 183

ADC: Analog-to-Digital Converter
AU: Astronomical Unit
BH: Black Hole
BIFO: Bi-directional First-Input First-Output buffer
BS: Beam Splitter
CIC: Cascade Integrator Comb filter
DAC: Digital-to-Analog Converter
DC: Direct Current or zero Fourier frequency
DDS: Direct Digital Synthesizer
DF2T: Direct Form II Transposed
DMA: Direct Memory Access
DRS: Disturbance Reduction System
DSP: Digital Signal Processing
EMRI: Extreme Mass-Ratio Inspiral
EOM: Electro-Optic Modulator
EPD: Electronic Phase Delay
ESA: European Space Agency
FIR: Finite Impulse Response filter
FPGA: Field-Programmable Gate Array
FSR: Free Spectral Range
FTP: File Transfer Protocol
GR: General Relativity
GUI: Graphical User Interface
GW: Gravitational Wave
IEEE: Institute of Electrical and Electronic Engineers
IIR: Infinite Impulse Response filter
IMS: Interferometric Measurement System
IQ: In-phase/Quadrature
LIGO: Laser Interferometric Gravitational-Wave Observatory
LISA: Laser Interferometer Space Antenna
LO: Local Oscillator
LSB: Least-Significant Bit
LSD: Linear Spectral Density
LUT: Look-Up Table
MI: Michelson Interferometer
MSB: Most-Significant Bit
NaN: Not a Number
NASA: National Aeronautics and Space Administration
NCO: Numerically Controlled Oscillator
Nd:YAG: Neodymium-doped Yittrium Aluminum Garnet
NPRO: Non-Planar Ring Oscillator
NS: Neutron Star
OB: Optical Bench

PC: Personal Computer
PCI: Peripheral Component Interconnect interface
PD: Photodiode
PDF: Probability Density Function
PDH: Pound-Drever-Hall
PLL: Phase-Lock Loop
PM: Phase Meter
PZT: Piezoelectric actuator
RF: Radio Frequency
RMS: Root Mean Square
SC: Space Craft
SDRAM: Synchronous Dynamic Random Access Memory
SiC: Silicon Carbide
SMBH: Super-Massive Black Hole
SR: Special Relativity
TCP/IP: Transmission Control Protocol / Internet Protocol
TDI: Time-Delay Interferometry
TOA: Time Of Arrival
TT: Transverse-Traceless
UF: The University of Florida
ULE: Ultra-Low Expansion
VCO: Voltage-Controlled Oscillator
VIM: Velocity Interface Module
VME: Virtual Machine Environment
WD: White Dwarf
ZOH: Zero-Order Hold

O : D'Alembertian operator
A'" : tensor amplitude
A(t): amplitude
b : bit width
c : speed of light
c(s): control signal
C(t) : Dirac delta-function comb
ds2 : differential spacetime interval
e(s) : error signal
E(t) : time-component of electric field
Edig : digitization noise energy
ESN : shot-noise energy
erfc(x) : complimentary error function
f : Fourier frequency
fe : cutoff frequency
f : interferometer null frequency
fNyq : Nyquist frequency
fuG : unity-gain frequency
F : Fourier transform
fs : sampling frequency
g ,,: metric tensor for general relativity
G : Newton's gravitational constant
G(s) : filter or system transfer function
G : Einstein curvature tensor
h : gravitational wave strain, Planck's constant
h+ : + polarization tensor
h : x polarization tensor
h+ : strain amplitude in + polarization
hx : strain amplitude in x polarization
hp : metric perturbation
hp : trace-reversed metric perturbation
hT : metric perturbation in TT gauge
hj : phase change due to GWs on light propagating from SCi to SCj
h(n) : impulse response function
H(s) : feedback transfer function
I : quadrupole moment tensor
I(t) : in-phase component, intensity
k : wave number
k" : 4-D wavevector
k : 3-D wavevector
L : optical path length
LGW : gravitational wave luminosity
L : Laplace transform

M : total memory
M(t) : mixer output
Me : Solar mass
n : index
N : photon number
Nchan : number of channels
p(s) : free-running noise
P : signal power
Pip : probability of cycle slip
Q : reduced quadrupole moment, quadrature component
R : decimation rate, reference signal, rectangular impulse
R~v : Riemann curvature tensor
s : Laplace complex frequency variable
Sij : PM signal from PDmain on OBij
S+ : common-arm error signal
S_ : differential-arm error signal
Sdirect : direct arm-locking error signal
T : sampling interval, measurement time
TCL(S) : closed-loop transfer function
ToL(S) : open-loop transfer function
Ts,(s) : arm-locking sensor transfer function
T,, : matter stress-energy tensor
Udig : digitization noise
Udigl : single-channel digitization noise
ULSB : LSB amplitude
UsN : shot-noise amplitude spectral density
w(n): window function
x(s) : input signal
x(n) : discrete input signal
y(s) : output signal
y(n) : discrete output signal
z : z-domain variable
Z : z transform
F~, : Christoffel symbol
6(x) : Dirac delta function
6ij : Kronecker delta function
61ig : digitization noise in in-phase component
6Qdig : digitization noise in quadrature component
vdig: digitization noise in frequency correction
AT : arm length difference
S: electric field amplitude
: signal phase
e : PM phase error
i : PM input phase
: PM model phase

o : PM output phase
r : PM residual phase
S: phase difference between Oi and Oj
rp, : special relativity metric
A : wavelength
v : signal frequency
cor. : PM correction frequency
vij : frequency difference between vi and vj
v, : PM model frequency
vff : PM offset frequency
p : Laplace or z-domain pole
- : average arm length
,ij : light travel time from SCi to SCj
Tmax : maximum delay time in EPD unit
TRT : round-trip light travel time
uw : angular frequency
cGw : gravitational wave angular frequency
( : Laplace or z-domain zero

Abstract of 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



James Ira Thorpe

December 2006

Chair: Guido Mueller
Major Department: Physics

The Laser Interferometer Space Antenna (LISA) is a collaboration between the

National Aeronautics and Space Administration (NASA) and the European Space Agency

(ESA) to design and build a space-based interferometric detector of gravitational waves.

The LISA sensitivity band will range from 3 x 10-5 Hz to 100 mHz, a regime currently

inaccessible to ground-based detectors.

The LISA detector will consist of a constellation of three identical spacecraft arranged

in a triangular formation 5 x 106 km on a side. Each spacecraft will contain a pair

of freely-falling proof-masses that will act as the geodesic-tracking test particles of

general relativity. The separation between the proof-masses will be monitored using laser

interferometry with a precision of ~ 10 pm, allowing for the detection of gravitational

waves with strain amplitudes in the range of 10-21.

The author is part of a group at the University of Florida that is developing a

laboratory-based simulator of LISA interferometry. This dissertation describes the

simulator in detail, emphasizing the electronic components designed and constructed

by the author. These include a phase meter capable of measuring the phase of a

cavity-stabilized laser beat-note with a noise floor of better than 10-5 cycles/v/Hz

from 1 Hz 10 kHz and an electronic phase delay unit capable of delaying signals with

frequencies up to 25 MHz for more than 300 s with or without a frequency offset.

Also described is a set of experiments made using the simulator that investigate

arm-locking, a proposed method for reducing the phase-noise of the LISA lasers. A laser

beat note was successfully stabilized to a 1.065 ms delay with a bandwidth of ~ 10 kHz.

The residual frequency noise was less than 200 mHz//Hz from 10 mHz through 100 Hz.


1.1 Motivation for Gravitational Wave Astronomy

The history of science is rife with examples of new technologies leading to breakthroughs

in our understanding of the natural world. This is particularly true in the fields of

astronomy and ;it!, Iili\ i -. The study of the heavens is undoubtedly one of the planet's

oldest sciences and for countless millenia it proceeded with one instrument: the human

eye. Fortunately, the eye is quite a good instrument and a great deal was learned about

the universe using it.

The invention of the optical telescope brought an improvement in angular resolution

over the eye, allowing Galileo to observe moons orbiting Jupiter. These observations

helped to cement the Copernican model of a heilocentric universe. While telescopes

improved the spatial resolution of astronomical observations, the advent of photography

widened the spectral window of these observations to include wavelengths at which the

human eye is insensitive.

The 20th century saw an explosion of new spectral windows opened to the heavens.

The universe can now be observed in radio, microwave, infrared, optical, ultra-violet,

X-ray, and 7-rays. Each of these new spectral windows produced surprising and significant

results that altered our understanding of the universe. For example, microwave astronomy

led to the detection of the cosmic microwave background and the validation of the Big

Bang theory while X-ray observations provided the first evidence of the existence of black


Nearly all of our information about the universe outside our own solar system comes

from some form of electromagnetic radiation. Despite the wide range of observable

frequencies (there are more than eighteen decades of frequency between a 1 MHz

radio wave and a 10 GeV gamma ray), all electromagnetic observations have common

characteristics. Electromagnetic radiation gives us direct information about the particles,

atoms, or molecules that generate it and interact with it. It is fundamentally a probe

of microscopic physics. Only through the association of radiating material with a

macroscopic object, such as the corona of a star or an accretion disk around a black

hole, can we make observations of macroscopic objects.

Gravitational waves' (GWs) represent an entirely new potential source of information

about the universe. A prediction of general relativity, GWs are disturbances in spacetime,

the combined fabric of space and time that is the arena for ]ph\ ii, in relativity. GWs

are thought to be produced by a variety of astroiph\ -il ,l systems, ranging in mass from

solar-mass neutron stars to black holes millions or billions of times larger at the centers

of colliding galaxies. In addition, the Big Bang may have produced GWs which would

exist today as a cosmological background. Unlike electromagnetic radiation, GWs couple

directly to large-scale objects. This makes them ideal for probing gravity, the dominant

force over macroscopic distances.

The ability to detect GWs will provide more than opening a new spectral window;

it is more akin to providing an entirely new --.i-''" with which we can learn about the

universe. If electromagnetic observations are our eyes, GWs are our ears. While it is risky

to make grand predictions about what we may learn, it certainly seems that we ought to

try and listen.

1.2 Gravitational Wave Detectors

The previous Section makes clear the motivations for trying to detect GWs. The

reason why it has yet to be done is that it is extremely difficult. The effect of a GW

passing through a detector is a tidal distortion characterized by a strain amplitude (change

in length over length) on the order of 10-21. A km-scale detector must detect length

changes on the order of 10-18 m, 1000 times smaller than the classical radius of a proton.

1 For two excellent extended introductions to gravitational wave astronomy, see Schutz
[1] and Hughes [2].

Despite these difficulties, there has been significant effort over the past half-century

to build GW detectors. The types and sizes of the detectors vary, with different detectors

optimized to observe GWs in different frequency bands. While no confirmed direct

detections have been made, measurements of orbital decay in binary pulsars [3] have

provided extremely convincing circumstantial evidence that GWs exist and generally

behave as expected.

One proposed GW detector is the Laser Interferometer Space Antenna (LISA),

which will consist of three separate spacecraft forming a triangular detector with sides of

5 Gm = 5 x 109 m. To measure GWs, LISA must detect length changes in these arms with

a precision of ~ 10 pm. Achieving this level of precision over such vast distances requires a

number of novel techniques and technologies.

1.3 LISA at the University of Florida

The author is part of a group in the Department of Physics at the University of

Florida (UF) that is developing a laboratory-based simulator of LISA interferometry. The

purpose of this simulator is to provide an arena in which the interferometric techniques

of LISA can be studied and developed. It also provides a source of LISA-like signals with

which to test prototype components. A long-term goal of the simulator is to have the

ability to inject model GW signals into the apparatus and produce LISA-equivalent science

signals with realistic instrumental noise. Such signals would be valuable for evaluating

data analysis techniques.

The remainder of this dissertation is divided into four parts. Chapter 2 presents

an overview of GWs including their theoretical origins, properties, likely sources, and

potential detection methods. Chapter 3 describes LISA in detail, with an emphasis on

the interferometry. Chapter 4 describes the development of the UF LISA interferometry

simulator, focusing on the electronic components of the simulator that were designed and

built by the author. Chapter 5 presents a series of experiments using the simulator that

investigate a laser phase-noise stabilization technique known as arm-locking that has been

proposed for LISA.


2.1 Overview

2.1.1 Relativity

The theoretical framework of Special and General Relativity represents our best

understanding of the macroscopic universe In both cases, the three usual dimensions of

space and the one dimension of time are combined into a single four-dimensional entity

known as spacetime. In Special Relativity (SR), spacetime is a passive background in

which pi\ -i -, occurs. Points in spacetime are known as events, and phi\ -i is concerned

with the relation between events. For example, the set of events that mark the position of

a particle in three-dimensional space as time evolves is known as that particle's worldline.

The worldline of a particular particle may be affected by non-gravitational phenomena

such as electromagnetic or nuclear forces.

As with Newtonian ph\ -i.' it is useful to define a coordinate system, or frame, which

can be used to label and compare events. In Special Relativity (SR), there exists a special

class of frames known as the Lorentz or inertial frames, in which free particles move in

straight lines with uniform velocity. A set of coordinate transformations, known as the

Lorentz transformations, relate the coordinates of an event in one inertial frame to the

coordinates of the same event in another inertial frame. Physically observable quantities

are independent of the particular frame used.

The four-dimensional Cartesian coordinate system x" = (t, x, y, z) can be used to

describe an inertial frame in SR2 The interval, or distance between events separated by

1 Much of the theoretical development in this Chapter follows Schutz [4]. Other portions
were adapted from Misner, et al. [5] and Shapiro & Teukolsky [6].
2 Unless otherwise noted, I will adopt the "natural ulir" of G = c = 1 for this Chapter.

the four-vector VT, can be computed as

As2 -= ,V1V", (2-1)

where As is the interval, 9r,, is the metric tensor, and the Einstein summation convention

(Xa y = 3 =o~xoa) applies. The metric tensor for an inertial frame in SR can be written

using the Cartesian coordinates described above as

-1 0 0 0

0 1 00
TI = (2-2)
0 0 1 0

0 001

Spacetime intervals are split into three classes according to their sign. Four-vectors

with negative intervals are known as time-like. The worldlines of all massive particles have

time-like intervals. The magnitude of the interval is equal to the proper time, the time

elapsed in an inertial frame comoving with the particle. Four-vectors with null intervals

are known as light-like, because photons and other massless particles have worldlines with

null intervals. Four-vectors with positive intervals are known as space-like.

In the language of differential geometry, spacetime is a four-dimensional Riemannian

manifold for which the distance between points on the manifold is given by a rank-2

metric tensor. In SR the manifold is "flat," meaning that every inertial frame is valid

over all of spacetime and the metric rl, can be used to compute the spacetime interval

between distant events. The straight lines that describe the worldlines of free particles are

special curves known as geodesics, which have the property that the interval along them is


In General Relativity (GR), the manifold is curved in an additional dimension or

dimensions. Because of the curvature, the global inertial frames of SR do not exist in GR.

However, since spacetime is smooth, it appears to be flat over small distances. At each

point in spacetime, a local inertial frame can be constructed in which the p]l\ -i4 of SR

apply locally. The differential interval between two nearby points separated by dx" can be

computed using a differential form of (2-1)

ds2 =gpdx"dx", (2-3)

where the metric is labeled g, to distinguish it from the SR metric. Since spacetime is

curved, the local inertial frames will differ between adjacent patches and consequently g9,

will be a function of position within spacetime. To compute the interval between distant

events, (2-3) must be integrated along the path between the two events. As in SR, the

worldlines of free particles in GR will follow geodesics. Due to the curvature of spacetime,

geodesics in GR will not generally be straight lines in a local inertial frame. Variational

principles can be used to produce equations describing geodesics in GR. If the worldline

is described by a set of events parametrized by a scalar parameter A, x"(A), then the

geodesic equation can be written as

P" + FX"P 0, (2-4)

where the dot denotes derivation with respect to A and F, is a combination of derivatives

of the metric known as a Christoffel symbol,

1PM W ^ (^V p + gpVV 9,c), (2-5)


9Yp = a (2-6)

The geodesic equation describes the behavior of free particles in a curved spacetime.

GR connects this with gravity by specifying that the source of the spacetime curvature is

matter. More specifically it is the energy density of all forms of matter and non-gravitational

forces. This relationship is expressed mathematically by the Einstein equations,

Gpl = s87TT,,, (2-7)

where T,, is the stress-energy tensor of matter and non-gravitational forces and G,, is

the Einstein curvature tensor, a particular combination of metric derivatives. The form of

G,, is imposed by the constraints placed by conservation laws for energy and momentum

on T,,. In order for (2-7) to be generally valid, G.,. iIti satisfy the same constraints.

The Einstein curvature tensor is a combination of metric derivatives that satisfies the

constraints on T,.

GR is sometimes summed up by stating that spacetime tells matter how to move

(geodesic equation) and matter tells spacetime how to curve (Einstein equations).

2.1.2 Weak-field GR and Gravitational Waves

The Einstein field equations are a set of ten coupled non-linear partial differential

equations. Only a handful of analytic solutions are known. It is often useful to consider

approximations to the full theory, which are more amenable to analytical study. One such

approximation is the weak-field limit, in which the GR metric is equal to the SR metric

plus some small perturbation, h,.,

g, -=rlyw + hl, (2-8)

1 h,, < 1 (2-9)

The linearized theory is developed by truncating the full GR equations to first order in

h,,. In doing so, it is useful to exploit the gauge freedom of GR. Gauge freedom refers

to the ability to make changes to tensors such as h., without affecting the observable

quantities, such as the spacetime interval between two events, that are computed from

these tensors. The linearized Einstein equations are typically written using the Lorentz

gauge, which requires

S 0. (2-10)

With this condition, the linearized Einstein equations become

OD" -167T", (2-11)

where h"l h/"-l Vl"hPp is the trace-reverse of h/, and H is the D'Alembertian operator.

The D'Alembertian operator is also known as the wave operator, since it gives the wave

equation when applied to a function,

/f = (- + 2)f, (2-12)

where V2 is the Laplacian in the three spatial dimensions. The form of (2-11) indicates

that there will be a set of wave solutions to the linearized Einstein equations. These

solutions are known as Gravitational Waves (GWs).

2.1.3 Properties of Gravitational Waves

The properties of GWs can be deduced from the homogeneous version of (2-11),

which corresponds to linearized GR in a vacuum (T" = 0). The general solution of the

homogeneous wave equation is a superposition of plane waves of the form

h"w (x") A""exp(ikzx), (2-13)

where A/" is an amplitude tensor and kV is the four-dimensional analog of the wave vector

in classical radiation theory. In order for the expression in (2-13) to satisfy (2-11), kV

must be a null or light-like four-vector,

klk, = 0. (2-14)

The dispersion relation for GWs can be found by expressing kV in a 3 + 1 (three spatial

coordinates plus one time coordinate) coordinate system and identifying the time

component, ko, as the angular frequency of the wave, w. The condition in (2-14) can

then be written as
= k (2-15)

where k is the spatial component of the wave vector. From (2-15) it can be seen that

both the phase and group velocities of GWs are 1 in natural units, which corresponds to

the speed of light.

In order to be a solution to (2-11), the expression in (2-13) must also satisfy the

Lorentz gauge condition in (2-10). This places restrictions on AP"; requiring it to be

orthogonal to k,

A~"k, 0. (2-16)

The Lorentz gauge condition in (2-10) describes a class of gauges. Additional restrictions

on h can be obtained by choosing a particular gauge within this class. For GWs in

vacuum, the transverse-traceless (TT) gauge is useful. The TT gauge conditions are

h = 0 (2-17)


ho = 0. (2-18)

Within the TT gauge, there is an inertial frame of the background spacetime (the r,, in

(2-8)) for which the wave is traveling in the z-direction. In this frame, there will only be

two independent components of h ,

0 0 0 0

0 hxx hxy 0
h 0 (2-19)
0 hy -hx 0

0 0 0 0

The hTT refers to transverse-traceless gauge and the overbar has been dropped since a

traceless tensor is its own trace-reverse. The two independent components of h are

interpreted as two orthogonal polarization states for GWs. It is common to rewrite (2-19)

using two scalar polarization states and two unit polarization tensors,

hP = h+h++ hhx (2-20)

where the polarization tensors are

00 0 0

1 0 1 0 0
h+ = 0 (2-21)
2 0 -1 0

0 0 0 0

0 0 0 02

1 0010
h = (2-22)
v2 0100


The two scalar polarization states, h+ and h. each will obey a scalar wave equation

analogous to (2-13),

h+/x = A+/x exp(ik,x"). (2-23)

2.1.4 Interaction with Matter

The proceeding Sections described GWs as mathematical solutions of GR. How they

manifest themselves ]pli\ -;. 1lv can be examined by studying how these solutions effect the

motion of free particles. Recall that the geodesic equation (2-4) describes the motion of

free particles. A related equation, known as the geodesic deviation equation, describes the

evolution of the 4-vector linking two nearby geodesics,

P' = RP V apV1U73 (2-24)

where (" is the 4-vector linking the two geodesics, the dots denote differentiation to

a parameter of the geodesic (such as proper time), R a,3 is a combination of metric

derivatives known as the Riemann tensor, and Va and U3 are the four-velocities of the

particles on the geodesics. Consider a frame in which there are two neighboring free

particles, both initially at rest, separated by a distance E in the x direction. For this case,

(0, E,0,0) (2-25)


V U3 = (1,0, 0,0). (2-26)

These expressions can be substituted into (2-24) to obtain

-R" oo. (2-27)

To first order in h", the Riemann tensor is given by

R" va3 = f11Ma(h3,,voa + ha,73 hha,vc3 hv3,ca)- (2-28)

Using the expressions for h"" in the TT gauge, the four equations in (2-27) can be reduced

to two

S- + (2-29)

= x. (2-30)

All other components of "' are zero. The tidal effects of GWs on freely-falling particles

will be restricted to the plane normal to the wave's propagation direction. The motion

in this plane will be oscillatory, with an angular frequency equal to that of the GW. A

similar analysis can be made for two particles initially separated by a distance E in the

y-direction. The results are

S= -, + (2-31)

S= Fh,. (2-32)

The expressions (2-29) through (2-32) can be used to determine the tidal distortions of an

initially-circular ring of freely-falling test particles in the x y plane as a GW traveling in

the z-direction passes by. The result is shown schematically in Figure 2-1.

o o 0 Q 0 0 o -

P =0 = 2 w 2.

O + O 0 + O O + O + 0
0 O O O 0 Q
.w = 0 0.= w= -^


Figure 2-1. Distortion of an initially-circular ring of freely-falling test particles by a
GW propagating into the plane for (a), the + polarization and (b), the x
polarization. (fw refers to the phase of the GW.

2.1.5 Generation of Gravitational Waves

The general solution of the inhomogeneous wave equation (2-11) for hyv at an event

with coordinates (t, x') is given by the integral of the retarded Green's function over the
past light cone of the event,

4f T (t- \31- ?/| ,i )

The integrand can be simplified in the case that the plane for (a), thsource is compact and far from the

field point x',
hP/(t, x) T (t r, yi)d3y, (2-34)
~w~os,, ~= ~r

where r is the distance between the source and the field point and the integral is over the

source where lyil < r. Further simplifications can be made by exploiting the restrictions

placed on T,, by conservation laws. The laws of conservation of energy and conservation

of momentum can be expressed together as

TP",, 0. (2-35)

When (2-35) is applied to (2-34), the result to first order in (1/r) is

hT 0 (2-36)

^hik, ( jk (t r), (2-37)

where Ijk is the quadrupole moment tensor of the source mass distribution and the dots

indicate derivatives with respect to t. The quadrupole moment tensor is defined as

jjk pxJxkd 3x, (2-38)

where p TOO is the energy-density of the source distribution in its rest frame and the

integral is over the entire source. Oftentimes, it is useful to express (2-36) and (2-37) in

the TT gauge. In a coordinate frame at the observation point with the z-axis oriented

along the propagation direction of the wave, the expressions become

S= [Q (t r) -) Qy(t r) (2-39)

hx QX(t- r), (2-40)

where Qij is the reduced quadrupole moment tensor, defined as

Qj I- 6jIk k (2-41)
ii 3

and 6ij is the identity tensor. The expressions in (2-36) through (2-41) are known as the

quadrupole approximation and can be used to describe the gravitational radiation in many

]pli\ .i1 systems. This is the topic of Section 2.2.

2.1.6 Energy Carried by Gravitational Waves

Like their electromagnetic counterparts, GWs carry energy. In GR, gravitational

energy results from curvature, which is a global phenomenon. Consequently, the energy

density cannot be assigned to a specific point and can only be computed as an average

energy density over a region large enough to define the curvature. To compute the energy

for a GW, the quadratic contributions from h., to the left-hand side of the Einstein

equation, (2-7) can be moved to the right-hand side and treated as a stress-energy source

term. The result3 is
T(Gw) 1 lb. -a, h- ",a -ha ha \ (2 42)
TP 327r ,,h -2 -h -h (2-42)

where the angle brackets denote an average over several wavelengths. The expression in

(2-42) can be simplified if the TT gauge conditions ((2-17) and (2-18)) are applied,

T() ( 1 (h TT)ijIl(hTT) i (2-43)

Using (2-43) and the quadrupole approximation described in Section 2.1.5, the GW

luminosity for a compact source can be estimated as

LGW .ij (2-44)

To compute the luminosity in ]pli\ -I .1 units, (2-44) is multiplied by the conversion factor

Lo = cs/G C 3.6 x 1052 W. (2-45)

3 For details see Section 35.7 of Misner, et al. [5]

This luminosity is an upper limit that is never reached since LGW < 1 in natural units.

Nevertheless, in certain astropil\ -i. ,l systems, GWs carry a tremendous amount of energy

and play an important role in the system dynamics.

2.2 Sources of Gravitational Waves

Section 2.1.5 described the quadrupole approximation for the generation of gravitational

waves. It was found that a mass distribution with a time varying quadrupolar (or higher)

moment would generate GWs. In theory, GWs could be generated by a laboratory

apparatus such as the one in Figure 2-2. Consider a uniform beam of mass M and length

L lying in the x y plane. The z-axis passes through the center of the beam and the beam

makes an angle ( with the x-axis.


x M

Figure 2-2. A hypothetical laboratory generator of GWs consisting of a bar of length L
and mass M

Under the assumption that the cross-sectional dimensions of the beam are small

compared to its length, the reduced quadrupole tensor can be written as

cos2 cos sin 0
Qi = 2 cos (sin sin2 0 (2-46)

0 0 1

If the beam is assumed to rotate about the z-axis with angular frequency U, the second

and third time derivatives of Qij are

-cos(20) -sin(20) 0
ML 2L,)2
Qij= 6 sin(2Q) cos(2Q) 0 (2-47)

0 0 0

sin(2Q) cos(2Q) 0

0 0 0

The quadrupole formulas for h, (2-39) and (2-40) can be used to estimate the strain

amplitude from (2-47). It is clear from (2-47) that the quadrupole moment oscillates

with an angular frequency of 2L. This is to be expected from the symmetry of the system

under a rotation of r radians about the z-axis. The GWs will have this same frequency so

that IGW = 2o;.

Since the quadrupole approximation is valid only in the far-field, (2-39) and (2-40)

must be evaluated at a source distance of at least one GW wavelength (r = c/2w). The

strain amplitude at that distance can be estimated as

h(lab) _2ML2U3. (2-49)

For a beam with M 104 kg, L 10 m, and w 60 rad/s, (2-49) gives h(lab) 10-42. The

GW luminosity from the beam can be estimated using (2-44) and (2-48) as

LGw (lab) -M2L426. (2-50)

Using the same parameters, this gives LGW (lab) 10-33 W. It is clear from the small size

of h(lab) and Low (lab) that GWs are not relevant for laboratory systems. What is needed

to generate physically meaningful GWs is larger masses and higher velocities. Both can be

found in astrophysical systems.

A binary star system is an example of an astroihl\-l ,il system with a time-varying

quadrupole moment. It is well-known that the 2-body problem in GR has no analytic

solution. However, for most systems it is appropriate to use Newtonian mechanics to

describe the orbital motion and then use the quadrupole approximation to compute the

GW amplitudes and strains. Consider two point masses with masses mi and m2 in a

circular orbit of radius a in the x y plane, as shown in Figure 2-3.


2a'---- y

Figure 2-3. A binary star system as generator of GWs

If the system center of mass is placed at the origin, the position vectors of the two

masses will be

T1 = a (cos sin Q) (2-51)


X2 = a-(-cos, -sin ), (2-52)

where / = mml2/(mnl+m 2) is the reduced mass and 0 is the orbital phase angle, measured

from the positive x-axis to mi. The reduced quadrupole moment for this system is

cos2 (- cos sin 0

Qj pa2 cos sin sin2 0 (2-53)

0 0

This is the same form as that for the beam in (2-46) as should be expected from the

similarity of the geometries. The time derivatives of Qij can be obtained from (2-47) and

(2-48) with the substitutions M -+ p and L -+ 2a as well as a division by 3 that results

from the distribution of mass within the uniform beam. For the binary system, Kepler's

law gives a relation between the orbital frequency and the binary separation,

2 = (2-54)

where M = mu + m2 is the total mass. As with the rotating beam, the gravitational

waves will be emitted at a frequency equal to twice the orbital frequency. Using the above

relationships, the GW amplitude and luminosity for a binary system can be estimated as

h' 2M (2-55)
r a


LGw(binary) 32 M (2-5
5 a (25

The forms of (2-55) and (2-56) demonstrate that the largest and most energetic GWs

will be generated in binaries with large mass and small separations. Ideal candidates for

such binaries are binaries where one or both members is a compact object such as a white

dwarf (WD), neutron star (NS), or black hole (BH).

For example, a NS-NS binary (mil mi2 1.4 Msn = 2.8 x 1030 kg) with an orbital

separation of 2a = 500 km would produce a GW luminosity of LGw 1044 W at

a frequency QGW = 370 Hz. At a distance of 1 Mpc = 3 x 1022 m, this would produce

an energy flux of ~ 9 mW/m2 at Earth, about three times brighter than the visible light

flux from the full Moon. The corresponding GW strain amplitude at Earth would be

h(binary) = 2.3 x 10-21, thought to be within the range of GW detectors.

The energy carried away from the binary in the form of GWs causes the overall

energy of the binary to decrease with time. Consequently, the orbital radius must decrease

while the orbital frequency increases. The decrease in orbital radius increases the GW

energy output (Low (binary) o a -5), causing the system to radiate more strongly. The

resulting GWs increase in both frequency and amplitude with time, a waveform known as

a chirp. So long as no other pl-\ -i' ,.1 effects conspire to prevent it, the orbit will continue

to decay until the two objects merge. The chirp waveform can be estimated using an

adiabatic approximation in which the GWs are calculated from the Keplerian orbits and

the orbital parameters are changed to match the energy loss. For a circular orbit, the total

orbital energy is

E(binary) 1 pM (2-57)
2 a

Taking a time derivative of (2-57) and equating it with (2-56) results in a differential

equation for a,
64 pM2 (
a-~^ ~ *^ 0
a 5 a3 (2-58)

This equation can be solved to yield

a(t) ao(1 t/tmerge)1/4, (2-59)

where ao is the orbital radius at time t = 0 and tmerge is the time of merger, given by

5 a04
tmerge = 5 (2-60)
256 (2 1-p0)

As the orbital radius decreases, the accuracy of the Newtonian adiabatic approximation

worsens. This is precisely the regime in which the GW luminosity is the largest, so it

is important that more accurate methods be applied to predict GW waveforms. These

include analytic treatments with relativistic corrections to the orbits as well as numerical

simulations that incorporate the full Einstein equations [7].

In addition to binary systems, several other types of astrop1i,\-i i.11 sources of GWs

are thought to exist. Rapidly rotating NSs with a slight ;,-\ r!-r,-iriy will produce GWs.

The energy lost through GW emission will cause their rotation rate to decrease, much

as the electromagnetic radiation from pulsars cause spin-down. The waveforms for such

sources can be calculated in a manner similar to that for the binary systems [8]. Stellar

core collapse associated with supernovae are also a likely source of GWs, although in

order to generate GWs, there must be an ;-i\ ir,.tii'-i; flow of mass. The difficulty in

modeling supernovae makes detailed predictions of their GW signatures hard to obtain

[9]. Finally, there is also a possibility of a cosmological background of GWs analogous to

the cosmic microwave background for electromagnetic radiation. This cosmic gravitational

wave background would be a stochastic signal, the level of which can be estimated from

cosmological arguments [2].

GW sources can be separated by frequency band, much as electromagnetic sources are

separated into radio, visible, gamma-ray, etc. In general, larger masses translate to lower

frequencies. The merger of a stellar-mass binary will occur in the ~ 1 kHz band, where as

the merger of two supper-massive black holes (SMBHs), with masses 106 Msun 109 Msun,

will occur in the 1 mHz band. A GW spectrum -iI.2-.t,.,l by Hughes [2] is contained in

Table 2-1.

Table 2-1. Suggested frequency bands for GWs

Band Frequency Range Persistent Srcs. Transient Srcs.

Ultra-low Frequency 10-18 Hz ~ 10-13 Hz ? ?

Very-low Frequency 10-9 Hz 10-7 Hz ?


Low-Frequency 10-6 Hz ~ 1 Hz EMRI SMBM


High-Frequency 1 Hz ~ 10 kHz

(SMBI Super-Massive Binary Inspiral, BI stellar-mass Binary Inspiral, EMRI -
Extreme Mass-Ratio Inspiral, SMBM Super-Massive Binary Merger, BM stellar-mass
Binary Merger, SN Supernovae, RNS Rotating/pulsating Neutron Stars, CGBR -
Cosmic Gravitational Wave Background)

As with electromagnetic sources, the same phi\ -i .i1 object may radiate in different

bands at different epochs within its evolution. A stellar-mass binary in the early stages

of inspiral will exist as a persistent source in the low-frequency band. As it evolves, the

frequency will increase until it merges in the high-frequency band.

