LASER NOISE MITIGATION THROUGH TIME DELAY INTERFEROMETRY FORSPACE-BASED GRAVITATIONAL WAVE INTERFEROMETERS USING THE UF LASERINTERFEROMETRY SIMULATOR

By
SHAWN J. MITRYKA DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2012

c 2012 Shawn J. Mitryk2

I dedicate this work to my parents, Jeff and Terry Mitryk, whom provided the opportunityand never questioned my desire to fulfill my dreams, whatever they may be.

3

ACKNOWLEDGMENTS
I’d like to give a special thanks to everyone who helped me with these projects.
Guido Mueller, whom managed to be an research advisor, a career mentor, and a close
colleague, all at the same time, provided insight and direction towards my educationand scientific pursuits. Ira Thorpe was essential to my development as a effective andcreative problem-solver and provided a base for my future experimental investigations.
Thanks to Dylan Sweeney, Yinan Yu, Pep Sanjuan, and Syed Azer for allowing me topick their brains and for assistance in decipering complications with my experiments.
Also, I’d like to specifically thank my high-school physics teacher, Mr. Steven Desanto,
whom noted my talents in the sciences and particularly, in physics, and inspired me tocultivate them. And thanks to everyone else who worked in the UF-LISA project as wellas those that came to visit for keeping my days in the lab interesting.

4TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

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

9

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
CHAPTER1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1
1.2
1.3
1.4
1.5

2

Gravitation and General Relativity . . . . . . . . . . . . . . . . . . . . . .
The Gravitational Wave Spectrum . . . . . . . . . . . . . . . . . . . . . .
Space-Based Gravitational Wave Astronomy . . . . . . . . . . . . . . . .
Details of the Dissertation’s Content . . . . . . . . . . . . . . . . . . . . .
Note to the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

GENERAL RELATIVITY AND GRAVITATIONAL WAVES . . . . . . . . . . . . . 21
2.1 Introduction to General Relativity . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Proper Time Invariance . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Metrics, Black Holes, and Curvature . . . . . . . . . . . . . . . . .
2.1.5 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Einstein Field Equations Revisited . . . . . . . . . . . . . . . . . .
2.2 Gravitational Wave Derivation . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Polarized Plane Wave Solutions . . . . . . . . . . . . . . . . . . . .
2.2.2 Spacetime Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Weak-field GR Multipole Expansion . . . . . . . . . . . . . . . . . .
2.3 Gravitational Wave Sources and Detection Methods . . . . . . . . . . . .
2.3.1 Strain Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Gravitational Wave Evidence . . . . . . . . . . . . . . . . . . . . .
2.3.3 The Gravitational Wave Spectrum . . . . . . . . . . . . . . . . . . .
2.3.4 LISA Gravitational Wave Sources . . . . . . . . . . . . . . . . . . .
2.3.4.1 Compact Binaries . . . . . . . . . . . . . . . . . . . . . .
2.3.4.2 Binary Black Hole Mergers . . . . . . . . . . . . . . . . .
2.3.4.3 Extreme Mass Ratio Inspirals . . . . . . . . . . . . . . . .
2.3.4.4 Other Suggested Sources . . . . . . . . . . . . . . . . . .

3

15
16
17
20
20

21
21
21
22
23
27
28
29
29
30
31
35
35
35
36
37
38
38
39
39

THE LASER INTERFEROMETER SPACE ANTENNA . . . . . . . . . . . . . . 40
3.1 LISA Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 The Disturbance Reduction System . . . . . . . . . . . . . . . . . . . . . 42

5

3.3 The Interferometry Measurement System . . . . . . . . . . . . . . . . . .
3.4 µCycle Accuracy Phase Measurement . . . . . . . . . . . . . . . . . . . .
3.4.1 Photodetector Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1.1 Shot-Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1.2 Dark Current Noise . . . . . . . . . . . . . . . . . . . . .
3.4.1.3 Johnson-Nyquist Noise . . . . . . . . . . . . . . . . . . .
3.4.1.4 Flicker Noise . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1.5 Heterodyne Phase Measurements . . . . . . . . . . . . .
3.4.2 µCycle Phasemeter . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2.1 Clock Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2.2 ADC Quantization Noise . . . . . . . . . . . . . . . . . .
3.4.2.3 ADC Amplitude Noise . . . . . . . . . . . . . . . . . . . .
3.4.2.4 Clock-ADC Timing Jitter . . . . . . . . . . . . . . . . . . .
3.4.2.5 Demodulation Noise Coupling . . . . . . . . . . . . . . .
3.4.3 Heterodyne Time-Delay Interferometry . . . . . . . . . . . . . . . .
3.4.3.1 Fiber Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3.2 Spacecraft/Proof-Mass Motion . . . . . . . . . . . . . . .
3.4.3.3 Inter-Spacecraft Motion . . . . . . . . . . . . . . . . . . .
3.4.3.4 Basic TDI-Combinations and Considerations . . . . . . .
4

42
47
48
48
49
49
50
51
51
52
53
53
54
55
58
60
61
62
65

TIME DELAY INTERFEROMETERY . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Laser Noise Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Laser Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Clock Noise Transfers . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Ranging Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Fractional Delay Filtering and Interpolation . . . . . . . . . . . . . .
4.2 Laser Pre-stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Pound-Drever-Hall Locking . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Arm Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 TDI Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 TDI Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Sagnac Combinations . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2.1 TDI Six-Pulse Combinations . . . . . . . . . . . . . . . .
4.3.2.2 TDI Symmetric-Sagnac Combination . . . . . . . . . . .
4.3.3 Michelson X-combinations . . . . . . . . . . . . . . . . . . . . . . .
4.3.3.1 TDI-X0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3.2 TDI-X1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3.3 TDI-X2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 LISA Orbital Dynamics and TDI Data Analysis . . . . . . . . . . . .
4.4 Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Pseudo-random Noise (PRN) Code Ranging . . . . . . . . . . . .
4.4.2 Time-delay Interferometry Ranging (TDIR) . . . . . . . . . . . . . .
4.4.2.1 TDI Ranging Tone . . . . . . . . . . . . . . . . . . . . . .
4.4.2.2 TDI Ranging Parameter Search Algorithm . . . . . . . . .

6

69
70
70
71
72
73
74
75
77
77
79
79
80
80
81
82
84
84
85
86
88
89
90

5

THE LISA PHASEMETER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 Digitial Signal Processing (DSP) System . . . . . . . . . . . . . . . . . . . 94
5.2 µCycle Phase Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.2 Phasemeter Readouts . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2.1 Phase Quantization Noise . . . . . . . . . . . . . . . . . 101
5.2.2.2 CIC Downsampling and Aliasing . . . . . . . . . . . . . . 102
5.2.3 Phasemeter Test-Measurements . . . . . . . . . . . . . . . . . . . 103
5.2.4 Phasemeter Noise Model . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.5 Differential and Entangled Measurements . . . . . . . . . . . . . . 107
5.2.5.1 Digitally Split Differential Noise . . . . . . . . . . . . . . . 107
5.2.5.2 Electronically Split Differential Noise . . . . . . . . . . . . 109
5.2.5.3 Entangled Phase Noise . . . . . . . . . . . . . . . . . . . 113
5.2.6 Applications in LISA and LIGO . . . . . . . . . . . . . . . . . . . . 115
5.3 ADC Noise Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Timing Jitter Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.1 Phase Dispersion Mitigation . . . . . . . . . . . . . . . . . . . . . . 126
5.4.2 Absolute Timing Jitter Extraction . . . . . . . . . . . . . . . . . . . 127
5.5 Phasemeter Performance Review . . . . . . . . . . . . . . . . . . . . . . . 128

6

THE UNIVERSITY OF FLORIDA LISA INTERFEROMETRY SIMULATOR . . . 131
6.1 The LISA Laser Test-bench . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 The Electronic Phase Delay (EPD) Unit . . . . . . . . . . . . . . . . . . . 134
6.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.2.1 Time-changing Time Delay . . . . . . . . . . . . . . . . . 142
6.2.2.2 Gravitational Wave Injection . . . . . . . . . . . . . . . . 142
6.3 UFLIS Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3.1 Arm-Locking Stabilization . . . . . . . . . . . . . . . . . . . . . . . 144
6.3.2 TDI Simulation Outline . . . . . . . . . . . . . . . . . . . . . . . . . 148

7

TIME-DELAY INTERFEROMETERY SIMULATONS . . . . . . . . . . . . . . . 150
7.1 Transponder TDI Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.1.1 Static-Arm Transponder Simulation . . . . . . . . . . . . . . . . . . 152
7.1.2 Dynamic-Arm Transponder Simulation . . . . . . . . . . . . . . . . 154
7.2 LISA-like (Master-Slave Phase Locked Laser) TDI Simulations . . . . . . 157
7.2.1 Static-Arm LISA-like Simulation . . . . . . . . . . . . . . . . . . . . 158
7.2.2 Dynamic-Arm LISA-like Simulation . . . . . . . . . . . . . . . . . . 162

8

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

APPENDIXATIME VARYING FRACTIONAL DELAY INTERPOLATION FUNCTION . . . . . 172

7

B

TDI 2.0 COMBINATION FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . 174

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8

LIST OF TABLES
Table

page

2-1 Primary LISA verification binary sources . . . . . . . . . . . . . . . . . . . . . . 36
2-2 Gravitational wave frequency range of emission . . . . . . . . . . . . . . . . . . 37
3-1 LISA characteristics and requirements . . . . . . . . . . . . . . . . . . . . . . . 43
4-1 Orbital dynamics approximations for TDI generations . . . . . . . . . . . . . . . 79
6-1 TDI experimental characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7-1 Transponder TDI-ranging estimation . . . . . . . . . . . . . . . . . . . . . . . . 155
7-2 LISA-like TDI-ranging estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9

LIST OF FIGURES
Figurepage

1-1 NASA-ESA LISA solicitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2-1 LISA/LIGO differential arm-length changes due to GW polarization . . . . . . . 30
2-2 A theoretical model of a binary star system . . . . . . . . . . . . . . . . . . . . 33
2-3 An outline of GW detection methods and associated frequency ranges . . . . . 37
3-1 Diagram of the LISA orbital dynamics . . . . . . . . . . . . . . . . . . . . . . . 41
3-2 Diagram of the LISA constellation. . . . . . . . . . . . . . . . . . . . . . . . . 45

3-3 Diagram of the interferometry measurement system . . . . . . . . . . . . . . . 59
4-1 Model of the PDH locking scheme. . . . . . . . . . . . . . . . . . . . . . . . . . 74
4-2 Diagrammatic models of the TDI-X and Sagnac combinations . . . . . . . . . . 79
4-3 Flow chart of the ranging-tone minimization process . . . . . . . . . . . . . . . 93
5-1 A model of the LISA phasemeter . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5-2 CIC filter transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5-3 Theoretical PM readout digitization limitations . . . . . . . . . . . . . . . . . . . 102
5-4 Phasemeter noise caused by CIC filter aliasing . . . . . . . . . . . . . . . . . . 103
5-5 Software verification of PM performance . . . . . . . . . . . . . . . . . . . . . . 104
5-6 Hardware verification of PM performance . . . . . . . . . . . . . . . . . . . . . 105
5-7 Verification of the PM feedback transfer function . . . . . . . . . . . . . . . . . 106
5-8 Phasemeter noise model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5-9 Experimental models of ADC noise estimation measurements . . . . . . . . . . 109
5-10 PM/ADC Quantization and differential noise . . . . . . . . . . . . . . . . . . . . 110
5-11 ADC timing jitter noise limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5-12 ADC amplitude noise limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5-13 Entangled phase measurement results . . . . . . . . . . . . . . . . . . . . . . . 114
5-14 N-stage CIC filter aliasing and entangled phase measurements . . . . . . . . . 115
5-15 Model for estimating ADC phase and amplitude noise . . . . . . . . . . . . . . 117

10

5-16 Common source ADC phase and amplitude noise results . . . . . . . . . . . . 118
5-17 Zero-crossing timing jitter estimate . . . . . . . . . . . . . . . . . . . . . . . . . 121
5-18 ADC timing jitter comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5-19 Timing jitter calibration model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5-20 Temperature dependent phase dispersion . . . . . . . . . . . . . . . . . . . . . 125
5-21 RF transformer phase loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5-22 Replacement RF transformer dispersion mitigation . . . . . . . . . . . . . . . . 127
5-23 Phasemeter noise model, estimation, and limits . . . . . . . . . . . . . . . . . . 129
5-24 Comparison of the PM noise characteristics . . . . . . . . . . . . . . . . . . . . 130
6-1 UFLIS laser benchtop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6-2 Measurements of commonly used sources . . . . . . . . . . . . . . . . . . . . 133
6-3 Model of the electronic phase delay (EPD) unit . . . . . . . . . . . . . . . . . . 135
6-4 EPD unit data-packing scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6-5 Noise limitations of the EPD unit . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6-6 EPD Unit’s phase-noise transmission accuracy . . . . . . . . . . . . . . . . . . 141
6-7 Time series of the interpolated delay difference . . . . . . . . . . . . . . . . . . 143
6-8 Interpolated delay differential measurements and corrections . . . . . . . . . . 144
6-9 Time series of an arbitrary gravitational wave . . . . . . . . . . . . . . . . . . . 145
6-10 Spectral correction of an arbitrary gravitational wave . . . . . . . . . . . . . . . 146
6-11 Long-arm hardware-based single-arm-locking experiment . . . . . . . . . . . . 147
7-1 Model of the TDI-Transponder experimental benchtop . . . . . . . . . . . . . . 151
7-2 Raw static transponder experimental results . . . . . . . . . . . . . . . . . . . . 153
7-3 Corrected static transponder experimental results . . . . . . . . . . . . . . . . . 156
7-4 Ranging tone cancellation spectral results . . . . . . . . . . . . . . . . . . . . . 157
7-5 Dynamic transponder experimental results . . . . . . . . . . . . . . . . . . . . . 158
7-6 TDI laser phase noise suppression . . . . . . . . . . . . . . . . . . . . . . . . . 159
7-7 Model of the LISA-like TDI experimental benchtop . . . . . . . . . . . . . . . . 160