2.3 Detection of Gravitational Waves

With the knowledge that GWs represent a set of solutions to the Einstein equations

(Section 2.1) and the existence of several plausible mechanisms for their generation

(Section 2.2), it is a reasonable assumption that most of the universe is bathed in

gravitational radiation. The obvious question is how can this radiation be detected. A

number of techniques for detecting GWs have been proposed or implemented. These

techniques generally fall into two categories: direct techniques which measure the

amplitudes of the waves themselves and indirect techniques which infer the presence

of the waves from their effects on a well-understood ]ph\ -i1 ,1 system.

2.3.1 Indirect Detection

Thus far, only indirect detections of GWs have been made. In 1975, Russel Hulse

and Joseph Taylor discovered a pulsar known as PSR 1913+16 [3]. A pulsar is a

rapidly-rotating NS with a highly-beamed radio emission. As the pulsar rotates, its

radio beam sweeps across Earth, producing a pulse in a radio detector. Pulsars are

some of the most stable oscillators in the universe and provide a unique opportunity for

precision measurement of the motion of a distant compact object. After observing PSR

1913+16 for some time, it was determined that it was in a binary orbit with a radio-quiet

companion, likely a second NS. The mass of the companion and the orbital parameters

(radius, eccentricity, orbital phase, etc.) were extracted by fitting the pulse arrival times to

an orbital model. Once the binary system was characterized, the expected GW luminosity

could be computed from a modification of (2-56)4 With LGw known, an energy balance

4 For elliptical orbits, (2-56) is modified by an enhancement factor f(e)
1+(732/24 +(37/96)e that depends on the orbital eccentricity, e. The system also radiates
preferentially at periastron, meaning that GW emission tends to circularize orbits [10].

could be used to determine the predicted rate of orbital decay as in (2-59). Figure 2-4

shows a plot of the observed shift in orbital phase (versus a non-decaying orbit) of PSR

1913+16 from 1975 to 1988 along with the predictions of GR.

-2 -

m 2

shift dueto Gs igure 5fo Tyo ad W
s -6



-10 i I I ,I ,

75 80 85 90

Figure 2-4. Observed shift of periastron for PSR 1913+16. The solid line is the predicted
shift due to GW emission (Figure 5 from Taylor and Weisberg [11], used by
permission of the American Astronomical Society)

The stunning agreement provides excellent circumstantial evidence for the existence of

GWs and provided Hulse and Taylor with the 1993 Nobel Prize in plin\ i -. In addition to

PSR 1913+16, several other binary pulsars have been observed. The observations of each

have thus far been in agreement with the predictions of GR [12].

2.3.2 Direct Detection

While the measurements of binary pulsars provide extremely strong evidence for the

existence of GWs, they do not allow the information carried by the waves themselves to be

extracted. What is needed is a method to directly measure the GW strain h(t). This will

allow for comparison with predicted models of h(t), providing tests of the models as well

as providing a means to measure parameters of the systems generating the waves. In the

following subsections, the most common methods for measuring GWs are discussed. For a

more exhaustive list of potential detectors see Misner, et al. [5], Chapter 37. Doppler-tracking of spacecraft

Beginning in the 1960s, man-made probes began to leave Earth orbit and travel

towards the outer planets. The radio communications systems on these spacecraft (SC)

provide a means to make precise measurements of the spacetime interval between the

SC and a receiver on Earth. Since both Earth and a SC in cruise phase appoximate

freely-falling particles, this is a measurement of geodesic separation and will be effected

by GWs. The Doppler tracking technique [13, 14] begins with an ultra-stable oscillator

of frequency vo which is used to drive an Earth-based transmitter. This signal travels

to the distant SC, which receives a Doppler-shifted version of the signal a time 71 later.

A phase-lock-loop (PLL) on-board the SC is used to fix the SC's local oscillator to the

incoming signal. The SC then transmits this signal back to Earth, where it is received

after an additional delay 72 and with a two-way Doppler shift Av. For a coordinate system

in which a GW propagates in the z-direction with the x y axes oriented parallel to the +

polarization (See Figure 2-5), the response of the Doppler shifts to GWs can be written as

Av(t) 1 1 + Ph(t
^A(t) p~h~(t) ph[t (1 + p)7] + ( 71 72)
vo 2 2
+N(t) + N2(t 1) + N(t 71 72), (2-61)

where the polar angles to the SC are (0, Q), = cos(0), N1, N2, and N3 are noise terms


h(t) h+(t) cos(2Q) + h (t) sin(2Q). (2-62)

The response to GWs in (2-61) is sometimes called a three-pulse response, since an

impulse in h(t) will show up in the signal at three distinct times. For long wavelengths

(AGw > 71,2), the three pulses will interfere destructively. This sets the lower frequency

limit for the Doppler-tracking technique.

Earth .... Y

Figure 2-5. Concept for Doppler-tracking detection of GWs.

At high frequencies, the noise terms, which include scintillation in Earth's atmosphere,

scintillation in the interplanetary medium, mechanical motion of the antennae, and

thermal noise in the receivers, begins to dominate the signal. For typical experiments, this

sets the frequency range to be roughly 10-4 Hz < few < 10-1 Hz [13]. The sensitivity of

the signal is set by the remaining noise level in the system.

An improvement upon the simple Doppler tracking can be made by flying an

additional ultra-stable oscillator on the SC and making a separate measurement of the

one-way Doppler shift between Earth and the SC. Since the noise terms that enter into

this measurement will be related to the noise terms in (2-61) under time shifts of '-, it is

possible to create a linear combination of the Earth-SC and SC-Earth Doppler shifts that

partially cancels the noise terms. It is expected that this two-way Doppler technique could

provide amplitude sensitivity of 10-18 at frequencies around 1mHz, corresponding to a

strain spectral density amplitude of 3 x 10-20 / /Hz [13]. Experiments using the one-way

technique have been performed using the Pioneer SC, Galileo, Mars Surveyor, and most

recently Cassini [15]. Pulsar timing

Another technique is to use a distributed array of pulsars as a timing network for

GW detection [16, 17]. Pulsars are among the most precise clocks in the universe, a fact

that made the indirect detection of GWs using binary-pulsars possible. In the direct

pulsar-timing technique, the pulse time-of-arrivals (TOAs) are measured for each pulsar

and used to generate a model of the pulsar's environment. This model includes all known

effects on the TOAs, including relative motion between Earth and the pulsar, detector

systematics, and the effect of GW emission on the pulsar orbit if it is in a binary system.

A set of residual TOAs is then formed by subtracting the model TOAs from the observed

TOAs. For a perfect model, the residuals will be zero. If a GW disturbs spacetime

between the pulsar and Earth, the signal will show up in the residual TOAs. Roughly

speaking, the relationship between TOA residuals and GW amplitude sensitivity is

hpT MR QGW (2-63)

where R is the rms of the TOA residuals in the absence of a GW, and cGW is the

frequency of the GW. The best current measurements have R ~ 200 ns, which allows

for a GW sensitivity of ~ 10-15 in the extremely low frequency band, QGw ~ 1 nHz. In

theory, a single pulsar measurement would be capable of detecting GWs in this manner.

In practice, multiple pulsars are needed to reduce the possibility of signals due to noise in

the residuals. For N completely uncorrelated sets of residuals with equal R, the sensitivity

in (2-63) will increase with VNV. The Parkes Pulsar Timing Array [18] is a US-Australian

collaboration with a goal of observing 20 pulsars with 100 ns residuals over a period of

10 yrs. This would give an amplitude sensitivity of ~ 10-16 at Gw = 1 nHz. Sources

in the extremely low frequency band include inspiraling SMBH binaries and stochastic

sources such as the cosmological background. Resonant mass detectors

The earliest GW detectors were resonant mass detectors, or "bars", first conceived by

Joseph Weber in the 1960s [19]. They consist of large masses (bars) suspended in such

a way as to minimize damping. A passing GW will deposit some energy into mechanical

vibrations of the bar. If this excess energy can be measured, the GW can be detected. The

challenge for bar detectors is distinguishing the small amount of energy added to the bar

by a GW from the large amount of energy already present in the bar and the read-out


The amount of energy deposited in the bar can be increased by increasing the bar's

mass. Todnv- bars have masses of several thousand kilograms. To prevent vibrations from

the outside world from disturbing the bar, it must be mechanically well-isolated.

Cooling the bars to cryogenic temperatures reduces the thermal noise present in

the bars. The remaining thermal noise can be mitigated by using materials with a high

mechanical quality factor, or Q. This places most of the thermal energy in a narrow

frequency band, leaving lower noise in the remainder of the band.

The second major challenge for bar detectors is measuring the energy within the bar

without disturbing it. Most modern bar detectors use mechanically resonant read-out

systems, which consist of smaller masses coupled to the main bar in such a way that

ti'.-v are resonant with the GW frequency of interest. The motion of these smaller

masses is measured with electromechanical transducers built from SQuID (Sub-Quantum

Interference Device) electronics.

Figure 2-6 shows a 1996 sensitivity curve for ALLEGRO, a bar detector in Baton

Rouge, LA [20]. To produce the curve, the spectral density of the detector noise was

scaled to equivalent GW strain amplitude. Detectable events would have a strain

amplitude above the curve. The curve for ALLEGRO shows two narrow bands of

maximum sensitivity, corresponding to resonances within the detector. In this sensitivity

curve the maximum sensitivity reaches 10-21 //Hz in a narrow band near 921 Hz. In

general, size restrictions and limits on vibrational isolation limit the observational window

of bar detectors to the high frequency regime (fMw > 100 Hz, see Table 2-1). Persistent

sources in this frequency band include rotating NSs and cosmological background. For

GWs originating from optically-observed pulsars, the frequency of the GWs is known and

the resonances of the bars can be tuned to search for it. Transient sources include the final

merger of stellar mass binaries, supernovae, and other unmodeled sources.

Measured strain noise spectral density of the LSU antenna




890 900 910
frequency (Hz)

920 930 940

Figure 2-6. Sensitivity of ALLEGRO bar detector 1996 (courtesy of W.O. Hamilton)

To date, no confirmed detections of GWs have been made with bar detectors. A

key technique for distinguishing GW signals from spurious noise bursts is coincidence

measurements between multiple bar detectors. This can also help provide direction

information for the observed GW, since bar detectors have nearly uniform antenna

sensitivity patterns. A number of major bar experiments are underway around the world,

some of which are listed in Table 2-2 .

Table 2-2. Operational GW bar detectors

Name Location Bar Temperature Operational Date

ALLEGRO Baton Rouge, USA 4.2 K 1991

ALTAIR Frascati, Italy 2 K 1980

AURIGA Lengaro, Italy 0.2 K 1997

EXPLORER Geneva, Switzerland 2.6 K 1989

NAUTILUS Rome, Italy 0.1 K 1994

NIOBE Perth, Australia 5.0 K 1993

5 Table 2-2 is an adaptation of a similar table by Johnston [21].

In addition, there are plans to build larger detectors, including ones with spherical or

nearly-spherical geometries. Spherical detectors both increase mass for a given volume as

well as provide additional resonant modes that can be used to determine GW polarization

and direction from a single detector [22]. Interferometric detectors

The tidal motion induced by GWs (Figure 2-1) is ideally suited to be detected with

a Michelson Interferometer (-!l). Consider a simple MI consisting of a light source,

beam-splitter (BS), two mirrors (M[,M ,), and a photodetector (D) oriented along the

x y axes as shown in Figure 2-7. The optics are assumed to be free to move along the

interferometer axes.




Figure 2-7. A Michelson interferometer as a detector of GWs. (LS = light source, BS =
beam splitter, MAy = mirrors, D = photodetector.

If a GW propagating in the z-direction passes through the detector, the mirrors will

respond as the masses in Figure 2-1, with the BS at the origin. The distances between the

BS and the mirrors in the x, y arm will then be

Lx(t) = Lo[1 + h+(t) cos(2b) h (t) sin(2b)] (2-64)


Ly(t) = Lyo0[ h (t) cos(2 ) + h (t) sin(2 )] (2-65)

where Lxo and Lyo are the nominal lengths of the x, y arms, h+, (t) are the polarization

amplitudes of the GW, and ) is the angle between the x-axis and the h+ polarization


Interferometry is a technique for measuring changes in the arm-lengths given in (2-64)

and (2-65) The light entering the BS can be described by an electric field oscillating at a

given frequency with a phase 0(t). As each light beam makes its out-and-back trip along

the arms, it will gain a phase of

i(t) = 2kLi(t) (2-66)

where i = x, y and k = 27/A is the wavenumber of the incoming light. When the light is

recombined at the BS, the two beams will have a phase difference given by

AQ(t) 2kAL(t), (2-67)

where AQ(t) (t) Oy(t) and AL(t) Lx(t) Ly(t). A number of techniques can be

applied to measure the phase difference in (2-67), which can then be used to extract h+(t)

and h (t).

For a more general relationship between the GW propagation direction, polarization,

and the detector plane, the expressions in (2-64) and (2-65) will include a functions of sky

position known as antenna patterns for each polarization. Except for certain orientations

such as a GW propagating along the x or y axes, the antenna patterns are nearly uniform.

Interferometric GW detectors are sensitive to GWs over the entire sky.

The observable frequency band for interferometric GW detectors is limited by

their size and by noise sources. The expression for phase accumulation in (2-66) is

valid only when the round-trip time is short compared to the GW period. For longer

arms (or shorter GW periods), the sign of the GW strain will reverse as the light is

propagating, causing the round-trip phase change to average towards zero. This effect acts

as a low-pass filter with a sinc(2Lfew) transfer function. There is no fundamental limit to

the low-frequency response of a interferometric GW detector. However, practical limits are

set by instrumental noise sources.

The many challenges that must be overcome in order to convert the concept in (2-7)

into a working detector can be broken into two areas: the ability to build freely-falling

test masses and the ability to make precision distance measurements between these

test masses. The former is known in the ground-based interferometer community as

displacement noise while the latter is called readout noise.

The chief source of displacement noise in interferometers is often seismic noise.

An Earth-bound laboratory is not a freely-falling frame, but a free-fall condition in

one dimension can be approximated by suspending the test masses on pendula. This

constrains the test mass motion in the vertical and transverse directions. Longituinally,

for small displacements at frequencies above the natural frequencies of the pendula, the

masses are free to move.

Vibrations can couple into the test masses through the pendula, spoiling the free-fall

condition. Other sources of displacement noise are radiation pressure noise from the

light on the mirrors, internal vibrations of the mirrors driven by thermal energy, and

gas pressure noise. Significant effort is required to suppress these noise sources to a

level sufficient for GW detection. The ultimate limit on displacement noise may be

gravity gradient noise, which describes the time-dependent portions of the Newtonian

gravitational field. For Earth-bound detectors with current technologies, gravity gradient

noise limits the useful band to roughly 10 Hz and above.

The other class of noise sources for interferometric GW detectors is readout noise.

Readout noise includes shot noise and phase noise in the light source. Shot noise can

be reduced by increasing the power of the light source, however a p.,n.ltrv is paid in

displacement noise through increased radiation pressure noise. The shot-noise/radiation-pressure

noise limit represents the theoretical maximum sensitivity for a given detector operating

over a broad frequency band.

Laser phase noise is a particularly important noise source for interferometric

detectors. As shown in (2-67), the detector output is derived from the phase difference

between the two arms, A0. ('Ii!'5-'s in A0 can result from two sources, a change in AL or

a change in k

6(AQ) = 2k 6(AL) + 26k AL, (2-68)

where 6 denotes a time-dependant change from the nominal value. The first term

corresponds to motion of the mirrors and a potential GW detection, while the second

is caused by frequency (or phase) fluctuations in the light source. Note that the phase

noise term is proportional to AL, so that in the case where the arm-lengths are exactly

equal, it vanishes. To first order, an equal-arm interferometer is insensitive to phase noise

of the light source.

In the past decade, several kilometer-scale interferometric detectors have been built

around the globe (see Table 2-3, based on data from Johnston [21]). Multiple detectors are

necessary for performing correlations in order to reduce the occurrences of false detections.

In addition, the relative timing between events as measured by widely-seperated detectors

can provide information on the direction to the GW source.

Table 2-3. M. i, .r ground-based GW interferometers.

A representative example of a modern interferometric GW detectors are the LIGO

(Laser Interferometric Gravitational Wave Observatory) detectors in the US [23]. Figure

2-8 shows an aerial photo of the 4 km LIGO detector at Hanford, WA. The effective length

Project Location Arm Length Status

LIGO USA(2) 4km in operation

VIRGO Italy 3 km commissioning

GE0600 Germany 600 m in operation

TAMA300 Japan 300 m under construction

of the arms is enhanced by placing Fabry-Perot cavities in each arm. A frequency and

intensity stabilized Nd:YAG laser operating at 1064 nm provides ~ 6 W of power to the

interferometer, which is enhanced by a factor of ~ 40 by a power-recycling mirror. The

test masses are 10 kg fused silica optics suspended from a wire pendulum with a 0.75 Hz

resonance frequency. Additional passive and active vibration isolation further reduce

seismic coupling into the test masses.

Figure 2-8. Aerial photograph of the LIGO interferometric GW detector (courtesy LIGO
Scientific Collaboration)

Figure 2-9 shows a sensitivity curve for LIGO during the science runs known as

S4 (Spring 2005) and S5 (Nov. 2005 present) along with the design goal for LIGO

sensitivity. Aside from a few narrow peaks and a slight excess at low frequencies, the

LIGO detectors are now performing at their design sensitivity. The peak sensitivity of

~ 3 x 10-23 / O/Hz occurs just above 100 Hz. At higher frequencies the sensitivity degrades

with a slope of roughly one (sensitivity is proportional to fl). At lower frequencies the

sensitivity decreases sharply, reaching ~ 10- "l //z at 10 Hz. This sharp decrease

in sensitivity can be attributed to displacement noise, mainly from vibration coupling

through the pendula as well as gravity gradient noise. This sets the LIGO observing band

to the "high-ft.r' IIi. y" band as defined by Table 2-1. LIGO is currently searching for

GWs from many of the predicted sources in this band. While no confirmed detections have

been made, the data has allowed important upper limits to be set [25-27].

Strain Scnsiti' ity for the LIGO 4km Interferometers
S5 Perfomince Jne 2006 LIGO-G060293-01-Z
1C 6
-- i I n. ,, H. *..It ., I rrI.- I rl-' .2 iI
'[, .r I .i j .- r ) 1i I 1 '. ir .il-q I. -i L l.-- i i i


IL Itl% j
Frequency [Hz]

Figure 2-9. Sensitivity curves for the LIGO detectors during the S5 science run (courtesy
LIGO Scientific Collaboration [24])

The sharp drop in sensitivity below ~ 30 Hz is sometimes called the "low-frequency

wall" and represents a limit for current ground-based interferometric GW detectors. While

future technology may allow for some improvement at low frequencies, it is unlikely that

ground-based interferometers will be able to access GW sources in the low-frequency band.

To do this, the detector must leave the noisy environment of Earth. One possibility is to

place LIGO-type detectors on celestial bodies with less seismic activity, such as the Moon.

Alternatively, the detector itself can be placed in space to avoid seismic noise completely.

This is the goal of the Laser Interferometer Space Antenna (LISA), the subject of the

remainder of this dissertation.


3.1 Introduction

As mentioned in Chapter 2, current understanding of GW detection suggests that

gravity gradient noise will prevent Earth-based GW detectors from observing sources

in the low-frequency band (10-6 Hz ~ 1 Hz, see Table 2-1). One way to access the

many interesting sources in this band is to move the detector into space. The Laser

Interferometer Space Antenna [28] (LISA) is a joint project of the National Aeronautics

and Space Administration (NASA) and the European Space Agency (ESA) that plans

to launch a space-based instrument capable of detecting gravitational radiation in

the frequency band of 3 x 10-5 Hz to 10-1 Hz with a strain sensitivity in the range of

10-21/ Hz.

3.2 Sources

Three known types of sources populate the LISA observational window: galactic

binaries, extreme mass-ratio inspirals (EMRIs), and SMBH mergers (see Figure 3-1).

Galactic binaries refer to the early inspiral phase of stellar-mass compact objects. Since

these objects will be far from merger while in the LISA band, th-v can be treated as

persistent sources. So many of these sources are thought to exist that the LISA sensitivity

will likely be limited by a confusion background of galactic binaries in some frequency

regimes. The fact that GW sources are treated as "noise" to some in the LISA community

is indicative of the difference in source abundance between the low-frequency and

high-frequency GW bands. There are also a handful of binary systems that have been

observed electromagnetically and should produce isolated GW signals in the LISA band.

These "verification binaries" will provide an instant test of the instrument, as well as

serving as valuable calibration sources throughout the mission lifetime [29].

An EMRI refers to a small (1 M. 10 M.) compact object falling into a SMBH

(106 M. ~ 109 M.). In an EMRI, the spacetime is dominated by the SMBH and the

smaller object serves as a "test particle", tracing out the geodesics of the spacetime

near the SMBH. This will allow for the first time precision tests of GR in highly-curved

spacetimes, a major goal of LISA science [2].


0.. Coalescence of
E Massive Black Holes
< 10-20- /
> \Resolved
SGalactic Binaries

> Unresolved
E I Galactic
3 Binaries LISA

10-24 I I I
10-4 102 100
Frequency [Hz]

Figure 3-1. Sources in the LISA observational window (Courtesy NASA)

A merger of two SMBHs may occur during galactic collisions, when the SMBHs at the

center of each parent galaxy inspiral into one another and merge. These events would be

among the most energetic in the universe and would be visible to redshifts of z ~ 5 10


LISA will also have an opportunity to search for a cosmological background of GWs

produced by the Big Bang. However, most models of such backgrounds that exist predict

that thi will lie below the LISA band.

3.3 Mission Design

The LISA mission concept calls for three individual spacecraft (SC) arranged in a

triangular constellation approximately 5 Gm (1 Gm = 109 m) on a side. The center of

the constellation will follow a circular heliocentric orbit with a radius of 1 AU, offset in

orbital phase from Earth by approximately 20, as shown in Figure 3-2. The plane of the

constellation is inclined with respect to the ecliptic plane by 60 and the constellation

revolves in its plane with a period of one year.

E10' km
Earth '
Relative orbits
'of spacecraft


Figure 3-2. Orbital configuration of LISA constellation (Courtesy NASA)

Each SC contains two optical benches at the center of which is a 4-cm cube

of gold-platinum alloy known as the proof mass. Like the mirrors in ground-based

interferometers, the proof mass will represent the geodesic-tracking free particle in GR.

Passing GWs will modulate the proper distance between the six proof-masses, an effect

that will be measured using laser interferometry.

As with ground-based detectors, the challenges of LISA naturally divide into two

areas: building a proof mass that approximates a freely-falling test particle and measuring

the distance between the proof masses with a precision sufficient to detect the minute

length changes caused by GWs. Accomplishing these tasks is the goal of the two

major LISA instrumental systems, the Disturbance Reduction System (DRS), and the

Interferometric Measurement System (IMS).

3.4 The Disturbance Reduction System (DRS)

A test mass in GR is completely isolated, its motion dictated solely by the geometry

of the space-time in which it exists. Real objects in the universe can only approximate

an ideal test mass; they are subject to electromagnetic interactions, particle interactions,

and other spurious forces. Isolating the LISA proof mass from these other forces is

the function of the disturbance reduction system (DRS). The LISA DRS is based on a

technology called drag-free control, in which the SC is used as a shield that flies around

the proof mass. Upon reaching their designated orbits, the SC will carefully release the

proof masses so that they are freely-floating inside a small enclosure. Capacitive and

optical sensors will monitor the position and orientation of the proof-mass and feed

this information to a controller. The controller will keep the proof masses centered in

their enclosures by utilizing one of two actuators: electrostatic plates to push the proof

mass and micro-Newton thrusters to move the SC. With an appropriately-designed

controller, the proof masses will act much as the suspended mirrors in ground-based GW

interferometers: free to move along the sensitive axis but constrained in other directions.

Design and construction of the DRS is an extremely challenging aspect of LISA

technology. In order to reach the desired strain sensitivity, the residual acceleration of the

proof-mass in the sensitive direction must be less than ~ 10-15 (m/s2)//Hz. Dozens of

potential noise sources such as electrostatic noises, thermal noises, and SC gravity gradient

noise can spoil this goal and must be addressed. Much effort has been made to design

ground-based experiments which can be used to investigate various aspects of the DRS

[30]. Most of these use model proof masses suspended on torsion pendula, providing a

similar system with a reduced number of degrees of freedom. In addition, an on-orbit

test of the DRS technology will come with the LISA Pathfinder mission, a technology

demonstrator mission planned for launch in 2009 [31].

3.5 The Interferometric Measurement System (IMS)

The other main function in LISA, measuring the distance between the proof-masses,

is the role of the interferometric measurement system (IMS). The goal is to measure length

changes on the order of h x L 10-21/v/Hz 5 Gm 10 pm/ /Hz between pairs of

proof-masses. While LISA is often colloquially referred to as "a Michelson interferometer

in i'," '", in reality LISA operates quite differently from a Michelson interferometer or

from any of the ground-based GW interferometers such as LIGO. Rather than utilizing

many optical elements to generate a single electronic readout, LISA makes a series of

one-way interferometric measurements between pairs of optical components and then

combines the results electronically to generate useful signals. The technologies required for

this approach differ greatly from those required for traditional interferometry.

3.5.1 IMS Overview

Figure 3-3 contains a schematic of the LISA IMS, consisting of the three SC each

with two identical optical benches. Each optical bench contains a proof mass, an infrared

laser light-source, photoreceivers, and optics. Referring to the notation in the figure, OBij

is the optical bench on SCi oriented towards SCj. The two optical benches on each SC

are connected to one another via an optical fiber. Optical benches on opposite ends of

a LISA "arm" are connected via a two 40 cm telescopes and a 5 Gm free-space link. Due

to diffraction losses over the long arms, only about 100 pW of light are received from the

~ 1W of light produced at the far SC.

Interferometry is used to make three types of measurements: distance between the

proof mass and the optical bench, the distance between optical benches on different SC,

and the phase difference between the lasers on adjacent optical benches.


-?F > 0B3
/'- .,

SC OB3 21 OB SC2

Sc, Sc2

Figure 3-3. Diagram of the LISA IMS. OBij refers to the optical bench on SCi oriented
towards SCj. Tij is the light travel time from SCi to SCj.

The basic premise of optical interferometry is to use the phase of a light field to make

measurents of distance. In general, a light field at a particular frequency can be described

by the real-part of a complex electric field given by

E(t) = Eo exp {i[2rvt + 0(t)]} exp[k k z] (3-1)

where R[Eo] describes the electric field amplitude and its transverse variations (spatial

mode and polarization), v is the optical frequency, and 0(t) is the optical phase, and k

is the wavevector. In (3-1), all of the variation of the light field is contained in 0(t), v is

assumed to be constant. An alternative view descirbes all variation in E(t) as Ht.-'I'l II y

noise", or a change in v rather than 0. These two equivalent descriptions are easily related.

If the frequency is described by v(t) = vo + 6v(t), the equivalent phase noise is
(t) = 27 / v(7r)d7 + (0) (3-2)

and the equivalent noise spectra is

(i) (f (3-3)

where f is the Fourier frequency and the tilde indicates a frequency spectrum.

Since electromagnetic waves are linear, a superpostion of two optical signals can

be descirbed by adding their electric fields. A photodiode (PD) can be used to measure

the intensity of the combined beam, which is proportional to the squared magnitude of

the total electric field. If the two signals have frequencies v1,2 and phases 01,2(t), the PD

output will be a signal of the form

S(t) o sin[27~r12t + (2(t)] (3-4)

where v12 v1 v2 is the difference frequency between the two light beams and 12(t) -

01(t) 02(t) is the difference phase between the two beams. This signal is commonly
referred to as a "beat note".

In interferometry it is the phase of the beat note that contains the distance

information that is of interest. Unfortunately, it also contains noise from the light source

itself. In LISA the interfered light fields are produced by independent light sources,

meaning that the size of the fluctuations in the beat note 012(f) will be roughly equivalent

to the fluctuations in the individual lasers.

012(f) 2 Wf) 2v i (f) (3-5)

The concept behind LISA interferometry is to measure these large phase fluctuations

with high precision and then make combinations of different signals which will cancel the

laser phase noise while leaving the phase fluctuations induced by GWs. The key to this

approach is the ability to make phase measurements with a precision of ~ 1 pcycle//Hz of

laser noise that may be greater than 106 cycles/vHz in the LISA band. This is the task of

an instrument known as the phasemeter (PM). A detailed discussion of PMs, focusing on

two prototypes designed for the UF LISA simulator, is presented in Section 4.5.

3.5.2 The Optical Bench

A conceptual LISA optical bench is shown in detail in Figure 3-4. Three different

infrared laser beams enter the optical bench. The local beam (red), is produced by the

laser associated with the optical bench. The adjacent beam (blue) is produced by the laser

associated with the neighboring optical bench on the same SC, and reaches the optical

bench through an optical fiber. The far beam (green), is produced by the laser associated

with the optical bench on the far SC. These beams are interfered at three PDs, PDmaini

PDbackl, and PDback2.

In the baseline design of Figure 3-4, PDmain is used to interfere the incoming beam

with the local beam, producing a signal containing the one-way motion between the local

and far optical benches. In the "cross-over" design option, the incoming beam is interfered

with the adjacent beam rather than the local beam. Since the local and adjacent beams

will general have different carrier frequencies, this will reduce the effect of stray light.

At PDbackl, the adjacent beam is interfered with a local beam that has reflected off

the proof mass, producing a signal that contains the motion between the proof-mass and

the local optical bench.

The signal at PDback2 is similar to that in PDbackl except that the local beam has

not reflected off the proof mass. This signal serves as a reference signal to compare the

two lasers on adjacent benches. It can also be used as the error signal in a phase-lock loop

(PLL) in which the phase of the laser on one bench is forced to track the phase of the

laser on the adjacent bench. This is the LISA equivalent of a beam-splitter in a traditional


PDack2 PDac. PDma,
Adjacent Bench '
l "* To/From
I ,. Telescope

Local Laser

Figure 3-4. Diagram of a LISA optical bench. Light from the local laser (red) enters from
the fiber coupler on the bottom, light from the adjacent optical bench (blue)
enters from the left, and light from the far SC (green) enters from the right.

These three measurements are repeated on all six optical benches, resulting in

eighteen independent measurements that must be properly combined in order to extract

the distances between the proof masses.

Consider the two "backside" PDs (PDbackl and PDback2) on the optical bench in

Figure 3-4. A PM can be used to extract the phase of the beat signals. The PM outputs

of the backside PDs are:

Sbackl(t) = l(t) + kxpm(t) a(t) na(t) (3-6)


Sback2(t) = 1(t) (t) na(t), (3-7)

where 01(t) and Q,(t) are the phases of the light fields on the local and adjacent benches

respectively, k 2rv/c is the wavenumber of the light, xpm(t) is the motion of the proof

mass in the sensitive direction relative to the optical bench, and nai(t) is the phase noise

accumulated during travel from the adjacent bench to the local bench. This will include

both noise in the fiber as well as the relative motion between the benches. By taking the

linear combination Sbackl(t)- Sback2(t), the laser phase noise and fiber noise cancel, leaving

only the term proportional to the proof mass motion.

The measurement between optical benches on opposite ends of an arm is accomplished

using the signals from PDmain. The PM signals Sij(t) from PDmain on OBij are

S12(t) 12(t) 21(t 721) + h21(t) (3-8)


S21(t) 21(t) 12(t 712) + h12(t), (3-9)

where .(t) is the phase of the laser associated with OBij, hij(t) is change in phase due to

a GW for a beam traveling from SCi to SCj, and Tij is the light travel-time between SCi

and SCj. The information of interest in Sij(t), the GW signal hij(t), will be overwhelmed

by the laser phase noise '. (t). Unlike the situation with the back-side interferometers,

it is not possible to form a linear combination of Sij(t) and Sji(t) that eliminates ,(t)

while retaining hij(t). The reason is that the phase noise terms enter with time delays due

to the large separation between the SC.

3.5.3 Time Delay Interferometry

While it is not possible to create a signal free of laser phase noise using the PM

signals on one arm, it is possible to do so, or nearly so, by using PM signals from multiple

1 Other noise sources, such as relative motion of the SC due to non-gravitational
effects, will also enter the PM signals at levels much higher than the GW signal. They
are typically smaller than the laser phase noise and can be treated in a similar fashion.

arms with appropriate time di-lnvis. This process is known as time-delay interferometry

[32] (TDI), and is key to the success of LISA. The three LISA arms provide a total of

six one-way links, each of which can be potentially utilized to form a TDI signal. The

resulting space of possible signals is large and is typically broken into several classes [33].

Some of the most basic TDI combinations are the Michelson-like combinations, typically

referred to as X, Y, and Z. The three letters refer to the three-fold symmetry of LISA:

the X combination is the Michelson-like combination with SC1 as the "-. ,,ii.-r" SC, the

Y combination is the Michelson-like combination with SC2 as the corner SC, and the Z

combination is the Michelson-like combination with SC3 as the corner SC.

To form the X combination, the two lasers on-board SC1 (Figure 3-3) are phase-locked

using the signals on PDback2 so that (12(t) 013(t) = 1(t). This is the LISA equivalent of

the beam-splitter in a true Michelson interferometer.