11

7-8 Static LISA-like experimental results . . . . . . . . . . . . . . . . . . . . . . . . 161
7-9 Cross-correlated TDI combinations - Static Transponder and LISA-like . . . . . 162
7-10 Dynamic LISA-like experimental results . . . . . . . . . . . . . . . . . . . . . . 165
7-11 Cross-correlated TDI combinations - Dynamic Transponder and LISA-like . . . 166
7-12 Confusion noise time-series comparison . . . . . . . . . . . . . . . . . . . . . . 167
8-1 Compiled results and comparison with TDI for LISA . . . . . . . . . . . . . . . 169

12

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
LASER NOISE MITIGATION THROUGH TIME DELAY INTERFEROMETRY FORSPACE-BASED GRAVITATIONAL WAVE INTERFEROMETERS USING THE UF LASERINTERFEROMETRY SIMULATOR
By
Shawn J. Mitryk
May 2012
Chair: Guido Müller
Major: Physics
The existence of gravitational waves was theorized in 1916 by Albert Einsteinin accordance with the linearized theory of general relativity. Most experiments
and observations to date have supported general relativity, but now, nearly 100
years later, the scientific community has yet devise a method to directly measuregravitational radiation. With the first attempts towards a gravitational wave measurementin the 1960s, many methods have been proposed and tested since then, all failingthus far to provide a positive detection. The most promising gravitational radiation
detection method is through the use of a space-based laser interferometer and withthe advancement of modern technologies, these space-based gravitational wave
measurements will eventually provide important scientific data to physics, astro-physics,
and astronomy communities.
The Laser Interferometer Space Antenna (LISA) is one such space-based laserinterferometer. LISA’s proposed design objective is to measure gravitational radiationin the frequency range from 30 µHz to 1 Hz using a modified Michelson interferometer.
The interferometer arms are 5 Gm in length measured between each of the 3 spacecraftin the interferometer constellation. The differential arm-length will be measured to an√√
accuracy of 18 pm/ Hz resulting in a baseline strain sensitivity of 3.6 × 10−21 / Hz.
Unfortunately, the dynamics of the spacecraft orbits complicate the differential

13arm-length measurements. The arms of the interferometer change in length resulting
in time-dependent, unequal arm-lengths and laser Doppler shifts. Thus, to cancel thelaser noise, laser beatnotes are formed between lasers on separate SC and, using theseone-way laser phase measurements, one can reconstruct an equal-arm interferometerin post-processing. This is commonly referred to as time-delay interferometry (TDI) andcan be exploited to cancel the laser phase noise and extract the gravitational wave (GW)
induced arm-length strain.
The author has assisted in the development and enhancement of The University
of Florida Laser Interferometry Simulator (UFLIS) to perform more accurate LISA-likesimulations. UFLIS is a hardware-in-the-loop simulator of the LISA interferometry
system replicating as many of the characteristics of the LISA mission as possible. This
includes the development of laser pre-stabilization systems, the modeling of the delayedinter-SC laser phase transmission, and the µcycle phase measurements of MHz laser
beatnotes.
The content of this dissertation discusses the general GW detection methods andpossible GW sources as well as the specific characteristics of the LISA mission’s design.
A theoretical analysis of the phasemeter and TDI performance is presented alongwith experimental verification measurements. The development of UFLIS is described
including a comparison of the UFLIS noise sources with the actual LISA mission. Finally,
the enhanced UFLIS design is used to perform a second-order TDI simulation withartificial GW injection. The results are presented along with an analysis of relevant LISA
characteristics and GW data-extraction methods.

14

CHAPTER 1
INTRODUCTION
1.1 Gravitation and General Relativity
Although a first-order theory of gravity was established in 1687 with the publicationof ’Principia [1]’ by Issac Newton and started the field of physics as we know it today,physicists are still trying to define the details of gravity’s interactions. In 1916, over 200
years later, Albert Einstein published his theory of general relativity (GR) which defined
the interaction of matter with the curvature of space and time [2, 3]. This provided
physicists with new insights on gravity and re-defined our concepts on the structureof the universe. Even though Einstein’s theories have yet to be dis-proven, modern
discoveries are raising new questions and testing the limits of Einstein’s equations.
For example, the standard-model of particle-physics does not include an explanation
for gravity, although some physicists have proposed the existence of a ’graviton,’ [4] thecarrier of the gravitational force, despite the fact that it would extremely difficult to detectbecause of how weakly gravity interacts.1 Proposals to explain an apparent ’missing
matter’ in galaxies and the accelerated expansion of the universe state that dark-matterand dark-energy [6–8] dominate over ’light’ matter and largely determine the past andfuture evolution of the universe. If dark-matter particles2 are discovered, the existenceof dark-matter raises new questions about the composition of matter in the universe and,
if not discovered, may indicate a need to modify Einstein’s theories. Furthermore, theinability to quantize gravity and create a unified theory of forces which is consistent withrelativity is arguably the greatest dilemma of modern-day physics.
That said, Einstein’s theories on general relativity have, at this point, been
supported by all experimental tests and observations from the bending of light by a1

Although, LISA could improve the upper bound on it’s possible mass [5].

2Dark matter particles interact gravitationally but not electromagnetically.

15

massive body [9] to the prediction and observed evidence for black holes. Nonetheless,
new questions are being raised. How do physicists explain both quantum mechanics
and relativity simultaneously? Is there some way to find a ’grand unification’ theory to
explain both phenomenon? Is the theory of general relativity a complete explanation ofgravity or does it have to be modified in some way? And finally, the question which willbe the indirect focus of this dissertation, can we use space-based laser interferometry
to measure a gravitational waves (GWs) [10], the space-time strain caused by the
motion of massive bodies, accurately enough to learn more about the details of generalrelativity?
1.2 The Gravitational Wave Spectrum
Observationally, the question of whether or not GWs exist has been answered.
In 1993, the Nobel Prize in Physics was awarded to Russell Hulse and Joseph Taylor
for the indirect detection of GWs by demonstrating that the rotational energy loss ofthe binary pulsar system PSR B1913+16 equaled the rate predicted through the GW
energy loss given by GR [11]. Unfortunately, scientists have yet to achieve a direct
detection of GWs. The first efforts, using resonant Weber bars [12], were not nearlysensitive enough to make a positive detection and have a primary design disadvantage
in that they can only measure at a single resonant frequency. More modern attempts
to measure GWs include the use of ground based interferometers such as the Laser
Interferometer Ground-Based Observatory (LIGO) [13, 14], pulsar timing analysis using
radio telescopes such as the Square Kilometer Array (SKA) [15, 16], and cryogenic
resonant Weber bars such as ALLEGRO [17, 18] and miniGRAIL [19]. Thus far, allattempts have failed to make a positive direct GW detection. On the other hand, LIGO
and other GW collaborators have been able to set upper limits on the distribution andamplitude of many proposed GW sources [20–22].
The justification for the many assorted efforts to measure GWs goes beyond
bragging rights for a first detection. As a compliment to the electromagnetic observations

16

of the universe from microwaves to x-rays, GWs provide a whole new spectrum throughwhich to observe astro-physical events. The measurements of GWs will provide a mapof black-hole space-times and verify relativistic black-hole models [23, 24], significantly
improve the constraints on the Hubble constant [25], and provide early warning systems
for electromagnetic observations and cross-correlated parameter estimation [26, 27].
Most importantly, the measurements of a GW signal could give scientists the necessary
information to be able to narrow down the many, and growing, modern theories of gravity
that are being used to explain the aforementioned complications to Einstein’s GR.
Fortunately, physicists are working on a more promising detection method:
space-based laser interferometers such as the Laser Interferometer Space Antenna(LISA) [28, 29], shown in Figure 1-1, and it’s conceptual successors like the Next
Gravitational Wave Observatory (NGO) [30].
1.3 Space-Based Gravitational Wave Astronomy
Space-based GW detectors such as LISA, NGO, or the Deci-hertz Interferometer
Gravitational Wave Observatory (DECIGO) [31] have many added benefits overground-based observatories. The gravitationally ’quiet’ environment of space allows
space-based detectors to get away from the seismic and gravity gradient noise that
limit the low-frequency detection capabilities of ground-based detectors. In addition,
space-based satellites allows for Gigameter (Gm) baseline arm-lengths, in comparison
to the few kilometer (km) arm-lengths of ground-based detectors, decreasing therequirements on the differential arm-length measurement resolution to obtain an
equivalent strain precision.
The GW-frequency measurement band of space-based detectors, from about
30 µHz to 1 Hz for LISA and from about 10 mHz to 100 Hz for DECIGO, has a numberof scientifically interesting GW sources. This includes, but is not limited to, compact
galactic binaries, extra-galactic binary black hole mergers, and extreme mass ratio
inspirals (EMRIs) [33]. Measurements of GWs from compact binary systems could17

Figure 1-1. NASA-ESA LISA solicitation: This solicitation, released by NASA, shows amodel of the LISA interferometry setup, including the 3 space-craft and theinter-space-craft laser links used to measure gravitational radiation. At thebottom of the solicitation is a simulation of a LISA-like GW signal including
the expected instrumental noise; the objective of this thesis is to re-create a
hardware-in-the-loop simulation much like the one shown. The figures at in
the top, left corner show the expected LISA sources including black holebinaries mergers, extreme-mass ratio inspirals, compact star binaries, andfinally, the gravitational background, and possibly, unknown,
gravitational-quantum effects. [32]18

be used to estimate the density of these binary systems within our galaxy. The GW
measurement of a binary black hole merger along with an electromagnetic observation,
could be used to put a more accurate constraint on the Hubble constant [25]. GWs fromextreme mass ratio inspirals provide a perfect experimental test-bench to map out the
space-time curvature around a black hole and verify Schwarzschild and Kerr black holemetric solutions [23, 24].
Space based gravitational wave detectors have their own complications though. The
proof-masses, the objects with which the gravitationally induced motion is measured,
must be kept in a gravitational free-fall and shielded from any other non-gravitational
forces, such as those produced by electromagnetic radiation from the sun. Also,
because the proof-masses must be in free-fall, their independent geodesic orbits
cause the arm-lengths of the interferometer to change over time. As a result, thecommon-mode laser phase noise rejection inherent in most interferometers, is no longer
maintained.
Space-based interferometers must make use of a sequence of laser phase noise
stabilization techniques in combination with one-way laser phase measurements andthe post-processing removal of laser phase noise to accurately extract GW signals
from the photo-detector (PD) signals [29, 34]. Laser pre-stabilization techniques
include various methods of locking the laser frequency to a stable reference; this
could be an ultra low expansion (ULE) glass cavity or the interferometer arm-length
itself. Once the lasers have been pre-stabilized, one-way laser phase measurements
can be taken along the individual arms of the interferometer. By combining theseone-way laser phase measurements in particular time-shifted and time-scaled linear
combinations, one can cancel the common-mode laser phase noise and extract thephase modulated gravitational wave. This method of laser noise cancellation in an
unequal-arm interferometer is known as time delay interferometry (TDI) [35].

19

1.4 Details of the Dissertation’s Content
To begin, we will discuss the basics of general relativity, derive the GW propagation
characteristics, outline GW detection methods, and provide an example of an interferometric GW measurement of a binary source. We will also give a brief introductionof the LISA design sensitivity and examine the expected LISA sources including
their expected strain amplitude, frequency, and detection rates (chapter 2). We willthen explain the details of the Laser Interferometer Space Antenna including theDisturbance Reduction System and the Interferometry Measurement System (chapter 3).
Chapter 4 provides a more in-depth explanation of laser phase mitigation methods of
the LISA Interferometry Measurement System (IMS) and a theoretical analysis of TDI.
Chapter 5 analyzes the design, performance, and limiting noise sources of the UF-LISA
phasemeter (PM) as well as presents a few experiments and results for ADC noise
mitigation in LISA. The University of Florida Laser Interferometry Simulator (UFLIS)
along with the electronic phase delay (EPD) unit and the pre-stabilized UF-LISA laser
bench-top is then explored in detail. Some basic measurements are then presented
to demonstrate the capabilities of UFLIS (chapter 6). Finally, the complete LISA
constellation is simulated to test TDI 1.0 and 2.0 (chapter 7) linear combinations andverify laser noise cancellation, ranging estimation, and GW extraction.
1.5 Note to the Reader
The reader should note that, although this dissertation focuses on the designand science of the LISA mission as well as references LISA mission characteristics
and publications, the specifications for the development of a future, space-based GW
interferometer may differ from those referenced [36]. This may include changes in thearm-length, relaxed sensitivity requirements on the components, and a reduction in
the number of laser links between SC. Nonetheless, the measurements and science
presented in this dissertation still hold true and can be applied to any space-based
interferometry mission.

20

CHAPTER 2
GENERAL RELATIVITY AND GRAVITATIONAL WAVES
The following introduction to GR as it pertains to gravitational wave emission anddetection was derived from a combination of [37], [38], and [39] along with additional
calculations and conceptual relationships provided by the author.
2.1 Introduction to General Relativity
2.1.1 Newtonian Gravity
Sir Issac Newton proposed that the gravitationally attractive force between two
massive objects, ma and mb , is given by:−→
Gma mbF ab = −r̂ab ,|→
r ab |2(2–1)→−
where F ab is the force vector between the objects, −r→
ab is the distance vector connecting
the two objects, and G is the gravitational constant. One can also write the gravitational
potential, V(r), surrounding an object of mass M as:
GM
→
,
V(|−
r ij |) = −|→r ij |(2–2)→→
where −
r ij is the distance vector connecting the center of mass of the object, −x i , to→a point of interest, −x j . These relationships became universally accepted based on
confirmed observations and predictions throughout the 1700s and 1800s, giving riseto the Newtonian interpretation of gravity. However, some phenomena, such as theprecession of the planet Mercury, were not entirely explained by Newton’s Laws.
2.1.2 Einstein Field Equations
It was not until 200 years later when the Theory of General Relativity, as proposedby Albert Einstein in 1916 [3], revealed the true nature of gravity by defining theinteraction of space and time with energy and mass. More specifically, GR asserted thatthe curvature of a tensor spacetime potential, Gµν , defines the gravitational force on an
object while, at the same time, the mass/energy distribution of the object, Tµν , defines

21

how the spacetime around the object will distort. This is summed up by Einstein’s tensor
field equations:
Gµν =

8πG
Tµν ,
c4

(2–3)

where Gµν is the symmetric, second-rank, covariant, Einstein tensor, Tµν is thesymmetric, second-rank, covariant, energy-momentum (or stress-energy) tensor, and cis the speed of light. Generally, Gµν , can be interpreted as the curvature and dynamics
of spacetime while the Einstein stress-energy tensor, Tµν , defines the distribution
and momentum of mass/energy being acted on, while simultaneously, changing thespacetime around it. This self-interactive nature of the 10 independent second-order,
differential Einstein’s equations makes them extremely hard to solve with only a few
specific cases having a complete analytic solution.
2.1.3 Proper Time Invariance
To better understand the Einstein field equations and eventually be able to computethe strain and frequency of a GW source, we must first introduce 4-vectors, vectortransformations, and explain the spacetime metric, gµν . Consider a 4-vector for aparameterized curve, xµ (λ) = [ct(λ), x(λ), y(λ), z(λ)], in a flat, non-moving coordinate
basis, commonly referred to as Minkowski space:


−1
0
0
0




 0 +1 00 ηµν = 



0
0
+1
0




0
0
0 +1

(2–4)

such that we can write the ’spacetime distance’ separating two infinitesimal points along
the curve, dxµ /dλ, as:
ds2 = ηµν dxµ dxν = dx2 + dy2 + dz2 − c2 dt2 ,

22

(2–5)

using Einstein summation notation. This ’spacetime distance’ can be related to theproper time and is invariant under transformation to a new vector basis xµ́ such that:ds2 = −c2 dτ 2 = dś2 = ηµ́ν́ dxµ́ dxν́ .

(2–6)

Solving for dτ , we can calculate the instantaneous change in the proper time in any
reference frame using the equation:
1
dτ =
c

r

−ηµν

dxµ dxν
dλτ .
dλ dλ

(2–7)

Using the fact that this quantity should not change as a function of the selectedreference-frame, we can define a transformation from one vector space, xµ , to anothervector space, xν́ , using transformation matrices, Λν́µ . For example, it can be shown thatthe transformation to a reference frame moving in the x-direction at a velocity v withrespect to a stationary reference frame takes the form of the Lorentz transformation:
cosh(α) − sinh(α) 0 0 − sinh(α) cosh(α) 0 0 


(2–8)
Λν́µ = 
,


0
0
1
0




0
0
0 1
where α = tanh−1 (v/c). This transformation is a member of the Poincaré transformationgroup which keeps the proper time interval invariant under reference frame transformations in a flat spacetime and is the basis for Special Relativity.
2.1.4 Metrics, Black Holes, and Curvature
Transformations in a ‘curved’ spacetime can be defined by replacing the Minkowski
spacetime metric, ηµν , in (2–6), with a more generalized spacetime metric, gµν , such thatthe invariance equation becomes:
ds2 = gµν dxµ dxν .

23

(2–9)

The spacetime metric, gµν , is a covariant, second-rank, tensor which contains all theinformation about the strain (current density) and dynamics (relative parameterized
density) of spacetime.
Some examples of curved analytic metrics which help reveal the nature of themetric definition of spacetime include the theoretical models of stationary, non-rotating,black hole, known as the Schwarzschild metric:
 


2GM
− 1 − c2 r
0
0
0






2GM −100 
0
1 − c2 rgµν = 
00
+1 0 



0
0
0 +1

and that of a rotating black hole, known as the Kerr metric: hi
2GM sin(θ)− 1 − 2GMr
0
0
−
c2 ρ2
c2 ρ2




2
ρ


0
0
0


∆
gµν = 
.


ρ2
0
0
0 r2




2GM sin(θ)
2
1
2
2 2
2
0 0 ρ2 r2 [(r + a ) − a ∆ sin (θ)]− c2 ρ2 ‘(2–10)(2–11)

These relationships are written in spherical coordinates where M is the mass of theblack hole, a is the ratio of the angular momentum to the mass (J/M) pointing in theẑ-direction, and2GMr+ a2 ,
2c(2–12)

ρ(r, θ) = r2 + a2 cos2 (θ).

(2–13)∆(r) = r2 −Computing ds2 as given by (2–9) using the Schwarzschild metric (2–10), we obtain:




−1
2GM
2GM
2
ds = − 1 − 2
dt + 1 − 2
dr2 + r2 dθ2 + r2 sin(θ)2 dφ2 ,crcr2which is a more common way to write out the metric equation.

24(2–14)

To gain a better intuition on what the metric represents, we compare theseblack hole metrics to the flat spacetime Minkowski metric (2–4). It will be shown inchapter 2.1.5 that the first-order gravitational acceleration, written in terms of thespacetime metric, can be written as:
d2 xi∂i ∗ h00=,(dx0 )22(2–15)

where hµν is the deviation from the Minkowski metric,hµν = ηµν − gµν .

(2–16)

Referencing the Schwarzschild metric for a stationary black hole, as in (2–10), we
see that the dt2 and dr2 terms in the metric converge to -1 and +1, respectively, while thegravitational acceleration approaches zero in the limit that r → ∞ or M → 0, resultingin Minkowski-like flat spacetime. But as r is decreased, the spacetime curvature due tothe black hole increases, thus increasing the gravitational acceleration. A useful way toanalyze these black holes is to consider the radius at which spacetime goes from being’time-like’, ds2 > 0, to ’space-like’, ds2 < 0, also known as the event horizon. Solving forthe inversion point, or ds2 = 0, we obtain the Schwarzschild radius:
RS =

2GM
.
c2

(2–17)

Now, changing our focus to the Kerr metric, given by (2–11), we see that the gθθand gφφ terms are no longer equal to 1, indicating that the gravitational field is no longer
spherically symmetric due to the black hole’s angular momentum. Additionally, if wecompute the gravitational acceleration near a rotating black hole, as given by (2–15),
we will see that the curvature is greatest near the poles of the rotating black hole andthat it does not approach infinity as r approaches zero. This is because the black hole’s
angular momentum causes the Schwarzchild-singularity to be stretched into a circularloop. Finally, one of the most interesting aspects of the Kerr black hole metric is the25non-zero gφt and gtφ terms which represent the angular dynamics of space-time due tothe rotation of the black hole, also referred to as frame dragging. This frame-dragging
concept has recently been confirmed by Gravity Probe B [40] using the Earth as thesource of rotational space-time frame dragging.
Another useful way to analyze the curvature of spacetime given a particular metric,gµν , is through the use of the Riemann and Ricci Tensors. The Riemann Tensor defines
how a vector changes as it is parallel transported around a closed curve defined bytranslation vectors, Aµ and Bν , and can be given by the definition:
δxρ = Rρσµν xσ Aµ Bν .

(2–18)

We will refrain from going into the details of the derivation and simply provide adefinition for the Riemann tensor in terms of partial derivatives of a spacetime metric,gµν , which may be given by:Rρσµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ ,(2–19)

where Γσµν are the Christoffel coefficients, also written in terms of partial derivatives ofthe metric, gµν , as:Γσµν =gσρ
(∂µ gνρ + ∂ν gρµ − ∂ρ gµν ),2(2–20)∇µ xν = ∂µ xν + Γνµλ xλ .

(2–21)

and are defined by:

One can easily see that this curvature tensor is defined entirely as a function of
the metric and it’s partial derivatives. Although this is the generalized definition for thecurvature on a given manifold, in General Relativity we will primarily be concerned with a26

contracted version of the Riemann tensor, known as the Ricci Tensor:
Rµν = Rλµλν .

(2–22)

By trace-reversing the Ricci Tensor, we finally arrive at the Einstein tensor:
1Gµν = Rµν − Rgµν
2

(2–23)

R = gµν Rµν .

(2–24)

where, R is the Ricci scalar, or:

The Einstein tensor is a symmetric, second-rank tensor which is equal to the nulltensor when space-time is flat and has zero divergence such that, ∇µ Gµν = 0.
2.1.5 Geodesics
Now that curvature and the metric have been sufficiently discussed, the nextquestion becomes how does this spacetime curvature affect the motion of a particle?
An object in free-fall within a given metric will follow a path known as a geodesic. The
geodesic equation gives a parameterized space-time solution for what is considered a’straight’ line, or the shortest path between two points, within a given space-time metric,or comparatively, Newton’s Laws for GR. The geodesic equation of motion is written as:
ν
µ
d2 xσ
σ dx dx
+
Γ
= 0.
µν
dλ2
dλ dλ

(2–25)

By considering a ’static’ metric and non-relativistic motion, the interesting components of the Christoffel coefficients simplify to:Γµ00 =−gµλ ∂λ g00.
2(2–26)

Then, by evaluating (2–20) with gµν = ηµν − hµν , the geodesic equation reduces to theaforementioned gravitational acceleration within a given spacetime metric, (2–15).

27

2.1.6 Einstein Field Equations Revisited
At this point we have seen how a massive object distorts the local space-time
metric as well as how this metric affects the motion of a nearby particle, but the basisof gravitational wave generation exists in the dynamics of Einstein’s equations, when
massive objects move through spacetime causing the metric to change. The basis forthe derivation of Einstein’s field equations lies in the attempt to dynamically equate theparameterized divergence from a flat Minkowski metric, gµν − ηµν , or equivalently, theEinstein tensor, Gµν , to the matter energy distribution tensor, Tµν . Essentially, we want
the Einsteinian equivalent of Poisson’s equation for gravitation:∇2 Φ(r, θ, φ) = 4πGρ(r, θ, φ),(2–27)

where Φ is the ’gravitational potential’ and ρ is the mass density. One can make acomparison between Poisson’s equation and Einstein’s field equations, again written as:
Gµν =

8πG
Tµν ,
c4

(2–28)

to better understand what the terms represent. A static, first-order comparison ofthese equations would equate the metric, gµν to the gravitational potential, Φ, theEinstein Tensor, Gµν to the Laplacian of the gravitational potential, ∇2 Φ, and thestress-energy tensor, Tµν , to the mass density distribution, ρ. Although, when we
take all the dynamics of the Einstein equations into consideration, we are required
to solve 10 second-order inter-dependent differential equations, which have very few
analytic solutions. Fortunately, if we consider a ’weak-field’ expansion, the linearizionof Einstein’s equations will provide us with a first-order dynamic solution for the change
in the metric due a change in the mass distribution. The solution of this weak field
expansion is manifested as polarized plane GWs which propagate outward at the speedof light from the generation source.

28

2.2 Gravitational Wave Derivation
2.2.1 Polarized Plane Wave Solutions
We begin the weak-field solution for GR by defining a metric expansion as a
Minkowski metric with a small perturbation added to it:gµν = ηµν + hµν + O[hnµν ],(2–29)

while assuming hµν << 1 so that the expansion converges. Evaluating this in Einstein’s
equation, (2–23), we obtain a relationship for the linearized field equations:−hµν,αα − ηµν hαβ,αβ + hµα,αν + hνα,αµ = 16πTµν ,(2–30)

where hµν is the trace-reversed hµν , given byηµν hαα,hµν = hµν −2(2–31)and hµν,α = ∂α hµν . After applying the gauge conditions such that:µαh ,α = 0,(2–32)

we are left with a concise form for the linearized field equations:hµν ,αα = 16πTµν .

(2–33)

Although the weak-gravitational field equations have an extra degree of freedom, using
Maxwell’s electromagnetic field equations as our guide, we expect a gravitational analog
to the polarized plane-wave solutions of electrodynamics. As well, we should be able tocalculate the luminosity and frequency of a particular radiation source using a multi-pole
expansion, as is commonly done in electrodynamics.
The GW solutions resulting from (2–32), after transforming hµν to the transversetraceless gauge, hTTµν , where the temporal components are zero, can be broken down
into two basis polarization states, h+ and h× . Note that hTTµν is it’s own trace-reverse,

29

TT

such that hTTµν = hµν , eliminating the trace-reversed basis needed to form the linearized
field equations. Written explicitly in Cartesian coordinates, a GW propagating in theẑ-direction will take the form: 
0 0   0
0
0
 0 0

 
 0 hxx hxy 0   0 ℜ[h+ e−iω(ct−z) ] ℜ[h× e−iω(ct−z) ] TThµν = = 
 0 hyx hyy 0   0 ℜ[h× e−iω(ct−z) ] −ℜ[h+ e−iω(ct−z) ]

 
0 0
0 0
0
0
0



0 

0 



0 

0

(2–34)

where h+/× is the GW polarization amplitude and ω is the GW frequency. The complete

analysis of the polarization states show that a ring of particles laid in the x/y plane will be
modulated with the patterns shown in Figure 2-1, hence the assigned names.

Figure 2-1. LISA/LIGO differential arm-length changes due to GW polarization: The
affects of the h+/× strain polarizations on a ring of particles is depicted. The
LIGO (blue) and LISA (red) detectors are overlaid on the ring to show thepolarization affects on the differential arm-length changes for each of thedetectors types. The LIGO detectors are only sensitive to a single GW
polarization, in this case, h+ , while the LISA detector
is sensitive to both√polarizations but with a reduction factor of 2/ 3 due to the 60o angle
between the differential arms. [29]2.2.2 Spacetime Strain
Another interpretation of a GW’s affect on matter is represented in terms of aspace-time strain which is computed from hµν . In this weak field transverse-traceless30gauge, the geodesic equation can be reduced to∂ 2 µ xµ ∂ 2 µ
x =
h σ.
∂t2
2 ∂t2

(2–35)

Considering two objects separated in the x-direction by a distance xo , the geodesic
equation is then calculated to be:x(t) ∂ 2 x
x(t) ∂ 2 x
∂2
x(t)
=
h
+
h y.
x
∂t2
2 ∂t2
2 ∂t2

(2–36)

To simplify the analysis, we take h× = 0 and h+ > 0. The distance between these two
objects in a ’flat’ spacetime is given by:
x(t) = xo 1 +h+ e−iω(t).
2(2–37)

Assuming h+ << 1, we can define the relationship:
δx= h+ .
xo(2–38)

where δx is the amplitude of the separation distance ranging from xo − (δx/2) toxo + (δx/2). Thus, in this way, the metric can be interpreted as a space-time strain whichchanges the distance between two objects by it’s magnitude. The measurement of thischange in distance is the basis for interferometric detection of GWs.
2.2.3 Weak-field GR Multipole Expansion
The magnitude of a GW source can be calculated, as is commonly done inelectrodynamics, in terms of Green’s functions given by:
16πG
hµν (x ) = − 4cσZG(xσ − yσ )Tµν (yσ )d4 y.