To approximate the mirrors, the far SC (SC2 and SC3) are configured as optical

transponders. The PM signals at PDmain on OBjl will be

Sjl(t) = ji(t) 01(t 'Tj) + h(t), j =2, 3. (3-10)

A PLL is used to adjust the phase of the laser on the far SC so that Sji(t) 0.


jl(t) =- 1(t- 7Tl)-1 hj (t). (3-11)

This is the optical equivalent of the radio transponders used in the Doppler-tracking

experiments discussed in Section When the two beams return to SCI, the PM

signal at PDmain on OBlj will be

Sj(t) = 1(t) jl(t Tjl) + h 1(t)

1(t) l(t Tl, Ti) + lj(t Tjl) + j(t). (3-12)

The X combination is formed from S12(t) and S13(t) as follows:

X(t) = S12(t) S13(t) S12(t- 713 31) + S13(t- 712 21). (3-13)

Substituting (3-12) into (3-13) and simplifying gives

X(t) = h21t) h31(t) + h12(t 21)- h13(t 731)

-h21(t- 13 31) + h3(t 12 721)

-h12(t- 21 ~13 31) + h31(t- 12 31). (3-14)

The X combination completely removes the contributions from laser phase noise, while

leaving the contributions from the GW signals. The response of X to GWs is sometimes

referred to as a "four-pulse" response since an impulse in h(t) will result in an impulse

in X(t) at four distinct times: t, t + 7, t + 27, and t + 37, where 7 is an average of the

four one-way light travel times. It is important to note that the quality of the PLLs in

the transponders is not critical. In actual practice, the errors in the PLLs on each of the

SC will be monitored and added into the TDI signals. A similar procedure can be used to

measure and correct for any residual phase noise between Q12(t) and (13(t) using the signal

at PDback2[34]. Visualizing TDI

The Michelson-like TDI variables can be visualized using the "rabbit-ear" diagram in

Figure 3-5. Time delaying the PM signals by an amount 7 can be interpreted as sending

them along a virtual path with a light-travel time of 7. In the X combination, one beam

(red) makes a ]phi\ -. i1 trip (solid line) from SC1 to SC2 and back and then makes a
virtual trip (dashed line) from SC1 to SC3 and back. The other beam (blue) does the

reverse. When th,-v return to SCI, both have traveled the same distance and consequently

the phase noise is common and cancels out. This can be viewed as synthesizing an

equal-arm Michelson interferometer or zero-area Sagnac interferometer from the individual

PM signals.




Figure 3-5. The "rabbit-ear" diagram for the first-generation TDI X combination. Each
light beam originates at SC1 and takes ]ph i\ -i, (solid) as well as virtual
(dashed) trips to the far SC. The total round-trip path length for the two
beams is identical and the laser phase noise in each beam at SC1 is common. Extensions to TDI

If the LISA constellation were static, the TDI variables such as X would perfectly

cancel laser phase noise. However, as the orbits of the individual SC evolve, the

constellation will change shape, causing the yij to differ by up to one percent between

arms. Aditionally, the rotation of the constellation causes an .i, rir11' try in the light travel

time for a single arm, Tij / ji. With time-dependant values of Ti, laser phase noise is

no longer completely canceled in the first-generation TDI variables such as X. Instead,

it couples into the measurement at a level proportional to the relative velocity between

the SC. One way to surmount this problem is to utilize the ",'- ..I. I-generation" TDI

variables [35], which include four additional terms and cancel out the relative velocities

of the SC. These are sometimes referred to as the '-- i;lit-pul-," TDI variables, since an

impulse in h(t) will arrive at eight separate times in the signal. The cancellation of the

second-generation TDI variables is also not perfect, since there is a relative acceleration

between the SC as well. However, the residual laser phase noise after second-generation

TDI is applied is low enough so that satisfies the LISA error budget. The zero-signal TDI variable

One important TDI variable for LISA is the symmetric Sagnac variable ( [36]. A

Sagnac interferometer consists of two beams from a common source propagating in

opposite directions around a closed loop. The phase difference between the two signals is

proportional to the area enclosed by the loop and the rotation speed of the loop. In LISA

a Sagnac signal can be generated either Iph\ -i, .lly through a combination of PLLs on

appropriate benches or virtually through an appropriate combination of time-delayed PM

signals. In either case, the rotation in the constellation results in a path length difference

between the two beams of approximately 14 km, which is equivalent to a time-difference

of a 47 s. This will cause laser phase noise and other instrumental noise sources to

couple into (. The GW signal, on the other hand, will not be present in ( since the tidal

distortions caused by GWs are area-preserving. What ( provides is a measurement channel

containing instrumental noise but no signal. This is essential for distinguishing between

possible signals and instrumental noise since, unlike ground-based detectors, LISA will

not have other detectors with which to perform correlations. In a sense, the three arms

in LISA are equivalent to two co-located interferometers and the ( variable measures the

uncorrelated instrumental noise. Limitations and noise sources

In addition to the restrictions placed on TDI by the motion of the constellation,

there are sources of error that occur when the variables are formed in the first place.

In order to form a variable such as X in (3-13), one must know the values of ij. This

requires an independent measurement of the range between the SC. Errors in this ranging

measurement will degrade the noise cancellation in TDI. It is estimated that a ranging

accuracy of 20 m to 200 m is needed to sufficiently suppress laser phase noise [34]. This

requirement is dependant on the initial laser phase noise, and can be relaxed if the input

laser phase noise can be reduced through stabilization.

Another source of error in forming the TDI variables is obtaining the PM signals at

the proper times. Once nyi is known from ranging measurements, signals such as Sij(t- ij)

must be formed. The original TDI concept called for a triggered PM fed by a real-time

ranging system in order to obtain the time delayed signals at the appropriate times. This

approach has since been abandoned in favor of a PM with a fixed sampling frequency of

~ 10 Hz. The PM data and ranging data are telemetered to the ground and the delayed

combinations are formed by time-shifting the various PM signals. In order to achieve the

required timing accuracy, the PM signals must be accurately interpolated within a small

fraction of a sample period. This can be accomplished efficiently using a technique known

as fractional delay filtering [37], which is discussed in Section A.4.6.

3.5.4 Arm-locking

In the previous Section, it was demonstrated how TDI will be utilized by LISA to

suppress laser phase noise in the GW measurement. The requirements on TDI and its

associated tasks (ranging and interpolation) are strongly tied to the input laser phase

noise. LISA will use lasers that are intrinsically stable, but the laser phase noise will still

be large compared with what is required. The stability of the lasers can be improved by

several orders of magnitude by using an optical cavity [38, 39] or molecular transition

[40, 41] as a frequency reference. The current LISA baseline calls for each laser to have an

optical cavity system capable of providing a frequency stability of

Jv(f) < (30 Hz//Hz) x 1 + (1mHz/f)4 (3-15)

or better in the LISA band (3x10-5Hz-0.1Hz). With this frequency noise, second-generation

TDI, and a ranging/interpolation error of < 30 m, the IMS will meet its displacement

noise requirements [34, 42]. Improvement in 6v(f) over (3-15) will allow a relaxation of

ranging/interpolation requirements and possibly the use of first-generation TDI variables.

Arm-locking [43] is a technique whereby some combination of the LISA arms is used

as a frequency reference for laser stabilization. Although the lengths of the LISA arms

change over a period of a year by as much as a few percent, in the LISA band they are

extremely stable. If this stability can be transferred to the laser phase, the requirements

on TDI can be relaxed considerably. Closed-loop system dynamics

Before examining arm-locking in detail, it is useful to review the basics of closed-loop

system dynamics. Figure 3-6 shows a Laplace-domain representation of a generic

closed-loop system. The input x(s) is combined with the control signal c(s) to produce the

error signal e(s). The error signal propagates through the system, with transfer function

G(s), to produce the output signal y(s). The controller, with transfer function H(s), forms

c(s) from y(s).

X(S) + G(s) > y(s)


Figure 3-6. Diagram of a closed-loop SISO system with negative feedback. Signals: x(s) -
input, e(s) error, y(s) output, c(s) control. G(s) is the system transfer
function. H(s) is the controller transfer function

This type of system can be referred to as a single-input single-output (SISO)

closed-loop system with negative feedback [44]. The signals and the transfer functions

G(s) and H(s) are complex-valued functions of the complex Laplace variable, s = a+2rif.

The advantage of expressing the closed-loop system in the Laplace-domain is that the

differential equations that relate the time-domain signals reduce to algebraic equations

relating the Laplace-domain signals. The error, output, and control signals can be

expressed in terms of the input signal as

e(s) 1
C() (3-16)
x(s) 1+ ToL(s)'

y(s) G(s)
x(s) 1+ ToL(s)'
c(s) TL (s)
x(s) + ToL(s)'
where TOL(S) -- G(s)H(s) is the open-loop transfer function.

In the case of a control-loop used for stabilization, x(t) is a noise input to the system

and y(t) is the noise in the system output. The goal of the control system is to reduce

the magnitude of y(t) for a given x(t). Note that (3-17) contains the term 1 + TOL(s) in

the denominator. The magnitude of this term indicates the performance of the loop as

a suppressor of noise. If |1 + ToL(s)I > 1, the closed-loop value for y(s) will be smaller

than the open-loop value, given by (3-17) with H(s) = 0. If I1 + ToL(s) < 1, then the

closed loop values for y(s) will be greater than or equal to the open-loop value, a condition

known as noise enhancement.

The performance of a particular stabilization system can be evaluated using a Nyquist

plot, a plot of ToL(s) in the complex s-plane. As shown in Figure 3-7, the Nyquist plot

has two regions separated by a circle of unit radius centered on the point (-1, 0) marked

with an x. If ToL(s) lies inside the circle, the closed-loop system will enhance the noise. If

it lies outside the circle, the closed-loop system will suppress the noise. The degree of noise

enhancement or suppression is related to the distance from the point (-1, 0). The closer

ToL(s) lies to (-1,0) the larger the noise enhancement or smaller the noise suppression.

If ToL(s) reaches the point (-1, 0), known as a pole, the expressions in (3-16) (3-18)
become infinite.

In a Nyquist plot, TOL(s) is plotted as a curve parameterized by the Fourier frequency

(s -- 27iif). For most systems, the curve will be a spiral with frequency increasing

clockwise. Since the gain of most systems eventually decreases with frequency, the spiral

will be an inward spiral. Because of this tendency to spiral inwards, any curve that

encircles the point (-1,0) will eventually become arbitrarily close to it, causing the system

to become unstable. This behavior is summarized by the Nyquist stability criterion [44]:

for closed-loop stability, the open-loop transfer function ToL(f) must not encircle the point

(-1, 0) in the complex plane.



ToL (f)

Figure 3-7. Generic Nyquist plot for open-loop transfer function ToL(f). The shaded
region indicates noise enhancement. The pole at (-1, 0) is marked by an x.
Frequency increases clockwise along the curve of ToL(f). Steady-state arm-locking performance

In order to transfer the stability of the LISA arm to the laser phase, an error signal

must be derived from the PM signals. Ideally, this signal would be directly proportional

to the laser phase. The most basic arm-locking scheme is single-arm locking, where the

round-trip length of a single LISA arm is used as a frequency reference. Consider the PM

signal from PDmain on OB12 with SC2 acting as a transponder, (3-12). In the absence of

GW signals (h2(t) h21(t) = 0), this signal can be simplified to

S12 1 (t) l(t- RT), (3-19)

where TRT 712 + 721 is the round-trip light travel time between SCI and SC2. The

sensor's transfer function can be computed as

en2() = RT. (3-20)
Qi(s) 1

The Nyquist plot of T,,s(f), contained in Figure 3-8(a) is a circle, the result of a

vector sum of the two terms in (3-20). The first term is a unit vector along the positive

real axis. The second term is a rotating unit vector making an angle 27rf- with the

negative real axis. As f increases, T,,,(f) traces out a clockwise circle, reaching the origin

at f = f n/T RT, n = 0, 1, 2.... These are the "null fr' i, '- for which the signals

01(t) and 01(t TRT) are in phase and cancel. As the curve passes through the origin, the

phase of Tse(f) shifts discontinuously from -900 to +90.

Im Im Im

XPRe Pt--Re Re

(a) (b) (c)

Figure 3-8. Representative Nyquist plots for single-arm arm-locking: (a) sensor
(1 e-RT), (b) system (1 e-TRT), (c) open-loop 81/2(1 e--RT). The red
curves in (b) and (c) have higher gains than the blue curves.

In order to compute the system transfer function G(s), Tse(s) must be combined

with the actuator transfer function. Most laser actuators are frequency actuators,

producing a change in laser frequency that is proportional to the controller input. A

frequency actuator can be represented as a phase actuator in the Laplace domain with an

additional 1/s in its transfer function. The system transfer function (sensor + actuator)

for single arm-locking is then

G(s) (1G e-SRT), (3-21)

where Go is an overall constant gain factor. A Nyquist plot of G(s) for two different

values of Go is shown in Figure 3-8(b). Comparing the curves in Figure 3-8(b) with Figure

3-8(a), the effect of the actuator transfer function can be seen as a clockwise rotation of

900 coupled with a decrease in magnitude as frequency increases. This causes the system

to enter the noise enhancement region (inside the dashed circle).

An alternative view of G(s) is the Bode plot in Figure 3-9. Here the interferometer

nulls and phase discontinuities are clearly seen at multiplies of 1/tRT 30 mHz.


1 10 100
Frequency (mHz)


1 10 100
Frequency (mHz)

Figure 3-9. Bode plot of Ts,(f) with TRT 33s, the round-trip delay in a single LISA

As Go is increased (red curve vs. blue curve in Figure 3-8(b)), the system passes

closer to the instability point at (-1, 0). This is a marginally-stable condition where

an increase in gain produces an increase in noise suppression at some frequencies but a

corresponding increase in noise enhancement at other frequencies. Any additional phase

loss will rotate the system further, allowing it to encompass the (-1, 0) point and become


One solution to this problem is to design a controller with a transfer function whose

magnitude drops below unity before f =f 1/TT. This allows large gains for

f < fi while avoiding the instabilities at f > fl. Unfortunately for single-arm locking,

fl a 30 mHz lies right in the middle of the LISA measurement band.

An alternative approach is to utilize a controller that provides some phase advance

in the vicinity of the interferometer nulls, rotating the Nyquist plot away from the point

(-1, 0). This can be achieved with a transfer function of the form

H(s)= HosP, (3-22)

where Ho is a gain constant and 0 < p < 1. This form of H(s) will produce a phase

advance of p x 900. A Nyquist plot of ToL(s) = G(s)H(s) with p = 1/2 is shown in Figure

3-8(c). As compared to Figure 3-8(b), the curve is rotated 450 counter-clockwise. As the

gain is increased (red curve vs. blue curve), the curve approaches the line Re[ToL(s)]

Im[ToL(s)]. With this type of controller, the gain can be arbitrarily increased (assuming

no additional phase loss) without increasing the level of noise enhancement.

In the Bode representation (Figure 3-9), the phase response of the open-loop transfer

function is equal to the sum of the system phase response and the controller phase

response. The phase advance in the arm-locking controller lifts the phase minima at the

null frequencies away from -1800. This provides some positive phase margin and hence


The price paid for the phase advance of the controller in (3-22) is a reduced slope

in the magnitude of ToL(f). For a given controller bandwidth, this limits the gain at low

freuqencies. Alternatively, a higher bandwidth is required to reach a given low-frequency


A general arm-locking controller will have a transfer function similar to that in Figure

3-10. The frequency response can be divided into three distinct regions. For f < fl,

the controller can have a transfer function with a steep slope, allowing for large gains for

f < fl. From slightly below fl to the unity-gain frequency2 fuG, the controller must be

of the form in (3-22) so that it provides the necessary phase advance. For f > fuc, the

controller response can begin to roll off.



O - - - ---- - - - -


Figure 3-10. Bode plot for a generic arm-locking controller. The controller must provide a
phase advance between fl and fuc

For a general arm-locking controller with transfer function H(s), the closed-loop

suppression of phase noise can be computed from (3-16) and (3-21),

TCL (S) O -(3-23)
p(s) 1 + H(s)Gos-1(1 e-)

where p(s) is the laser phase noise prior to arm-locking and O(s) is the laser phase noise

after arm locking. The magnitude of the closed loop suppression, TcL(f)|, is plotted for a

generic arm-locking system in Figure 3-11.

For f < fi, the suppression can be large, due to the steep roll-off of the controller.

At a frequency just below fl, the closed-loop suppression crosses the 0 dB line, indicating

noise enhancement. This corresponds to the curve of ToL(f) entering the dashed circle

2 Strictly speaking, arm lock loops have many unity-gain frequencies, one for each time
TOL crosses the dashed circle in the Nyquist plot. There will be two of these points for
each f, where GGoH(f,)/2rifl > 0. Here we refer to the unity-gain frequency between
the interferometer nulls IGoH(fuG)/2wifuGI = 0.

in Figure 3-8(c). The level of noise enhancement increases until it reaches a maximum,

corresponding to the closest approach to the point (-1,0) in the Nyquist plot.

"servo bump"

dI I


Figure 3-11. Closed-loop noise suppression for a generic arm-locking loop

At f = fl, the closed-loop suppression is again OdB, corresponding to the curve

passing through the origin in the Nyquist plot. This behavior is repeated at each f, with

the heights of the noise-enhancement peaks and the depths of the noise-suppression valleys

decreasing as f increases. Near f = fuc, a "servo liiii1," may occur, caused by the loss of

phase in the controller as its magnitude flattens out. Transient response

The noise suppression curve in Figure 3-11 represents a steady-state limit that is

reached only after any transient responses decay. For a qualitative understanding of

arm-locking transients, consider (3-19), the single-arm error signal in the time-domain.

Under the assumption of high gain, the arm-locking controller will enforce the condition

1(t) {1 (t- TRT). (3-24)

If the controller is suddenly switched on at t = RT, the phase noise Q(t) for 0 < t < TRT

will effectively become "frozen" in the system. Under the idealized conditions of infinite

gain and no losses, the pattern of noise would repeat indefinitely. In the frequency domain,

this would correspond to all of the transient signal power being concentrated in the

frequency bins near f j fn, producing noise peaks similar to the ones in Figure 3-11.

For a real system, the relationship in (3-24) becomes more approximate as additional

uncorrelated noise mixes into the system. As it does so, the transients decay and the

system approaches its steady-state limit. The time constants for this decay are a critical

measure of arm-locking performance. If the time-constants are too large, valuable

observing time might be wasted while waiting for the noise to die down. It could also

limit the flexibility of mission managers to unlock and re-lock the constellation as needed.

For a given controller, it is possible to make an analytical estimate of the transient

frequencies and time-constants [45]. This is done using a Laplace-domain anwh -'i, that

properly accounts for the integration constants in the Laplace-transform of the constituent

signals. For the system described above,

f(trans) n Inm{9(f)-1}
'I (3-25)

(trans) TRT
tan Re{g(f)} (3-26)
Re{0(fn)-l (
where fT a is the frequency and r7tan) is the time constant of the nth transient, and

g(s) GoH(s)/s is the ratio of the open-loop transfer function to the interferometer

transfer function. These expressions are valid to first order in IG(s)-1l. In order to

compute the transient response for a given initial condition, the initial phase from 0 <

t < rnT is expressed as a Fourier sum of signals with frequencies fT ans and amplitudes

Atran). After the loop is closed, the amplitudes will then decay with time constants

Numerical simulations of arm-locking [43] suggest that the transient response may

be suppressed by slowly ramping the loop gain to its steady-state value rather than

suddenly turning it on at t = RT. The analytical treatment above does not apply

for time-dependant gains, but the result can be intuitively understood as follows. For

the initially low gains (small Go), the transient time constants will be small (3-26).

The system will reach steady state quickly, but the steady state will only be a slight

improvement over the free-running condition. The gain is then incrementally increased,

increasing the time constants and decreasing the steady-state noise. However, the input

noise is now slightly lower than the free-running case, corresponding to smaller initial

Atans). This process is then repeated until the final gain is reached. The overall time to

reach steady state is decreased since

1 -exp i ) > [1 -exp(-i)] (3-27)
i i=1

where Tr represent the values of Tra) for each quasi-steady-state value of the gain Go(i). Alternative arm-locking schemes

The single-arm locking discussed in is feasible, but its performance is less than

ideal. Since 1/tRT is in the LISA band, extremely large bandwidths would be needed to

achieve sufficient suppression. A suppression in laser phase noise by a factor of 104 at

10 mHz would require a unity-gain frequency of 1 MHz for a controller with p = 1/2.

Regardless of the controller shape or bandwidth, there will still be noise peaks in the

"stabilized" spectrum that are larger than the un-stabilized noise in the same frequency


The fundamental reason that these peaks are present is that the single-arm sensor

signal, (3-19), contains no information about phase noise with Fourier frequencies f = fn.

Suppression of noise at those frequencies requires an error signal that is sensitive to noise

at those frequencies. One way to obtain such an error signal is to utilize additional LISA

arms. As mentioned in Section, the orbital dynamics of the constellation cause the

LISA arms to differ by up to one percent. Consequently the values of f, for one arm will

be slightly different than those for another arm.

Consider the LISA constellation arranged as in the Michelson X TDI combination.

SC1 is designated the "master" SC and its two lasers are phase-locked to generate a single

light source with phase Q(t). The far SC are configured as transponders, so that the PM

signals from PDain on SC1 are

Slj(t) = 0(t) 0(t Tlj), j = 2, 3. (3-28)

If we define the average arm-length and arm-length difference as

712 713 (329)
7 2(3-29)


AT = 712 713 (3-30)

then (3-28) can be re-written as

S12(t) = (t) t ( + (-3 (3-31)


S13(t) 0(t) t- (- (3-32)

The transfer functions of Slj in the Laplace domain can be found using (3-20) with

S- i7 Ar/2. The signals for each of the two arms can either be added or subtracted.

The former produces the common-arm error signal, S+, while the latter produces the

difference-arm error signal, S_. In the Laplace domain, these two signals are

S+(s) 2 [1 e-S cosh s ) (3-33)


S_ (s) 2e- sinh (sA (3-34)

Common-arm locking uses S+ as an error signal. Figure 3-12 shows a schematic

Nyquist plot of S+. The signal consists of two terms added vectorially, a vector of length

2 along the positive real axis and a vector of length 2 Icos(TrfAr)| making an angle of

27wf> with the negative real axis. The curve will make its closest approach to the origin

for frequencies f, n/r. Unlike the case of single-arm locking, the error signal will not

pass through the origin and consequently the error signal phase will not reach -900. This

may relax the constraints on the controller, allowing the gain to rolloff more steeply in

the vicinity of the minima. For frequencies where f = m/Ar, m = 1, 2, 3..., '+ has a

true null. At these frequencies, a controller must provide the same phase advance as the

single-arm controller. For a one-percent arm-length difference in LISA, the first null would

occur at 3 Hz, which is above the LISA measurement band.


/ 2cos(7fAr) \A\

Aminmin I
p Re [TOL

Figure 3-12. Nyquist plot for common arm-locking

The depth of the minima in S+ and the corresponding minimum phase will depend

on the value of cosine term as f approaches fn. Since AT < 7 in LISA, it is reasonable

to approximate the Nyquist plot of S+ as a series of circles centered on the point (2, 0)

with the radius given by 2 |cos(TrfATr). The minimum amplitude and phase can then be

estimated geometrically as

Ami 2 [1 cos (7nr/T)] (3-35)


Omin w sin-1 [cos (TwAr/r)]. (3-36)

The first concern is at n = 1, since the cosine function approaches unity at DC. For

Ar/r = 0.01, A,ji t 10-3 and Omi, m -880. The additional 20 of phase margin provided

by common arm-locking would not cause a significant improvement in either stability

or performance over single arm-locking. If Ar were increased to 0.17, the minimum

amplitude would increase to A 0.1 while the minimum phase would increase to -720. This

might provide some improvement, but it would also cause the location of the first true null

to move from ~ 3 Hz to ~ 300 mHz.

Given the proceeding analysis of common-arm locking, it seems doubtful that it would

be of much use in LISA. However, an improvement over common-arm locking known as

direct arm-locking [42] looks to be more promising. The direct arm-locking error signal is

formed from the common and difference error signals as

Sdirect(t) =S+(t) J S_ (t)dt. (3-37)

The sign of the integrated term depends on which arm is longer and is chosen so that the

term is positive. In the Laplace domain, the direct arm-locking error signal can be written


Sdirect(s) = 2 ( e-8 cosh sinh Ar (3-38)
e S rosh 2 2 sinh 2 )
An ideal arm-locking error signal would only retain the first term in the curly brackets.

The second term will go to zero when the magnitude of the square-bracketed term is zero.

Figure 3-13 contains a plot of the magnitude of the square-bracketed term versus Fourier

frequency f. For f < 1/AT, the term drops to zero. At these frequencies, the response

of the direct arm-locking error signal is nearly flat, greatly relaxing the restrictions on

controller shape. This allows for increased noise suppression at these frequencies and the

removal of the frequency noise peaks.

As f increases, the square-bracketed term in (3-38) approaches unity, allowing the

overall error signal to come closer to zero at the frequencies f, = n/r This results in

a loss of phase at these points and a corresponding increase in noise in the closed-loop

system. Just before f reaches 1/AT, the term actually exceeds unity. In the Nyquist

representation of Figure 3-12, this results in the system entering the noise enhancement

region on the left-hand side of the complex plane. The maximum amplitude of 1.06 is

reached at f 0.87/Ar. The direct arm-locking controller must provide additional phase

advance at these frequencies in order to maintain the stability of the system.

E 0.5

0 1 2 3 4

Figure 3-13. Magnitude of square-bracketed term in (3-38)

One minor problem with direct arm-locking arises in the fact that the arm-length

difference is not constant and at times the arms are equal. This causes problems with

scaling S_(t) by 1/A7 in the direct arm-locking error signal (3-37). The arms will only be

equal occasionally (a few times a year) and it should be possible to address the problem

by occasionally switching the location of the master SC, reverting to single arm-locking,

or simply waiting for the arm-length difference to drift through zero while some scheduled

maintenance activity is taking place.

A final possibility for an arm-locking error signal is to use the entire constellation in a

Sagnac mode, much as for the ( variable discussed in Section 3.5.3. Starting at the master

SC, one beam is sent via phase-locks on the far SC on a counter-clockwise loop around

the constellation. The other beam is sent on a clockwise loop. This produces two signals

similar to those in (3-31) and (3-32), except that -r 50 s and Ar 7 47 ps. These signals

can also be combined to form the common and differential signals, which can be used to

form a direct arm-locking error signal.

The small size of AT pushes the location of the first instability to 21 kHz, well

above the LISA band. Furthermore, the rotation of the constellation is ,1i n., in the

same direction, meaning that Ar will be constant. Sagnac arm-locking would allow for

extremely large gains in the LISA band, reducing the residual phase noise considerably.

The downside is that it would require all six links to be operational and quiet, whereas the

other arm-locking configurations would still be viable if one or more links were inoperative

or degraded. GW signals

At first glance, there appears to be one major drawback to arm-locking: the

arm-locking error signals are sensitive to phase changes caused by GWs as well as those

caused by laser noise. If the signal is suppressed along with the noise, nothing is gained.

However, it is important to remember that the GW signals in LISA are extracted from

the TDI signals, not the individual PM outputs. The cancellation of laser frequency noise

in TDI works for any disitrbution of laser noise. If some of the laser noise happens to be

correlated with the GW signals, it will not effect the cancellation. Interaction with pre-stabilization system

As mentioned earlier, the lasers on LISA will be externally stabilized to a local

frequency reference, most likely a stable optical cavity. Ideally, arm-locking should

cooperate with the local frequency stabilization, so that the maximum stability can be

achieved. The problem in doing this is that the lock points for the local reference and the

arm-locking reference will not generally be the same. For example, an optical cavity has

a series of lock points separated by a free-spectral range (FSR) of c/2L, where L is the

cavity length. For a 30 cm cavity, this corresponds to an FSR of 500 MHz. The linewidth

of the cavity stabilized laser will be in the range of 10 Hz over 1000 s timescales. Over

longer time scales, thermal and mechanical effects in the cavities can cause the lock points

to drift by many MHz with slopes of 100 Hz/s or more [46].

A single-arm arm-lock controller will also have a series of lock points separated by an

FSR of c/(2 x 5 Gm) = 30 mHz, meaning that there will be an arm-locking point near the

cavity lock point. Unfortunately the drifts in the cavity will cause the cavity lock point to

move over thousands of arm-locking FSRs.

One possibility is to simply replace the pre-stabilization with the arm-locking loop.

While this has the advantage of simplicity, it would require a much larger gain in the

arm-locking loop. In addition, the PMs would have to be capable of handling the larger

phase noise without loosing accuracy.

Several possibilities for combining arm-locking with laser pre-stabilization have

been proposed [42]. Each of these involve creating a "loop-within-a-loop" using an

additional actuator. The tuneable-cavity approach, shown in Figure 3-14, replaces the

fixed optical cavity with a PZT-actuated tuneable cavity. The laser is locked to the cavity

using a standard locking scheme such as Pound-Drever-Hall (PDH) [47], providing the

pre-stabilization. The arm-locking error signal is used to actuate the cavity, keeping it

"l I, 1:., I" to the arm-length. The concern with this approach is that placing a PZT in the

cavity will degrade the cavity's length stability. It remains to be seen how severe this

degradation will be.

PM .------- .-----Fromfar SC
From far SC


Tuneable Cavity

Figure 3-14. Combining pre-stabilization and arm-locking with a tuneable cavity

A second approach involves using a fixed cavity but modifying the PDH scheme to

use a sideband lock. The PDH locking scheme utilizes an electro-optic modulator (EOM)

to place RF sidebands on the laser beam entering the cavity. The error signal is typically

generated by making the carrier beam nearly resonant with the cavity and observing the

phase of the reflected sidebands. It is also possible to generate an error signal by making

one of the sidebands resonant in the cavity. In the sideband-lock approach (Figure 3-15),

one RF sideband is locked to the cavity, providing stability to the carrier as well. The

carrier is used to generate the arm-locking error signal, which is fed back to the local

oscillator (LO) driving the EOM. This approach provides tunability while avoiding the

need for a tuneable cavity. However, sideband locking is known to introduce additional

noise sources and is not generally utilized in ultra-stable applications.

LO PM L-------- --- ------------
LO I -From far SC

EOM To far SC

Fixed Cavity

Figure 3-15. Combining pre-stabilization and arm-locking using a sideband cavity lock

A final option for combining arm-locking with laser pre-stabilization is the offset

phase-lock approach, shown in Figure 3-16. Here an additional low-power laser is locked

to a fixed cavity, providing a stable reference. The main high-power laser is phase-locked

to this reference laser with an offset frequency provided by a tuneable LO. For a high-gain

PLL, the phase noise of the main laser will be the same as that of the reference laser. The

arm-locking error signal is derived from the main laser, and is used to generate a feedback

to the LO. Since low-noise high-gain PLLs are an existing LISA technology requirement

(transponder locks), no new technologies are needed. However the additional laser is a

potential source of added mass and power consumption.


From far SC


STo far SC

Fixed Cavity



Figure 3-16. Combining pre-stabilization and arm-locking using a reference laser and an

offset PLL


4.1 Background

A group of researchers at the University of Florida (UF), including the author, are

developing a laboratory-based model of LISA interferometry. The purpose of this model

is to provide an arena for studying measurement techniques and technologies under

conditions similar to those present in LISA. Examples include laser pre-stabilization

systems, phase-lock-loops, phasemeters, TDI, arm-locking, inter-SC ranging, and laser

communication, all of which were discussed in Chapter 3. A long-term goal is to inject

GW signals into the simulator and produce data streams with LISA-like noise sources that

can be used by data analysis groups for mock data challenges.

There are two aspects of LISA interferometry that the simulator seeks to re-create:

noise sources and transfer functions. For the most part, this can be accomplished by

building a bench-top model of the IMS with a one-to-one correspondence between

parts. For example, the optical cavity pre-stabilization system is built with a laser, an

ultra-stable high-finesse cavity, and appropriate electronics. Standard components are used

in favor of space-qualified versions to reduce cost, but the noise characteristics and transfer

functions are essentially the same.

There is one aspect of LISA that cannot be simply copied on the ground: the size of

the constellation. There is no viable way to optically delay a laser beam for 16 s. Delays

in the microsecond or perhaps millisecond regime can be achieved using long optical fibers

[48], but eventually loses and fiber noise will overwhelm the signal. To model this aspect

of LISA interferometry, which is essential for studying TDI, arm-locking, and ranging, the

UF simulator uses a novel technique known as Electronic Phase Delay (EPD) [49].

4.2 The EPD Concept

The motivation behind EPD comes from the fact that nearly all of the information of

interest for interferometry is contained in the phase of the laser beams. Recall from (3-1)

that the time-varying component of a light field can be described by a complex electric

field with a time-varying phase,

E(t) Eo exp {i[27vt + Q(t)]}. (4-1)

In LISA, the phase can include contributions from GW signals, laser phase noise, motion

of the optical bench, etc. If 0(t) can be extracted from (4-1) and delayed, the result is

interferometrically equivalent to delaying E(t). Furthermore there is a limited bandwidth

over which variations in 0(t) are important. For most components, this is the LISA band,

but for some components, such as data communication and clock transfer, this may be as

high as a few GHz.

The infrared lasers used in LISA will have wavelengths of ~ 1 pm, which corresponds

to an optical frequency of v w 280 THz. At such high frequencies, it is impossible to

measure the oscillations in (4-1) directly. However, if two beams with different frequencies

are interfered on a photodiode (PD) to produce a beat note, the resulting intensity signal

is given by

I(t) o sin[2rAvt + Aq(t)], (4-2)

where Av is the difference frequency between the beams and AQ(t) is the difference in

phase between the two beams. Unlike v, Av can be made arbitrarily small by carefully

tuning the lasers. If the two lasers have similar noise characteristics but are independent,

then the noise spectrum of A(/f) will be approximately equal to V/2Q(f). In terms of

phase, I(t) is an analogue of E(t) at a lower frequency. If Av can be brought within the

bandwidth of a photoreceiver and a digital signal processing (DSP) system, it will be

possible to measure I(t), store it in a digital delay buffer, and regenerate it at a later time.

This is the concept behind the EPD technique.

As an illustration of how EPD is applied to model LISA interferometry, consider the

measurement of a single LISA arm between optical benches on different SC, shown in

Figure 4-1(a). A laser on one SC (L1), produces a light field with frequency vi and phase

Q (t). This light field traverses the 5 Gm to the other SC, incurring a delay of r t 16s and

a frequency shift, vDoppIer, caused by the relative motion between the SC. At the other SC,

the incoming light-field is interfered on a PD with the light from a local laser (L2), which

has a frequency v2 and phase 02(t). The frequency and phase of the resulting beat signal


VLISA V1 V2 + VDoppler




Delay PD

(a) LISA

[I y `t(c 20 (t) 1o(t r)

(b) EPD Analog

Figure 4-1. The EPD technique applied to a single LISA arm

The EPD equivalent of 4-1(a) is shown in Figure 4-1(b). The light field from L1 is

first interfered with a reference laser (Lo), which has phase (o(t). This beat signal has a

frequency v1o vi vo and phase Oio(t) 01(t) Oo(t). So long as (o(f) < ~i(f) and the

phases are uncorrelated, io(f) 1i(f). The L1 Lo beat signal is the EPD analog of the

optical signal from L1 in Figure 4-1(a).

OLISA = 2(t) 01(t T)

02 (t)- 0 (t T)

The Li Lo beat signal is used as an input to the EPD unit, which digitizes the

signal, stores it in a memory buffer for a time 7, and regenerates the delayed signal. The

output of the EPD unit is a signal with phase 1io(t -). Some versions of the EPD

hardware (see Section 4.6) are also capable of putting a frequency shift on the beat signal,

mimicking the Doppler shifts present in LISA.

The optical signal from L2 in Figure 4-1(a) is modeled by a beat between L2 and Lo

in Figure 4-1(b). This signal has a frequency v20o 2 vo and phase 20 (t) = 2(t) o(t).

As with the L1 Lo beat, the noise characteristics of this signal will be the same as that

for the original optical signal so long as Lo is independent from L2 and has equal or lesser

phase noise.

The PD in Figure 4-1(a) is replaced by an electronic mixer in Figure 4-1(b). The

mixer performs a similar operation on the two electronic signals as the PD does on the two

optical signals. The mixer output contains two terms, one with a frequency equal to the

difference frequency of the two input signals and one with a frequency equal to the sum of

the two input signals. A low-pass filter is used to remove the high-frequency term, leaving

a signal with a frequency and phase given by

VEPD = V10 20 + VDoppler (4-5)


QEPD 20(t) 10(t ). (4-6)

A comparison of (4-5) and (4-6) with (4-3) and (4-4) indicates that the EPD model

produces a signal that is of the same form as the LISA arm. With the restrictions on Lo

mentioned above, the noise characteristics will be similar as well. More complex models of

LISA can be built up in a similar fashion. Table 4-1 lists the major components in LISA

and their EPD equivalents.

As a final note, although Figure 4-1(b) is drawn with the same reference laser being

used to generate 1io(t) and )20(t), it is not a requirement of the EPD technique. Provided

all reference lasers had the appropriate noise characteristics, it would be possible to use a

different reference laser for each beat note. Of course, using the same reference laser at all

beat notes is more cost effective. Since the various beat notes are time-delayed before thi

are mixed, the noise from the common reference laser will not cancel out.

Table 4-1. Major LISA IMS components/signals and their EPD equivalents

4.3 Optical Components

The UF LISA interferometry simulator is designed to study primarily the SC to

SC interferometry in LISA. While in theory it would be possible to include the backside

interferometry by including a model proof mass suspended on a torsion pendulum [50],

this would significantly complicate the experiment. Furthermore, such a combined

experiment is not necessary at this stage since the SC to proof-mass and SC to SC

interferometry are treated as separate measurements in LISA.

4.3.1 Layout

In the current optical layout of the simulator, shown in Figure 4-2, each of the three

SC is modeled by an independent Nd:YAG non-planar ring oscillator (NPRO) laser,

denoted as L1 through L3 in the figure. A fourth NPRO, Lo, is used as a reference laser.

The lasers L1 and Lo are each locked via the PDH method [47] to independent optical

cavities housed in a thermally-isolated vacuum chamber. Beat notes between the far SC

lasers (L2 and L3) are made with Lo, allowing two complete LISA arms to be modeled


LISA Component EPD Equivalent

laser field beat note with reference laser

optical delay electronic delay

photodiode electronic mixer

optical beat note mixer output

Figure 4-2. Optical layout of the UF LISA interferometry simulator. L1 L3 represent
SC1 SC3 in LISA. Lo is the reference laser.

4.3.2 Pre-stabilization

The function of the optical cavities in Figure 4-2 is to provide LISA-like laser phase

noise for the simulator. The dominant noise source for cavity length in the LISA band is

thought to be thermally-driven expansion. Consequently, the LISA cavities will likely use

spacers of ultra-low expansion glass such as Dow-Corning's ULE or Schott's Zerodur. If

the pre-stabilization requirements are relaxed due to improvements in TDI or arm-locking,

it may be possible to utilize other materials such as Silicon Carbide (SiC). In parallel with

the interferometry experiments, the UF group is studying the stability of various materials

and bonding techniques [52]. Consequently, the optical cavities used for pre-stabilization in

the simulator are occasionally changed.

Figure 4-3 shows a spectrum1 of the beat note frequency between L1 and Lo for

two pairs of optical cavities (Zerodur-Zerodur and Zerodur-SiC), along with the LISA

pre-stabilization requirement given by (3-15). As can be seen, the Zerodur-SiC frequency

noise rises steeply above the Zerodur-Zerodur frequency noise at frequencies below

S10 mHz.

1 All spectra were computed using the MATLAB routine "mypsd.m", written by the
author. It implements Welch's method of overlapped-average periodograms as described
by Heinzel et al. [53].

In addition, both beat notes exhibit long term drifts of tens or hundreds of MHz.

For the Zerodur-Zerodur beats, the slope of the drift is typically ~ 1 Hz/s, comparable

to the changing Doppler shift present in LISA. For the SiC-Zerodur beats, the slopes

can be much larger, typically in the range of 100 Hz/s ~ 400 Hz/s. Over short time

periods (hours to dan;) the drifts are monotonic but over longer time periods (days to

weeks) the behavior becomes more complex. Similar drifts on the order of a few Hz/s

were seen between a pair of ULE cavities in an other experiment [46], and the cause is not

completely understood. For short-term interferometry experiments, the short-term drifts

are of the most concern.

LISA Requirement


1001 10 100

Figure 4-3. Frequency noise in the L1 Lo beat note

As mentioned in Section 3.5, the noise in a sinusoidal signal can be placed either in

the frequency or in the phase. Figure 4-4 shows the same data as Figure 4-3 converted to

phase noise using (3-3).

4.4 Electronic Components

In addition to the optical components described in the previous Section, several

electronic components are key to the operation of the simulator. These can be broken

down into three primary categories: the control filters, phasemeters (PMs), and the EPD

unit. Control filters are used to provide actuator signals for the laser-pre-stabilization,

phase-lock loops, and arm-locking loops. PMs are used to measure the phase of the various

LISA Requirement

frequency (mHz)

Figure 4-4. Data from 4-3 converted into phase noise by dividing by 2r f, where f is the
Fourier frequency

beat signals in the optical layout, providing inputs to the control loops as well as the

science signals.

Both the control filters and the PMs are critical components for LISA as well. Since

the simulator versions and their LISA counterparts are subjected to similar signals, the

simulator provides an excellent arena for evaluating potential designs. The EPD unit is of

course unique to the simulator, and must reproduce the LISA arm as faithfully as possible.

The addition of noise not present in LISA or elimination of noise present in LISA would

limit the accuracy of the simulator.

Several types of electronic architectures are employed in the UF simulator. These

include both analog systems and digital systems, which are reviewed in Appendix A. For

the most demanding electronic ii-\ -.t iii'. such as the PMs and EPD unit, the UF group

selected and purchased a high-speed digital signal processing system from the Pentek

Corporation in Upper Saddle River, NJ. An overview schematic of the system is shown in

Figure 4-5 below.

The system consists of three individual products, the model 4205 carrier board, the

model 6256 digital downconverter, and the model 6228 digital upconverter. The model

41211, carrier board is housed in a VME crate and contains a 1 GHz, 32-bit PowerPC

microprocessor. The processor is connected via a PCI bus to several components,

including SDRAM, serial ports, Ethernet ports, and the VME backplane. In addition

the model 4205 provides four specialized high-speed interfaces known as velocity interface

module (VIM) connectors.

Model 6256 Digital Downconverter Model 6228 Digital Upconverter





PCI Interface

--------------------- --' -- ---
Model 4205 Carrier Board

Figure 4-5. Overview of the DSP system from Pentek Corporation

The VIM interface consists of a 32-bit data interface and a 32-bit control/status

interface. The data interface is connected to the main PCI bus via a bi-directional

first-input first-output buffer (BIFO) and a direct-memory access (DMA) controller. The

DMA controller allows data to be read from the BIFO and directly deposited into memory

or another location on the PCI bus without the need for processor intervention. The speed

of the VIM interface is limited by the speed of the PCI bus, which is clocked at 66 MHz.

The control/status interface is connected to the processor via a separate 33 MHz PCI bus.

The 6256 digital downconverter contains four 14-bit ADCs that can be clocked at

frequencies up to 105 MHz. The ADCs are connected to front-panel connectors via RF

transformers with a high-pass frequency response. The -3 dB point of the transformers is

at 400 kHz. Signals at lower frequencies cannot be measured with the 6256. The full-scale

input of the ADCs is reached with a +4 dBm signal at the front-panel input, which has an

impedence of 50 2.

Data from the ADCs is passed into one of two FPGAs, where it can be processed.

The processed data is connected via two of the VIM interfaces to the BIFOs on the

1211-.. The VIM control/status interface also connects with the FPGA and can be used to

configure the board.

The 6228 digital upconverter connects to the 1211i, via the second pair of VIM

modules. Data from the BIFOs passes into an FPGA where it is processed. The processed

data is fed to two two-channel 16-bit DACs that can be clocked at frequencies up to

500 MHz. The DACs produce a full-scale output of -2 dBm and are coupled to the

front-panel via RF transformers with a 50 Q output impedance and a -3 dB point at

400 kHz. As with the 6256, the VIM control/status interface connects with the FPGA and

can be used to configure the board.

This arrangement provides a powerful and flexible system for digital signal processing.

The FPGAs on the 6256 and 6228 can be used to perform high-speed processing with

fixed-point arithmetic, while the microprocessor can be used for floating-point processing

at lower speeds. This system is used for the EPD units (Section 4.6), the hardware PM

(Section 4.5.5), and one instance of the arm-locking control filter (Section 5.4).

4.5 Phasemeters

4.5.1 Overview

The phase of a sinusoidal signal can be specified by the time at which the signal

has a specific value and specific first derivative (i.e. positive zero-crossing). Since this

is fundamentally a timing measurement, all phasemeters (PM)s must be based on a

reference clock. The phase measured by the PM is the difference phase between the signal

and the reference clock. Any single phase measurement will be limited in accuracy by

the phase stability of the reference clock. With multiple measurements using the same

reference clock, the phase noise of the reference clock can be measured and canceled. In

the remainder of this Section, phase noise in the reference clock is not explicitly included.

A simple PM that is often used in the laboratory is the analog mixer. The signal to

be measured, S(t), can be described by a sinusoid with amplitude A, angular frequency w,

and phase 0(t),

S(t) A sin[wt + 0(t)]. (4-7)

This signal is mixed with a reference signal, R(t), with the same frequency and a constant


R(t) = cos[wt]. (4-8)

The mixer output, M(t), is the product of (4-7) and (4-8),

M(t) = S(t) x R(t)

A sin[wt + p(t)] cos[wt]
A sin[(t)] sin[2wt + Q(t)]}. (4-9)

The mixer output is then low-pass filtered to remove the 2w term. If the phase noise is

small and the signal amplitudes are constant, then the resulting signal is proportional

to 0(t). Some analog mixers are specifically designed to compensate for the sinusoidal

response and can produce a linear phase response for |I(t)l < 70.

The filtered mixer PM is an example of a general type of PM called in-phase/quadrature

or IQ PMs. Mathematically, a sinusoidal wave at a given frequency contains two pieces of

information. These can be expressed as the amplitude and phase, as was done in (4-7) or

as the in-phase, I(t), and quadrature, Q(t), components:

A(t) sin[wt + 0(t)] = I(t) sin[wt] + Q(t) cos[wt]. (4-10)

Trigonometric identities can be used to derive the following relationships between the IQ

and amplitude-phase formalisms:

I(t)= A(t) cos[ (t)], (4-11)

Q(t)= A(t) sin[(t)], (4-12)

A(t) I(t)2 (t)2, (4-13)

0(t) arctan[Q(t)/I(t)]. (4-14)

Comparison of (4-12) with (4-9) reveals that the filtered mixer PM described above

measures Q(t)/2 rather than 0(t). A complete IQ PM can be built by extending the

filtered mixer concept to include two demodulations, one with cos[wt], which produces

Q(t)/2, and one with sin[wt], which produces I(t)/2. The relations in (4-11) (4-14) can
then be used to compute 0(t) and A(t).

In addition to direct mixing, I(t) and Q(t) can be measured in a number of other

ways. One technique involves sampling the signal of interest with a sampling frequency

equal to four times the carrier frequency of the signal [54]. Each set of four data points

can be manipulated to measure the phase at a rate of one-half the carrier frequency.

Another approach is to use integral transforms of the time series data to extract the phase


One type of PM that is distinct from the IQ type is the counter/timer PM [56]. In

this technique, the number of zero-crossings in a time interval T is counted, providing a

crude estimate of the signal frequency. This estimate is then corrected by measuring the

additional time between first and last zero-crossings and the time-interval boundaries.

A combination of these two measurements gives an estimate of the phase accumulated

during the interval T. This approach is used in frequency counters such as the ones used

to measure the beat note stabilities discussed in Section 4.3.2.

4.5.2 Phasemeters for LISA-like signals

The accuracy requirement for the LISA PM is set by the error budget for the IMS to

be ~ 10-6cycles/v/Hz in the LISA measurement band (Shaddock [57] gives 3 pcycles/v/Hz

at 5 mHz). Reaching this level of performance requires careful suppression of other noise

sources that present themselves as phase noise. Examples include phase and amplitude

noise in the reference signal, residual signals from the second harmonic term in (4-9),

and, for digital systems, digitization and quantization effects. It has been shown that

commercial digital radio receivers are capable of meeting the LISA phase accuracy

requirements for low-noise signals at fixed frequencies [59].

The input signals to the PM in LISA differ from those for a commercial radio receiver

in two important ways: large shifts of the carrier frequency, and large intrinsic phase noise

on the signal. The relative motion between the SC will cause the one-way Doppler shifts

to vary by up to 30 MHz over the course of a year. If the frequencies of the SC lasers are

held fixed, the beat frequencies will also vary over a range of 30 MHz. This range can be

reduced by periodically adjusting the laser frequencies during the course of the year. One

proposed frequency plan will keep the beat notes in the range 2 MHz 20 MHz.

Of comparable size to the Doppler shifts are the drifts in the optical reference cavities

discussed in Section 4.3.2. These drifts will be more difficult to model and must be taken

into account when specifying the PM range requirements.

In addition to the frequency drifts, which can be considered as noise in the beat

note below the LISA band, there is also a large amount of phase noise in the LISA

measurement band, as evidenced by Figure 4-4. This large phase noise poses a problem

to the IQ phasemeter. As shown in (4-11) and (4-12), I(t) and Q(t) are periodic in 0(t).

Therefore the IQ phase measurement in (4-14) is a measurement of 0(t) modulo 27 and

only gives 0(t) if I\(t) I < r rad. In other words, the rms phase noise must be less than half

a cycle in the measurement band or the phase noise measurement will --vi .'"'. The LISA

laser phase noise requirement (Figure 4-4) corresponds to an rms phase noise of greater

than 106 cycles in the LISA band, meaning that the phase noise will constantly wrap

between 1/2 cycles. Addressing phase wrapping is a major challenge to the designers of

the LISA PM.

4.5.3 An IQ phasemeter with a tracking LO

One way to address both the frequency drifts and the large laser phase noise is

to demodulate the incoming signal with a local oscillator that tracks the phase of the

incoming signal to within 1/2 cycle. This is the approach taken by researchers at the Jet

Propulsion Laboratory who are designing the LISA PM [57, 58]. A schematic of such a

system is shown in Figure 4-6 below.

A, sin(2rvt+ ,)

----------- ----- -^ -J t- x2 > -
sin Gls) x2 A

Cos | G~s) x2 01
O Scaling


Phase Reconstruction

Figure 4-6. Schematic of a IQ phasemeter with feedback

The input signal is a sinusoid with frequency vi, amplitude Ai(t), and phase (t(t).

The frequency is assumed to be fixed and any frequency noise is converted to phase noise

using (3-2). The input signal is demodulated with two signals from a local oscillator (LO),

a cosine and a sine with model phase, '. (t). The sine term is filtered by a low-pass filter

with a transfer function G(s) and scaled by 2 to form the signal I(t),

I(t) = 2G(t) 0 A (cos [i(t) (t)] cos [4 t + o(t) + (t)]) (4-15)
2 I

where G(t) is the filter's impulse-response function, and 0 denotes convolution. The cosine
term is similarly filtered and scaled to form Q(t),

Q(t) = 2G(t) 0 At) (sin [pi(t) .. (t)] + sin [47uit + Oi(t) + .. (t)]) (4-16)
(t) 22(1)X G ""

With a properly designed filter, the high-frequency terms in (4-15) and (4-16) can be

eliminated while retaining the first terms. The additional scaling by a factor of 2 produces

the standard definitions of I(t) and Q(t).
The I(t) and Q(t) signals can be used to compute the residual phase, defined as

Qr(t) Qi(t) /.. (t), and the output amplitude, Ao(t), using (4-13) and (4-14). If ,(t)
is small enough to linearize the equations without introducing unacceptable errors, the

relations become

01(t) Q(t)/I(t) (4-17)


Ao(t) a I(t). (4-18)

The residual phase is used as an error signal for the LO tracking loop. It is filtered

by a control filter with transfer function H(s), forming the control signal for the LO. For

most types of oscillators, the control signal is proportional to the oscillator's frequency.
This adds an implicit factor of 1/s into the controller transfer function.

In most cases, the approximate frequency of the beat signal will be known, and an

offset frequency, voff, can be added to the frequency correction, v.orr(t), provided by the
control filter. The model frequency, v,(t), is the sum of voff and v.orr(t) and is integrated

to form the model phase, '., (t),

Vm(t) = Voff + Vcorr(t) (4-19)

'. (t) m(t)dt. (4-20)

Full Text








ManyofmycolleagueshereatUFandelsewhereweretrulyinstrumentalintheproductionofthiswork:ShawnMytrik,myright-handmanwiththeelectronics;VolkerQuetschke,theresidentcomputerwizard;RodrigoDelgadillo,MichaelHartman,andGabrielBoothe,fortheirhoursoffaithfulserviceinthelab;DanielShaddock,whoprovidedinsightintoarm-locking,phasemeters,andotheraspectsofLISA;RachelCruz,whobuilttheopticssideoftheLISAsimulator;andespeciallymyadvisor,GuidoMueller,forguidingmetothispoint.Mostimportantly,Iwouldliketothankmywife,Suzanne,whoputupwithnearlythreeyearsofseperationwhileIcompletedthisdegree. 4


page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 KEYTOABBREVIATIONS ............................... 13 KEYTOSYMBOLS .................................... 15 ABSTRACT ........................................ 18 CHAPTER 1INTRODUCTION .................................. 20 1.1MotivationforGravitationalWaveAstronomy ................ 20 1.2GravitationalWaveDetectors ......................... 21 1.3LISAattheUniversityofFlorida ....................... 22 2GRAVITATIONALWAVES ............................. 24 2.1Overview .................................... 24 2.1.1Relativity ................................ 24 2.1.2Weak-eldGRandGravitationalWaves ............... 27 2.1.3PropertiesofGravitationalWaves ................... 28 2.1.4InteractionwithMatter ......................... 30 2.1.5GenerationofGravitationalWaves ................... 32 2.1.6EnergyCarriedbyGravitationalWaves ................ 34 2.2SourcesofGravitationalWaves ........................ 35 2.3DetectionofGravitationalWaves ....................... 41 2.3.1IndirectDetection ............................ 41 2.3.2DirectDetection ............................. 42 ................ 43 ......................... 44 ................... 45 ................... 48 3THELASERINTERFEROMETERSPACEANTENNA ............. 54 3.1Introduction ................................... 54 3.2Sources ...................................... 54 3.3MissionDesign ................................. 55 3.4TheDisturbanceReductionSystem(DRS) .................. 56 3.5TheInterferometricMeasurementSystem(IMS) ............... 57 3.5.1IMSOverview .............................. 58 5


........................... 60 3.5.3TimeDelayInterferometry ....................... 62 ........................ 64 ...................... 65 ................. 66 ................ 66 3.5.4Arm-locking ............................... 67 ................ 68 ........... 70 ...................... 75 ............... 77 .......................... 82 .......... 82 4THEUFLISAINTERFEROMETRYSIMULATOR ............... 86 4.1Background ................................... 86 4.2TheEPDConcept ............................... 86 4.3OpticalComponents .............................. 90 4.3.1Layout .................................. 90 4.3.2Pre-stabilization ............................. 91 4.4ElectronicComponents ............................. 92 4.5Phasemeters ................................... 95 4.5.1Overview ................................. 95 4.5.2PhasemetersforLISA-likesignals ................... 98 4.5.3AnIQphasemeterwithatrackingLO ................. 99 4.5.4ASoftwarePhasemeter ......................... 103 ............................. 103 ............................ 105 4.5.5AReal-timeHardwarePhasemeter .................. 108 ....................... 108 ........................ 113 .............. 115 .......... 117 ............ 120 ....... 123 ................... 127 4.6EPDUnit .................................... 129 4.6.1Second-generationEPDunit ...................... 130 4.6.2Third-generationEPDunit ....................... 132 5ARM-LOCKINGINTHEUFLISAINTERFEROMETRYSIMULATOR .... 136 5.1Introduction ................................... 136 5.2ElectronicModel ................................ 137 5.2.1Method .................................. 137 6


.................................. 138 5.2.3Discussion ................................ 139 5.3InitialOpticalModel .............................. 140 5.3.1Method .................................. 140 5.3.2Results .................................. 143 .............. 144 ....................... 147 ....................... 149 5.3.3Discussion ................................ 150 5.4ImprovedOpticalModel ............................ 151 5.4.1SystemCharacterization ........................ 152 5.4.2FilterDesign ............................... 155 5.4.3Results .................................. 158 6CONCLUSION .................................... 159 6.1PhasemetersandEPDUnits .......................... 159 6.2Arm-Locking .................................. 160 APPENDIX ADIGITALSIGNALPROCESSING ......................... 162 A.1Introduction ................................... 162 A.2Sampling ..................................... 162 A.2.1Aliasing ................................. 164 A.2.2Upconversion .............................. 165 A.3DigitalSignals .................................. 168 A.3.1BinaryFractions ............................. 169 A.3.2MultiplicationandOtherOperations ................. 169 A.3.3Floating-pointRepresentations ..................... 171 A.3.4DigitizationNoise ............................ 172 A.4DigitalFiltering ................................. 174 A.4.1Time-domainResponse ......................... 175 A.4.2FrequencyResponse ........................... 175 A.4.3DesignMethods ............................. 178 A.4.3.1FIRFilters-windowedimpulseresponsemethod ..... 178 A.4.3.2IIRFilters-bilineartransformmethod ........... 180 A.4.4RealizationandPracticalities ...................... 181 A.4.4.1Filterstructures ........................ 181 A.4.4.2Latency ............................ 182 A.4.5CICFilters ................................ 183 A.4.6Fractional-DelayFilters ......................... 184 REFERENCES ....................................... 186 BIOGRAPHICALSKETCH ................................ 191 7


Table page 2-1SuggestedfrequencybandsforGWs ......................... 40 2-2OperationalGWbardetectors ............................ 47 2-3Majorground-basedGWinterferometers ...................... 51 4-1MajorLISAIMScomponents/signalsandtheirEPDequivalents ......... 90 4-2ReconstructionalgorithmsforthehardwarePM .................. 114 4-3Beatnotefrequenciesandamplitudesforopticalentangled-phasemeasurement 123 4-4ProgressionofEPDunits .............................. 130 A-1IEEEstandardoatingpointrepresentations .................... 171 8


Figure page 2-1Tidaldistortionofaninitially-circularringoffreely-fallingtestparticles .... 32 2-2AhypotheticallaboratorygeneratorofGWs .................... 35 2-3AbinarystarsystemasageneratorofGWs .................... 37 2-4ObservedshiftofperiastronforPSR1913+16 ................... 42 2-5ConceptforDoppler-trackingdetectionofGWs .................. 44 2-6SensitivityofALLEGRObardetector ....................... 47 2-7AMichelsoninterferometerasadetectorofGWs ................. 48 2-8AerialphotographoftheLIGOGWdetector .................... 52 2-9SensitivitycurvesfortheLIGOobservatory .................... 53 3-1SourcesintheLISAobservationalwindow ..................... 55 3-2OrbitalcongurationoftheLISAconstellation ................... 56 3-3DiagramoftheLISAIMS .............................. 58 3-4DiagramofaLISAopticalbench .......................... 61 3-5rabbit-eardiagramfortherst-generationTDIXcombination ......... 65 3-6Diagramofaclosed-loopSISOsystemwithnegativefeedback .......... 68 3-7GenericNyquistplot ................................. 70 3-8Nyquistplotsforsingle-armarm-locking ...................... 71 3-9BodeplotofTsen(f)withRT=33s ........................ 72 3-10Bodeplotforagenericarm-lockingcontroller ................... 74 3-11Closed-loopnoisesuppressionforagenericarm-lockingloop ........... 75 3-12Nyquistplotforcommonarm-locking ........................ 79 3-13Magnitudeofsquare-bracketedtermin( 3 ) ................... 81 3-14Combiningpre-stabilizationandarm-lockingwithatuneablecavity ....... 83 3-15Combiningpre-stabilizationandarm-lockingusingasidebandcavitylock .... 84 3-16Combiningpre-stabilizationandarm-lockingusinganosetPLL ........ 85 9


................. 88 4-2OpticallayoutoftheUFLISAinterferometrysimulator .............. 91 4-3FrequencynoiseintheL1L0beatnote ...................... 92 4-4PhasenoiseintheL1L0beatnote ........................ 93 4-5OverviewoftheDSPsystemfromPentekCorporation .............. 94 4-6SchematicofaIQphasemeterwithfeedback .................... 99 4-7LaplacedomainmodelofthesysteminFigure 4-6 ................. 101 4-8BodeplotofG(s)forthesoftwarePM ....................... 104 4-9BodeplotofH(s)forthesoftwarePM ....................... 105 4-10ExpectedbehaviorofthesoftwarePM ....................... 106 4-11ObservedbehaviorofthesoftwarePM ....................... 107 4-12Schematicofthereal-timehardwarePM ...................... 108 4-13SchematicofaDirectDigitalSynthesizer ...................... 109 4-14CICdecimationlterneartherstaliasingband ................. 110 4-15PassbandatnessoftheCIClterinthehardwarePM .............. 111 4-16FeedbacklterforhardwarePMtrackingloop ................... 112 4-17PackingformatforPMdatatransferredovertheVIMinterface ......... 113 4-18VCOphasenoisemeasuredbyfourchannelsofthehardwarePM ........ 117 4-19Linearspectraldensityoflaserbeatnotephase .................. 118 4-20Qualitativeamplitudespectrumofinterferringbeamswithshotnoise ...... 119 4-21TimeseriesforentangledphasetestusingVCOs .................. 122 4-22LSDofanentangledphasetestusingVCOs .................... 122 4-23AnalogelectronicsusedtopreparebeatsignalsforPM .............. 124 4-24Linearly-detrendedphaseforopticalentangled-phasemeasurement ....... 124 4-25Quadratically-detrendedphaseforopticalentangled-phasemeasurement .... 125 4-26Linearspectraldensityforopticalentangled-phasemeasurement ......... 126 4-27Noisesuppressioninopticalentangled-phasemeasurement ............ 127 10


......... 128 4-29SchematicoftheNCOusedinthe3rd-generationEPDunit ............ 132 4-30DetrendedphaseofVCOsignalinEPDtest .................... 134 4-31LinearspectraldensitiesinEPDtestwithVCOsignals .............. 134 5-1Experimentalsetupforelectronicarm-lockingexperiment ............. 137 5-2Transferfunctionsforelectronicarm-lockingexperiment ............. 138 5-3Linearspectraldensityofarm-lockedVCOsignal ................. 139 5-4Closed-loopnoisesuppressionforelectronicarm-lockingexperiment ....... 140 5-5Experimentalarrangementfortheinitialopticalarm-lockingexperiments .... 141 5-6Laplace-domainmodelofthesysteminFigure 5-5 ................. 142 5-7Bodeplotofcontrollerforinitialopticalarm-lockingsystem ........... 143 5-8Responseofinterferometertophasemodulation .................. 144 5-9TimeseriesofL2L0beatnoteforlockedandunlockedcases .......... 145 5-10DetrendedtimeseriesofL2L0beatnoteforlockedandunlockedcases .... 146 5-11Close-upoflockedcaseinFigure 5-9 ........................ 146 5-12Spectrumoflockedandunlockedfrequencynoise ................. 147 5-13Timeseriesofbeatnotephaseforunlockedandlockedcases ........... 148 5-14Phasenoisespectrafortheunlockedandlockedcases ............... 148 5-15Closed-loopnoisesuppressionforopticalarm-locking ............... 149 5-16Error-pointnoiseforlockedandunlockedcases .................. 150 5-17Modicationofelectronicsforimprovedopticalarm-locking ........... 151 5-18Rawphasetimeseriesofarm-lockingsystemcharacterizationdata ........ 153 5-19Detrendedtimeseriesofarm-lockingsystemcharacterizationdata ........ 154 5-20Linearspectraldensitiesofarm-lockingsystemcharacterizationdata ...... 154 5-21Magnitudeoftransferfunctionforarm-lockingsystem .............. 155 5-22TimeseriesoflteredandunlteredfrequencynoisefromVCOinput ...... 157 5-23Measuredanddesignedtransferfunctionsofarm-lockingcontrollter ...... 157 11


........................ 163 A-2Thephenomenonofaliasing ............................. 165 A-3Anoverviewoftheupconversionprocess ...................... 167 A-4AssumedPDFforquantizationerror ........................ 172 A-5Non-uniform,non-whitequantizationerror ..................... 173 A-6ComparisonofLaplaceandzdomains ....................... 177 A-7WindowedimpulseresponsemethodfordesigningFIRlters ........... 179 A-8Thedirect-formIlterstructure .......................... 181 A-9Thedirect-formIItransposedlterstructure .................... 182 A-10Magnituderesponseofagenerictwo-stageCIClter ............... 183 12


















1 ]andHughes[ 2 ]. 21


3 ]haveprovidedextremelyconvincingcircumstantialevidencethatGWsexistandgenerallybehaveasexpected.OneproposedGWdetectoristheLaserInterferometerSpaceAntenna(LISA),whichwillconsistofthreeseparatespacecraftformingatriangulardetectorwithsidesof5Gm=5109m.TomeasureGWs,LISAmustdetectlengthchangesinthesearmswithaprecisionof10pm.Achievingthislevelofprecisionoversuchvastdistancesrequiresanumberofnoveltechniquesandtechnologies. 22




2.1.1RelativityThetheoreticalframeworkofSpecialandGeneralRelativityrepresentsourbestunderstandingofthemacroscopicuniverse 4 ].OtherportionswereadaptedfromMisner,etal.[ 5 ]andShapiro&Teukolsky[ 6 ].2 24




2 ) 2 )mustbeintegratedalongthepathbetweenthetwoevents.AsinSR,theworldlinesoffreeparticlesinGRwillfollowgeodesics.Duetothecurvatureofspacetime,geodesicsinGRwillnotgenerallybestraightlinesinalocalinertialframe.VariationalprinciplescanbeusedtoproduceequationsdescribinggeodesicsinGR.Iftheworldlineisdescribedbyasetofeventsparametrizedbyascalarparameter,x(),thenthegeodesicequationcanbewrittenas 2g(g;+g;g;);(2)where 26


2 )tobegenerallyvalid,Gmustsatisfythesameconstraints.TheEinsteincurvaturetensorisacombinationofmetricderivativesthatsatisestheconstraintsonT.GRissometimessummedupbystatingthatspacetimetellsmatterhowtomove(geodesicequation)andmattertellsspacetimehowtocurve(Einsteinequations). 27