(2–39)

After applying boundary conditions in the far-field approximation, we can write theFourier domain metric perturbation as:
iωr Z

4Ge
e
hµν (ω, xi ) =
c4 r

31

e µν (ω, yi ),d3 yT(2–40)

or the time-domain perturbation as:
4Ghµν (ω, x ) = 4criZ

d3 yTµν (ct − r, yi ).

(2–41)

Using a multi-pole expansion of the radiating system in the far-field approximation,
we may write the observed strain from a quadrapole source as:hij =2G d2 Iij(tr ),
c4 r dt2(2–42)

where Iij is the quadrupole moment tensor:
ij

I =

Z

d3 xT00 xi xj ,(2–43)and tr = ct − r is the retarded time. The monopole and dipole terms do not contribute to
GW radiation while higher order terms fall off as (1/r)(n−1) and will be neglected in thisfar-field approximation.
One of the most common astrophysical GW sources are binary systems, where twomasses are orbiting a common center of mass. Given a binary system with two masses,
m1 and m2 , and a separation distance of 2a, we can write the quadrapole moment ofinertia of the system as:

2cos (ωt)cos(ωt)sin(ωt) 0 




2
Iij = µa2 cos(ωt)sin(ωt)
sin
(ωt)
0




0
0
0
where µ = (m1 m2 )/(m1 + m2 ) is the reduced mass and ω =p(2–44)(GM)/(a3 ) is the orbital

frequency given by Kepler’s Law, M = m1 + m2 = ω 2a3 /G [41]. Evaluating Iij in (2–42),32

the expected GW radiation strain can be written as: −cos(2ωtr ) −sin(2ωtr ) 0 
4Gµhij (t, xi ) = 4 ω 2 a2 −sin(2ωtr ) cos(2ωtr ) 0 


cr


0
0
0


 −cos(2ωtr ) −sin(2ωtr ) 0 2

4G m1 m2  −sin(2ωt ) cos(2ωt ) 0  .
=rr

c4 ra 000(2–45)(2–46)

m1
m2

2a



Figure 2-2. A theoretical model of a binary star system: Two masses, m1 and m2 rotate
about a common center of mass with a separation distance of 2a. As
rotational energy is lost through GW radiation the separation distance will
decrease (2–50) while the angular frequency increases (2–52).
The instantaneous power output through GW radiation, or rather the gravitational
luminosity, can be calculated by the time-averaging over a single orbital period of theradiating source, given by:
G
LGW = 5
5c



dI3ij
dt3r

2

(2–47)For a binary system, this results in a luminosity given by:
LBinary =32 µ2 M3 G4f(ǫ),
5 a5 c533(2–48)

where f(ǫ) is a correction function based on the eccentricity, ǫ, given by:f(ǫ) = [1 +73 2 37 4
ǫ + ǫ ][1 − ǫ2 ](−7/2) ,24
96(2–49)and may contribute an order of magnitude to the resulting luminosity [37].
As a result of the GW energy radiated by the binary source, the orbital kinetic
energy will decay resulting in a reduced orbital distance, increased luminosity,and increased orbital frequency. These values can be derived in the first order
Post-Newtonian approximation for a circular orbit (ǫ = 0) as:1/4
ta = ao 1 −
τo
Lo
L= 
5/4
1 − τtovu1
ω = ωo u3/4 ,tt
1 − τo(2–50)(2–51)(2–52)

respectively. In this equation, the time-scaling factor, τo , is the ’merger time’ until thebinary system inspirals into a single compact object:τo =5 c5 a4o,
256 G3 µM2(2–53)

resulting from initial system values of ao , Lo , and ωo .
Post-Newtonian and Numerical Relativity Solutions.
Higher order Post-Newtonian and Numerical Relativity (NR) solutions provide moreaccurate descriptions to these binary in-spiral systems by taking into account orbital
circularization and the relative angular momentum of the individual in-spiraling objects.
Qualitatively, gravitational radiation from an in-spiraling system is minimized through areduction in the eccentricity and circularization of the binary orbits. The relative angularmomentum between the stars and the angular momentum of the binary may cause theactual merger time to either lead or lag the non-rotating merger time, τo . This can also

34

cause an asymmetric gravitational radiation which imparts a linear momentum to thebinary system. These higher order solutions are necessary in order to accurately extract
gravitational wave signals from the detector through wave-form matching.
2.3 Gravitational Wave Sources and Detection Methods
2.3.1 Strain Estimation
Now that we have a measure for the expected GW radiation from astrophysicalquadrapole sources, we can estimate the GW strain from particular sources based on
the parameters determined through electromagnetic observations. For example, theprimary LISA ’verification’ binary, RX J0806.3+1527, a known AM CVn binary systemwith an orbital period of 321 s [42], is expected to have binary stars with m1 = 0.13MJ
and m2 = 0.5MJ with an observation distance of r = 300 pc, as seen in Table 2-1.
Using Kepler’s Law, we can calculate the separation distance between the binarystars as aRX−J = 37.97 × 106 m. Including the other known characteristics of the RX
J0806.3+1527 system, we can calculate the strain magnitude observed on Earth from
(2–45) as:|hµν | =4G2 µM
−21 m
=
1.608
×
10
.
c4 ra
m

(2–54)

A large number of known compact binaries have strain amplitudes on the order of
10−22 → 10−21 m/m in the frequency range from 0.1 → 10 mHz and are primary LISA
sources. Three other verification binaries and four other known, possible sources
are outlined in Table 2-1 [42]. Further details of these sources are discussed inchapter 2.3.4.
2.3.2 Gravitational Wave Evidence
The current evidence for GW radiation from astronomical sources is based on theobservation and orbital decay measurements of binary pulsar systems. The first orbital
decay measurements were performed by Hulse and Taylor using the PSR B1913+16
binary pulsar. In 1993, Hulse and Taylor were awarded the Nobel Prize in Physics for35

Table 2-1. Primary LISA verification binary sources: Here we list the top 8
most-probable verification binaries for the LISA-GW detector according to[42]. These binaries will also assist in constraining the Hubble constant [5].
The TDI experiments performed in chapter 7 will use the double white dwarf
RXJ0806.3+1527 binary as the simulated GW source.
Name f (mHz) SNR [mean : max]r (pc)m1
m2
RXJ0806.3+1527 6.22027[62 : 227]
300 − 1000
0.13
0.2 − 0.5
V407 Vul 3.51250[30 : 79]300 − 1000 0.068
0.7
ES Cet3.22[19 : 62]350 − 1000 0.062
0.7
AM CVn 1.94414[8 : 13]
606
0.14
0.85
HP Lib
1.813[< 3 : 5]
197
0.032
0.57
4U 1820-30
2.92[< 3 : 5]
8100
< 0.1
1.4
WZ Sge 0.04065[< 3 : 5]43< 0.11> 0.7
KPD 1930+2752 0.2434[< 3 : 5]
100
0.5
0.97
showing the orbital decay of the PSR B1913+16 system equaled that predicted byGR, thus providing indirect evidence of energy loss through gravitational radiation [11].
To date, the orbital decay of the PSR B1913+16 system matches that predicted by
GR to within 0.2% [43]. Other astronomical observations, such as the double binary
pulsar PSR J0737-3039, are providing even tighter constraints and further supporting
the validity of Einstein’s GR [44, 45]. Despite the certainty of GW existence, a direct
observation along with electromagnetic counterparts would provide physicists vital
information in support or opposition of modern post-Einsteinian theories.
2.3.3 The Gravitational Wave Spectrum
Astronomical GWs radiate from a wide variety of dynamic quadrapole sources
including binary inspirals of compact star and black hole mergers, asymmetric spinningcompact objects, super-nova star collapses, and black hole captures of compact objects.
It is also expected that there is a gravitational wave background, much like the cosmic
microwave background, resulting from quantum fluctuations shortly after the big-bang
and the sub-sequential expansion of the universe. Each of these systems produces
GWs within a characteristic frequency range from 10−18 → 109 Hz, each with an

36

Table 2-2. Gravitational wave frequency range of emission: This table outlines thegravitational wave emission spectrum, the GW emission sources, and theassociated detection techniques. (BBHM: Binary Black Hole Mergers; EMRI:
Extreme Mass Ratio Inspirals)
Freq. Range (Hz)Name
Sources
Detection Method
−18
−7
10
− 10
Very Low Primordial GWs,
Pulsar Timing
Freq.
Early Universe Dynamics−40
10 − 10
Low Freq. BBHM, EMRIs,
Space-based
Compact Binary Stars
Interferometers
101 − 104
High Freq. Supernova,
Ground-based
Compact Binary Inspirals
Interferometers
102 − 109
Very High Binary Inspiral Harmonics, Resonant Detectors
Freq.
Technological Applications
associated possible method of detection. The frequency ranges and their proposed
detection methods are outlined in Table 2-2 and depicted in Figure 2-3.

Figure 2-3. Outline of GW detection methods and associated frequency ranges: Very
low-frequency (e.g. pulsar timing - Square Kilometer Array); Low frequency(e.g. space-based laser interferometers - Laser Interferometer Space
Antenna, Next Gravitational Wave Observatory); High-frequency (e.g.
ground-based laser interferometer - Laser Interferometer Gravitational Wave
Observatory); Mid-frequency (e.g. space-based resonant cavities Deci-Hertz Gravitational Wave Observatory (DECIGO)); Very high frequency(e.g. GW amplification and detection through the measured deformations in
a resonant geometrical structures - Weber Bars/miniGRAIL:).
2.3.4 LISA Gravitational Wave Sources
The low-frequency/LISA measurement band, from 0.1 mHz to 1Hz contains awide-array of interesting gravitational wave sources.

37

2.3.4.1 Compact Binaries
The strongest of these sources are monotonic GWs resulting from interacting
and non-interacting binary star systems composed of binaries with one or both of thestars being a compact white dwarf (WD) or neutron star (NS) star. Although many
non-interacting WD/WD, WD/NS, NS/NS systems have been observed, none are known
to be in the LISA sensitivity range. At the same time, there are more than 10 known andelectromagnetically observed interacting AM-CVn (WD accretor) and compact X-ray (NSaccretor) binary systems which are expected to be excellent verification sources and are
well within the LISA strain sensitivity. The strongest of these verification systems wereintroduced in Table 2-1.
The science motivating LISA’s observations of these sources include estimations of
the populations of these binary systems within our galaxy and an improved constraint on
the Hubble constant (also referred to as redshift uncertainty). In fact, LISA is expected
to be sensitive to so many company binary systems that their populations are expected
to form a ’confusion noise background’ due to the inability to differentiate between
individual sources for frequencies below 2 mHz [46].
2.3.4.2 Binary Black Hole Mergers
One of LISA’s most interesting astronomical observations will be that of extragalactic near-equal-mass, 103 M⊙ - 107 M⊙ , binary black hole mergers. These systems
will appear, depending on the total mass of the system, near the low-frequency limitsof the LISA measurement band, from 0.1 mHz to 1 mHz, and will increase in frequency
and strain amplitude as the system evolves. The dynamics of these binary blackhole mergers is not well defined, requiring numerical relativity simulations to produce
theoretical merger wave-forms. Observations of these events will provide precision
tests of these extreme self-interacting spacetimes as well as details about black holeformation and evolution. Although no systems are known, event rate estimates range
from 1 to 100s per year depending on the population and LISA’s achieved strain

38

sensitivity. As an example, a recent massive black hole binary system has been found,
named NGC-3393 with binary system characteristics, m1 = 3 × 107 M⊙ , m2 = 1 ×
106 M⊙ , 2a ≃ 5 × 1018 m = 162 pc, r = 50 Mpc [47]. If this system were closer to it’s
merger time, the expected LISA sensitivity could provide a 105 to 106 signal-to-noise
ratio on the GW emission waveform just before the black-hole merger. As a resultof GW’s non-interacting properties, cross-correlations between the GW signal andelectromagnetic observations of these binary black hole mergers will allow LISA toestimate an absolute distance to the binary source, providing an improved constrain on
the Hubble constant with an error of 1%1 [25, 29].
2.3.4.3 Extreme Mass Ratio Inspirals
Binary black hole mergers with a large mass difference ratio, otherwise known as
extreme mass ratio inspirals (EMRIs), will be visible near the 3 mHz corner frequency of
the LISA sensitivity band (Figure 8-1) and are the primary driving motivation to improve
the peak LISA sensitivity. EMRIs consist of a ≃ 106 M⊙ massive black hole being closely
orbited by and, eventually merging with, a smaller 10 − 100M⊙ black hole. The smaller
black hole will act as a test-particle to provide a spacetime map in the vicinity of themassive Kerr black hole beyond the singularity’s event-horizon [23, 24].
2.3.4.4 Other Suggested Sources
Finally, LISA will provide some of the first direct tests of new physics including
attempts to probe the microwave-background-like, gravitational wave stochastic
background. This GW background is conjectured based on the inflation of first-orderphase transitions of the early universe, shortly after the big bang and is expected toresult in a white-noise GW background throughout the universe. For more information on
this and the other LISA-like tests of new physics, refer to [48].