2histhetrace-reverseofhandistheD'Alembertianoperator.TheD'Alembertianoperatorisalsoknownasthewaveoperator,sinceitgivesthewaveequationwhenappliedtoafunction, 2 )indicatesthattherewillbeasetofwavesolutionstothelinearizedEinsteinequations.ThesesolutionsareknownasGravitationalWaves(GWs). 2 ),whichcorrespondstolinearizedGRinavacuum(T=0).Thegeneralsolutionofthehomogeneouswaveequationisasuperpositionofplanewavesoftheform 2 )tosatisfy( 2 ),kmustbeanullorlight-likefour-vector, 2 )canthenbewrittenas 2 )itcanbeseenthatboththephaseandgroupvelocitiesofGWsare1innaturalunits,whichcorrespondstothespeedoflight. 28


2 ),theexpressionin( 2 )mustalsosatisfytheLorentzgaugeconditionin( 2 ).ThisplacesrestrictionsonA;requiringittobeorthogonalto!k, 2 )describesaclassofgauges.Additionalrestrictionson 2 ))forwhichthewaveistravelinginthez-direction.Inthisframe,therewillonlybetwoindependentcomponentsof 2 )usingtwoscalarpolarizationstatesandtwounitpolarizationtensors, 29


2 ), 2 )describesthemotionoffreeparticles.Arelatedequation,knownasthegeodesicdeviationequation,describestheevolutionofthe4-vectorlinkingtwonearbygeodesics, 30


2 )toobtain 2(h;+h;h;h;):(2)UsingtheexpressionsforhintheTTgauge,thefourequationsin( 2 )canbereducedtotwo 2"h+(2)and 2"h:(2)Allothercomponentsofarezero.ThetidaleectsofGWsonfreely-fallingparticleswillberestrictedtotheplanenormaltothewave'spropagationdirection.Themotioninthisplanewillbeoscillatory,withanangularfrequencyequaltothatoftheGW.Asimilaranalysiscanbemadefortwoparticlesinitiallyseparatedbyadistance"inthey-direction.Theresultsare 2"h+(2)and 2"h:(2) 31


2 )through( 2 )canbeusedtodeterminethetidaldistortionsofaninitially-circularringoffreely-fallingtestparticlesinthexyplaneasaGWtravelinginthez-directionpassesby.TheresultisshownschematicallyinFigure 2-1 (b)Figure2-1. Distortionofaninitially-circularringoffreely-fallingtestparticlesbyaGWpropagatingintotheplanefor(a),the+polarizationand(b),thepolarization.GWreferstothephaseoftheGW. 2 )for 32


2 )isappliedto( 2 ),theresulttorstorderin(1=r)is 2 )and( 2 )intheTTgauge.Inacoordinateframeattheobservationpointwiththez-axisorientedalongthepropagationdirectionofthewave,theexpressionsbecome 3ijIkk(2) 33


2 )through( 2 )areknownasthequadrupoleapproximationandcanbeusedtodescribethegravitationalradiationinmanyphysicalsystems.ThisisthetopicofSection 2.2 2 )canbemovedtotheright-handsideandtreatedasastress-energysourceterm.Theresult 32 2 2 )canbesimpliediftheTTgaugeconditions(( 2 )and( 2 ))areapplied, 32(hTT)ij;(hTT)ij;:(2)Using( 2 )andthequadrupoleapproximationdescribedinSection 2.1.5 ,theGWluminosityforacompactsourcecanbeestimatedas 5D...Qij...QijE:(2)Tocomputetheluminosityinphysicalunits,( 2 )ismultipliedbytheconversionfactor 5 ] 34


2.1.5 describedthequadrupoleapproximationforthegenerationofgravitationalwaves.Itwasfoundthatamassdistributionwithatimevaryingquadrupolar(orhigher)momentwouldgenerateGWs.Intheory,GWscouldbegeneratedbyalaboratoryapparatussuchastheoneinFigure 2-2 .ConsiderauniformbeamofmassMandlengthLlyinginthexyplane.Thez-axispassesthroughthecenterofthebeamandthebeammakesananglewiththex-axis. Figure2-2. AhypotheticallaboratorygeneratorofGWsconsistingofabaroflengthLandmassM 3cossin0cossinsin21 30001 31CCCCA:(2) 35


...Qij=ML2!3 2 )and( 2 )canbeusedtoestimatethestrainamplitudefrom( 2 ).Itisclearfrom( 2 )thatthequadrupolemomentoscillateswithanangularfrequencyof2!.Thisistobeexpectedfromthesymmetryofthesystemunderarotationofradiansaboutthez-axis.TheGWswillhavethissamefrequencysothatGW=2!.Sincethequadrupoleapproximationisvalidonlyinthefar-eld,( 2 )and( 2 )mustbeevaluatedatasourcedistanceofatleastoneGWwavelength(r=c=2!).Thestrainamplitudeatthatdistancecanbeestimatedas 3ML2!3:(2)ForabeamwithM=104kg,L=10m,and!=60rad=s,( 2 )givesh(lab)1042.TheGWluminosityfromthebeamcanbeestimatedusing( 2 )and( 2 )as 15M2L4!6:(2)Usingthesameparameters,thisgivesLGW(lab)1033W.Itisclearfromthesmallsizeofh(lab)andLGW(lab)thatGWsarenotrelevantforlaboratorysystems.WhatisneededtogeneratephysicallymeaningfulGWsislargermassesandhighervelocities.Bothcanbefoundinastrophysicalsystems. 36


2-3 Figure2-3. AbinarystarsystemasgeneratorofGWs Ifthesystemcenterofmassisplacedattheorigin,thepositionvectorsofthetwomasseswillbe m1(cos;sin)(2)and m2(cos;sin);(2)where=m1m2=(m1+m2)isthereducedmassandistheorbitalphaseangle,measuredfromthepositivex-axistom1.Thereducedquadrupolemomentforthissystemis 3cossin0cossinsin21 30001 31CCCCA(2)Thisisthesameformasthatforthebeamin( 2 )asshouldbeexpectedfromthesimilarityofthegeometries.ThetimederivativesofQijcanbeobtainedfrom( 2 )and 37


2 )withthesubstitutionsM!andL!2aaswellasadivisionby3thatresultsfromthedistributionofmasswithintheuniformbeam.Forthebinarysystem,Kepler'slawgivesarelationbetweentheorbitalfrequencyandthebinaryseparation, a3(2)whereM=m1+m2isthetotalmass.Aswiththerotatingbeam,thegravitationalwaveswillbeemittedatafrequencyequaltotwicetheorbitalfrequency.Usingtheaboverelationships,theGWamplitudeandluminosityforabinarysystemcanbeestimatedas a(2)and 52M3 2 )and( 2 )demonstratethatthelargestandmostenergeticGWswillbegeneratedinbinarieswithlargemassandsmallseparations.Idealcandidatesforsuchbinariesarebinarieswhereoneorbothmembersisacompactobjectsuchasawhitedwarf(WD),neutronstar(NS),orblackhole(BH).Forexample,aNS-NSbinary(m1m21:4MSun=2:81030kg)withanorbitalseparationof2a=500kmwouldproduceaGWluminosityofLGW(binary)1044WatafrequencyGW=370Hz.Atadistanceof1Mpc=31022m,thiswouldproduceanenergyuxof9mW=m2atEarth,aboutthreetimesbrighterthanthevisiblelightuxfromthefullMoon.ThecorrespondingGWstrainamplitudeatEarthwouldbeh(binary)=2:31021,thoughttobewithintherangeofGWdetectors.TheenergycarriedawayfromthebinaryintheformofGWscausestheoverallenergyofthebinarytodecreasewithtime.Consequently,theorbitalradiusmustdecreasewhiletheorbitalfrequencyincreases.ThedecreaseinorbitalradiusincreasestheGWenergyoutput(LGW(binary)/a5),causingthesystemtoradiatemorestrongly.TheresultingGWsincreaseinbothfrequencyandamplitudewithtime,awaveformknownas 38


2M a:(2)Takingatimederivativeof( 2 )andequatingitwith( 2 )resultsinadierentialequationfora, 5M2 256a40 7 ].Inadditiontobinarysystems,severalothertypesofastrophysicalsourcesofGWsarethoughttoexist.RapidlyrotatingNSswithaslightasymmetrywillproduceGWs.TheenergylostthroughGWemissionwillcausetheirrotationratetodecrease,muchastheelectromagneticradiationfrompulsarscausespin-down.Thewaveformsforsuchsourcescanbecalculatedinamannersimilartothatforthebinarysystems[ 8 ].StellarcorecollapseassociatedwithsupernovaearealsoalikelysourceofGWs,althoughinordertogenerateGWs,theremustbeanasymmetricowofmass.Thedicultyin 39


9 ].Finally,thereisalsoapossibilityofacosmologicalbackgroundofGWsanalogoustothecosmicmicrowavebackgroundforelectromagneticradiation.Thiscosmicgravitationalwavebackgroundwouldbeastochasticsignal,thelevelofwhichcanbeestimatedfromcosmologicalarguments[ 2 ].GWsourcescanbeseparatedbyfrequencyband,muchaselectromagneticsourcesareseparatedintoradio,visible,gamma-ray,etc.Ingeneral,largermassestranslatetolowerfrequencies.Themergerofastellar-massbinarywilloccurinthe1kHzband,whereasthemergeroftwosupper-massiveblackholes(SMBHs),withmasses106MSun109MSun,willoccurinthe1mHzband.AGWspectrumsuggestedbyHughes[ 2 ]iscontainedinTable 2-1 Table2-1. SuggestedfrequencybandsforGWs Band FrequencyRange PersistentSrcs. TransientSrcs. Ultra-lowFrequency ? ? Very-lowFrequency SMBICGBR ? Low-Frequency BIEMRICGBR SMBM High-Frequency RNSCGBR BMSN (SMBI=Super-MassiveBinaryInspiral,BI=stellar-massBinaryInspiral,EMRI=ExtremeMass-RatioInspiral,SMBM=Super-MassiveBinaryMerger,BM=stellar-massBinaryMerger,SN=Supernovae,RNS=Rotating/pulsatingNeutronStars,CGBR=CosmicGravitationalWaveBackground) Aswithelectromagneticsources,thesamephysicalobjectmayradiateindierentbandsatdierentepochswithinitsevolution.Astellar-massbinaryintheearlystages 40


2.1 )andtheexistanceofseveralplausiblemechanismsfortheirgeneration(Section 2.2 ),itisareasonableassumptionthatmostoftheuniverseisbathedingravitationalradiation.Theobviousquestionishowcanthisradiationbedetected.AnumberoftechniquesfordetectingGWshavebeenproposedorimplemented.Thesetechniquesgenerallyfallintotwocategories:directtechniqueswhichmeasuretheamplitudesofthewavesthemselvesandindirecttechniqueswhichinferthepresenceofthewavesfromtheireectsonawell-understoodphysicalsystem. 3 ].Apulsarisarapidly-rotatingNSwithahighly-beamedradioemission.Asthepulsarrotates,itsradiobeamsweepsacrossEarth,producingapulseinaradiodetector.Pulsarsaresomeofthemoststableoscillatorsintheuniverseandprovideauniqueopportunityforprecisionmeasurementofthemotionofadistantcompactobject.AfterobservingPSR1913+16forsometime,itwasdeterminedthatitwasinabinaryorbitwitharadio-quietcompanion,likelyasecondNS.Themassofthecompanionandtheorbitalparameters(radius,eccentricity,orbitalphase,etc.)wereextractedbyttingthepulsearrivaltimestoanorbitalmodel.Oncethebinarysystemwascharacterized,theexpectedGWluminositycouldbecomputedfromamodicationof( 2 ) 2 )ismodiedbyanenhancementfactorf(e)=1+(73=24)e2+(37=96)e4 10 ]. 41


2 ).Figure 2-4 showsaplotoftheobservedshiftinorbitalphase(versusanon-decayingorbit)ofPSR1913+16from1975to1988alongwiththepredictionsofGR. Figure2-4. ObservedshiftofperiastronforPSR1913+16.ThesolidlineisthepredictedshiftduetoGWemission(Figure5fromTaylorandWeisberg[ 11 ],usedbypermissionoftheAmericanAstronomicalSociety) ThestunningagreementprovidesexcellentcircumstantialevidencefortheexistenceofGWsandprovidedHulseandTaylorwiththe1993NobelPrizeinphysics.InadditiontoPSR1913+16,severalotherbinarypulsarshavebeenobserved.TheobservationsofeachhavethusfarbeeninagreementwiththepredictionsofGR[ 12 ]. 42


5 ],Chapter37. 13 14 ]beginswithanultra-stableoscillatoroffrequency0,whichisusedtodriveanEarth-basedtransmitter.ThissignaltravelstothedistantSC,whichreceivesaDoppler-shiftedversionofthesignalatime1later.Aphase-lock-loop(PLL)on-boardtheSCisusedtoxtheSC'slocaloscillatortotheincomingsignal.TheSCthentransmitsthissignalbacktoEarth,whereitisreceivedafteranadditionaldelay2andwithatwo-wayDopplershift.ForacoordinatesysteminwhichaGWpropagatesinthez-directionwiththexyaxesorientedparalleltothe+polarization(SeeFigure 2-5 ),theresponseoftheDopplershiftstoGWscanbewrittenas wherethepolaranglestotheSCare(;),=cos(),N1,N2,andN3arenoisetermsand 2 )issometimescalledathree-pulseresponse,sinceanimpulseinh(t)willshowupinthesignalatthreedistincttimes.Forlongwavelengths(GW1;2),thethreepulseswillinterferedestructively.ThissetsthelowerfrequencylimitfortheDoppler-trackingtechnique. 43


ConceptforDoppler-trackingdetectionofGWs. Athighfrequencies,thenoiseterms,whichincludescintillationinEarth'satmosphere,scintillationintheinterplanetarymedium,mechanicalmotionoftheantennae,andthermalnoiseinthereceivers,beginstodominatethesignal.Fortypicalexperiments,thissetsthefrequencyrangetoberoughly104HzfGW101Hz[ 13 ].Thesensitivityofthesignalissetbytheremainingnoiselevelinthesystem.AnimprovementuponthesimpleDopplertrackingcanbemadebyyinganadditionalultra-stableoscillatorontheSCandmakingaseparatemeasurementoftheone-wayDopplershiftbetweenEarthandtheSC.Sincethenoisetermsthatenterintothismeasurementwillberelatedtothenoisetermsin( 2 )undertimeshiftsof,itispossibletocreatealinearcombinationoftheEarth-SCandSC-EarthDopplershiftsthatpartiallycancelsthenoiseterms.Itisexpectedthatthistwo-wayDopplertechniquecouldprovideamplitudesensitivityof1018atfrequenciesaround1mHz,correspondingtoastrainspectraldensityamplitudeof31020=p 13 ].Experimentsusingtheone-waytechniquehavebeenperformedusingthePioneerSC,Galileo,MarsSurveyor,andmostrecentlyCassini[ 15 ]. 16 17 ].Pulsarsareamongthemostpreciseclocksintheuniverse,afactthatmadetheindirectdetectionofGWsusingbinary-pulsarspossible.Inthedirect 44


2 )willincreasewithp 18 ]isaUS-Australiancollaborationwithagoalofobserving20pulsarswith100nsresidualsoveraperiodof10yrs.Thiswouldgiveanamplitudesensitivityof1016atGW=1nHz.SourcesintheextremelylowfrequencybandincludeinspiralingSMBHbinariesandstochasticsourcessuchasthecosmologicalbackground. 19 ].Theyconsistoflargemasses(bars)suspendedinsuchawayastominimizedamping.ApassingGWwilldepositsomeenergyintomechanicalvibrationsofthebar.Ifthisexcessenergycanbemeasured,theGWcanbedetected.Thechallengeforbardetectorsisdistinguishingthesmallamountofenergyaddedtothebar 45


2-6 showsa1996sensitivitycurveforALLEGRO,abardetectorinBatonRouge,LA[ 20 ].Toproducethecurve,thespectraldensityofthedetectornoisewasscaledtoequivalentGWstrainamplitude.Detectableeventswouldhaveastrainamplitudeabovethecurve.ThecurveforALLEGROshowstwonarrowbandsofmaximumsensitivity,correspondingtoresonanceswithinthedetector.Inthissensitivitycurvethemaximumsensitivityreaches1021=p 2-1 ).PersistentsourcesinthisfrequencybandincluderotatingNSsandcosmologicalbackground.ForGWsoriginatingfromoptically-observedpulsars,thefrequencyoftheGWsisknownandtheresonancesofthebarscanbetunedtosearchforit.Transientsourcesincludethenalmergerofstellarmassbinaries,supernovae,andotherunmodeledsources. 46


SensitivityofALLEGRObardetector1996(courtesyofW.O.Hamilton) Todate,noconrmeddetectionsofGWshavebeenmadewithbardetectors.AkeytechniquefordistinguishingGWsignalsfromspuriousnoiseburstsiscoincidencemeasurementsbetweenmultiplebardetectors.ThiscanalsohelpprovidedirectioninformationfortheobservedGW,sincebardetectorshavenearlyuniformantennasensitivitypatterns.Anumberofmajorbarexperimentsareunderwayaroundtheworld,someofwhicharelistedinTable 2-2 Table2-2. OperationalGWbardetectors Name Location BarTemperature OperationalDate ALLEGRO BatonRouge,USA 1991 ALTAIR Frascati,Italy 1980 AURIGA Lengaro,Italy 1997 EXPLORER Geneva,Switzerland 1989 NAUTILUS Rome,Italy 1994 NIOBE Perth,Australia 1993 2-2 isanadaptationofasimilartablebyJohnston[ 21 ]. 47


22 ]. 2-1 )isideallysuitedtobedetectedwithaMichelsonInterferometer(MI).ConsiderasimpleMIconsistingofalightsource,beam-splitter(BS),twomirrors(Mx,My),andaphotodetector(D)orientedalongthexyaxesasshowninFigure 2-7 .Theopticsareassumedtobefreetomovealongtheinterferometeraxes. Figure2-7. AMichelsoninterferometerasadetectorofGWs.(LS=lightsource,BS=beamsplitter,Mx;y=mirrors,D=photodetector. IfaGWpropagatinginthez-directionpassesthroughthedetector,themirrorswillrespondasthemassesinFigure 2-1 ,withtheBSattheorigin.ThedistancesbetweentheBSandthemirrorsinthex;yarmwillthenbe 48


2 )and( 2 ).ThelightenteringtheBScanbedescribedbyanelectriceldoscillatingatagivenfrequencywithaphase(t).Aseachlightbeammakesitsout-and-backtripalongthearms,itwillgainaphaseof 2 ),whichcanthenbeusedtoextracth+(t)andh(t).ForamoregeneralrelationshipbetweentheGWpropagationdirection,polarization,andthedetectorplane,theexpressionsin( 2 )and( 2 )willincludeafunctionsofskypositionknownasantennapatternsforeachpolarization.ExceptforcertainorientationssuchasaGWpropagatingalongthexoryaxes,theantennapatternsarenearlyuniform.InterferometricGWdetectorsaresensitivetoGWsovertheentiresky.TheobservablefrequencybandforinterferometricGWdetectorsislimitedbytheirsizeandbynoisesources.Theexpressionforphaseaccumulationin( 2 )isvalidonlywhentheround-triptimeisshortcomparedtotheGWperiod.Forlongerarms(orshorterGWperiods),thesignoftheGWstrainwillreverseasthelightispropagating,causingtheround-tripphasechangetoaveragetowardszero.Thiseectactsasalow-passlterwithasinc(2LfGW)transferfunction.Thereisnofundamentallimitto 49


2-7 )intoaworkingdetectorcanbebrokenintotwoareas:theabilitytobuildfreely-fallingtestmassesandtheabilitytomakeprecisiondistancemeasurementsbetweenthesetestmasses.Theformerisknownintheground-basedinterferometercommunityasdisplacementnoisewhilethelatteriscalledreadoutnoise.Thechiefsourceofdisplacementnoiseininterferometersisoftenseismicnoise.AnEarth-boundlaboratoryisnotafreely-fallingframe,butafree-fallconditioninonedimensioncanbeapproximatedbysuspendingthetestmassesonpendula.Thisconstrainsthetestmassmotionintheverticalandtransversedirections.Longituinally,forsmalldisplacementsatfrequenciesabovethenaturalfrequenciesofthependula,themassesarefreetomove.Vibrationscancoupleintothetestmassesthroughthependula,spoilingthefree-fallcondition.Othersourcesofdisplacementnoiseareradiationpressurenoisefromthelightonthemirrors,internalvibrationsofthemirrorsdrivenbythermalenergy,andgaspressurenoise.SignicanteortisrequiredtosuppressthesenoisesourcestoalevelsucientforGWdetection.Theultimatelimitondisplacementnoisemaybegravitygradientnoise,whichdescribesthetime-dependentportionsoftheNewtoniangravitationaleld.ForEarth-bounddetectorswithcurrenttechnologies,gravitygradientnoiselimitstheusefulbandtoroughly10Hzandabove.TheotherclassofnoisesourcesforinterferometricGWdetectorsisreadoutnoise.Readoutnoiseincludesshotnoiseandphasenoiseinthelightsource.Shotnoisecanbereducedbyincreasingthepowerofthelightsource,howeverapenaltyispaidindisplacementnoisethroughincreasedradiationpressurenoise.Theshot-noise/radiation-pressurenoiselimitrepresentsthetheoreticalmaximumsensitivityforagivendetectoroperatingoverabroadfrequencyband. 50


2 ),thedetectoroutputisderivedfromthephasedierencebetweenthetwoarms,.Changesincanresultfromtwosources,achangeinLorachangeink ()=2k(L)+2kL;(2)wheredenotesatime-dependantchangefromthenominalvalue.ThersttermcorrespondstomotionofthemirrorsandapotentialGWdetection,whilethesecondiscausedbyfrequency(orphase)uctuationsinthelightsource.NotethatthephasenoisetermisproportionaltoL,sothatinthecasewherethearm-lengthsareexactlyequal,itvanishes.Torstorder,anequal-arminterferometerisinsensitivetophasenoiseofthelightsource.Inthepastdecade,severalkilometer-scaleinterferometricdetectorshavebeenbuiltaroundtheglobe(seeTable 2-3 ,basedondatafromJohnston[ 21 ]).Multipledetectorsarenecessaryforperformingcorrelationsinordertoreducetheoccurrencesoffalsedetections.Inaddition,therelativetimingbetweeneventsasmeasuredbywidely-seperateddetectorscanprovideinformationonthedirectiontotheGWsource. Table2-3. Majorground-basedGWinterferometers. Project Location ArmLength Status LIGO USA(2) inoperation VIRGO Italy commisioning GEO600 Germany inoperation TAMA300 Japan underconstruction ArepresentativeexampleofamoderninterferometricGWdetectorsaretheLIGO(LaserInterferometricGravitationalWaveObservatory)detectorsintheUS[ 23 ].Figure 2-8 showsanaerialphotoofthe4kmLIGOdetectoratHanford,WA.Theeectivelength 51


Figure2-8. AerialphotographoftheLIGOinterferometricGWdetector(courtesyLIGOScienticCollaboration) Figure 2-9 showsasensitivitycurveforLIGOduringthesciencerunsknownasS4(Spring2005)andS5(Nov.2005-present)alongwiththedesigngoalforLIGOsensitivity.Asidefromafewnarrowpeaksandaslightexcessatlowfrequencies,theLIGOdetectorsarenowperformingattheirdesignsensitivity.Thepeaksensitivityof31023=p 2-1 .LIGOiscurrentlysearchingforGWsfrommanyofthepredictedsourcesinthisband.Whilenoconrmeddetectionshavebeenmade,thedatahasallowedimportantupperlimitstobeset[ 25 27 ]. 52


SensitivitycurvesfortheLIGOdetectorsduringtheS5sciencerun(courtesyLIGOScienticCollaboration[ 24 ]) Thesharpdropinsensitivitybelow30Hzissometimescalledthelow-frequencywallandrepresentsalimitforcurrentground-basedinterferometricGWdetectors.Whilefuturetechnologymayallowforsomeimprovementatlowfrequencies,itisunlikelythatground-basedinterferometerswillbeabletoaccessGWsourcesinthelow-frequencyband.Todothis,thedetectormustleavethenoisyenvironmentofEarth.OnepossibilityistoplaceLIGO-typedetectorsoncelestialbodieswithlessseismicactivity,suchastheMoon.Alternatively,thedetectoritselfcanbeplacedinspacetoavoidseismicnoisecompletely.ThisisthegoaloftheLaserInterferometerSpaceAntenna(LISA),thesubjectoftheremainderofthisdissertation. 53


2-1 ).Onewaytoaccessthemanyinterestingsourcesinthisbandistomovethedetectorintospace.TheLaserInterferometerSpaceAntenna[ 28 ](LISA)isajointprojectoftheNationalAeronauticsandSpaceAdministration(NASA)andtheEuropeanSpaceAgency(ESA)thatplanstolaunchaspace-basedinstrumentcapableofdetectinggravitationalradiationinthefrequencybandof3105Hzto101Hzwithastrainsensitivityintherangeof1021=p 3-1 ).Galacticbinariesrefertotheearlyinspiralphaseofstellar-masscompactobjects.SincetheseobjectswillbefarfrommergerwhileintheLISAband,theycanbetreatedaspersistentsources.SomanyofthesesourcesarethoughttoexistthattheLISAsensitivitywilllikelybelimitedbyaconfusionbackgroundofgalacticbinariesinsomefrequencyregimes.ThefactthatGWsourcesaretreatedasnoisetosomeintheLISAcommunityisindicativeofthedierenceinsourceabundancebetweenthelow-frequencyandhigh-frequencyGWbands.TherearealsoahandfulofbinarysystemsthathavebeenobservedelectromagneticallyandshouldproduceisolatedGWsignalsintheLISAband.Thesevericationbinarieswillprovideaninstanttestoftheinstrument,aswellasservingasvaluablecalibrationsourcesthroughoutthemissionlifetime[ 29 ].AnEMRIreferstoasmall(1M10M)compactobjectfallingintoaSMBH(106M109M).InanEMRI,thespacetimeisdominatedbytheSMBHandthesmallerobjectservesasatestparticle,tracingoutthegeodesicsofthespacetime 54


2 ]. Figure3-1. SourcesintheLISAobservationalwindow(CourtesyNASA) AmergeroftwoSMBHsmayoccurduringgalacticcollisions,whentheSMBHsatthecenterofeachparentgalaxyinspiralintooneanotherandmerge.Theseeventswouldbeamongthemostenergeticintheuniverseandwouldbevisibletoredshiftsofz510[ 2 ].LISAwillalsohaveanopportunitytosearchforacosmologicalbackgroundofGWsproducedbytheBigBang.However,mostmodelsofsuchbackgroundsthatexistpredictthattheywillliebelowtheLISAband. 3-2 .Theplaneofthe 55


Figure3-2. OrbitalcongurationofLISAconstellation(CourtesyNASA) EachSCcontainstwoopticalbenchesatthecenterofwhichisa4-cmcubeofgold-platinumalloyknownastheproofmass.Likethemirrorsinground-basedinterferometers,theproofmasswillrepresentthegeodesic-trackingfreeparticleinGR.PassingGWswillmodulatetheproperdistancebetweenthesixproof-masses,aneectthatwillbemeasuredusinglaserinterferometry.Aswithground-baseddetectors,thechallengesofLISAnaturallydivideintotwoareas:buildingaproofmassthatapproximatesafreely-fallingtestparticleandmeasuringthedistancebetweentheproofmasseswithaprecisionsucienttodetecttheminutelengthchangescausedbyGWs.AccomplishingthesetasksisthegoalofthetwomajorLISAinstrumentalsystems,theDisturbanceReductionSystem(DRS),andtheInterferometricMeasurementSystem(IMS). 56


30 ].Mostoftheseusemodelproofmassessuspendedontorsionpendula,providingasimilarsystemwithareducednumberofdegreesoffreedom.Inaddition,anon-orbittestoftheDRStechnologywillcomewiththeLISAPathndermission,atechnologydemonstratormissionplannedforlaunchin2009[ 31 ]. 57


3-3 containsaschematicoftheLISAIMS,consistingofthethreeSCeachwithtwoidenticalopticalbenches.Eachopticalbenchcontainsaproofmass,aninfraredlaserlight-source,photoreceivers,andoptics.Referringtothenotationinthegure,OBijistheopticalbenchonSCiorientedtowardsSCj.ThetwoopticalbenchesoneachSCareconnectedtooneanotherviaanopticalber.OpticalbenchesonoppositeendsofaLISAarmareconnectedviaatwo40cmtelescopesanda5Gmfree-spacelink.Duetodiractionlossesoverthelongarms,onlyabout100pWoflightarereceivedfromthe1WoflightproducedatthefarSC.Interferometryisusedtomakethreetypesofmeasurements:distancebetweentheproofmassandtheopticalbench,thedistancebetweenopticalbenchesondierentSC,andthephasedierencebetweenthelasersonadjacentopticalbenches. Figure3-3. DiagramoftheLISAIMS.OBijreferstotheopticalbenchonSCiorientedtowardsSCj.ijisthelighttraveltimefromSCitoSCj. 58


3 ),allofthevariationofthelighteldiscontainedin(t),isassumedtobeconstant.AnalternativeviewdescirbesallvariationinE(t)asfrequencynoise,orachangeinratherthan.Thesetwoequivalentdescriptionsareeasilyrelated.Ifthefrequencyisdescribedby(t)=0+(t),theequivalentphasenoiseis 2f(3)wherefistheFourierfrequencyandthetildeindicatesafrequencyspectrum.Sinceelectromagneticwavesarelinear,asuperpostionoftwoopticalsignalscanbedescirbedbyaddingtheirelectricelds.Aphotodiode(PD)canbeusedtomeasuretheintensityofthecombinedbeam,whichisproportionaltothesquaredmagnitudeofthetotalelectriceld.Ifthetwosignalshavefrequencies1;2andphases1;2(t),thePDoutputwillbeasignaloftheform 59


4.5 3-4 .Threedierentinfraredlaserbeamsentertheopticalbench.Thelocalbeam(red),isproducedbythelaserassociatedwiththeopticalbench.Theadjacentbeam(blue)isproducedbythelaserassociatedwiththeneighboringopticalbenchonthesameSC,andreachestheopticalbenchthroughanopticalber.Thefarbeam(green),isproducedbythelaserassociatedwiththeopticalbenchonthefarSC.ThesebeamsareinterferedatthreePDs,PDmain,PDback1,andPDback2.InthebaselinedesignofFigure 3-4 ,PDmainisusedtointerferetheincomingbeamwiththelocalbeam,producingasignalcontainingtheone-waymotionbetweenthelocalandfaropticalbenches.Inthecross-overdesignoption,theincomingbeamisinterferedwiththeadjacentbeamratherthanthelocalbeam.Sincethelocalandadjacentbeamswillgeneralyhavedierentcarrierfrequencies,thiswillreducetheeectofstraylight. 60


Figure3-4. DiagramofaLISAopticalbench.Lightfromthelocallaser(red)entersfromthebercoupleronthebottom,lightfromtheadjacentopticalbench(blue)entersfromtheleft,andlightfromthefarSC(green)entersfromtheright. Thesethreemeasurementsarerepeatedonallsixopticalbenches,resultingineighteenindependentmeasurementsthatmustbeproperlycombinedinordertoextractthedistancesbetweentheproofmasses.ConsiderthetwobacksidePDs(PDback1andPDback2)ontheopticalbenchinFigure 3-4 .APMcanbeusedtoextractthephaseofthebeatsignals.ThePMoutputsofthebacksidePDsare: 61




32 ](TDI),andiskeytothesuccessofLISA.ThethreeLISAarmsprovideatotalofsixone-waylinks,eachofwhichcanbepotentiallyutilizedtoformaTDIsignal.Theresultingspaceofpossiblesignalsislargeandistypicallybrokenintoseveralclasses[ 33 ].SomeofthemostbasicTDIcombinationsaretheMichelson-likecombinations,typicallyreferredtoasX,Y,andZ.Thethreelettersrefertothethree-foldsymmetryofLISA:theXcombinationistheMichelson-likecombinationwithSC1asthecornerSC,theYcombinationistheMichelson-likecombinationwithSC2asthecornerSC,andtheZcombinationistheMichelson-likecombinationwithSC3asthecornerSC.ToformtheXcombination,thetwolaserson-boardSC1(Figure 3-3 )arephase-lockedusingthesignalsonPDback2sothat12(t)13(t)=1(t).ThisistheLISAequivalentofthebeam-splitterinatrueMichelsoninterferometer.Toapproximatethemirrors,thefarSC(SC2andSC3)areconguredasopticaltransponders.ThePMsignalsatPDmainonOBj1willbe .WhenthetwobeamsreturntoSC1,thePMsignalatPDmainonOB1jwillbe 63


3 )into( 3 )andsimplifyinggives TheXcombinationcompletelyremovesthecontributionsfromlaserphasenoise,whileleavingthecontributionsfromtheGWsignals.TheresponseofXtoGWsissometimesreferredtoasafour-pulseresponsesinceanimpulseinh(t)willresultinanimpulseinX(t)atfourdistincttimes:t,t+ 34 ]. 3-5 .TimedelayingthePMsignalsbyanamountcanbeinterpretedassendingthemalongavirtualpathwithalight-traveltimeof:IntheXcombination,onebeam(red)makesaphysicaltrip(solidline)fromSC1toSC2andbackandthenmakesavirtualtrip(dashedline)fromSC1toSC3andback.Theotherbeam(blue)doesthereverse.WhentheyreturntoSC1,bothhavetraveledthesamedistanceandconsequentlythephasenoiseiscommonandcancelsout.Thiscanbeviewedassynthesizinganequal-armMichelsoninterferometerorzero-areaSagnacinterferometerfromtheindividualPMsignals. 64


Therabbit-eardiagramfortherst-generationTDIXcombination.EachlightbeamoriginatesatSC1andtakesphysical(solid)aswellasvirtual(dashed)tripstothefarSC.Thetotalround-trippathlengthforthetwobeamsisidenticalandthelaserphasenoiseineachbeamatSC1iscommon. 35 ],whichincludefouradditionaltermsandcancelouttherelativevelocitiesoftheSC.Thesearesometimesreferredtoastheeight-pulseTDIvariables,sinceanimpulseinh(t)willarriveateightseparatetimesinthesignal.Thecancellationofthesecond-generationTDIvariablesisalsonotperfect,sincethereisarelativeacceleration 65


36 ].ASagnacinterferometerconsistsoftwobeamsfromacommonsourcepropagatinginoppositedirectionsaroundaclosedloop.Thephasedierencebetweenthetwosignalsisproportionaltotheareaenclosedbytheloopandtherotationspeedoftheloop.InLISAaSagnacsignalcanbegeneratedeitherphysicallythroughacombinationofPLLsonappropriatebenchesorvirtuallythroughanappropriatecombinationoftime-delayedPMsignals.Ineithercase,therotationintheconstellationresultsinapathlengthdierencebetweenthetwobeamsofapproximately14km,whichisequivalenttoatime-dierenceof47s.Thiswillcauselaserphasenoiseandotherinstrumentalnoisesourcestocoupleinto.TheGWsignal,ontheotherhand,willnotbepresentinsincethetidaldistortionscausedbyGWsarearea-preserving.Whatprovidesisameasurementchannelcontaininginstrumentalnoisebutnosignal.Thisisessentialfordistinguishingbetweenpossiblesignalsandinstrumentalnoisesince,unlikeground-baseddetectors,LISAwillnothaveotherdetectorswithwhichtoperformcorrelations.Inasense,thethreearmsinLISAareequivalenttotwoco-locatedinterferometersandthevariablemeasurestheuncorrelatedinstrumentalnoise. 3 ),onemustknowthevaluesofij.ThisrequiresanindependentmeasurementoftherangebetweentheSC.ErrorsinthisrangingmeasurementwilldegradethenoisecancellationinTDI.Itisestimatedthatarangingaccuracyof20mto200misneededtosucientlysuppresslaserphasenoise[ 34 ].This 66


37 ],whichisdiscussedinSection A.4.6 38 39 ]ormoleculartransition[ 40 41 ]asafrequencyreference.ThecurrentLISAbaselinecallsforeachlasertohaveanopticalcavitysystemcapableofprovidingafrequencystabilityof 34 42 ].Improvementinf(f)over( 3 )willallowarelaxationofranging/interpolationrequirementsandpossiblytheuseofrst-generationTDIvariables. 67


43 ]isatechniquewherebysomecombinationoftheLISAarmsisusedasafrequencyreferenceforlaserstabilization.AlthoughthelengthsoftheLISAarmschangeoveraperiodofayearbyasmuchasafewpercent,intheLISAbandtheyareextremelystable.Ifthisstabilitycanbetransferredtothelaserphase,therequirementsonTDIcanberelaxedconsiderably. 3-6 showsaLaplace-domainrepresentationofagenericclosed-loopsystem.Theinputx(s)iscombinedwiththecontrolsignalc(s)toproducetheerrorsignale(s).Theerrorsignalpropagatesthroughthesystem,withtransferfunctionG(s),toproducetheoutputsignaly(s).Thecontroller,withtransferfunctionH(s),formsc(s)fromy(s). Figure3-6. Diagramofaclosed-loopSISOsystemwithnegativefeedback.Signals:x(s)=input,e(s)=error,y(s)=output,c(s)=control.G(s)isthesystemtransferfunction.H(s)isthecontrollertransferfunction Thistypeofsystemcanbereferredtoasasingle-inputsingle-output(SISO)closed-loopsystemwithnegativefeedback[ 44 ].ThesignalsandthetransferfunctionsG(s)andH(s)arecomplex-valuedfunctionsofthecomplexLaplacevariable,s=+2if.Theadvantageofexpressingtheclosed-loopsystemintheLaplace-domainisthatthedierentialequationsthatrelatethetime-domainsignalsreducetoalgebraicequationsrelatingtheLaplace-domainsignals.Theerror,output,andcontrolsignalscanbe 68


1+TOL(s);(3) 1+TOL(s);(3)and 1+TOL(s);(3)whereTOL(s)G(s)H(s)istheopen-looptransferfunction.Inthecaseofacontrol-loopusedforstabilization,x(t)isanoiseinputtothesystemandy(t)isthenoiseinthesystemoutput.Thegoalofthecontrolsystemistoreducethemagnitudeofy(t)foragivenx(t).Notethat( 3 )containstheterm1+TOL(s)inthedenominator.Themagnitudeofthistermindicatestheperformanceoftheloopasasuppressorofnoise.Ifj1+TOL(s)j>1,theclosed-loopvaluefory(s)willbesmallerthantheopen-loopvalue,givenby( 3 )withH(s)=0.Ifj1+TOL(s)j1,thentheclosedloopvaluesfory(s)willbegreaterthanorequaltotheopen-loopvalue,aconditionknownasnoiseenhancement.TheperformanceofaparticularstabilizationsystemcanbeevaluatedusingaNyquistplot,aplotofTOL(s)inthecomplexs-plane.AsshowninFigure 3-7 ,theNyquistplothastworegionsseparatedbyacircleofunitradiuscenteredonthepoint(1;0)markedwithan.IfTOL(s)liesinsidethecircle,theclosed-loopsystemwillenhancethenoise.Ifitliesoutsidethecircle,theclosed-loopsystemwillsuppressthenoise.Thedegreeofnoiseenhancementorsuppressionisrelatedtothedistancefromthepoint(1;0).ThecloserTOL(s)liesto(1;0)thelargerthenoiseenhancementorsmallerthenoisesupression.IfTOL(s)reachesthepoint(1;0),knownasapole,theexpressionsin( 3 )-( 3 )becomeinnite.InaNyquistplot,TOL(s)isplottedasacurveparameterizedbytheFourierfrequency(s!2if).Formostsystems,thecurvewillbeaspiralwithfrequencyincreasing 69


44 ]:forclosed-loopstability,theopen-looptransferfunctionTOL(f)mustnotencirclethepoint(1;0)inthecomplexplane. Figure3-7. GenericNyquistplotforopen-looptransferfunctionTOL(f).Theshadedregionindicatesnoiseenhancement.Thepoleat(1;0)ismarkedbyan.FrequencyincreasesclockwisealongthecurveofTOL(f). 3 ).IntheabsenceofGWsignals(h12(t)=h21(t)=0),thissignalcanbesimpliedto 70


3-8 (a)isacircle,theresultofavectorsumofthetwotermsin( 3 ).Thersttermisaunitvectoralongthepositiverealaxis.Thesecondtermisarotatingunitvectormakinganangle2fwiththenegativerealaxis.Asfincreases,Tsen(f)tracesoutaclockwisecircle,reachingtheoriginatf=fnn=RT,n=0;1;2:::.Thesearethenullfrequenciesforwhichthesignals1(t)and1(tRT)areinphaseandcancel.Asthecurvepassesthroughtheorigin,thephaseofTsen(f)shiftsdiscontinuouslyfrom90to+90. (b) (c)Figure3-8. RepresentativeNyquistplotsforsingle-armarm-locking:(a)sensor(1esRT),(b)system1 InordertocomputethesystemtransferfunctionG(s),Tsen(s)mustbecombinedwiththeactuatortransferfunction.Mostlaseractuatorsarefrequencyactuators,producingachangeinlaserfrequencythatisproportionaltothecontrollerinput.AfrequencyactuatorcanberepresentedasaphaseactuatorintheLaplacedomainwithanadditional1=sinitstransferfunction.Thesystemtransferfunction(sensor+actuator) 71


3-8 (b).ComparingthecurvesinFigure 3-8 (b)withFigure 3-8 (a),theeectoftheactuatortransferfunctioncanbeseenasaclockwiserotationof90coupledwithadecreaseinmagnitudeasfrequencyincreases.Thiscausesthesystemtoenterthenoiseenhancementregion(insidethedashedcircle).AnalternativeviewofG(s)istheBodeplotinFigure 3-9 .Heretheinterferometernullsandphasediscontinuitiesareclearlyseenatmultipliesof1=RT30mHz. BodeplotofTsen(f)withRT=33s,theround-tripdelayinasingleLISAarm AsG0isincreased(redcurvevs.bluecurveinFigure 3-8 (b)),thesystempassesclosertotheinstabilitypointat(1;0).Thisisamarginally-stableconditionwhereanincreaseingainproducesanincreaseinnoisesuppressionatsomefrequenciesbutacorrespondingincreaseinnoiseenhancementatotherfrequencies.Anyadditionalphaselosswillrotatethesystemfurther,allowingittoencompassthe(1;0)pointandbecomeunstable. 72


3-8 (c).AscomparedtoFigure 3-8 (b),thecurveisrotated45counter-clockwise.Asthegainisincreased(redcurvevs.bluecurve),thecurveapproachesthelineRe[TOL(s)]=Im[TOL(s)].Withthistypeofcontroller,thegaincanbearbitrarilyincreased(assumingnoadditionalphaseloss)withoutincreasingthelevelofnoiseenhancement.IntheBoderepresentation(Figure 3-9 ),thephaseresponseoftheopen-looptransferfunctionisequaltothesumofthesystemphaseresponseandthecontrollerphaseresponse.Thephaseadvanceinthearm-lockingcontrollerliftsthephaseminimaatthenullfrequenciesawayfrom180.Thisprovidessomepositivephasemarginandhencestability.Thepricepaidforthephaseadvanceofthecontrollerin( 3 )isareducedslopeinthemagnitudeofTOL(f).Foragivencontrollerbandwidth,thislimitsthegainatlowfreuqencies.Alternatively,ahigherbandwidthisrequiredtoreachagivenlow-frequencygain.Ageneralarm-lockingcontrollerwillhaveatransferfunctionsimilartothatinFigure 3-10 .Thefrequencyresponsecanbedividedintothreedistinctregions.Forf

3 )sothatitprovidesthenecessaryphaseadvance.Forf>fUG,thecontrollerresponsecanbegintorollo. Figure3-10. Bodeplotforagenericarm-lockingcontroller.Thecontrollermustprovideaphaseadvancebetweenf1andfUG 3 )and( 3 ), 1+H(s)G0s1(1es);(3)wherep(s)isthelaserphasenoisepriortoarm-lockingand(s)isthelaserphasenoiseafterarmlocking.Themagnitudeoftheclosedloopsupression,jTCL(f)j,isplottedforagenericarm-lockingsysteminFigure 3-11 .Forff1,thesuppressioncanbelarge,duetothesteeproll-oofthecontroller.Atafrequencyjustbelowf1,theclosed-loopsuppressioncrossesthe0dBline,indicatingnoiseenhancement.ThiscorrespondstothecurveofTOL(f)enteringthedashedcircle 74


3-8 (c).Thelevelofnoiseenhancementincreasesuntilitreachesamaximum,correspondingtotheclosestapproachtothepoint(1;0)intheNyquistplot. Figure3-11. Closed-loopnoisesuppressionforagenericarm-lockingloop Atf=f1,theclosed-loopsuppressionisagain0dB,correspondingtothecurvepassingthroughtheoriginintheNyquistplot.Thisbehaviorisrepeatedateachfnwiththeheightsofthenoise-enhancementpeaksandthedepthsofthenoise-suppressionvalleysdecreasingasfincreases.Nearf=fUG,aservobumpmayoccur,causedbythelossofphaseinthecontrollerasitsmagnitudeattensout. 3-11 representsasteady-statelimitthatisreachedonlyafteranytransientresponsesdecay.Foraqualitativeunderstandingofarm-lockingtransients,consider( 3 ),thesingle-armerrorsignalinthetime-domain.Undertheassumptionofhighgain,thearm-lockingcontrollerwillenforcethecondition 75


3-11 .Forarealsystem,therelationshipin( 3 )becomesmoreapproximateasadditionaluncorrelatednoisemixesintothesystem.Asitdoesso,thetransientsdecayandthesystemapproachesitssteady-statelimit.Thetimeconstantsforthisdecayareacriticalmeasureofarm-lockingperformance.Ifthetime-constantsaretoolarge,valuableobservingtimemightbewastedwhilewaitingforthenoisetodiedown.Itcouldalsolimittheexibilityofmissionmanagerstounlockandre-locktheconstellationasneeded.Foragivencontroller,itispossibletomakeananalyticalestimateofthetransientfrequenciesandtime-constants[ 45 ].ThisisdoneusingaLaplace-domainanalysisthatproperlyaccountsfortheintegrationconstantsintheLaplace-transformoftheconstituentsignals.Forthesystemdescribedabove, RTImfG(fn)1g 43 ]suggestthatthetransientresponsemaybesuppressedbyslowlyrampingtheloopgaintoitssteady-statevalueratherthansuddenlyturningitonatt=RT.Theanalyticaltreatmentabovedoesnotapplyfortime-dependantgains,buttheresultcanbeintuitivelyunderstoodasfollows.For 76


3 ).Thesystemwillreachsteadystatequickly,butthesteadystatewillonlybeaslightimprovementoverthefree-runningcondition.Thegainisthenincrementallyincreased,increasingthetimeconstantsanddecreasingthesteady-statenoise.However,theinputnoiseisnowslightlylowerthanthefree-runningcase,correspondingtosmallerinitialA(trans)n.Thisprocessisthenrepeateduntilthenalgainisreached.Theoveralltimetoreachsteadystateisdecreasedsince isfeasible,butitsperformanceislessthanideal.Since1=RTisintheLISAband,extremelylargebandwidthswouldbeneededtoachievesucientsuppression.Asuppressioninlaserphasenoisebyafactorof104at10mHzwouldrequireaunity-gainfrequencyof1MHzforacontrollerwithp=1=2.Regardlessofthecontrollershapeorbandwidth,therewillstillbenoisepeaksinthestabilizedspectrumthatarelargerthantheun-stabilizednoiseinthesamefrequencybins.Thefundamentalreasonthatthesepeaksarepresentisthatthesingle-armsensorsignal,( 3 ),containsnoinformationaboutphasenoisewithFourierfrequenciesf=fn.Suppressionofnoiseatthosefrequenciesrequiresanerrorsignalthatissensitivetonoiseatthosefrequencies.OnewaytoobtainsuchanerrorsignalistoutilizeadditionalLISAarms.AsmentionedinSection ,theorbitaldynamicsoftheconstellationcausetheLISAarmstodierbyuptoonepercent.Consequentlythevaluesoffnforonearmwillbeslightlydierentthanthoseforanotherarm.ConsidertheLISAconstellationarrangedasintheMichelsonXTDIcombination.SC1isdesignatedthemasterSCanditstwolasersarephase-lockedtogenerateasingle 77


3 )canbere-writtenas 3 )with! coshs sinhs 3-12 showsaschematicNyquistplotofS+.Thesignalconsistsoftwotermsaddedvectorially,avectoroflength2alongthepositiverealaxisandavectoroflength2jcos(f)jmakinganangleof2f withthenegativerealaxis.Thecurvewillmakeitsclosestapproachtotheorigin 78


.Unlikethecaseofsingle-armlocking,theerrorsignalwillnotpassthroughtheoriginandconsequentlytheerrorsignalphasewillnotreach90.Thismayrelaxtheconstraintsonthecontroller,allowingthegaintorollomoresteeplyinthevicinityoftheminima.Forfrequencieswheref=m=;m=1;2;3:::,S+hasatruenull.Atthesefrequencies,acontrollermustprovidethesamephaseadvanceasthesingle-armcontroller.Foraone-percentarm-lengthdierenceinLISA,therstnullwouldoccurat3Hz,whichisabovetheLISAmeasurementband. Figure3-12. Nyquistplotforcommonarm-locking ThedepthoftheminimainS+andthecorrespondingminimumphasewilldependonthevalueofcosinetermasfapproachesfn.Since )](3)and )]:(3)Therstconcernisatn=1,sincethecosinefunctionapproachesunityatDC.For= =0:01,Amin103andmin88.Theadditional2ofphasemarginprovided 79


42 ]lookstobemorepromising.Thedirectarm-lockingerrorsignalisformedfromthecommonanddierenceerrorsignalsas ZS(t)dt:(3)Thesignoftheintegratedtermdependsonwhicharmislongerandischosensothatthetermispositive.IntheLaplacedomain,thedirectarm-lockingerrorsignalcanbewrittenas coshs 3-13 containsaplotofthemagnitudeofthesquare-bracketedtermversusFourierfrequencyf.Forf1=,thetermdropstozero.Atthesefrequencies,theresponseofthedirectarm-lockingerrorsignalisnearlyat,greatlyrelaxingtherestrictionsoncontrollershape.Thisallowsforincreasednoisesupressionatthesefrequenciesandtheremovalofthefrequencynoisepeaks.Asfincreases,thesquare-bracketedtermin( 3 )approachesunity,allowingtheoverallerrorsignaltocomeclosertozeroatthefrequenciesfn=n= .Thisresultsinalossofphaseatthesepointsandacorrespondingincreaseinnoiseintheclosed-loopsystem.Justbeforefreaches1=,thetermactuallyexceedsunity.IntheNyquistrepresentationofFigure 3-12 ,thisresultsinthesystementeringthenoiseenhancement 80


Magnitudeofsquare-bracketedtermin( 3 ) Oneminorproblemwithdirectarm-lockingarisesinthefactthatthearm-lengthdierenceisnotconstantandattimesthearmsareequal.ThiscausesproblemswithscalingS(t)by1=inthedirectarm-lockingerrorsignal( 3 ).Thearmswillonlybeequaloccasionally(afewtimesayear)anditshouldbepossibletoaddresstheproblembyoccasionallyswitchingthelocationofthemasterSC,revertingtosinglearm-locking,orsimplywaitingforthearm-lengthdierencetodriftthroughzerowhilesomescheduledmaintenenceactivityistakingplace.Analpossibilityforanarm-lockingerrorsignalistousetheentireconstellationinaSagnacmode,muchasforthevariablediscussedinSection 3.5.3 .StartingatthemasterSC,onebeamissentviaphase-locksonthefarSConacounter-clockwiselooparoundtheconstellation.Theotherbeamissentonaclockwiseloop.Thisproducestwosignalssimilartothosein( 3 )and( 3 ),exceptthat 81


46 ]. 82


42 ].Eachoftheseinvolvecreatingaloop-within-a-loopusinganadditionalactuator.Thetuneable-cavityapproach,showninFigure 3-14 ,replacesthexedopticalcavitywithaPZT-actuatedtuneablecavity.ThelaserislockedtothecavityusingastandardlockingschemesuchasPound-Drever-Hall(PDH)[ 47 ],providingthepre-stabilization.Thearm-lockingerrorsignalisusedtoactuatethecavity,keepingitlockedtothearm-length.TheconcernwiththisapproachisthatplacingaPZTinthecavitywilldegradethecavity'slengthstability.Itremainstobeseenhowseverethisdegradationwillbe. Figure3-14. Combiningpre-stabilizationandarm-lockingwithatuneablecavity AsecondapproachinvolvesusingaxedcavitybutmodifyingthePDHschemetouseasidebandlock.ThePDHlockingschemeutilizesanelectro-opticmodulator(EOM)toplaceRFsidebandsonthelaserbeamenteringthecavity.Theerrorsignalistypically 83


3-15 ),oneRFsidebandislockedtothecavity,providingstabilitytothecarrieraswell.Thecarrierisusedtogeneratethearm-lockingerrorsignal,whichisfedbacktothelocaloscillator(LO)drivingtheEOM.Thisapproachprovidestunabilitywhileavoidingtheneedforatuneablecavity.However,sidebandlockingisknowntointroduceadditionalnoisesourcesandisnotgenerallyutilizedinultra-stableapplications. Figure3-15. Combiningpre-stabilizationandarm-lockingusingasidebandcavitylock Analoptionforcombiningarm-lockingwithlaserpre-stabilizationistheosetphase-lockapproach,showninFigure 3-16 .Hereanadditionallow-powerlaserislockedtoaxedcavity,providingastablereference.Themainhigh-powerlaserisphase-lockedtothisreferencelaserwithanosetfrequencyprovidedbyatuneableLO.Forahigh-gainPLL,thephasenoiseofthemainlaserwillbethesameasthatofthereferencelaser.Thearm-lockingerrorsignalisderivedfromthemainlaser,andisusedtogenerateafeedbacktotheLO.Sincelow-noisehigh-gainPLLsareanexistingLISAtechnologyrequirement(transponderlocks),nonewtechnologiesareneeded.Howevertheadditionallaserisapotentialsourceofaddedmassandpowerconsumption. 84


Combiningpre-stabilizationandarm-lockingusingareferencelaserandanosetPLL 85


48 ],buteventuallylosesandbernoisewilloverwhelmthesignal.TomodelthisaspectofLISAinterferometry,whichisessentialforstudyingTDI,arm-locking,andranging,theUFsimulatorusesanoveltechniqueknownasElectronicPhaseDelay(EPD)[ 49 ]. 3 )thatthetime-varyingcomponentofalighteldcanbedescribedbyacomplexelectric 86


4 )anddelayed,theresultisinterferometricallyequivalenttodelayingE(t).Furthermorethereisalimitedbandwidthoverwhichvariationsin(t)areimportant.Formostcomponents,thisistheLISAband,butforsomecomponents,suchasdatacommunicationandclocktransfer,thismaybeashighasafewGHz.TheinfraredlasersusedinLISAwillhavewavelengthsof1m,whichcorrespondstoanopticalfrequencyof280THz.Atsuchhighfrequencies,itisimpossibletomeasuretheoscillationsin( 4 )directly.However,iftwobeamswithdierentfrequenciesareinterferedonaphotodiode(PD)toproduceabeatnote,theresultingintensitysignalisgivenby 4-1 (a).AlaserononeSC(L1),producesalighteldwithfrequency1andphase1(t).Thislighteldtraversesthe5GmtotheotherSC,incurringadelayof16sand 87


(b)EPDAnalogFigure4-1. TheEPDtechniqueappliedtoasingleLISAarm TheEPDequivalentof 4-1 (a)isshowninFigure 4-1 (b).ThelighteldfromL1isrstinterferedwithareferencelaser(L0),whichhasphase0(t).Thisbeatsignalhasafrequency1010andphase10(t)1(t)0(t).Solongas~0(f)~1(f)andthephasesareuncorrelated,~10(f)~1(f).TheL1L0beatsignalistheEPDanalogoftheopticalsignalfromL1inFigure 4-1 (a). 88


4.6 )arealsocapableofputtingafrequencyshiftonthebeatsignal,mimickingtheDopplershiftspresentinLISA.TheopticalsignalfromL2inFigure 4-1 (a)ismodeledbyabeatbetweenL2andL0inFigure 4-1 (b).Thissignalhasafrequency2020andphase20(t)2(t)0(t).AswiththeL1L0beat,thenoisecharacteristicsofthissignalwillbethesameasthatfortheoriginalopticalsignalsolongasL0isindependentfromL2andhasequalorlesserphasenoise.ThePDinFigure 4-1 (a)isreplacedbyanelectronicmixerinFigure 4-1 (b).ThemixerperformsasimilaroperationonthetwoelectronicsignalsasthePDdoesonthetwoopticalsignals.Themixeroutputcontainstwoterms,onewithafrequencyequaltothedierencefrequencyofthetwoinputsignalsandonewithafrequencyequaltothesumofthetwoinputsignals.Alow-passlterisusedtoremovethehigh-frequencyterm,leavingasignalwithafrequencyandphasegivenby 4 )and( 4 )with( 4 )and( 4 )indicatesthattheEPDmodelproducesasignalthatisofthesameformastheLISAarm.WiththerestrictionsonL0mentionedabove,thenoisecharacteristicswillbesimilaraswell.MorecomplexmodelsofLISAcanbebuiltupinasimilarfashion.Table 4-1 liststhemajorcomponentsinLISAandtheirEPDequivalents.Asanalnote,althoughFigure 4-1 (b)isdrawnwiththesamereferencelaserbeingusedtogenerate10(t)and20(t),itisnotarequirementoftheEPDtechnique.Provided 89


Table4-1. MajorLISAIMScomponents/signalsandtheirEPDequivalents LISAComponent EPDEquivalent lasereld beatnotewithreferencelaser opticaldelay electronicdelay photodiode electronicmixer opticalbeatnote mixeroutput 50 ],thiswouldsignicantlycomplicatetheexperiment.Furthermore,suchacombinedexperimentisnotnecessaryatthisstagesincetheSCtoproof-massandSCtoSCinterferometryaretreatedasseparatemeasurementsinLISA. 4-2 ,eachofthethreeSCismodeledbyanindependentNd:YAGnon-planarringoscillator(NPRO)laser,denotedasL1throughL3inthegure.AfourthNPRO,L0,isusedasareferencelaser.ThelasersL1andL0areeachlockedviathePDHmethod[ 47 ]toindependentopticalcavitieshousedinathermally-isolatedvacuumchamber.BeatnotesbetweenthefarSClasers(L2andL3)aremadewithL0,allowingtwocompleteLISAarmstobemodeled[ 51 ]. 90


OpticallayoutoftheUFLISAinterferometrysimulator.L1L3representSC1SC3inLISA.L0isthereferencelaser. 4-2 istoprovideLISA-likelaserphasenoiseforthesimulator.ThedominantnoisesourceforcavitylengthintheLISAbandisthoughttobethermally-drivenexpansion.Consequently,theLISAcavitieswilllikelyusespacersofultra-lowexpansionglasssuchasDow-Corning'sULEorSchott'sZerodur.Ifthepre-stabilizationrequirementsarerelaxedduetoimprovementsinTDIorarm-locking,itmaybepossibletoutilizeothermaterialssuchasSiliconCarbide(SiC).Inparallelwiththeinterferometryexperiments,theUFgroupisstudyingthestabilityofvariousmaterialsandbondingtechniques[ 52 ].Consequently,theopticalcavitiesusedforpre-stabilizationinthesimulatorareoccasionallychanged.Figure 4-3 showsaspectrum 3 ).Ascanbeseen,theZerodur-SiCfrequencynoiserisessteeplyabovetheZerodur-Zerodurfrequencynoiseatfrequenciesbelow10mHz. 53 ]. 91


46 ],andthecauseisnotcompletelyunderstood.Forshort-terminterferometryexperiments,theshort-termdriftsareofthemostconcern. FrequencynoiseintheL1L0beatnote AsmentionedinSection 3.5 ,thenoiseinasinusoidalsignalcanbeplacedeitherinthefrequencyorinthephase.Figure 4-4 showsthesamedataasFigure 4-3 convertedtophasenoiseusing( 3 ). 92


Datafrom 4-3 convertedintophasenoisebydividingby2f,wherefistheFourierfrequency beatsignalsintheopticallayout,providinginputstothecontrolloopsaswellasthesciencesignals.BoththecontrolltersandthePMsarecriticalcomponentsforLISAaswell.SincethesimulatorversionsandtheirLISAcounterpartsaresubjectedtosimilarsignals,thesimulatorprovidesanexcellentarenaforevaluatingpotentialdesigns.TheEPDunitisofcourseuniquetothesimulator,andmustreproducetheLISAarmasfaithfullyaspossible.TheadditionofnoisenotpresentinLISAoreliminationofnoisepresentinLISAwouldlimittheaccuracyofthesimulator.SeveraltypesofelectronicarchitecturesareemployedintheUFsimulator.Theseincludebothanalogsystemsanddigitalsystems,whicharereviewedinAppendixA.Forthemostdemandingelectronicsubsystems,suchasthePMsandEPDunit,theUFgroupselectedandpurchasedahigh-speeddigitalsignalprocessingsystemfromthePentekCorporationinUpperSaddleRiver,NJ.AnoverviewschematicofthesystemisshowninFigure 4-5 below.Thesystemconsistsofthreeindividualproducts,themodel4205carrierboard,themodel6256digitaldownconverter,andthemodel6228digitalupconverter.Themodel4205carrierboardishousedinaVMEcrateandcontainsa1GHz,32-bitPowerPC 93


Figure4-5. OverviewoftheDSPsystemfromPentekCorporation TheVIMinterfaceconsistsofa32-bitdatainterfaceanda32-bitcontrol/statusinterface.ThedatainterfaceisconnectedtothemainPCIbusviaabi-directionalrst-inputrst-outputbuer(BIFO)andadirect-memoryaccess(DMA)controller.TheDMAcontrollerallowsdatatobereadfromtheBIFOanddirectlydepositedintomemoryoranotherlocationonthePCIbuswithouttheneedforprocessorintervention.ThespeedoftheVIMinterfaceislimitedbythespeedofthePCIbus,whichisclockedat66MHz.Thecontrol/statusinterfaceisconnectedtotheprocessorviaaseparate33MHzPCIbus.The6256digitaldownconvertercontainsfour14-bitADCsthatcanbeclockedatfrequenciesupto105MHz.TheADCsareconnectedtofront-panelconnectorsviaRFtransformerswithahigh-passfrequencyresponse.The3dBpointofthetransformersisat400kHz.Signalsatlowerfrequenciescannotbemeasuredwiththe6256.Thefull-scale 94


4.6 ),thehardwarePM(Section 4.5.5 ),andoneinstanceofthearm-lockingcontrollter(Section 5.4 ). 4.5.1OverviewThephaseofasinusoidalsignalcanbespeciedbythetimeatwhichthesignalhasaspecicvalueandspecicrstderivative(i.e.positivezero-crossing).Sincethisisfundamentallyatimingmeasurement,allphasemeters(PM)smustbebasedonareferenceclock.ThephasemeasuredbythePMisthedierencephasebetweenthesignalandthereferenceclock.Anysinglephasemeasurementwillbelimitedinaccuracybythephasestabilityofthereferenceclock.Withmultiplemeasurementsusingthesame 95


4 )and( 4 ), Themixeroutputisthenlow-passlteredtoremovethe2!term.Ifthephasenoiseissmallandthesignalamplitudesareconstant,thentheresultingsignalisproportionalto(t).Someanalogmixersarespecicallydesignedtocompensateforthesinusoidalresponseandcanproducealinearphaseresponseforj(t)j70o.ThelteredmixerPMisanexampleofageneraltypeofPMcalledin-phase/quadratureorIQPMs.Mathematically,asinusoidalwaveatagivenfrequencycontainstwopiecesofinformation.Thesecanbeexpressedastheamplitudeandphase,aswasdonein( 4 )orasthein-phase,I(t),andquadrature,Q(t),components: 96


4 )with( 4 )revealsthatthelteredmixerPMdescribedabovemeasuresQ(t)=2ratherthan(t).AcompleteIQPMcanbebuiltbyextendingthelteredmixerconcepttoincludetwodemodulations,onewithcos[!t],whichproducesQ(t)=2,andonewithsin[!t],whichproducesI(t)=2.Therelationsin( 4 )-( 4 )canthenbeusedtocompute(t)andA(t).Inadditiontodirectmixing,I(t)andQ(t)canbemeasuredinanumberofotherways.Onetechniqueinvolvessamplingthesignalofinterestwithasamplingfrequencyequaltofourtimesthecarrierfrequencyofthesignal[ 54 ].Eachsetoffourdatapointscanbemanipulatedtomeasurethephaseatarateofone-halfthecarrierfrequency.Anotherapproachistouseintegraltransformsofthetimeseriesdatatoextractthephase[ 55 ].OnetypeofPMthatisdistinctfromtheIQtypeisthecounter/timerPM[ 56 ].Inthistechnique,thenumberofzero-crossingsinatimeintervalTiscounted,providingacrudeestimateofthesignalfrequency.Thisestimateisthencorrectedbymeasuringtheadditionaltimebetweenrstandlastzero-crossingsandthetime-intervalboundaries.AcombinationofthesetwomeasurementsgivesanestimateofthephaseaccumulatedduringtheintervalT.ThisapproachisusedinfrequencycounterssuchastheonesusedtomeasurethebeatnotestabilitiesdiscussedinSection 4.3.2 97


57 ]gives3cycles=p 4 ),and,fordigitalsystems,digitizationandquantizationeects.IthasbeenshownthatcommercialdigitalradioreceiversarecapableofmeetingtheLISAphaseaccuracyrequirementsforlow-noisesignalsatxedfrequencies[ 59 ].TheinputsignalstothePMinLISAdierfromthoseforacommercialradioreceiverintwoimportantways:largeshiftsofthecarrierfrequency,andlargeintrinsicphasenoiseonthesignal.TherelativemotionbetweentheSCwillcausetheone-wayDopplershiftstovarybyupto30MHzoverthecourseofayear.IfthefrequenciesoftheSClasersareheldxed,thebeatfrequencieswillalsovaryoverarangeof30MHz.Thisrangecanbereducedbyperiodicallyadjustingthelaserfrequenciesduringthecourseoftheyear.Oneproposedfrequencyplanwillkeepthebeatnotesintherange2MHz20MHz.OfcomparablesizetotheDopplershiftsarethedriftsintheopticalreferencecavitiesdiscussedinSection 4.3.2 .ThesedriftswillbemorediculttomodelandmustbetakenintoaccountwhenspecifyingthePMrangerequirements.Inadditiontothefrequencydrifts,whichcanbeconsideredasnoiseinthebeatnotebelowtheLISAband,thereisalsoalargeamountofphasenoiseintheLISAmeasurementband,asevidencedbyFigure 4-4 .ThislargephasenoiseposesaproblemtotheIQphasemeter.Asshownin( 4 )and( 4 ),I(t)andQ(t)areperiodicin(t).ThereforetheIQphasemeasurementin( 4 )isameasurementof(t)modulo2andonlygives(t)ifj(t)jrad.Inotherwords,thermsphasenoisemustbelessthanhalfacycleinthemeasurementbandorthephasenoisemeasurementwillwrap.TheLISAlaserphasenoiserequirement(Figure 4-4 )correspondstoanrmsphasenoiseofgreater 98