1

Current methods of constraining the Hubble constant provide an error of 5%.
39

CHAPTER 3
THE LASER INTERFEROMETER SPACE ANTENNA
A majority of the characteristics for this description can be found in the LISA
NASA/ESA Yellow-book [29].
3.1 LISA Overview
The LISA GW measurement scheme utilizes a modified-Michelson detection
technique by taking one-way measurements between freely-falling proof-masses
housed within three spacecraft (SC). These spacecraft follow independent, helio-centric
orbits trailing 20o behind the Earth1 while maintaining a nearly equilateral triangle
configuration which is off-set by 60o to the orbital plane of the center-of-mass as shownin Figure 3-1. Housed within each of these space-craft are two proof-masses which
are maintained in a gravitational free-fall by a disturbance reduction system (DRS).
Meanwhile, the interferometric distance between the proof-masses on opposite SC
are measured with an interferometry measurement system (IMS) as diagrammed in
Figure 3-2. The data is then sent to Earth to form the post-processed combinations
required to extract the GW signals. Using this design basis, the success of the LISA
mission towards detecting GWs depends on a series of requirements (Table 3-1) which
are defined to optimally measure the previously discussed GW sources while staying
within the bounds of cost and feasibility. Generally, the primary sensitivity limiting noise
sources in the LISA design are the DRS’s acceleration noise for f < 3 mHz and the IMS’s
sensitivity noise for f > 3 mHz.
The IMS’s interferometry sensitivity, δx̃IMS−δφ (f) adds to the DRS’s accelerationnoise, δãDRS (f), with the a scaling factor of 2/ω 2 , in root-sum-square to produce the1

The SC orbit the L5 Lagrange point to maintain the stability and reduce thedivergence of the orbits.

40

Figure 3-1. Diagram of the LISA orbital dynamics: Depicted is a diagram of the LISA
constellation’s orbital dynamics showing the triangular configuration of theSC, the geodesic path of an individual SC, and the relative angle between
Earth’s orbital plane and the LISA constellation.
effective differential length precision:sδx̃LISA (f) =(δx̃IMS−δφ (f))2 +

2δãDRS (f)(2πf)2

2(3–1)

When we take into consideration the sensitivity with respect to the GW sources,√
this length precision is scaled by a factor of 5 to account for a 1-year average over the√
4π 2 sky radians and a by factor of 2/ 3 to account for the non-orthogonal ≃ 60o angle
between the interferometer arms. In addition, GW’s which are smaller in wave-lengththan the LISA-arm have a reduced sensitivity due to a GW aliasing type effect resulting
in multiple GW oscillations between the two proof-masses. The combined effects ofthese scaling factors results in a LISA GW-to-length sensitivity function given by:
s

2
√ 2
f
T(f) ≃ 5 √1+
0.41fo3(3–2)

where fo = c/(2L) [29].
Using this sensitivity function, the expected length precision as a result of the IMS
and DRS noises, and the LISA arm-length, L = 5 Gm, we can calculate the effective
strain sensitivity:h̃LISA (f) = T(f)41δx̃LISA (f)
.
L

(3–3)

√
This sensitivity function is plotted explicitly in cycles/ Hz in Figure 8-1.
3.2 The Disturbance Reduction System
In order to accurately detect GWs, we must ensure that no other non-gravitational
forces such as electromagnetic interactions or solar radiation dominate the dynamics
of the proof-masses’ motion. The proof-masses, 46 mm3 cubes, will be composed ofa gold-platinum alloy and will be shielded from these non-gravitational forces by the
DRS. The SC and DRS themselves must largely follow the free-falling motion of these
proof-masses, thus, micro-newton thrusters are used to move the SC and track theproof-masses’ geodesic path. Capacitative sensors ensure that the proof-masses do
not hit the walls of the housing and, at times, intentionally actuate the proof-masses toaccount for the independent motion of the two proof-masses on each SC. The goal of allthese components working in collaboration is to keep the non-gravitational accelerationof the proof masses below:
2

3 fm/sδãDRS (f) = √
Hz

s

1+



f
8 mHz

4 s


0.1 mHz
1+
f

(3–4)

in the LISA measurement band. This defines the low-frequency sensitivity limit of
the LISA detector. Testing the ability to achieve this acceleration noise goal, the
LISA Pathfinder mission is being launched [49]. Collapsing a LISA-arm into a single
spacecraft, the LISA Pathfinder mission will attempt to measure the distance between
two free-falling test masses using heterodyned laser fields, providing an excellent
platform to test many LISA-like complications and characteristics.
3.3 The Interferometry Measurement System
The LISA IMS’s primary objective is to measure the differential distance between
the free-falling proof masses to an accuracy of:s4
2.8 mHz
18 pm
1+.
δx̃IMS−δφ (f) = √
f
Hz

42

(3–5)

Table 3-1. LISA characteristics and requirements: The requirements outlined in thistable [29] define the resulting LISA GW-sensitivity curve (Figure 8-1) based on
the pre-stabilized laser phase noise, the inter-SC ranging accuracy, thephasemeter measurement precision, the IMS’s TDI noise extraction capability,
and DRS isolated the acceleration noise of the proof-masses. Although afuture space-based GW mission may use different characteristics [36], therequirements specified in this description focus on a LISA-like scenario.
(fL = 0.1 mHz, fM = 2.8 mHz, fH = 8 mHz)
Characteristic
Laser Pre-stabilization
Phasemeter Precision
IMS Strain Sensitivity
DRS Acceleration Noise
Ranging Accuracy
Arm-Length
Light-Travel Delay
Relative Velocity

3 fm/s2√
Hz

Specification
q

fM 4
280
Hz
√
1
+
f
Hz q

1 µcycle
fM 4
√
1
+
f
Hz q
4
18
pm
f
M
√
1
+
f
rHz   q
1+

f

fH

4

1+

fL
f



δL = 1 meter, δτ = 3.3 ns
L = 5.0 ± 0.1 Gm
τ = 16.66 ± 0.33 s
v = ±20 m/s, β = ±66 ns/s

Each component in the IMS chain must be tested to ensure that this requirement issatisfactorily met [29]. This includes a laser pre-stabilization requirement to suppression
the inherent free-running laser phase noise, a PD/phasemeter differential phasemeasurement precision requirement to perform heterodyne time-delay interferometery,
and a ranging requirement to accurately shift and cancel the residual laser phase in
the TDI combinations as shown in Table. 3-1. These values are defined such that theresidual laser phase noise is cancelled beyond the shot-noise (3–8) and accelerationnoise (3–4) limits.
First, the λ = 1064 nm, ν = 282 THz, lasers must be pre-stabilized to an accuracy of:s
4
2.8 mHz280 Hz(3–6)
1+δν̃Pre−Stab (f) = √fHz
by locking to a frequency reference. A well-known method of frequency-referencing alaser is by Pound-Dever-Hall (PDH) locking to a ULE cavity [50–52]. Another method,which is also being used on the LISA-Pathfinder mission, is Mach-Zehnder locking.

43

This has an advantage over PDH locking in that it uses the two laser fields which are
already part of the LISA design. Either of these frequency reference methods could be
assisted or, possibly, completely replaced by using the LISA arm itself as the frequency
reference in a tracking technique known as arm-locking [53, 54]. Some of the work inthis dissertation (chapter. 6.3.1) will focus on the first long-arm hardware-in-the-loop
tests of this arm-locking technique.
The pre-stabilized laser fields are fiber-coupled through electro-optical modulators(EOM) to add the clock-noise transfers (chapter 4.3) and SC-to-SC laser communication
signals. Each of these modulated fields are then fiber-coupled onto an ULE opticalbench which distributes them to the back-link fiber, telescope, and the local opticalbench’s proof mass. The back-link fiber transmits the laser field to the adjacent opticalbench on the same SC while the telescope transmits the field to the the adjacent SC as
shown in Figure 3-2. The three laser fields on each optical bench (the local laser, theadjacent optical-bench’s laser, and the adjacent SC’s laser) are heterodyned to form
the three main beatnote observables as depicted with more detail in Figure 3-3. Theseobservables can be interpreted to represent the differential laser phase between
lasers on the same SC, χsr , the local-SC to local-proof-mass distance, bsr ,2 andthe local-SC to the far-SC distance, ssr . The subscripts allow us to differentiate
between the measurements on different spacecraft where the ’s’ subscript refers to
the sending SCs while ’r’ refers to the receiving SCr . The time-changing arm-lengths
and resulting inter-SC light-travel time-delays, τq = τq (0) + βq t, are indexed suchthat SCq is opposite ’Arm-q’ with q referring to clock-wise light propagation and q′

2

Other TDI descriptions [55] use τsr or zsr for these variables but we will use bsr to
avoid confusion with the light-travel time delay, τq (t).

44

3'

τ3

τ 2'
τ2
Figure 3-2. Diagram of the LISA constellation: A model of the complete LISA
constellation depicting the three space-craft, the six individual laser
benchtops, the inter-SC laser links, and the names of each of the arm-length
light travel times, τq/q′ , opposite SCq .

45

referring to counter-clock-wise propagation3 as shown in Figure 3-3. From each ofthese measurements we are able to re-construct the differential distance between the
proof-masses on opposite SC in post-processing (chapter 3.4.3) and, thus, determine
the GW strain.
The complications to this measurement scheme are many-fold. None of thecomponents in the laser transfer chain can exceed a specified requirement so that
added noise terms are not introduced into the interferometry measurement. This
includes the EOMs used to apply the clock transfers and SC-to-SC laser communications,the fibers used to transfer the laser field between optical benches, and the telescopes
used to transfer the laser fields between the SC. The fiber-backlink may introduce’non-reciprocal’ (different in counter-propagating directions) noise. The telescope length
may change over time, causing an apparent arm-length change. In addition, the angle
between the optical benches and telescope pointing direction must be actuated toaccount for the breathing of the constellation.
Another complication to the IMS measurement is the received power from adjacent
SC. Of the 2 W of laser power emitted from the local SC, the divergence of the laser
field over the 5 Gm arm-length results in only 100 pW being received on the adjacentSC’s photodetector. This results in a shot-noise limited heterodyned field which must be
measured to an accuracy of 1 µcycle.
Generally, the basis and details of the IMS system described here for the LISA
mission are analyzed and outlined in [29] and [34]. Although new proposals for a morecost effective design are being considered [30, 56, 57], they must still use the same setof principles to perform gravitational wave measurements. These changes may include

3

Note that the following description is done without a loss of generality bydifferentiating between independent τq (t) and τq′ (t), and accounting for different
counter-propagating time-delays along the same arm due to the constellation’s orbital
rotation.

46

shorter arm-lengths, the reduction to a two-arm measurement with only four inter-SC
laser links, a geocentric orbit, or the use of a single spherical proof-mass.
3.4 µCycle Accuracy Phase Measurement
The IMS differential path-length measurement precision, δx̃IMS−δφ (f) (3–5), is
primarily based on the cumulative error from the photo-detectors and phasemeters
used to measure the heterodyned MHz beatnotes. Independent of shot noises, thephotodetector error must be less than 3 pm while the phasemeter error must be less
than:
1 pm
δx̃PM (f) = √
Hz

s

1+



2.8 mHzf4

1 µcycle
≃ √
Hz

s

1+



2.8 mHz
f

4

.

(3–7)

Applying the expected shot-noise limitation:
7.7 pm,δx̃Shot−Noise (f) = √
Hz

(3–8)

√
and the 7 pm/ Hz path-length noise requirement, the overall root sum squared
differential phase of each pair of laser fields is measured to an accuracy of:s
4
2.8 mHz11.7 pm.
1+
δx̃δφ−Total (f) = √
f
Hz

(3–9)

Finally, the individual measurements are added in the linear time-shifted TDI combinations
to achieve the δx̃IMS−δφ (f) IMS requirement (3–5).
These low-frequency LISA band requirements are usually hindered by long-term
errors such as sampling biases, temperature dependent phase dispersion, andinterferometric length changes which couple into the phase measurements. For
example, if the voltage bias which is used in sampling and converting the beat signal
drifts over the course of the measurement, this will result in an un-accounted phase
coupling. Also, if the temperature of the filters or RF transformers used to prepare anddistribute the beat signal changes in time, this may result in a time-changing transfer

47

function phase response. The details of these noise sources are outlined in the following
sections.
3.4.1 Photodetector NoiseThe phase noise of a photo-detector with respect to the MHz heterodyned beatnote
can be determined by a linear combination of three independent noise sources: (1)shot noise, (2) Johnson-Nyquist noise, and (3) flicker noise. The shot noise limitation
is defined by the limited number of photons per second which are received on thephotodetector. Johnson-Nyquist noise results from temperature dependence ofresistive devices in the photo-detector electronics which may be influenced by bothinternal heating and fluctuations in the laser power. Flicker noise scales with a pink,
1/f, or more generally 1/f n , power spectrum which result from a combination of
long-term processes. This includes any non-shot-noise based relaxation processes
or fluctuations in the semi-conductor characteristics which range from white noise (1/f 0 )to Brownian-quantum noise (1/f 2 ) [58].
3.4.1.1 Shot-Noise
The theoretical basis for shot noise results from the inability to distinguish between
individual photons when incident on a photo-current producing semi-conductor. This
well-understood measurement limitation presents itself with a Poissonian distribution
and an uncertainty that scales with the square root of the number of photons in ∆t√measurement time, σN = N∆t, whereN∆t =PLaser∆t,
Ephoton(3–10)

for a total laser power on the photo-detector, PLaser , with an average photon energy
Ephoton = hν. Exploiting Heisenberg’s Uncertainty Principle, we can write the RMS phase
limitation as:
1
1==√
δφRMS−SN =
σN
N∆t

48

r

hν
PLaser ∆t.

(3–11)

The phase error, δ φ̃Shot−Noise (f), presents as a white noise specturm defined by
δφRMS−SN . Evaluating this with LISA-like values, P = 200 pW, ν = c/λ = 282 THz,√and ∆t = 1 s we obtain a phase noise of 5 µcycles/ Hz. This is far from a complete
derivation which should also consider the photon-electron conversion efficiencies, η, but
it gives us an idea of the expected phase noise.
For comparison to the dark-current, we can extend this result and write theshot-noise current error as:
δiRMS−SN =√σiSN =sZ

fBW(e2 η(f)N)df =

−fBW

p

2e2 fBW ηN(3–12)

where we have integrated the limited efficiency electro-current over the band-width ofthe photo-detector, fBW . [59]3.4.1.2 Dark Current NoiseThe semi-conductors used to convert the photons incident on a photo-diode to anelectro-current have a characteristic dark-current which results from randomly created
and destroyed current producing electrons in the semi-conductor. This random process
also results in a shot-noise like Poissonian error which can cause RMS white noise
current errors from 1 to 500 nA. [59, 60] These processes result in a current error givenby:δiRMS−Dark =√σiDark =sZ

fBW

eidark (f)df =

−fBW

p

2eidark fBW ,(3–13)

which can be linearly combined with the photo-detector’s shot noise to find the total
quantum noise:qpδiRMS−Total = δi2SN + δi2Dark = 2efBW (idark + [eηN]).