57 58 ].AschematicofsuchasystemisshowninFigure 4-6 below. Figure4-6. SchematicofaIQphasemeterwithfeedback Theinputsignalisasinusoidwithfrequencyi,amplitudeAi(t),andphasei(t).Thefrequencyisassumedtobexedandanyfrequencynoiseisconvertedtophasenoiseusing( 3 ).Theinputsignalisdemodulatedwithtwosignalsfromalocaloscillator(LO),acosineandasinewithmodelphase,m(t).Thesinetermislteredbyalow-passlterwithatransferfunctionG(s)andscaledby2toformthesignalI(t), 2(cos[i(t)m(t)]cos[4it+i(t)+m(t)]);(4) 99

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2(sin[i(t)m(t)]+sin[4it+i(t)+m(t)]):(4)Withaproperlydesignedlter,thehigh-frequencytermsin( 4 )and( 4 )canbeeliminatedwhileretainingtherstterms.Theadditionalscalingbyafactorof2producesthestandarddenitionsofI(t)andQ(t).TheI(t)andQ(t)signalscanbeusedtocomputetheresidualphase,denedasr(t)i(t)m(t),andtheoutputamplitude,Ao(t),using( 4 )and( 4 ).Ifr(t)issmallenoughtolinearizetheequationswithoutintroducingunacceptableerrors,therelationsbecome 100

PAGE 101

4 )doesnotcontainanyinformationofinterest.Consequently,inthephasereconstructiononlycorr(t)isincludedintheintegration 4-6 isshowninFigure 4-7 .Tobuildthemodel,itisassumedthatjr(t)j1,sothatQ(t)Ai(t)r(t)=2.ThemixingandlteringprocesscanthenbereplacedbyasubtractionofphasesfollowedbyalteringbyG(s). Figure4-7. LaplacedomainmodelofthesysteminFigure 4-6 101

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4-7 ,thefollowingrelationshipscanbederivedbetweenthephasevariables. 1+H0(s)G0(s);(4) 1+H0(s)G0(s);(4) 1+H0(s)G0(s);(4)whereG0(s)(Ai(t)=2)G(s)andH0(s)(1=s)H(s).ThePMerror,denedasthedierencebetweentheoutputphaseandtheinputphase, 1+H0(s)G0(s):(4)Givenaninputphasenoisespectrum,suchasthelasernoiseinFigure 4-4 ,therelationsin( 4 )and( 4 )canbeusedtodesigntheltersG(s)andH(s)sothatjr(t)j<1=2cycleandthephaseaccuracyrequirementsaremet.ThechieffunctionofG(s)istoeliminatetheeectofthesecondtermsin( 4 )and( 4 ).Infrequencyspace,thesetermswillbeapeakscenteredatafrequencyof2iwithlinewidthsrelatedtotheinputphasenoisespectrum,ei(f).Ingeneral,thefrequency2iwillbefarawayfromthemeasurementband(>10MHzvs.1mHzforLISA),sothedirecteectofthepeakisnotofmuchconcern.Theonlyrequirementisthatthepeakbereducedtomuchlessthan1=2cyclesothattheresidualphasewillnotwrap.Thepicturechangessomewhatifthesignalsin( 4 )and( 4 )aresampledataratelowerthan2(ordown-sampledinthecaseofadigitalPM).This 102

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1+G(f)H(f)ei(f)df0:5cycle;(4)whereLFisthelow-frequencylimitofthemeasurementband.Atlowfrequencies,thegainofH(f)mustincreaseatleastasfastastheinputphasenoise.ForthephasenoisesinFigure 4-4 ,thiscorrespondstoaslope,asoftware-basedPMwasbuiltusingMATLAB'sSIMULINKenvironment.ThePMwasdesignedtoanalyzetimeseriesdatafromalaserbeatnotedemodulatedtoapproximately10kHzandsampledatarateof80kHz.BuildingthePMinSIMULINKallowedforexibilityinthedesignoflters A.2.1 .4 103

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4-6 .ThelterG(s)isa16-tap(N=16)FIRlterwithapassbandof10kHzandastopbandof18kHz.ItwasdesignedinMATLABusingtheequirippledesigntechniquewhichspeciesthattherippleinthepassbandmatchthatinthestopband.ABodeplotofG(s)isshowninFigure 4-8 BodeplotofG(s)forthesoftwarePM ThefeedbacklterH(s)isdesignedtohaveanf1responsefor200Hzf20kHzandanf2responseelsewhere,includingthe1=sfromthefrequencytophaseconversionintheLO.ThiscanbeaccomplishedwithtwopolesatDC(oneintheLO)andapole-zeropairwiththezeroat200Hzandthepoleat20kHz, 4-9 .Theunity-gainfrequencyofthetrackingloopinthesoftwarePMisapproximately2kHz. 104

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BodeplotofH(s)forthesoftwarePM 4 )-( 4 )withG(s)andH(s)asspeciedinFigure 4-8 andFigure 4-9 ,thespectraldensitieser(f),fm(f),eo(f),andee(f)canbepredicted.ThesepredictionsareshowninFigure 4-10 .Totestthesepredictions,thesimulationwasrunandi(t),r(t),m(t),o(t),ande(t)wererecordedfora100ssimulationtime.Toreducethesizeofthegenerateddata,thesignalsweredown-sampledtoa10kHzdatarateusingacascaded-integrator-comb(CIC)decimationlter 4-11 A.4.5 105

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4-10 andFigure 4-11 revealsthat,withthenotableexceptionoftheee(f),theobservedbehaviormatchesthatoftheexpectedbehavior.Thefactthatforfrequenciesbelow20Hz,ee(f)actuallyexceedser(f)indicatesthattheproblemmaylieinthereconstructionofthemodelphase.Theintegrationofthefrequencycorrectioncorr(t)usedtoformm(t)isnotexactlythesameastheintegrationthattakesplaceinsidetheLO.InsidetheLO,theintegratorusesamodulo-1cycleaccumulatortocomputethephase,sincethephaseisonlyusedtocomputesinusoidswithvaluesthatrepeatevery2.Furthermore,theaccumulatorinsidetheLOincludestheosetfrequencyoff.Foridealarithmeticoperators,thesedierencesdonotmatter,butitispossiblethatthenumericalerrorsinthetwocasesdier.ThisseemsunlikelyinthecaseofthesoftwarePMsinceSIMULINKutilizesdouble-precisionoating-pointarithmetic. ExpectedbehaviorofthesoftwarePMforsignalwithinputphasenoisewithlinearspectraldensity(1cycle=p 106

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ObservedbehaviorofthesoftwarePM ThesoftwarePMwassuccessfullyusedforanumberofsimulatorexperiments,includinginvestigationsofTDI[ 51 ]andarm-locking[ 52 ].Theseexperimentsweremainlyproof-of-principlemeasurements,andtheexcessnoiseoorinFigure 4-11 wasnotmuchofaconcern.OneissuethatdidarisewerethesizeofthedatasetsrequiredtoreachtheLISAband.Asignaldigitizedwith16-bitresolutionatafrequencyof80kHzproducesdataatarateof156kB/s.FortheTDImeasurements,whichrequiredtwosignalsrecordedforseveralminutes,thedatalesreached100MB.ThiscreatedissueswithdatastorageaswellasthelengthoftimeittooktoprocessthedatathroughthesoftwarePM.Itislikelythatwithfurtherwork,theproblemswiththesoftwarePMcouldhavebeenaddressed.However,oncetheinitialsuccessofthePMconceptwasveried,thefocuswasshiftedtobuildingareal-timehardwarePM.ThisisthetopicofthenextSection. 107

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4.4 .ThedesignofthehardwarePM,showninFigure 4-12 ,issimilartothedesigndescribedin 4.5.3 .TotakeadvantageofthedierentarchitecturespresentinthePenteksystem,thePMfunctionisbrokenintotwoparts.Thefront-end,implementedonthemodel6256daughterboard,trackstheincomingsignalandgeneratesthesignalsI(t),Q(t),andcorr(t).Theback-end,implementedonthemodel4205carrierboard,usesthesesignalstoreconstructtheinputphaseandperformfurtherprocessing. Figure4-12. Schematicofthereal-timehardwarePM 4-13 .TheinputtotheDDSisthephaseincrementregister,afractionalnumberequaltothefractionofcyclestoadvanceperclockperiod.Thiscorrespondstothemodelfrequency,m.Forexample,aphaseincrementof0:1witha100MHzclockfrequencywouldcorrespondtom=10MHz.ThefrequencyresolutionoftheDDSis 108

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Figure4-13. SchematicofaDirectDigitalSynthesizer(DDS) Thephaseincrementisusedastheinputtoanaccumulator,whichperformsarunningsumofthephaseincrementvalueateachrisingclockedge.TheaccumulatorvariableisalsoU32.32andwrapstozeroonoverow.ThevalueoftheaccumulatorcorrespondstothephaseoftheDDSincycles.Theaccumulatoroutputissliced(re-quantizedtoalowerbitresolution)toaU10.10andusedtofeedtheaddressbitsoftwolook-uptables(LUTs).ThepurposeofthesliceristoreducetheamountofmemoryneededfortheLUTswhilestillpreservingthefrequencyresolutionofalargerphaseincrementwidth.ThepricepaidisanincreaseintheamplitudesofspuriousharmonicsintheoutputoftheLUT[ 60 ].ForthepurposesofthePM,a10-bitaddressdepthintheLUTresultsinsucientlylowharmonics.EachLUTcontainsonecycleofsineorcosinewaveformsinS14.13format.Theinputsignalsfromthemodel6256ADCs,expressedasS14.14,aremultipliedwiththeDDSoutputs.Thesesignalsarethenlteredbya2-stageCIClterwithadecimationrateof128.CIClters,discussedindetailinSection A.4.5 ,areanecienttypeofmulti-ratedigitalltersoftenusedtoachievelargesample-ratechangeswithminimalaliasing.Section A.4.5 derivesthemagnituderesponseforaCICdecimatorwith A.3.1 109

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4-14 containsaplotofjG(f)jforthehardwarePMaroundtherstaliasingband,centeredatfs=128=781:25kHz.Theamplitudeofthelterstaysbelow107forabandofhalf-width250Hzaroundthecentralnull.Nomorethan107ofthephasenoiseathigherfrequencieswillbealiasedintotheband0Hzf250HzbytheCIClter. Figure4-14. MagnituderesponseoftheCICdecimationlterinthehardwarePMneartherstaliasingbandat781:25kHz Theotherconcernforthemagnituderesponseofthedecimationlteristhepassbandatness.Figure 4-15 containsaplotof1jG(f)jnearDC.Ascanbeseenfromthegure,thepassbandatnessisbetterthan107upto135Hz.Thisensuresthatanycouplingbetweenphasenoisefrequencyandmeasuredphasenoiseamplitudewillbeminimal.ThephaseoftheCIClterislinear,withanequivalentgroupdelayof1:3s.AfterexitingtheCIClter,thedataarescaledbyafactoroftwoandformedintoS16.16wordscorrespondingtothestandarddenitionsofI(t)andQ(t)givenin( 4 )and( 4 ).ThehardwarePMusestheQ(t)signalratherthan(t)astheerrorsignalforthetrackingloopdrivingtheDDS.Thisisdonetoavoidtheneedtoperformdivisionorarctangentoperations,bothofwhichareproblematicforxed-pointsystems.Thedownsideofthisapproachisthatthetracking-loopgainscaleswiththesignalamplitude. 110

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PassbandatnessoftheCIClterinthehardwarePM Thisisnotaseriousproblemsolongasthegaininthetrackingloopiseasilyadjustableandthesignalamplitudesremainrelativelyconstantduringthemeasurementtime.Thefeedbacklterconsistsoftwoparallelpaths,aproportionalpathandanintegralpath.Theintegralpathconsistsofanaccumulatorthatperformsarunningsumoftheerrorsignal.TherunningsumoperationisrelatedtoatrueintegralbyT,theclockperiodoftheaccumulator 111

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4-16 showsaplotofjH(f)jcomputedusing( 4 )forH0=0:0005.AlsoshownisanobservedvalueforjH(f)jmadefrom5soflaserbeatnotedata.TocomputejH(f)j,themeasuredmodelphasespectrumwasdividedbythemeasuredresidualphasespectrum.From( 4 )and( 4 )itisclearthat~m=~r=jH(f)j. FeedbacklterforhardwarePMtrackingloop.Predictedresultswerecomputedfrom( 4 ),observedresultswerecomputedasaratioofmodelphasetoresidualphaseduringa5sdatarun. TheoutputsofthePMfrontendareI(t),Q(t),andcorr(t)signals(IQ).Theseoutputsareeithertransmittedasisatarateof781:25kHzoraredown-sampledinasecond2-stageCIClterto97:65625kHz(anadditionaldivisionby8).Thisreducesthe107anti-aliasingbandto30Hzandthe107atpassbandto17Hz.AsshowninFigure 4-5 ,themodel6256hastwoFPGAs,eachofwhichisassociatedwithtwoADCsandoneVIMinterface.Consequently,eachFPGAcontainstwoPM 112

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4-17 andtransferredacrosstheVIMinterfacetotheBIFOonthemodel4205carrierboard.Duetothepackingscheme,theaverageclockrateontheVIMisfourtimesthesamplingrateofthePMdata. Figure4-17. PackingformatforPMdatatransferredovertheVIMinterface Thecurrentfront-enddesignoccupiesapproximately10%oftheresourcesonthemodel6256FPGA(totalforbothchannels).ThereisroomtoaddseveraladditionalchannelstoallowforPMtrackingofmultipletonesforclocktransfer,datatransmission,etc.onthesameinputsignal.DoingsowouldrequireamodicationofthepackingformatinFigure 4-17 .TheprimarybottleneckisthespeedoftheVIMandtheprocessinginthePMback-end. 113

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4-2 .Thearctangentmodeisthemostaccurate,butisalsothemostcomputationallyintensive.Thenoarctangentmodeutilizestheapproximationtanxxforx1.Themodel-onlymode,thesimplestpossiblereconstructionmode,reliesonthelargegainofthetrackingloopinthefront-end.Theerrorinthemodel-onlymodeistheresidualphase,r=atan(Q=I). Table4-2. ReconstructionalgorithmsforthehardwarePM.RistheoveralldecimationratebetweentheDDSclockrateandtheIQdatarate. Mode Algorithm arctangent 114

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4-18 containsaspectrumoftheoriginalsignalsandofthreeresidualsfromthepairwisesubtractions.Thelowestresidual,labeledsamewasobtainedbysubtractingpairsofsignalsthatshareaVIMinterface(channel1/2andchannel3/4).TheresidualsbetweenpairsondierentVIMinterfaces(channel1/3,channel1/4,channel2/3,andchannel2/4)werelargerbyafactorof10athighfrequencies(curvelabeleddiffinFigure 4-18 ).Onepossibilityforthehighernoisewouldbeatime-lagbetweenthemeasuredphasesignalfromeachchannel.Theresidualsignalwithatimedelayis 4-18 ,theestimatewast6s.Thesourceofsuchalargetisnotreadilyapparent.It 115

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4-18 weresampledat98kHz,correspondingtoasampleperiodof10s.Theestimateddelaytcorrespondsto0:6samples.Shiftingthedatabylessthanasampleperiodcanbeaccomplishedusingfractionaldelayltering,discussedin A.4.6 .ThecurvelabeledshiftinFigure 4-18 wasobtainedbyshiftingthedatafromchannel3by0:6samplesusinga51-pointfractional-delaylterwithaLagrangewindow.Ascanbeseen,theshiftreducedtheresidualnoisetonearlythelevelofsame.ThecurveslabeledUdiginFigure 4-18 representanestimateofthedigitizationnoiselevelpresentinthePM.ThesourceofthedigitizationnoiseistheniteprecisionoftheIQdataproducedbythePMfront-end.Section A.3.4 derivesaformula( A )thatestimatesthelinearspectraldensityofdigitizationnoiseforaspecicsamplingrateandbitresolution.ApplyingthisformulafortheIQdatagives 3 ),producinga1=fnoisespectrum.TheIandQtermswillremainat.Thetotaldigitizationnoiseforasinglechannelisgivenby 4-18 .Theblackcurvecorrespondsto 116

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4-18 ),therequirementscouldbemet.Convertingtoa42-bitaccumulatorwillcausethefront-endtooccupyslightlymoreFPGAresourcesaswellasrequireare-designoftheVIMdatapackingschemeinFigure 4-17 VCOphasenoisemeasuredbyfourchannelsofthehardwarePM 4-2 )wasfocusedintoanopticalberandtransmittedtoasecondopticaltable.Atthesecondopticaltable,thelightfromtheberwassuperimposedonaphotodiodewiththelightfromalaserlockedtoahydroxide-bondedZerodurcavity.Thelightcomingfromtheberhadapowerof5Wwhilethelocallighthadapowerof56W. 117

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4-19 containsthelinearspectraldensityoftheinputphasenoiseaswellastheinterchannelerrorandthedigitizationnoiseforboth32-bitand42-bitDDS.Atfrequenciesbelow100Hz,theinterchannelerrorliesonthe32-bitdigitizationnoise,indicatingthatitisthelimitingfactorinthemeasurement. LinearspectraldensityofphasenoiseinlaserbeatnoteasmeasuredbythehardwarePM AnotherpotentialsourceofnoiseinthePMisshotnoise,thequantum-mechanicalvacuumuctuationsofthelightsources.Figure 4-20 showsaschematicrepresentationoftheamplitudespectrumoftheelectriceldsoftwolightsources.Eachsourcehasalargepeakatanfrequency!iwithanamplitudei,wherei=1;2.Thesepeaksrepresentthecoherentlaserlight,whichhasaphasei. 118

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hc(4)whereNiisthenumberofphotonsinthemeasurement,Piisthesignalpower,isthecarrierwavelength,Tisthetotalmeasurementtime,hisPlanck'sconstant,andcisthespeedoflight. Figure4-20. Qualitativeamplitudespectrumofinterferringbeamswithshotnoise IfthetwolightsourcesinFigure 4-20 aresuperimposed,thetotalintensitywillincludeabeatbetweenthetwocarriersat!12!1!2withanamplitudep 119

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4-19 canbeestimatedusing( 4 ),( 4 )and( 4 )withP1=5WandP2=56Wandameasurementtimeof85s.Theresultingestimateis~USN2:5109cycles=p 4-19 A.2 120

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4-21 showsthetime-seriesofanentangledphasetestusingtwoVCOsandafunctiongeneratorasoscillators.Thebeatfrequencieswere4:86MHz,5:78MHz,and10:64MHzandtheamplitudeswerescaledtomatchthe4dBmfull-scaleinputsofthePM.ThefunctiongeneratorhaslowerintrinsicphasenoisethantheVCOs,consequentlythephasenoiseinthethreesignalsisdominatedbytheVCOnoise.Thesignalsappeartobecorrelated,afactwhichisconrmedbytheplotof123(t),whichappearstobenearlyzeroonthescaleofupperpanelofthegure.Thelowerpanelshowsablow-upof123(t)withandwithoutadditionaltimeshiftingin23(t)(seebelow).Figure 4-22 containsthecorrespondingspectraldensitiesofthethreebeatsignalsandthenullcombination.Aswiththesingle-signaltestsinSection ,thereappearedtobeaslightdelaybetweenthesignalrecordedonchannel3andthesignalsrecorded 121

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LinearlydetrendedtimeseriesforanentangledphasetestusingtwoVCOsandafunctiongenerator.Bottompanelshowsclose-upofresidualnoiseinthenullcombination. onchannel1andchannel2.Whenadelayof1:95swasremovedusinga51-pointLagrange-windowedfractionaldelaylter,theresidualnoiseinthenullcombination(cyancurve)wasreducedtothelevelofthedigitizationnoisegivenbyp 4 ). LinearspectraldensitiesofanentangledphasetestusingtwoVCOsandafunctiongenerator 122

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4-2 )wasfocusedintoanopticalberandtransferredtoasecondopticaltable.Thelightexitingtheberwascombinedwithathirdbeamfromanadditionalcavity-stabilizedlaser.Theresultingphotodiodesignalcontainedthreebeatnotes,onefromeachpairoflasers.ThefrequenciesandpowersofeachsignalarelistedinTable 4-3 Table4-3. Beatnotefrequenciesandamplitudesforopticalentangled-phasemeasurement LaserPair Frequency(MHz) Power(dBm) O-S 28.7 -23.3 O-H 113.4 -21.3 H-S 142.1 -17.3 O:Zerodurcavitywithoptically-contactedmirrors H:Zerodurcavitywithhydroxide-bondedmirrors S:Siliconcarbidecavitywithoptically-contactedmirrors BeforethebeatsignalsinTable 4-3 couldbereadintothePM,theyhadtobeconditionedsothattheirfrequencieswere25MHzandtheiramplitudeswereapproximatelyequalto4dBm,thefull-scaleinputofthePM.ThiswasaccomplishedusingthesignalconditioningarrangementinFigure 4-23 .ThePDsignalwasrstampliedina10RFamplierandthensplitintotwopartsusinga50-50RFsplitter.EachofwhichwasdemodulatedwithaLOinordertobringthesignalswithinthefrequencyrangeofthePM.OneportionwasdemodulatedusingaLOat22MHzandlteredusingaLPFwithan11MHzcornerfrequency,producinga6:7MHzsignalfromtheO-Sbeat.Thissignalwasampliedinasecond10RFamplierandconnectedtoaPMinputchannel. 123

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AnalogelectronicsusedtopreparebeatsignalsinTable 4-3 forPM ThesecondportionwasdemodulatedbyanotherLOat130MHzandlteredwitha21MHzLPF,placingtheO-Hbeatat16:6MHzandtheH-Sbeatat12:1MHz.EachofthesesignalswasampliedinanadditionalRFamplierandconnectedtoPMinputchannels.ThetwoLOsusedforthedemodulationsaswellasthePMclockwereeachlockedtoaRubidium-stabilized10MHzreferencesignalinordertoreducethecouplingofLOphasenoiseintothemeasurement. Linearly-detrendedphaseforopticalentangled-phasemeasurement 124

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4-24 showsthelinearly-detrendedphaseoutputforthethreechannelsandthenullcombination.ThemostobviousfeatureisthelargeparabolictracksofOSandHS.Thisisaresultofafrequencydrift,discussedin 4.3.2 ,thatispresentbetweentheSiCcavityandtheZerodurcavities.AttothedatainFigure 4-24 givesadriftof248Hz=satthetimeofthemeasurements.NotethatthedriftbetweenthetwoZerodurcavitiesismuchsmaller,ontheorderof1Hz=s,despitethefactthatthetwoZerodurcavitiesarelocatedindierentvacuumchambersonoppositeendsofthelaboratory.Thelargefrequencydriftsdonotposeaproblemfordataanalysis,buttheydolimittheamountoftimeforwhichthePMcanstaylocked(seeSection ).Forthethree-signalmeasurements,thelongestsetsofdatawiththreesimultaneously-lockedsignalslastedaround30s.IndividualmeasurementsusingtheO-Hbeatappearedtobeabletolastindenitely. Quadratically-detrendedphaseforopticalentangled-phasemeasurement Whenthelinearfrequencydriftofthebeatnotesisremoved,theresultisthetimeseriesinFigure 4-25 .Thenoiseofthethreebeatsignalsisapproximatelyofthesameamplitude,withthatofOSbeingslightlylowerthantheothertwo.Thissuggeststhat 125

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4-25 containsablow-upoftheresidualnoiseinthenullcombinationbothwithandwithoutanadditionalshiftinOH(t).Figure 4-26 showsthelinearspectraldensitiesofthethreeindividualbeatsaswellasthenullcombination.Asintheearliermeasurements,atimedelaywaspresentinthethirdPMchannel,whichwasusedtomeasureOH.Themagnitudeofthistimedelaywasestimatedbyplottingthenoisesuppressioninthenullcombinationandmakingatusingtheexpressionin( 4 ).Theresultwasatimedelayof2:18s,showninFigure 4-27 .Tocorrectforthisdelay,theOHdatawasshiftedby2:18susinga51-pointLagrange-windowedfractional-delaylter.TheresultsarethecyancurvesinFigure 4-26 andFigure 4-27 .Theeectoftheshiftissignicant,increasingthenoisesuppressionbyafactorofmorethan100near10Hz. Linearspectraldensityforopticalentangled-phasemeasurement 126

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4-26 istheexpecteddigitizationnoise,givenbyp 4 ).UnliketheVCO-measurementdata,theresidualphase-noiseinthenullcombinationdoesnotreachthedigitizationnoise.Itinsteadfollowsa1=fslopewithanamplitudearound4:4timeslargerthanUdig.Onepossiblesourceforthisnoisewouldberelativephasenoisebetweenthetwooscillatorsusedtodemodulatethebeatsignals. Noisesuppressioninnullcombinationforopticalentangled-phasemeasurement 4-28 showsahistogramfor5sofresidualphaseinthehardwarePMforabeatnotebetween 127

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2#;(4)whereAistheamplitudeofthet,ristheresidualphase,isthemean,andisthestandarddeviation.Usingthisdistribution,theprobabilityofacycleslipcanbecomputedusingthecomplimentaryerrorfunction ;(4) erfc(x)2 4-28 ,thebesttGaussianhasanamplitudeof17%,ameanofzero,andastandarddeviationof7:4millicycles.Forsuchasmallstandarddeviation,theprobabilityofacycleslipiseectivelyzero. HistogramofresidualphaseforlaserbeatnoteinhardwarePM Inadditiontobeingcausedbyhigh-frequencyphasenoise,cycleslipscanalsobecausedbylargefrequencydriftsintheinputsignals,suchasthoseontheSiCbeatnotes.Atlowfrequencies,thegainofthetrackingloopinthePMfront-endincreasesasf2,withonepoweroffcomingfromtheintegratorinthefeedbacklterandthesecondpowercomingfromtheimplicitintegrationintheDDS.Alinearfrequencydriftcorrespondstoaquadraticincreaseofthephasewithtime,whichinturncorresponds 128

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Nchanbfs;(4)whereMistheamountofmemoryinbytes,Nchanisthenumberofdelaychannels,bisthewidthofthesamplewordsinbytes,andfsisthesamplingrate.TheEPDunitsweredevelopedinthreegenerations,summarizedinTable 4-4 .TheinitialprototypewasbuiltusingaDAP-5216aDSPcardbyMicrostar,Inc.TheDAP-5216acontains16-bitADCsand16-bitDACsclockedat200kHzand64MBofSDRAM.Thisallowedsignalsinthe10kHzregimetobedelayedbytensofseconds.This1st-generationEPDunitwasusedforseveralearlysimulatorexperiments[ 61 62 ]. Table4-4. ProgressionofEPDunits EPDUnit Hardware SamplingFrequency #ofChan. Max.Delay Microstar 2 Pentek 4 Pentek 4 4.4 anddiagrammedinFigure 4-5 .TheinputdataisdigitizedbytheADCsonthemodel6256downconverter.TheADCdatafromeachpairofchannelsispackedintoa32-bitwordandtransferredviatheVIMinterfacetotheBIFOonthemodel4205carrierboard.The4205readsthedataotheBIFOinblocksof1024wordsandplacesitinamemorybuerintheSDRAM.Theoutputofthememorybueristransferredviathe 130

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51 63 ]andinvestigationsofarm-lockingwithopticalsignals[ 64 ].Thedownsideofthe2nd-generationsystemisthat,despitehaving1GBofSDRAM,itisunabletodelaysignalsformorethan2:5s.Thisisbecausethedataratesforthehighsamplingfrequenciesaresolarge.ThisproblemisespeciallyannoyingbecausethereisverylittleinformationofinterestintheLISAsignalsathighfrequencies. 131

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SchematicoftheNCOusedinthe3rd-generationEPDunit The3rd-generationEPDunitaddressthisproblembyincorporatingaPMandannumerically-controlledoscillator(NCO)intothedelayprocess. .The100MHzsamplingfrequenciesallowsforsignalswithfrequenciesofupto30MHz.TheIQdataforeachchannelispackedusingtheformatinFigure 4-17 andtransferredacrosstheVIMatarateof97:65625kHz.The4205readsthisdataooftheBIFOinblocksof1024andunpacksitintoseparatestreamsforeachchannel.ThesestreamsarestoredinamemorybueronSDRAMforthespecieddelaytime.ThedelayeddataisrepackedandtransferredacrosstheVIMtotheFPGAonthe6228.The6228unpacksthedataandfeedstheIQdatatoaNCO,shownschematicallyinFigure 4-29 .TheNCOconvertstheIQdatafromthePMintosinusoidalsignalswithanosetfrequencyoffatasamplingrateof100MHz.TheoutputoftheNCOsisfedintotheDACs,reproducingthesignals.BydecimatingtheIQdatabeforestorage,the3rd-generationEPDunitgreatlyincreasesthemaximumdelaytimewhilesimultaneouslyincreasingthemaximumcarrierfrequency.ThepricepaidisinthebandwidthoftheEPDunit:anycomponentsofthesignalabove50kHzwillnotbereproduced.Thisisabovethebandwidthofmostoftherelevantsignalswiththeexceptionofthemodulationtonesandpossiblytheactive 132

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4-30 .Tochecktheeectivenessofthedelayunit,thedelayedsignalwasshiftedintimebyvariousamountsusingacombinationofsimpleintegersamplingpointshiftingandfractionaldelaylteringandthensubtractedfromtheinputsignal.Thedelaytimewasoptimizedbyminimizingthepowerspectrumoftheshifteddierence.Forthisdataset,adelaytimeof=2:039663sproducedtheminimalerror. 133

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DetrendedphaseofVCOsignalinEPDtest.Expectedtimedelay:=2s. ThelinearspectraldensityoftheinputsignalsandtheshiftedandsubtractedcombinationisshowninFigure 4-31 .Forfrequenciesbelow100Hz,thenoiseinthesubtractedcombinationmatchesthedigitizationnoise,computedasp 4 ).ThisindicatesthattheEPDunitdoesnotaddanyadditionalnoisetothesignalbeyondthataddedbythePM. LinearspectraldensitiesinEPDtestwithVCOsignals.Optimaltimedelay:=2:039663s 134

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3.5.4 ).ArmlockingisafunctionoftheIMSasawhole,involvingsubcomponentssuchasphasemeters(PMs),laserpre-stabilizationsystems,andphase-lockloops(PLLs).Todate,arm-lockinghasbeenstudiedanalytically[ 43 45 ],throughtime-domainsimulations[ 42 43 65 ],andinseveralhardware-analogexperiments[ 48 61 64 66 ].Theparticularadvantageofthehardwareexperimentsisthat,inforcingonetoactuallybuildaworkingsystem,theycanexposeeectsthathavenotbeenincludedintheanalyticornumericalmodels.TheUFLISAinterferometrysimulatorisideallysuitedtostudyingarm-lockinginaLISA-likeenvironment.Itistheonlysysteminexistencethatcanprovidebothrealisticlasernoiseandrealisticdelaytimes.Forstudyingarm-locking,thelargedelaytimesareessential,becausetheysetthefrequencyscaleforthecontroller.Inparallelwiththedevelopmentoftheinterferometrysimulator(Chapter4),theauthorhasdevelopedaseriesofhardwaremodelsofsingle-armarm-lockingbasedontheEPDtechnique.Theinitialmodel,describedinSection 5.2 ,wasapurelyelectronicmodelusingaVCOinplaceofalaserbeatnote.Thisprovidedaproof-of-principlefortheEPDtechnique.Thenextiteration,describedinSection 5.3 ,incorporatedimprovedelectronicstoallowlockingofapre-stabilizedlaserbeatnotetoa1msdelay.ThesubsequentdevelopmentofthehardwarePM,describedinSection 4.5.5 ,allowedforachangeintheexperimentaltopologythatproducedtheimprovedopticalsystemdescribedinSection 5.4 .ThissystemiscapableofgeneratingLISA-likearm-lockingerrorsignalswithdelaysof1sormore.Unfortunately,atechnicalissuewiththeimplementationofthearm-lockinglterforthissystempreventedthesystemfrombeinglocked. 136

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5.2.1MethodTherstEPD-basedarm-lockingexperiment,describedindetailbyThorpe&Mueller[ 61 ],wasapurelyelectronicmodel.Theexperimentalapparatus,showninFigure 5-1 ,centersaroundaVCOwithanominalfrequencyof25kHz.TheVCOsignalissplitintotwoportions,oneofwhichisdelayedinanEPDunit.Fortheseexperiments,theEPDunitwastherst-generationversiondescribedinSection 4.6 ,withasamplingfrequencyof200kHz.Thedelayedandpromptsignalsweremixedinananalogmixerandthemixeroutputwaslow-passlteredbyasinglepoleat300Hz,generatinganerrorsignaloftheform cycle[(t)(t)](5)where(t)isthephaseoftheVCOandisthedelaytimeoftheEPDunit.Thiserrorsignalhasthesameformasthesingle-armarm-lockingerrorsignalgivenin( 3 ).Fortheseexperiments,thedelaytimewassetto500ms,sothatthenullsintheinterferometerresponseoccurredatfnn2Hz: Experimentalsetupforelectronicarm-lockingexperiment ThecontrollerfortheelectronicmodelwasimplementedusingaPC-basedDSPsystemfromNationalInstruments.Theerrorsignalwasdigitizedat1kHzwith16-bitresolutionandstreamedintoNationalInstrumentsLabVIEWsoftware.Thelterwasan 137

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5-2 alongwithaBodeplotofthearm-lockingsystem(interferometerplus1=sactuator).Anoverallgainof200hasbeenremovedfromthecontrollertransferfunctiontoallowthetwocurvestooverlapintheplot.NotefromthelowerpanelofFigure 5-2 thatthecontrollerprovidesaphaseadvanceinthevicinityoftheinterferometernulls. Systemandcontrollertransferfunctionsforelectronicarm-lockingexperiment 4.5 .Figure 5-3 showsthelinearspectraldensity(LSD)ofthearm-locked A.4 .FordetailsonthebilineartransformdesignmethodforIIRlters,seeSection A.4.3.2 138

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Linearspectraldensityofarm-lockedVCOsignal Ideally,theunlockedspectrumwouldbeincludedinFigure 5-3 aswell.However,thermsphasenoiseintheunlockedVCOexceeds1cycle,preventingthecrudemixerPMfromworkingproperly.Asanalternative,theclosed-loopnoisesuppressionwasmeasuredin-loopbyinjectingasinusoidalsignalintotheVCOcontrolinputandobservingthecorrespondingsignalintheerrorsignaloutput.TheresultsofthismeasurementareshowninFigure 5-4 ,alongwithatmadeusingtheknownopen-looptransferfunctionsofthecomponents.Thetparameterswereandanoverallgainfactorandthebesttvalueswere=500:9msandH0G0=200. 5-4 indicatethattheEPD-basedsystemisareasonablehardwareanalogueforLISA.Theprimarylimitationsofthissystemistheshortdelaytime,thelowcontrollerbandwidth,andthelimiteddynamicrangeofthemixerPMs.Thelattertwolimitationsprecluded 139

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Closed-loopnoisesuppressionforelectronicarm-lockingexperiment 5.3.1MethodAftertheinitialsuccessoftheelectronicarm-lockingmodel(Section 5.2 ),thenexteortwasbuildinganarm-lockingsystemincorporatingopticalsignals[ 64 ].TheopticalcomponentsofthesystemwerecomposedfromtheopticalbenchshowninFigure 4-2 .Forthearm-lockingexperiments,theseopticswerearrangedasshowninFigure 5-5 below.Laser1(L1)isstabilizedtoanopticalcavityconsistingofaZerodurspacerwithoptically-contactedmirrors.Laser2(L2)isphase-lockedtoL1withafrequencyosetprovidedbyaVCOwithanominalfrequencyof25MHz.AsdiscussedinSection ,thisisoneapproachthatcouldbeutilizedonLISAtocombinearm-lockingwithpre-stabilization.Theround-tripdelayoftheLISAarmismodeledusingtheEPDtechnique(Section 4.2 ).ThereferencelaserisLaser0(L0),whichislockedtotheSiCcavity.Asmentioned 140

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Experimentalarrangementfortheinitialopticalarm-lockingexperiments inSection 4.3.2 ,theZerodur-SiCbeatnote,denotedasS20(t),exhibitsalargelineardrift.Atthetimeoftheseexperiments,theslopeofthisdriftwas200Hz=s.TomodeltheLISAarm,S20(t)isrstampliedina10RFamplierandsplitintotwoequalpartsusinga50-50RFsplitter.ThisproducestwoidenticalcopiesofS20(t)withapowerofapproximately2:8dBm.OneofthesignalsisdelayedintheEPDunit,representingtheround-triplighttraveltimeinthearm.Thisexperimentutilizedthe2nd-generationEPDunit,describedinSection 4.6 .ThesecondcopyofS20(t)isampliedbyanadditional10amplierandfedintotheLOportofanRFmixer.Themixeroutputisthenlteredbyasinglepolewithacornerfrequencyof300kHztoremovethehigherharmonics.Thelteroutputwasamplied,producinganoverallerrorsignalof cyclemod[(t)(t);1cycle]:(5)Asintheelectronicarm-lockingsystem,thismixer-lterarrangementactsasacrudePMwhichgivesanunambiguousphaseresponseonlywhenj(t)(t)j0:5cycles.Forpre-stabilizedlasernoise,thiswillonlyoccurformeasurementtimesof100msor 141

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Laplace-domainmodelofthesysteminFigure 5-5 less.Consequently,thedelaytimeintheEPDunitwassetto1mssothat( 5 )wouldapproximatethesingle-armtransferfunctionin( 3 ).Thelteredmixeroutputwasusedastheerrorsignalforthearm-lockingcontroller,whichwasimplementedonaNationalInstrumentsDSPboardwithanFPGAprocessor.Theerrorsignalwasdigitizedat200kHzwith16-bitresolutionandlteredusingasecond-orderIIRlterwithtransferfunctionH(s).Thelteroutputisup-convertedina16-bitDACandusedtoadjustthefrequencyoftheVCOinthePLL,completingthearm-lockingloop.Figure 5-6 containsaLaplace-domainmodelofthesysteminFigure 5-5 .TheunlockedphasenoisesofL1,L0,andtheVCOare~p1(s),~p0(s),and~pVCO(s),respectively.AnanalysisofthesysteminFigure 5-6 showsthatthephasenoiseofS20is 1+H(s)s1[1exp(s)]:(5)In( 5 ),itisassumedthatthegaininthePLLislargeenoughtoeectivelyeliminatethefree-runningnoiseofL2.Comparing( 5 )withtheclosed-looptransferfunctionforarm-locking,( 3 )indicatesthatthetwosystemsareidenticalsolongasthebeat-notephasenoise,[~p1(s)~p0(s)]+~pVCO,issimilartotheLISAlaserphasenoise,~p(s). 142

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5-7 containsaBodeplotforthecontrollerwithandwithouttheintegrators.Thecurvewithouttheintegratorsisameasurementmadeusinganetworkanalyzerandincludestheeectsoflatencyinthelter.Thecurvewiththeintegratorsincludedcombinesthetheoreticaltransferfunctionoftheintegratorsandthemeasuredlatencyintheltersystem.Withtheintegratorson,thelterhasaslopeoff2below100Hz.From300Hzto30kHz,thelterhasaslopeoff1=2,providingtherequiredphaseadvanceinthevicinityofthefnfrequencies. Bodeplotofcontrollerforinitialopticalarm-lockingsystem 5-5 wasmadebyreplacingtheL2L0beatnotewithafunctiongenerator.Theresponseoftheinterferometerwasmeasuredbyinjectingsinusoidalphasemodulationatagivenfrequencyusingthefunctiongeneratorandobservingthemixeroutputsignal.TheresultsofthesemeasurementsareshowninFigure 5-8 .Theobserveddatawasusedtottothesingle-armerrorsignal 143

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3 )todeterminetheactualtimedelayandanoverallgainparameter.Thebesttparameterswereadelayof1:065msandagainof8mV=deg. Responseofinterferometertophasemodulation Oncethesystemcharacterizationwascomplete,thefunctiongeneratorsignalwasreplacedbytheL2L0beatnoteandtheerrorsignalwasconnectedtothecontroller.Theeectivenessofthearm-lockingsystemwasevaluatedusinganout-of-loopmeasurementsystemconsistingofabeatnotebetweenL2andL0ataPDseparatefromtheoneusedtogeneratetheerrorsignal.Thestabilityofthebeatnoteforboththelockedandunlockedcaseswasmeasuredusingtwoinstruments:acommercialfrequencycounterandthesoftwarePMdescribedinSection 4.5.4 .Thefrequencycounterallowedforlong-durationmeasurementstoprobelow-frequencieswhilethePMallowedthehigh-frequencyregimetobestudied. 5-9 containsatimeseriesofthelockedandunlockedfrequencynoiserecordedusingthefrequencycounteratarateof0:5sample/s.Theunlockedcaseclearlyexhibitsa 144

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5-10 showsthesamedatawithalineardriftof167Hz/sremovedfromtheunlockeddata. TimeseriesofL2L0beatnoteforlockedandunlockedcases Theresidualunlockedfrequencynoisedriftsoverapproximately1kHzin1000s,withasmallerhigh-frequencycomponent.Onthisscale,thelockedtimeseriesconsistsofaaseriesofatplateausseparatedbydistinctverticalshifts.Theseplateausarearesultofthemod(1cycle)characteroftheerrorsignal( 5 ),whichhaslockpointsseperatedinfrequencyspaceby1=.Excessnoisecancausethearm-lockingsystemtoslipfromonelockpointtoanother,aneventdubbedafringe-slip.CloseexaminationoftheplateausinFigure 5-9 showsthattheirlevelsdierby1=1:065ms939Hz:Betweenthefringe-slips,thelockedfrequencyappearsnearlyconstantonthescaleofFigure 5-9 .Figure 5-11 showsaclose-upofthelockedfrequencydatabetween1000sand2000s.Duringthistimeperiod,thebeatnotefrequencyremainedwithinroughly250mHz. 145

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TimeseriesofL2L0beatnotewithlineartrendremovedfromtheunlockedcase Close-upoflockedcaseinFigure 5-9 from1000sto2000s ThespectraofthelockedandunlockedfrequencynoiseareshowninFigure 5-12 .Forthelockedcase,onlydatabetweenfringe-slipswasincluded.Thelockedfrequency 146

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Spectrumoflockedandunlockedfrequencynoise.Fringe-slipshavebeenremovedfromthelockedfrequencynoise. 4.5.4 .TheL2L0beatsignalfromtheout-of-loopphotodiodewasdemodulatedwithalocaloscillatortoafrequencyof10kHz.Thissignalwasdigitizedatarateof80kHz.TherecordeddatawasthenprocessedoineinthesoftwarePM,producingthephasetimeseriesshowninFigure 5-13 .Thereductioninphasenoiseinthelockedcaseisclear.ThelinearspectraldensitiesofthePMsignalsareshownintheright-handsideofFigure 5-14 .Ontheleft-handsideofFigure 5-14 aretheequivalentphase-noisespectraobtainedfromascalingofthefrequencynoisespectrainFigure 5-12 .Althoughthetwodatasetsdonotoverlap,theyareclearlyconsistentwithoneanother. 147

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Timeseriesofbeatnotephaseforunlockedandlockedcases Phasenoisespectrafortheunlockedandlockedcases.Thespectraontheright-handsidearefromthePMdata.Thespectraontheleft-handsidearethefrequencynoisespectrafromFigure 5-12 scaledtophasenoise. Anestimateoftheclosed-loopnoisesuppressionofthearm-lockingloopcanbemadebydividingthelockedspectraldensitybytheunlockedspectraldensity,asshownin 148

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5-15 .AlsoshowninFigure 5-15 isthetheoreticalclosed-loopsuppressionthatwascalculatedusingthemeasuredopen-looptransferfunctionsoftheinterferometer,controllter,andVCO.Themeasuredclosed-loopsuppressiongenerallyfollowstheshapeofthepredictedtransferfunction,butdeviatesatboththeupperandlowerfrequencies.ThedeviationattheupperfrequenciesislikelyaresultoflimitationsinthesoftwarePM,whichhasalimitedresolutionabove1kHz(seeSection 4.5.4 ).Atlowerfrequencies,thesuppressionappearstohitanoiseooraround40dB. Closed-loopnoisesuppressionforopticalarm-locking 5-15 .ApredictionbasedonthemeasuredtransferfunctionsofthesystemcomponentsestimatesfUG12kHz.Figure 5-16 containsameasurementoftheerrorpointnoise,recordedasthevoltageoutofthemixerinFigure 5-5 ,forthelockedandunlockedcases.Theerror-pointnoiseforthelockedcaseisclearlylowerthantheunlockedcaseatlow-frequencies.Athigherfrequencies,thereisabroad 149

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Error-point(mixeroutput)noiseforlockedandunlockedcases 150

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4.6 )allowsforanimprovementontheopticalmodeldescribedabove.TheopticalcongurationremainsthesameasinFigure 5-5 whiletheelectronicsaremodiedasshowninFigure 5-17 Figure5-17. Modicationofelectronicsforimprovedopticalarm-locking Asintheoriginalexperiment,theL2L0beatnoteisdividedintotwoportions,oneofwhichenterstheEPDunit.TheEPDunitdelaysthesignalbyanamountwhilealsoimpartingaxedfrequencyshiftof.TheoutputoftheEPDunitismixedwiththeoriginalbeatnoteandtheoutputislow-passltered,producingasignalwithafrequencyandaphase20(t)20(t).Thisisadirectanalogueofthebeatsignalforsingle-armlockinginLISA,whichwillhaveanearlyconstantfrequencygivenbytheDopplershiftsandanyosetinthePLLatthefarSC.AsinLISA,thephaseofthemixeroutputsignalcanberead-outwithareal-timePMsetwithanosetfrequencyequalto.Thisprovidesanerrorsignalproportionalto20(t)20(t)solongasthephasedierenceremainsinthelinearrangeofthe 151

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5-17 ,ameasurementofthesystemtransferfunctionwasmade.TheL1L2beatnotewasphase-lockedtoa10MHzLOsignalusingananalogPLL.ThisplacedtheL2L0beatnoteatapproximately98MHz,withanamplitudeof41dBm.TheL2L0beatnotewasdemodulatedwithaxed90MHzLOsignalandampliedusingtwoRFamplierswithamplitudegainsof10each.Theampliedsignalwassplitintothreeequalportions,eachwithanamplitudeofroughly4dB.OneportionofthesignalwasfedintotheEPDunit,whichwassetwithadelayof1sandaDopplershiftof4MHz.ThesecondportionofthesignalwasampliedbyathirdRFamplierandfedintotheLOportofanRFmixer.TheotherportofthemixerwasconnectedtotheoutputoftheEPDunit,whichhadanamplitudeofapproximately10dBm.Themixeroutputwaslteredwitha5MHzcornerfrequencyandamplied,producinga4MHzsignalwithanamplitudeofroughly0dBm.ThissignalcorrespondstotheoutputofthemixerinFigure 5-17 .ThemixeroutputsignalandthethirdportionoftheL2L0beatnotewereeachfedintothehardwarePMdescribedinSection 4.5.5 .Figure 5-18 showstherawtimeseriesofthephasedataforbothsignals.TheL2L0beatnotefollowsaquadratictrend 4.3.2 .Ithappenedthatthesemeasurementsweretakenduringatimesmalldriftsbetweenthecavities,whichallowedthePMtoremainedlockedforlongtimeperiods. 152

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Rawphasetimeseriesofarm-lockingsystemcharacterizationdata.BluecurveisL2L0beatnote,redisinterferometeroutput,S(t). Foraninputwithaquadratictrendinphase,theinterferometeroutput,S(t),willhavealineartrendinphase.Using( 5 ), 5-18 .S(t)doesindeedshowalineartrend,althoughtheslopeis1092Hzratherthan50Hz.PartofthediscrepancycanbetracedtoaroundingerrorinthefrequencyosetregisterofthePM,whichusesaU16.16binaryfractiontorepresentoff.Theroundingerrorfora4MHzsignalisapproximately671Hz.Theadditional370Hztrendisunaccountedfor.Removingthequadratictrendfrom20(t)andthelineartrendfromS(t)yieldsthetimeseriesinFigure 5-19 .Overlongtimeperiods,S(t)isquieterthan20(t)sincevariationswithperiodslongerthan1sarecommontoboththepromptanddelayedsignals.Figure 5-20 showsthelinearspectraldensities~20(f)and~S(f).Clearlyvisiblein~S(f)isaatteningatlowfrequenciesaswellasnullsatmultiplesof1==1Hz.An 153

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5-20 ,asshowninFigure 5-21 Detrendedtimeseriesofarm-lockingsystemcharacterizationdata.Bluecurveisthequadratically-detrendedL2L0beatnote,redisthelinearly-detrendedinterferometeroutput,S(t). Linearspectraldensityofarm-lockingsystemcharacterizationdata 154

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5-21 isathetheoreticalsingle-armtransferfunctiongivenin( 3 ).Themagnitudeisgivenby 5 )forthedatainFigure 5-21 obtainedadelayof=1:039s.Theadditional39msofdelayisconsistentwiththeadditionaldelayobservedinthe3rd-generationEPDexperimentsdescribedin 4.6.2 .Theadditionalroll-ointhemeasuredresponseatlowfrequenciesmayonlybeduetoascarcityofpointsinthespectrainFigure 5-21 .Themeasureddepthoftheinterferometernullsisaectedbythefrequency-resolutionofthespectra.Alongermeasurementtimewouldreducebothoftheseeects. Magnitudeoftransferfunctionforarm-lockingsystem.Fittedsystemhasadelayof=1:039s. 5-17 greatlyreducestherequirementsonthearm-lockinglter.UnlikethesystemdescribedinSection 5.3 ,theerrorsignalwillbepresentregardlessoftheamountofnoiseinthesystem.Consequently, 155

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5.4.1 above,acontrollterwithpolesat1Hz,10Hz,100Hz,and1kHzandzerosat3:163Hz,31:63Hz,316:3Hz,and3:163kHzwasdesignedusingthebilineartransformmethod(Section A.4.3.2 ).Thisproducesatransferfunctionthatapproximatess1=2between1Hzand10kHz.Thelterwasimplementedasatwo-stage,second-order-section,direct-formIIlterintheFPGAon-boardthemodel6256downconverter 5-22 showsatimeseriesoflteredandunlteredfrequencynoiseforaVCOinput.Thisdatawasusedtogenerateameasurementofthemagnitudeoftheltertransferfunctionbycomputingthelinearspectraldensityforeachsignalanddividingthetwospectra.ThisproducesthemeasuredresultshowninFigure 5-23 .Itisclearlyconsistentwiththepredictedresult,indicatingthatthelterisbehavingasexpected. A.4.4.1 .ForanoverviewofthePentekhardware,seeSection 4.4 156

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TimeseriesoflteredandunlteredfrequencynoisefromVCOinput Measuredanddesignedtransferfunctionsofarm-lockingcontrollter AsmentionedinSection 4.4 ,theDACoutputsonthe6228upconverteraretransformer-coupledwitha3dBpointat400kHz.ThereforetheycannotbeusedtogenerateaDCcontrolsignaltopasstoaVCO.Toavoidthisproblem,thelteroutputcanbeusedasafrequencyinputtoanNCOrunningontheFPGAinthe6228.TheNCOcanthenreplacetheVCOastheoscillatorinthePLLbetweenL1andL2inFigure 5-5 .ThisshouldbesimilartothesituationinLISA,wherethePM,controlelectronics,andNCOwillallbepartofacommonavionicssystem. 157

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5.4.1 indicatethatthesystemiscapableofproducingerrorsignalswithLISA-likenoiseandLISA-scaletimedelays.AspotentialPM/ltersystemsaredevelopedatUFandelsewhere,theEPD-basedopticalarm-lockingmodelwillbeavailabletoevaluatethem. 158

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A.2 .Digitalsignalsarealsoquantizedinamplitude,theycanonlytakeonalimitednumberofvalues.TheconsequencesofamplitudequantizationarediscussedinSection A.3 .Section A.4 providesanintroductiontodigitalltering,anextremelyexibleandpowerfultechniqueformanipulatingdigitalsignals.Emphasisisplacedonthetypesofdigitalltersusedindevelopingthesimulatorelectronics. A-1 (a).Bydenitionasacontinuoussignal,thevalueofx(t)isspeciedforallvaluesoft.Thesignalcanbedescribedinfrequencyspacebythespectrum~x(f)showninFigure A-1 (b),whichisrelatedtothetimeseriesviatheFouriertransform: 67 ] 162

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(b)originalsignal (c)samplingfunction (d)samplingspectrum (e)sampledsignal (f)sampledspectrumFigureA-1. Anoverviewofthesamplingprocess 163

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A-1 (c).Theresultingsignal,S(t),showninFigure A-1 (e)issometimescalledanimpulsetrain.ThespectrumofthesampledsignalcanbedeterminedusingthefactthattheFouriertransformofaproductoftwosignalsisequaltotheconvolutionoftheFouriertransformsofthetwosignals.Inthiscase, A-1 (d).Theconvolutionin( A )willcontainthreeterms.Therstterm,fromthe(f)in~C(f),placesacopyof~x(f)atDC.Thesecondterm,fromthe(fkfs),placescopiesof~x(f)ateachmultipleofkfs.Thenalterm,fromthe(fkfs),placesfrequency-reversedcopiesof~x(f)ateachmultipleofkfs.Theoverallspectrumofthesampledsignal,~s(f),isshowninFigure A-1 (f). A-1 (f),thesampledsignalisanexactreplicaofthetime-domainsignal.Ifthebandwidthof~x(f)islimitedtofs=2,asshowinFigure A-1 ,thentheimagesdonoteectthesignal.Ifhowever,thebandwidthof~x(f)exceedsfs=2,thenthefrequency-reversedimagefrom1=Twillbegintooverlapwiththeoriginalspectrum.Whenthisoccurs,thecontributionsfromtheDCandthe1=Timageoverlap,asshowninFigure A-2 .Thisphenomenonisknownasaliasingandisgenerallyundesirablebecausehigh-frequencynoisecomponentsintheoriginalsignalcanmapintolowerfrequenciesinthesampledsignal.Topreventaliasing,theinputsignalmustbeband-limitedbelow 164

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A.4.5 ,theanti-aliasinganddownsamplingfunctionscanbecombinedintooneoperation,improvingeciency. FigureA-2. Thephenomenonofaliasing.Theoriginalspectrum(solidblueline)andtheimagedspectrum(dashedblueline)overlap.Thecontributionsfromeachcannotbeseparatedintheresultingspectrum(red). A-1 (e).Theresultingoutputspectrawouldcontainthedesiredspectraplustheimages,andcouldbelteredtoeliminatetheimages.Thistechniqueisoftenusedintherelatedprocessofupsampling,increasingthesampleratefromalowratetoahigherrateinamulti-ratediscrete-timesystem.In 165

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u(t)=s(t)R(t);(A) A-3 .Inthefrequencydomain,theconvolutionimpliesamultiplicationofthespectra A-3 (d).Thespectrumoftheupconvertedsignal,showninFigure A-3 (f)tracks~s(f)atlowfrequencies,butbeginstofalloathigherfrequenciesduetothesincresponseof~R(f).Intheory,thispassbanddroopcanbecorrectedbyareconstructionlterwithasinc1frequencyresponse.Suchalterisdiculttobuildintheanalogdomain,especiallysinceforffNyq,theresponseoftheltershouldbezerotoeliminatethespectralimagesathigherfrequencies.Inmostsystemsthereconstructionlterissimilartotheanti-aliasinglter,withaatresponseinthepass-band.Theeectofthesincresponsecanbereducedataparticularfrequencybyincreasingtheupsamplingrate. 166

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(b)sampledspectrum (c)rectangularimpulse (d)spectrumof(c) (e)ZOHsignal (f)ZOHspectrumFigureA-3. Anoverviewoftheupconversionprocess Associatedwiththesincresponseinthemagnitudeof~R(f)isalinearphaseresponse, 167

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00011011,141+141 27:(A)Tocomputetheresultusingintegers,thetwointegersaresummed(141+141=282)andtheresultisexpressedmodulo255,(282mod255=27).The2NvaluesinanN-bitwordcanalsobeusedtorepresentsignedintegers.Forexample,an8-bitwordcouldbeusedtodescribetheintegersfrom-128to127.Inthis 168

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A-1 .Inadditiontothenumbersexplicitycovered,theyalsohaverepresentationsforandnot-a-number(NaN),whichsigniesamathematicalerrorsuchasdividebyzero. TableA-1. IEEEstandardoatingpointrepresentations type totalbits mantissa exponent range oat 32 23 8 double 64 52 11 Fornumbersofroughlythesamemagnitude,theprecisionoftherepresentationisconstant.Asthesizeofthenumberincreasesordecreases,theprecisioniscorrespondinglydecreasedorincreased.Thishappensindiscretesteps,whichcansometimesproducestrangeresults. 171

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A-4 FigureA-4. AssumedPDFforquantizationerror Ingeneral,theenergycontainedinthequantizationerrorisequaltothesecondmomentofthePDF, A-4 ,( A )canbeevaluatedas 172

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A )andsolvingfor~Udiggives A-4 andthespectrumofthenoiseiswhite.ForsignalswitharandomnoisecomponentontheorderofULSB,thisisareasonableassumption.Forsignalswithlowernoise(orfewerbits),oneorbothoftheseassumptionscanbreakdown.Figure A-5 (a)showsthePDFofthequantizationerrorforapuresinusoidalsignalwithunitamplitudeandfrequencyf=fs=10quantizedasaS6.5binaryfraction. Non-uniform,non-whitequantizationerrorfromasinusoidwithunitamplitudeandf=fs=10quantizedasaS6.5binaryfraction 173

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A-5 (a)fallsintoalimitedrangeofbinsandisnolongeruniform.Thelinearspectraldensity(LSD)ofthequantizationerrorisshowninFigure A-5 (b).Thepeaksatfs=10and2fs=10resultfromthefactthatthetimeseriesofthequantizationerrorisarepetitivesequence. A )isknownasacausallter,sincetheoutputonlydependsonpreviousoutputsandthecurrentandpreviousinput.Filtersthatoperateinreal-time,suchascontrolltersmustbecausal.Filtersusedforoinedataprocessingcanbeacausal,includingnegativeindiciesforithatamounttoknowledgeoffuturesamples. 68 ]foramorecompletetreatmentofdigitalltering 174

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A ),anFIRisalterwithaj=0,ornofeedback.ForanFIR, 175

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A-6 (a).Thefrequencyresponseofadigitalltercanbecomputedusingthez-transform,adiscreteanalogueoftheLaplacetransform. A )canbeexpressedinthez-domainas A ).Thepole-zerorepresentationofH(z)is 176

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A ).Thetwocanberelatedusingtherelationshipbetweenthez-transformvariableisrelatedtotheLaplacevariable, A )with( A ), A-6 (b).Thefrequenciesk=Twherek=0;1;2;3:::allmaptothepoint(1;0)onthez-plane.ForalterwithaknownH(z),thefrequencyresponsecanbecomputedusing( A ).Inaddition,theleft-handsideofthecomplexs-plane,wherepolesmustbelocatedinorderforasystemtobestable,mapstotheregioninsidetheunitcircleinthez-plane. (b)z-planeFigureA-6. ComparisonofLaplaceandzdomains.Theredlinerepresentsthefrequencyaxis. 177

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68 ]. A-7 (a), A-7 (a),theimpulseresponseisgivenby, A-7 (b).ThisinniteimpulseresponsecanbeapproximatedbytruncatingtheseriesafteratotalofNpoints.Theedgeeectscausedbythetruncationcanbemitigatedbymultiplyingtheinniteimpulseresponsewithawindowfunctionthatgoestozeroattheendpoints.Forexample,theinniteimpulseresponseinFigure A-7 (b)canbetruncatedtoa31-pointresponseusingtheHamming 178

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A-7 (d).ThefrequencyresponseofthiswindowedltercanbedeterminedbyperformingadiscreteFouriertransformofthewindowedimpulseresponsefunction.Thefrequency-responseofthe31-pointHanning-windowedsincisshowninFigure A-7 (c). WindowedimpulseresponsemethodfordesigningFIRlters TheactuallterresponseinFigure A-7 (c)exhibitssimilarlow-passcharacteristicsastheidealresponseinFigure A-7 (a).Itdiersfromtheidealresponseinseveralimportantways:theslopeofthemagnituderesponseforffcisnite,thepassband(0f
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A ).Thewindowedimpulseresponsewillgenerallyincludenegativevaluesofn,correspondingtoanacausallter.Anequivalentcausalltercanbeconstructedbyshiftingh[n]topositivevaluesofn.Thiswillproducealterwiththesamemagnituderesponsebutalinearphaseresponsecorrespondingtoadelayequaltothenumberofpointsshifted.Forexample,the31-pointlterin( A )willbeshiftedtotherightby15points,correspondingtoadelayof15=fs.Thiswillproduceaphaseresponseof2f(15=fs). A ),toasetofrecursioncoecientsforanIIRdigitallteristhebilineartransformmethod.Thebilineartransformmethodstartswiththerelationshipbetweensandz,givenin( A ).Thisrelationshipcanbeinvertedtogive A )issimpliedusingthebilinearapproximation, 180

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A )toreplacesin( A ).Expandallofthetermsinthenumeratoranddenominator,andalgebraicallymanipulateitintotheformof( A ).Therecursioncoecientscanthenbeextractedusing( A ).Forallbutthemostsimplelters,thismanipulationbecomesextremelycumbersometoperformanalytically.However,itiseasytoperformnumerically. A.4.4.1FilterstructuresOncetheltercoecientsareknown,adatasetcanbelteredusing( A ).Onewaytorealizethisequationindigitallogicwouldbethedirect-formIstructureinFigure A-8 .ThisimplementationrequiresN+Mmultipliers,N+M+1adders,andN+Mregistersforstoringthepriorinputandoutputdata. FigureA-8. Thedirect-formIlterstructure Thesameresultcanbeachievedwithanumberofotherlterstructures,someofwhicharemorecomputationallyecient.Forexample,thedirect-formIItransposed(DF2T)lterstructureisshowninFigure A-9 .TheDF2Tstructurereducesthenumberofregisterstomax(N;M).Algebraically,alllterstructuresareidentical,butbecausetheoperationsoccurindierentorder,eectssuchasquantizationnoisecanvaryfromstructuretostructure.ThisismostpronouncedinIIRlters,whicharemademoresensitivetoquantization 181

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FigureA-9. Thedirect-formIItransposed(DF2T)lterstructure AgoodchoiceforIIRltersisacascadeofsecond-order-sections(SOS).Asecond-ordersectionisalterwithN=M=2andcanprovideuptotwopole-zeropairs.Higher-orderlterscanbeformedbycascadingaseriesofSOSs.Byadjustingwhichpolesandzerosareplacedinwhichsectionandtheoverallorderofthesections,tradeoscanbemadebetweenthestabilityofthelteranditsdynamicrange.Thisissimilartotheprocessofdesigningananaloglterusingmultipleoperationalamplierstages. A ),isequivalenttoadelayofT=2. 182

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69 ],areaspecialtypeoflterusedformakinglargesampleratechangesinDSPsystems.Theyusenomultipliers,whichmakesthemcomputationallyecient.AnN-stageCICdecimatorconsistsofNintegrators(singlepoleatDC)followedbyadecimationofRandNdierentiators(singlezeroatDC).ThetransferfunctionofaCIClterinthez-domainisgivenby A )withz!exp(2if=fs).Themagnituderesponseis sin(f=fs)N:(A) Magnituderesponseofagenerictwo-stageCIClter TheCICmagnituderesponsehasnullsatf=(k=R)fs;k=1;2;3:::,whichareatthecentersofthealiasingbandsforresamplingatfs=R.ABodeplotforaCIClteris 183

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A-10 .ThephaseoftheCIClterislinear,withanequivalentgroupdelayof(R1)=fsforN=2,thecaseusedinthehardwarephasemeter. 37 ]areonetechniqueusedtointerpolatebetweensampleddatapoints.Thebasisforafractionaldelaylteristheideallow-passlterinFigure A-7 (a).AsdiscussedinSection A.4.3.1 ,thecorrespondingimpulseresponseforanideallow-passisthesincfunctiongivenin( A ).ForthecasewherethecutofrequencyistheNyquistfrequency,( A )simpliesto A )toproducealterwhichdoesnoteectthemagnitudeofasignalbutproducesalinearphaseresponsewhichcorrespondstoadelay.ForadelayofDsamples,where0:5D0:5,thefractional-delaylterkernelisgivenby A )mustbetruncatedtoanitelength.Thisisgenerallyaccomplishedwithawindowfunction.Thetruncation 184

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37 ]. 185

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IwasborninSantaFe,NewMexico,onJune2nd,1979.IspentmychildhoodandadolescenceinSantaFe,graduatingfromtheSantaFePreparatorySchoolin1997.Inthefallof1997,IenrolledatBucknellUniversityinLewisburg,PAasamechanicalengineeringstudent.Whiletakingmyfreshmanandsophomorephysicsclasses,IdiscoveredIenjoyedphysicsasmuchasmechanicalengineeringanddecidedtopursueadoublemajorinthetwoelds.In2001,IgraduatedsummacumlaudewithaB.S.inmechanicalengineeringandaB.A.inphysics.Inthefallof2001,IenrolledinthePh.D.programatthephysicsdepartmentattheUniversityofMaryland,CollegeParkasaNASALaboratoryforHigh-EnergyAstrophysics(LHEA)fellow.AspartoftheLHEAfellowship,IwasrequiredtoworkattheNASAGoddardSpaceightCenter(GSFC)duringthesummerof2002.IbeganworkingwithGuidoMuelleratGSFConlaserstabilizationfortheLaserInterferometerSpaceAntenna(LISA).Afterthesummer,IcontinuedtoworkonLISAasaLHEAfellow,obtainingaM.S.inphysicsinDecemberof2002.InordertocontinuemyworkonLISA,IelectedtotransfertotheUniversityofFloridainJanuaryof2004,againworkingwithGuidoMueller.AlongwithGuidoandRachelCruz,IworkedtobuilduptheLISAlabinthephysicsdepartmentatUF.InAugustof2004,IwasawardedtheUniveristyAlumniFellowship,whichhascontinuedtosupportmeformytenureatUF.InDecemberof2005,IwasawardedtheTomScottMemorialPrizeforbestexperimentalistgraduatestudentbythephysicsdepartment. 191