(3–14)

3.4.1.3 Johnson-Nyquist Noise
Thermal heating of the photo-diode, whether from internal or external sources,causes the shunt resistance’s Johnson-Nyquist noise to add errors to the output current.

49This noise couples as a function of the temperature with a magnitude given by:qR
rfBW
4RkB T
4kB TfBW
0
δiRMS−JN (T) ==R
R(3–15)

where R is the photo-diode’s shunt resistance, T is the temperature in Kelvin, and kB is
Boltzmann’s constant. [61]Explicitly stated, these shot noise, dark current noise, and Johnson-Nyquist noises
are all white noise sources such that the noise contribution increases linearly with the
integration time. LISA refers all measurements to a 1 second integration time and plots√
these white phase noise spectra in cycles/ Hz.
3.4.1.4 Flicker Noise
Flicker noise may result from a number of sources which are all characterized byhaving a 1/f n power noise spectrum where 0 < n < 2. These are largely influenced bylong-term non-quantum fluctuations. Long term temperature variations may modify thephoto-amplifier’s transfer function causing a time-changing phase response, and thus, along term temperature correlated phase error. Fluctuations in the individual laser powers
will also couple into the phase (chapter 3.4.1.5).
Long-term fluctuations in the laser field intensity might be attributed to longterm characteristic changes in the coherent field producing Nd:YAG laser crystals
including a temperature dependent cavity finesse and the availability of excited states
to produce stimulated emission. High-frequency components of the laser intensity will
also couple into the measurement and are generally based on the laser’s resonant
relaxation oscillation [62, 63]. This amplitude-phase coupling can also be introducedby other sources including a photo-detector polarization dependency, electronicnoise in the photo-detectors and ADCs resulting from LISA’s digitized demodulation
measurement scheme. The ill-defined nature of these long-term processes justifies
further investigation with lab-based differential PD measurements.

50

3.4.1.5 Heterodyne Phase Measurements
Consider two laser fields given by:
ǫ1 (t) = E1 (t)eiω1 (t)t

ǫ2 (t) = E2 (t)eiω2 (t)t .

(3–16)

Superimposing and combining these laser fields on a photo-detector, we obtain acurrent given by:ibeat (t) ∝(ǫ1 (t) + ǫ2 (t)) (ǫ1 (t) + ǫ2 (t))∗(3–17)∝|E1 (t)|2 + |E2 (t)|2 + E1 (t)E2 (t) cos([ω1 (t) − ω2 (t)]t)
In heterodyne interferometry, the DC portions of |E1 (t)|2 and |E2 (t)|2 are generally
AC coupled with RF transformers, although these may still introduce high-frequency
components. Combining the independent field terms, ETot (t) = E1 (t)E2 (t) = Eo (1 +δE(t)), and introducing the mean MHz beat frequency, ω∆ = hω1 (t)−ω(2)(t)i, we can pickout the, E1 (t)E2 (t) cos([ω1 − ω2 ]t) terms by demodulating this PD output with sin(ω∆ t).
Integrating the demodulated DC output, we can get a measure of the differential laser
phase:φ∆ (t) =Z t
0[ω1 (τ ) − ω2 (τ ) − ω∆ ]dτ .

(3–18)

Variations in the differential field intensity fluctuation, δE(t), at frequencies of ω∆ could
be coupled into the measurement such that complete description should be written as:eiω∆ τ + e−iω∆ τdτφ∆ (t) =[ω1 (τ ) − ω2 (τ ) − ω∆ ] + Eo δE(τ )
2
0


Z t

(3–19)

3.4.2 µCycle Phasemeter
The photo-currents produced by the photo-diodes will be sampled with Ω-bit
analog-to-digital converters (ADCs). The clocked sampling process which digitizes the
PD signals to perform the phasemeter measurements introduces it’s own independent
noise sources. The 2 − 20 MHz PD-beatnote is sampled with a 50 MHz clock reference51comparing φ∆ (t) to the clock phase, φCLK (t), which is not a perfect timing reference,
thus, introducing it’s own phase noise. The digitization process provides a limited
resolution with which to measure the beatnote phase, meanwhile, voltage-bias noisefrom the ADC voltage reference can also couple into the phase measurement. Timing
jitter, the error caused by the time-dependent delay between voltage conversion time√
and the clock’s rising-edge, couples into the measurement as a 1/ f phase noise andscales with the input beatnote frequency. These are all discussed in more detail below.
3.4.2.1 Clock Noise
Digitizing the 2 − 20 MHz PD signals with a 50 MHz clock, we sample the PD current
driven voltage bias across a load resistor at the ’rising-edge’ of the clock oscillation. The
PD voltage is compared to a voltage biased resistor bank within the ADCs resulting ina Ω-bit read-out of the load resistance voltage. Given an input oscillation with an ideal
amplitude written as:
xin (t) = Ain sin(2πfint + φin (t))(3–20)

we sample the input signal with a clock source given by:
xCLK (t) = AClk sin(2πfClkt + φClk (t)).

(3–21)

where φin (t) is the input phase noise with respect to the input frequency, fin , and φClk (t)is the clock phase noise with respect to fClk [64].
Using an ideal clock the phase error in the digitization conversion would be limitedby the bit-resolution, discussed in the following section. In practice, the phase noise in
the clock source with respect to the theoretical clock frequency is interpreted as input
phase noise scaled by the ratio of their frequencies such that:
φMeasured (t) =

Z t

fin
ωin (τ )dτ = φin(t) −φClk(t).
fClk
0 ωCLK (τ )52

(3–22)

In LISA, these clock noise terms must be accounted for by performing inter-SC
clock noise transfers and measuring the differential clock noise terms against the local
SC’s clock. Afterwards, the differential phase and differential clock measurements areinterpolated to compare them against one global master clock.
3.4.2.2 ADC Quantization Noise
When sampling a voltage signal, Vin (t), with a clock, fClk , the ADC compares the
input voltage against the ADC internal voltage bias reference. This comparison process
produces a series of samples given by:
xin [n] =Vin (n/fClk)
VBias(3–23)

where we have taken the bias voltage to be a constant, stable reference. This -1 to 1
ratio is converted to a series of Ω-bit values with a 21−Ω resolution [65]. Thus, given this
bit-accuracy we obtain a standard-deviation in the measured amplitude given by:δ φ̃ADC Amp.−Quant. (ω) =|VBias | 21−Ω√ .
|Vin | 6fs(3–24)√The extra factor of 1/ 6 comes from the white-noise properties of the quantization
probability density function [65, 66]. The factor of VBias /Vin accounts for the fact thatthe signal amplitude may not span the full Ω-bits of the ADC’s conversion-amplitude.
Additional quantization errors can and will be applied to this hard-ware based quantization
limit as these digitized signals are measured with bit-limited fixed point processors andrate-limited read-outs introducing their own quantization errors.
3.4.2.3 ADC Amplitude Noise
Extending this description, we consider fluctuations in the voltage bias against
which the photo-current is measured. More generally one can write the bias voltage as:
VBias (t) = VBiaso G(t) + VGnd (t)53(3–25)

where G(t) is the gain factor on the voltage bias with hG(t)i ≃ 1 and VGnd is the voltageof the ground reference in comparison to the input’s ’ideal’ ground. Ignoring the groundterm for the moment, this fluctuation in the bias voltage scales the amplitude error as:
δx̃ADC Amp−Quant. (ω) = δ G̃(ω)|VBiaso | 21−Ω
√
|Vin|
6fs

(3–26)

which shows how these gain factors can couple directly into the measured amplitude.
Since LISA measurements are focused on phase noise, the coupling of these amplitude
quantization and bias noise error sources into the phase measurements are analyzed inchapter 3.4.2.5
It is often difficult to distinguish between the ADC amplitude noise, the laser
intensity noise, and the PD electronic current amplitude noise since they all couple into
the phase measurement with the same characteristics. On the other hand, we can getsome measure of the ADC amplitude bias noise by verifying the inverse dependence on
the input voltage. Often-times, these low frequency amplitude variations are dominatedby temperature variations; thus, thermal correlation coefficients can be measured byobserving the phase variation with a change in temperature. Given these correlation
coefficient measurements we can define an electronic temperature environment stability
requirement.
3.4.2.4 Clock-ADC Timing Jitter
ADC timing jitter is defined as the time-changing delay between the well-definedrising edge of the clock and the actual sample triggering of the ADC input signal. Thistiming jitter value can have a frequency dependence, δ t̃(ω), which scales proportionately
with the beat-frequency, fin = hdφ/dti, to phase noise [67]:δ φ̃(ω) = fin δ t̃Jit (ω).

(3–27)

This frequency coupling can and will be exploited to estimate the ADC timing jitter.

54

3.4.2.5 Demodulation Noise Coupling
Now that we have seen how each of these individual noise sources scale with
their input characteristics, we evaluate how these measurements will couple into
the demodulated phasemeter readout. First, as an example of how this readout isperformed, we consider an input signal with some phase and amplitude variation whereδA(t), δφ(t) << 1, while ωin and ’A’ are constant, given by:
x(t) = A[1 + δA(t)] sin(ωin t + δφ(t)).

(3–28)

If we demodulate this with the in-phase and quadrature components of the input at thesame frequency, such that ωoff = ωin , we obtain following terms:
yIn−phase (t) = x(t) × sin(ωoff t)(3–29)

A[1 + δA(t)][cos(δφ(t)) − cos((ωin + ωoff )t + δφ(t))],2=

yQuad (t) = x(t) × cos(ωoff t)(3–30)

A[1 + δA(t)][sin(δφ(t)) + sin((ωin + ωoff )t + δφ(t))].
2=

After low-pass filtering to remove the sum, (ωin + ωoff ) term, we can write the result as afunction of the amplitude and phase error as:
A[1 + δA(t)][[cos(δφ(t))],2
A[1 + δA(t)][[sin(δφ(t))]
yQuad (t) =2yIn−phase (t) =(3–31)(3–32)

Based on the definitions of the in-phase and quadrature components, one can exactly
reconstruct the amplitude
q
ARead−Out (t) = 2 yIn−phase (t)2 + yQuad (t)2 = A[1 + δA(t)]55(3–33)

and phase,−1

φRead−Out (t) = tan
yQuad (t)= δφ(t).
yIn−phase (t)(3–34)On the other hand, using the small value assumption, δφ(t) << 1, we can make theapproximations, cos(δφ(t)) = 1 and sin(δφ(t)) = δφ(t), such that the amplitude and phaseread-outs smplify. Using the in-phase component only we could write the amplitude as:
ARead−Out (t) ≃ 2yIn−phase (t) = A[1 + δA(t)] cos(δφ(t)).

(3–35)

Using the same argument we can write two approximations for the phase, given by:yQuad (t)= tan(δφ(t)),
yIn−phase (t)(3–36)

2
yQuad (t) = (1 + δA(t)) sin(δφ(t)).
A(3–37)

δφ(t) ≃
or
δφ(t) ≃

These approximations are useful in reducing the load on the digital processing devices
but, as we can see from these calculations, they also tend to couple the amplitude andphase noise sources together which must be considered for LISA-like high-precisionphase measurements.
The details of the phasemeter phase tracking and read-out become more
complicated when we allow for A(t) and φ(t) to change arbitrarily, but this model
will give us an idea of how un-desired noise terms will couple into the demodulated
phase measurement. At this point we can see that, depending on the type of readout
approximation used, these error terms will couple in differently. Our focus will be on(3–37) since, as we will see in chapter 5.2.1, this is the phasemeter read-out scheme
we use to measure the input phase. Already we can see how the amplitude noise
couples into the phase (G(t) in (3–25) for example). Unfortunately, this fails to give acomplete understanding of the noise coupling since this first order approximation fails to56

consider how high-frequency terms couple into the measurement through the clocking
and feed-back tracking demodulation processes.
The clock-sampling demodulation will alias high-frequency terms into the digital
ADC output such that the input phase terms at nfClk ± fin will superimpose with a gainof 1 directly into the fin frequency bin. Following standard electronics procedures, RFtransformers are used to AC couple the input and low-pass anti-aliasing filters are used
to suppress any information at frequencies greater than fin.
Even though the input signal is AC coupled, we will include an offset error in theextended measurement-demodulation treatment such that we now write the input as:x(t) = δAOff (t) + A[1 + δA(t)] sin(ωint + δφ(t)).

(3–38)

Again, demodulating with the quadrature component we obtain:
yQuad (t) =A[1 + δA(t)][sin(δφ(t)) + sin(2ωo t + δφ(t))] + δAOff (t) cos(ωo t).
2(3–39)

where we have set ωo = ωin = ωoff .
Taking a deviation from the previous treatment we expand out these results to
explicitly show the DC, ωo , and 2ωo terms:
yQuad (t) =A[1 + δA(t)][sin(δφ(t)) + sin(2ωo t) cos(δφ(t)) + cos(2ωo t) sin(δφ(t)))]2+ δAOff (t)[cos(ωo t)].

(3–40)

Finally applying the δφ, δA, δAoff << 1 approximation, low-pass filtering the terms
which have no noise coupling, applying (3–37), and expanding explicitly, we obtain:
δφEst (t) =2
yQuad (t)A

(3–41)




δA(t)
δA(t)
+ δφ(t) 1 +cos(2ωo t)= δφ(t) 1 +
A
A
δA(t)
δAOff (t)+ δφ(t) cos(2ωo t) +sin(2ωo t) +cos(ωo t).
AA57(3–42)

From this we can read off the noise-coupling of each term. The term we areinterested in, δφ(t), is scaled by 1 + δA(t)/A. Generally, since δA(t)/A << 1, this effectis small, but may be a limiting noise source when attempting to form differential phasecancellation combinations. Although, since this term scales inversely with the amplitude,
it can be reduced, depending on the source, by increasing the signal power. If theamplitude noise comes from fluctuations in the laser power, for example, increasing
the signal power tends to also increase the noise such that δA(t)/A remains constant.
Next we notice both the δ φ̃(2ωo ) and δ Ã(2ωo ) terms of the input wave-form couple into
the phase measurement. Finally, we notice that the offset error, δ ÃOff (ωo ), term couples
directly into the measurement. This term would be of an error like VGnd (t) in (3–25).
We can mitigate the 2ωo terms by anti-alias low-pass filtering the input signal atfrequencies greater than ωo . Unfortunately, if these terms are generated by the ADC
after the anti-aliasing filter they will, again, couple into the phase data. Taking this into
account, extensive low-frequency testing of the ADCs must be performed to ensure
these errors do not limit the phase measurement sensitivity. We will see how theUF-phasemeter’s ADCs are tested and characterized in chapter 5.3.
3.4.3 Heterodyne Time-Delay Interferometry
Until now, we have not specifically stated how the one-way interferometry
observables, χsr , bsr , and ssr , are measured and mitigated. Here we derive the terms
on SC1 but the procedure used can be employed to derive the observables on theother SC. For the χsr and bsr observables, we consider the two local SC1 lasers, Laser21
and Laser31 , which transmit their fields to SC2 and SC3 , respectively, having a laser
frequencies, ω21 and ω31 = ω21 + ω∆ , and laser phases, φ21 (t) and φ31 (t), such that we
can could write the laser fields as:
ǫ21 (t) = E21 (t)ei(ω21 t+φ21 (t))

ǫ31 (t) = E31 (t)ei((ω21 +ω∆ )t+φ31 (t)) .

58(3–43)

Figure 3-3. Diagram of the interferometry measurement system: This model shows amore detailed depiction of the individual satallites shown in Figure 3-2. Here
we see the beam path and heterodyned laser fields resulting from the two
local lasers and the two laser fields being transmitted from the far SC. The
LISA observables, χsr , bsr , and ssr and how they are generated with theheterodyned laser fields is shown explicitly.

59

These laser noise sources are used to track the major phase noise coupling terms
through the interferometer although, from here on, we will ignore the amplitude terms
and take E21 (t) = E31 (t) = E.
3.4.3.1 Fiber Noise
The independent laser fields are passed through a fiber bank-link adding bothcommon-mode and non-reciprocal phase noise such that the fields after passing
through the back-link fibers can be written as:
ǫ21,fib (t) = Eei(ω21 t+φ21 (t)+φfib,C (t)+φfib,21 (t))(3–44)

ǫ31,fib (t) = Eei(ω31 t+φ31 (t)+φfib,C (t)+φfib,31 (t))(3–45)

where φi1,fib are the independent, counter-propagating fiber noise terms and φfib,C are
the common fiber noise terms.
Superimposing these laser fields from either side of the fiber onto a photo-detectorand AC coupling the PD output, we obtain photo-currents given by:PDχ21 (t) =q(ǫ21 (t) + ǫ31,fib (t))(ǫ21 (t) + ǫ31,fib (t))∗∝ cos(ω∆ t + φ31 (t) − φ21 (t) + φfib,C (t) + φfib,31 (t))
qPDχ31 (t) = (ǫ31 (t) + ǫ21,fib (t))(ǫ31 (t) + ǫ21,fib (t))∗∝ cos(ω∆ t + φ31 (t) − φ21 (t) − φfib,C (t) − φfib,21 (t))(3–46)(3–47)

Demodulating these signals with sin(ω∆ t) and measuring the phase terms we obtain
two phase read-outs given by:χ21 (t) = φ31 (t) − φ21 (t) + φfib,C (t) + φfib,31 (t) +f∆

δφCLK (t)fCLKf∆
χ31 (t) = φ31 (t) − φ21 (t) − φfib,C (t) − φfib,21 (t) +δφCLK (t)fCLK

(3–48)(3–49)

with the expected differential laser phase, φ31 (t) − φ21 (t), counter-propagating fiber
noise terms, and clock noise terms. The common fiber noise terms couple into each PD

60

with the opposite phase such that the sum of these observables gives a measure of thedifferential counter-propagating phase noise terms:
f∆φχ−Sum = χ21 + χ31 = 2 φ31 (t) − φ21 (t) +δφCLK (t) + φfib,31 − φfib,21
fCLK(3–50)

while the difference of these terms gives a measure of the common mode fiber terms:
φχ−Dif = χ21 − χ31 = 2φfib,C (t) + φfib,31 + φfib,21 .

(3–51)The individual counter-propagating phase noise terms must be smaller than 1 µcycle
since there is no way to differentiate between this and the independent laser noise
terms. In the next section we will see how these differential laser phase and common
mode fiber terms cancel when evaluating the proof-mass to spacecraft distance.
3.4.3.2 Spacecraft/Proof-Mass Motion
Now that we have a measure of the fiber noise, we use the same laser fields to
get a measure of the individual SC to proof-mass distance. If we define the distance
between the SC beam-splitter and the proof-mass as δd(t) we can covert this to laser
phase noise after reflecting off the proof mass by re-writing the fiber terms, ǫ21,fib (t) andǫ21,fib (t), as:δd31

ǫ21,mass (t) = Eei(ω21 t+φ21 (t)+φfib,C (t)+φfib,21 (t)+ λ )(3–52)

δd21

ǫ31,mass (t) = Eei(ω31 t+φ31 (t)+φfib,C (t)+φfib,31 (t)+ λ ) .

(3–53)

Demodulating these with the local bench’s laser field, we obtain the bsr signals:
PDb21 (t) =q(ǫ21 (t) + ǫ31,mass (t))(ǫ21 (t) + ǫ31,mass (t))∗δd2∝ cos(ω∆ t + φ31 (t) − φ21 (t) + φfib,C (t) + φfib,31 (t) +)λq
PDb31 (t) = (ǫ31 (t) + ǫ21,mass (t))(ǫ31 (t) + ǫ21,mass (t))∗∝ cos(ω∆ t + φ31 (t) − φ21 (t) − φfib,C (t) − φfib,21 (t) −

61

δd3
).
λ

(3–54)(3–55)

Measuring the beatnote phase with phasemeters, we can write the bsr observables as:
δd2f∆δφCLK (t)
+
λ
fCLK
f∆
δd3
δφCLK (t)+b31 (t) = φ31 (t) − φ21 (t) − φfib,C (t) − φfib,21 (t) −λfCLKb21 (t) = φ31 (t) − φ21 (t) + φfib,C (t) + φfib,31 (t) +(3–56)(3–57)

Linear combinations of these four signals, χ21 (t), χ31 (t), b21 (t), and b31 (t) provide us
with a local measure of the SC to proof mass distance and the differential laser phaseterms:
δd2 = λ [b21 − χ21 ] ,(3–58)

δd3 = λ [χ31 − b31 ] ,(3–59)

φ31 − φ21 =χ21 + χ31 φfib,21 − φfib,31f∆+−
δφCLK (t)22
fCLK

(3–60)

We notice in these equations that the SC-to-proof-mass distances, δd2 and δd3 , areindependent of any fiber noise terms. Meanwhile, the differential laser phase terms arelimited by the individual counter-propagating fiber terms φfib,31 and φfib,21 . The δd terms
are independent of clock noise since they are measured on the same spacecraft at thesame heterodyne offset frequency, ω∆ , and the common clock noise cancels in thesedifferential measurements.
3.4.3.3 Inter-Spacecraft Motion
Now that we have a method of evaluating the local-SC to local-proof-mass distance
we must measure the inter-SC arm-length terms to be able to construct the complete
differential inter-proof-mass interferometer. In this description we will focus on a single
one-way inter SC link between SC1 and SC2 where Laser12 , having a laser frequency
ω12 (t), is transmitted along Arm-3’ acquiring a time-delay, τ3′ (t), and then is heterodynedwith Laser21 , having a laser frequency ω21 (t). Inter-SC length changes at velocities of620 − 20 m/s for LISA4 , cause relativistic Doppler shifts and time-domain phase scaling of
the laser fields.
To calculate the field transformation, we first define the measured phase as theintegration of the laser beat frequency with respect to a retarded time, tr = t − x/c, suchthat both position and time changes affect the measured phase:Z tφ(t) =

ω(tr)dtr(3–61)

0

Taking SC2 as the stationary frame of reference we define the inter-SC light travel
time from SC2 to SC1 as τ3′ (t) = τ3′ (0) + β3′ t where β3′ = v3′ /c = dτ (t)/dt. A positive
velocity refers to an increasing inter-SC distance such that SC1 is moving away from theSC2 . In the stationary SC2 frame we can write the laser field generated by Laser21 as:ǫ12;∅ (x, t) = Ee−i(t− c ) ω12 (t− c )xx(3–62)and the measured phase evaluated at x=0:
φ12 (t) =

Z t

ω12

0




0
dτ
τ−
c

(3–63)

We can transform this to the moving SC1 reference frame through a relativistic
Lorentz transformation as:
′



′
′
iγ(1−β) t′ − xc ω12 (γ(1−β)(t′ − xc ))]′ǫ12;3′ (x , t ) = Ee

(3–64)

pwhere, for this arm, β = β3′ and γ = γ3′ = 1/ 1 − β 2 and tr = γ(1 − β)(t′ − x′ /c).

From this equation, we see the relativistic Doppler shift transforming ω12 → γ(1 − β)ω12 .

Integrating the beatnote frequency in this moving frame:′φ(t ) = γ(1 − β)

4

Z t′
0

ω12





Lq (0)dτγ(1 − β) τ −c(3–65)

Other missions may have differential velocities which are significantly larger[56]63such that we can write the measured phase relationship:φ′ (t′ );q = φ(γq (1 − βq )(t′ − τq (0)))(3–66)

where τq (0) = Lq (0)/c. In most LISA-like cases, we ignore the small factors of β 2 < 10−12
and so, will drop the γ factors in the analysis from this point forward. Notice the ’;’notation used to transform the laser field between moving frames in (3–64) and (3–66).
In general φsr;q(t) = φsr ((1 − βq )(t − τq (0))) where we have taken t → (1 − βq )(t − τq (0)).
We will use this same notation through the rest of the TDI analysis.
The field from the far SC is reflected through a transmitting telescope, transmitted to
the local SC acquiring a multi-second time-delay5 , and captured by the local receiving
telescope.
Heterodyning these laser fields we obtain a photo-current given by:
PDs21 (t) =q(ǫ21 (t) + ǫ12;3′ (t))(ǫ21 (t) + ǫ12;3′ (t))∗(3–67)∝ cos((ω21 − (1 + β3′ )ω12 )t ± (φ21 (t) − φ12 ((1 − β3′ )(t − τ3′ (0))))(3–68)

where we notice that the sign of the phase information depends on the relative
magnitude of the local laser frequency in comparison with the inter-SC Doppler
shifted laser’s frequency. Although this may be easily corrected in post-processing,
the relative laser frequencies and inter-SC Doppler shifts will change the sign on boththe observable and clock noise corrections terms in the TDI combinations and must be
kept in mind when forming the post-processed TDI combinations.
Generalizing to include the clock noise and GW terms we can write the s21 inter-SC
term as:s21 = φ21 (t) − φ12 ((1 − β3′ )(t − τ3′ (0))) + h21 (t) +5

τ = 5Gm/c = 16.7 s for LISA

64

f21
φClk:1(t),fClk(3–69)

where we have introduced the GW strain modulation of h(t) = δL(t)/Lo , and a local SCi
clock noise term, φClk:i(t), scaled by the PD beatnote frequency, f21 . We have also madeassumptions about the sign of the differential measured phase based on the relative
laser and Doppler frequencies for evaluating the TDI combinations.
3.4.3.4 Basic TDI-Combinations and Considerations
At this point, we have formed and evaluated all the observables required tocompletely re-construct the differential proof-mass interferometer. These signals, χsr ,bsr , and ssr , can be evaluated for each of the 3 SC and 6 inter-SC laser links resulting
in 18 different observables. When any of these observables are measured with aphasemeter, clock noise terms given by (fin /fClk )φClk:i(t) where is the phase noise ofclock located on SCi , are added to the digital signals and must be accounted for with theinter-SC side-band clock noise transfers [68–70]. These observables can also be used
to phase-lock the adjacent optical bench lasers or the adjacent inter-SC lasers into order
to transfer the laser stability and obtain common-mode laser noise cancellation to within
the accuracy of the phase lock loops (PLLs) [71].
The focus of this dissertation is to form the TDI combinations based on theinter-SC laser links, ssr . To simplify the analysis, from this point on we will makesome assumptions about the measured observables. First, assuming the local-SC
to local-proof-mass motion can be measured through heterodyne interferometry andthat the acceleration noise of the proof-mass can be effectively reduced with the DRS,
it is reasonable to imagine the proof-masses as being mounted directly to the SC,
thus setting δdsr = 0. Next, assuming the non-reciprocal fiber-backlink noise can be
reduced beyond the 1 µcycle requirement, we will have a reasonable measure of theadjacent bench-top differential laser noise and can ignore the χsr signals, referencing allmeasurements to a single Laseri on SCi with a laser phase φi (t).

65

In this specific case we can write a complete basis6 for all 6 inter-SC interferometry
measurements including clock noise terms as:
f21
φClk:1(t),(3–70)
fClk
f31
s31 (t) = φ1 (t) − φ3 ((1 − β2 )(t − τ2 (0))) + h31 (t) +φClk:1(t),fClk
f12
φClk:2 (Λ12 t),s12 (t) = φ2 (Λ12 t) − φ1 ((1 − β3 )(Λ12 t − τ3 (0))) + h12 (Λ12 t) +fClk
f32
s32 (t) = φ2 (Λ12 t) − φ3 ((1 − β1′ )(Λ12 t − τ1′ (0))) + h32 (Λ12 t) +φClk:2(Λ12 t),
fClk
f13
s13 (t) = φ3 (Λ13 t) − φ1 ((1 − β2′ )(Λ13 t − τ2′ (0))) + h13 (Λ13 t) +φClk:3(Λ13 t),
fClk
f23
s23 (t) = φ3 (Λ13 t) − φ2 ((1 − β1 )(Λ13 t − τ1 (0))) + h23 (Λ13 t) +φClk:3 (Λ13 t).
fClk
s21 (t) = φ1 (t) − φ2 ((1 − β3′ )(t − τ3′ (0))) + h21 (t) +where we have differentiated between the different clock phase noise terms, φClk:i , andabsolute clock frequency offsets7 :Λij =fClk:j.
fClk:i(3–71)

The clock phase noise terms are removed with the inter-SC clock noise transfers whenforming the TDI combinations while the clock frequency offsets are accounted for bytime-scaled interpolation [72, 73] of the ’far’ SC signals by the inverse clock ratio, Λij ,before forming the TDI combinations with respect to the local reference clock, in theabove example case, fClk:1.
Two useful combinations for further analysis will be the single-arm round-trip
sensor signals ∆sr and the counter-propagating Sagnac signals ζsr , as diagrammed in

6

Assuming SC-1 is our stationary frame of reference and Clock-1 is our absolute
clock frequency reference.
7

The clock frequency offsets from clocks on different SC cause an error in thereference to an absolute time which is defined by the frequency of the clock on SC1 .

66

Figure 4-2 which provide us with a measure of the constellation interferometry dynamics
referred to a single laser source.
The local sensor signals can be derived from the ssr data streams by time-scaling
the far SC measurements by the respective inverse clock ratio (Λij−1 ), then time-shifting
and time-scaling the result by the return trip’s delay transformation, ssr;q .
∆21 (t) = s21 (t) + s12;3′ (Λ12−1 t)= φ1 − φ2;3′ + h21 +(3–72)

f21
f12
φClk:1 + (φ2 − φ1;3 + h12 +φClk:2 );3′
fClk
fClk

= φ1 − φ1 ((1 − β3 )(((1 − β3′ )(t − τ3′ (0))) − τ3 (0)))(3–73)+ h21 (t) + h12 ((1 − β3′ )(t − τ3′ (0)))
+

f12
f21
φClk:1 (t) +φClk:2((1 − β3′ )(t − τ3′ (0)))
fClk
fClk

and, for Arm-3,∆31 (t) = s31 (t) + s13;2 (Λ13−1 t)= φ1 − φ3;2 + h31 +(3–74)

f13
f31
φClk:1 + (φ3 − φ1;2′ + h13 +φClk:3 );2
fClk
fClk

= φ1 − φ1 ((1 − β2′ )(((1 − β2 )(t − τ2 (0))) − τ2′ (0)))(3–75)+ h31 (t) + h13 ((1 − β2 )(t − τ2 (0)))
+

f31
f13
φClk:1(t) +φClk:3 ((1 − β2 )(t − τ2 (0)))
fClk
fClk

which cancel the far SC’s laser phase noise and reference all phase modulations to asingle laser source, φ1 (t). Here we can explicitly see the clock-noise coupling which
enters into the sensor signals’ data-streams with the form:∆ − Clks1 (t) =fs1
fClk:1

φClk:1(t) +f1s
φClk:s ((1 − βq )(t − τq (0))).
fClk

As we will see in Ch. 4.1.2 and Ch. 4.3, these terms can be accounted for by theclock-noise transfers in (4–3).

67(3–76)

Another set of useful combinations are (Ch. 4.3.2.1) the counter-propagatinground-trip Sagnac signals:
θ21 (t) = s21 + s32 (Λ12−1 t);3′ + s13 (Λ13−1 t);1′ 3′(3–77)= φ1 − φ1;2′ 1′ 3′ + [h21 + h32;3′ + h13;1′ 3′ ](3–78)

andθ31 (t) = s31 + s23 (Λ13−1 t);2 + s12 (Λ12−1 t);12(3–79)= φ1 − φ1;312 + [h31 + h23;2 + h12;12 ].

(3–80)

Finally, the last combination of interest is the fully symmetric Sagnac combination:
ζ = s31;1 + s12;2 + s23;3 − (s21;1′ + s32;2′ + s13;3′ )(3–81)= [(φ1 − φ3;2 );1 + (φ2 − φ1;3 );2 + (φ3 − φ2;1 );3 ](3–82)− [(φ1 − φ2;3′ );1′ + (φ2 − φ3;1′ );2′ + (φ3 − φ1;2′ );3′ ]which is independent of laser noise for a non-rotating8 constellation.
We have ignored the clock noise terms for the sake of simplicity since these Sagnac
combinations are not the focus of this dissertation.

8(τq (t) = τq′ (t))
68

CHAPTER 4
TIME DELAY INTERFEROMETERY
Now that we have developed a method of characterizing the signals with which tomeasure the inter-proof-mass noise terms the question enters: How do we combine
these signals to cancel the laser phase terms and extract the GW modulations?
4.1 Laser Noise Cancellation
The ability to cancel the laser noise the IMS is largely based on the accuracy withwhich we can measure the time-dependent distance1 between the space-craft andinterpolate the phase signals to account for this distance. To get an idea of why this is
the case, let’s consider two laser phase signals, φ(t) and φ((1 − β)(t − τ )) + g(t). Thesesignals are independently measured on different SC producing two digitally sampled
signals:x1 [n] = φ(fs n) +fin1
φClk1 (fs n),
fs

x2 [n] = φ((1 − β)(n(fs + ∆fs ) − τ ))+ h(n(fs + ∆fs )) +

(4–1)
(4–2)

fin2
φClk2 (n(fs + ∆fs )).
fs + f∆s

The (1 − β) factor is introduced to account for the laser phase time-scaling as aresult of SC motion while the ∆fs factor accounts for the small difference in the absolute
clock frequencies. Although this ∆fs factors will be small,2 they can cause accumulated
errors in the phase corrections if they are not accounted for.3 These can be related
to the Λij factors defined in (3–71) with fClki = fs and fClkj = fs + ∆fs . The ability
to extract g(t) from these signals depends on four things: (1) the initial laser-phase1

Or equivalently, the laser light travel time-delay: τ (t) = L(t)/c2

fs ≃ 50 MHz, ∆fs ≃ 1 − 10 Hz depending on clock tolerances.

Relativistic time-dilation causes the same effect but it is small, β 2 < 10−12 , and cansimply incorporated into the clock frequency error term.
369spectral density, φ̃(ω), (2) the ability to measure the differential clock noise terms, (3) the
accuracy of our knowledge of the time delay, τ , and (4) the ability to perform fractional
delay filtering and interpolation to time-shift and time-scale these signals to account forthe τ , f∆s , and (1 − β) factors.
4.1.1 Laser Noise
Obviously, the initial laser noise level, φ̃(ω), plays a large roll in our ability to extract
the g(t) terms4 from the above signals. The frequency references proposed to stabilize
the lasers in LISA include optical cavities, Mach-Zander interferometers, molecularresonators, and arm locking. Optical cavity references and the associated locking
methods are well tested and robust but the EOMs and cavities used as a frequency
reference add to weight of the SC. Mach-Zander interferometers fit well with the
LISA interferometry base design but, in comparison to cavities, lack low-frequency
stability [34]. Molecular hyper-fine resonance is beneficial in providing an absolute laser
frequency reference but is generally complicated in implementation. Arm-locking has
the greatest advantage for LISA since it requires no additional hardware and can be
implemented completely through digital signal readouts and controls which are alreadyin the LISA design. At the same time, arm-locking is at a great dis-advantage since it
has only been tested through electronic simulations and a possible risk of failure when
implemented in LISA. (chapter. 6.3.1)
4.1.2 Clock Noise Transfers
The laser fields on each SC are modulated with up-converted5 clock noiseside-bands [74] such that the side-band beat-notes between laser fields from adjacentSC, ssr , also produce differential clock noise terms. Given that the clock signals are
up-converted by a factor, Gup , and that the side-band beatnote produced when the clock

4
5√
Generally, h(ω) < 100µCycles/ Hz
From 50 MHz to 2 GHz, Gup ≃ 40
70

from SC2 is transmitted to SC1 is measured with a phasemeter at a frequency of f∆−Clk ,we obtain:f∆−ClkφClk−Sidebands (t) = Gup (φClk1 + φClk2;3 ) +φClk1 ,
fClk1



f∆−Clk= Gup
1+
φClk1 + φClk2;3
Gup fClk1≃ Gup (φClk1 + φClk2;3 ) .

(4–3)
(4–4)
(4–5)

As we’ll see in chapter. 4.3, these are the same terms which show up in the TDI
combinations and are used to correct for this clock noise coupling.6
4.1.3 Ranging Errors
Before we outline the methods of inter-SC ranging in chapter. 4.4, we derive ageneral relationship between the ranging error and the input laser noise. Given that
we have a measurement of some φ(t) as well as some time-delayed measurement
φdelayed (t) = φ(t − τ ) + g(t), we time-shift and subtract these signals to extract g(t).
Assuming we have some time-shifting error, τ + δτ , when we fractional-delay interpolate
the time-delayed signal, φdelayed (t), and subtract it from the input signal we estimate thenoise cancellation. Using the Taylor approximation we can write,
XErr ≃ φ(t) − φ(t − τ + τ + δτ ),X̃Err ≃ [e−iωt − e−iω(t+δτ ) ]φ̃,|X̃Err | ≃ ωδτ |φ̃|.
where ω is the angular Fourier-transformation frequency. Now we can estimate a
simplified but reasonable measure of the relationship between laser noise cancellation

6

For more information on the clock noise transfers and corrections, see [68–70, 75].

71

and the ranging precision as a function of frequency for ωδτ << 1 with the relationship:
XErr (ω) = ω δτ φ(ω)αφ (ω).

(4–6)

Thus, given a certain ranging capability we can derive the required laser prestabilization,or visa-versa, given a laser pre-stabilization level we can derive the required ranging
accuracy to meet the interferometry sensitivity specifications. The complete TDI
derivation includes complex phase factors, αφ (ω), which may range in magnitude from 0
to 4 depending on the location of the arm-length dependent ∆sr zeros (chapter. 4.4.2.1).
4.1.4 Fractional Delay Filtering and Interpolation
Depending on the finalized design, LISA will produce satellite to Earth data-streamsof the TDI observables at a fdata = 3 − 10 Hz data rate. Integer sample shifts of thesedata-sets will result in a shifting error of 0.33 − 0.1 s/s, respectively. But, given the
expected laser pre-stabilization level, LISA will have to time-shift and time-scale thesedata-streams to a 3.3 ns accuracy7 . Thus, assuming a 10 Hz data-rate, LISA requires afractional shifting accuracy of δτ /Tdata = 3.3 ns/0.1 s = 3.3 × 10−8.
Fractional delay interpolation [72] is used to interpolate the data-sets and apply
the fractional shift [73]. In LISA applications, the Lagrange filter is ideally suited fordata-interpolation due to the constant low-frequency phase loss8 . Generally we canwrite the interpolated data-set, s(n − D) where D is the fractional shift D = ∆τ /Tdata , asa function of the input data-set, the Lagrangian filter window, and the sinc(x) function as:(N−1)
2s(n − D) =7

δL= 1 meter, δτ = 3.3 ns

8

constant group delayXk= −(N−1)2s(n + k) w(k) sinc(D − k)

72

(4–7)

where tD = D + (N − 1)/2,



πN
N−1
w(n) =,
B(td , N)B N − 1, n +sin(πtd )2(4–8)

and [76]
B(n, k) =Γ(n)Γ(k).
Γ(n + k)(4–9)

In the following description, we default to a filter length of N = 51 which provides us
with a shifting accuracy of greater than 10−12 [73]. Generalizing this to incorporatetime-scaling as well as time-shifting, we can write the time-changing time delay as afunction of the integer, nD , and fractional shift, D(t) as:τ (t) = τ (0) + βt = nD ∗ Tdata + D(t) ∗ Tdata .

(4–10)In this manner we advance the fractional delay, D, as a function of the data-sample, n,for each iteration of the loop: D(n) = D(0) + nβ. The MATLAB code required and used toperform the data filtering and interpolation is outlined in Appendix A.
4.2

Laser Pre-stabilization

Based on the expected ranging accuracy, the current LISA design requires a laser
frequency stability of:
280 HzδωPre−Stabilization (f) = √
Hz

s

1+



2.8 mHzf

4.

(4–11)

This could be achieved through a single stabilization method or by some combination of these methods. The costs and benefits of each of these methods are outlined73

well in the LISA frequency control white paper [34]. The following discussion will focus
on two methods: (1) Pound-Drever-Hall cavity stabilization9 and (2) arm-locking10 .
4.2.1 Pound-Drever-Hall Locking

Figure 4-1. Model of the PDH locking scheme: A model of the Pound, Drever, Hall laser
frequency locking technique is depicted. The laser field is modulated withside-bands and aligned through a polarizing beam splitter incident on thecavity. The back reflected field is used to control the laser frequency after
demodulating the side-bands.
The application of Pound-Drever-Hall (PDH) laser stabilization [50] involves
stabilizing the frequency of a laser to the length of an ultra-low expansion (ULE) cavity
such that an integer multiple of the laser’s wavelength, λ = 1064 nm, equals the length ofthe cavity:
Nλ = L.

(4–12)

9This is used as the pre-stabilized input for the TDI simulations as well as a stablereference against which to compare other laser stability measurements
10

These experiments mutually provided a proof-of-concept for the UFLIS electronics
along with the verification of the arm-locking stabilization methods themselves.

74

This gives us a relationship between the laser frequency, f + δf(t), and the cavity length,
L + δL(t) given by [52]:
δf(ω)
δL(ω)
=
,
f
L

(4–13)

transferring the length stability of the cavity to the frequency stability of the laser.
The PDH frequency stabilization is accomplished by modulating the laser
phase using an electro-optical modulator with MHz side-bands [51] as shown in
Figure 4-1. The phase relationship of these off-resonance side-bands reflectedfrom the cavity are scaled by a complex reflection coefficient which is a functionof the cavities mirrors’ reflection and transmission coefficients. Demodulating the
PD signal against the modulation generating oscillator using an electronic mixer
cancels the common frequency noise of the local oscillator11 and results in anerror signal which is proportional to the phase offset of laser carrier phase to thecavities resonant length. This error signal can then be used with the appropriate
proportional-integrating-differentiating (PID) or finite-impulse-response (FIR) control
electronics to feedback to the piezo-electric transducer (PZT) and temperature controlsof the laser output frequency. For more information on this locking technique and it’sapplications in LISA, refer to [34, 51, 52].
4.2.2 Arm Locking
Arm locking is a laser stabilization technique proposed by Sheard, et. al. [53] which
exploits the long-term stability of the LISA arm-lengths12 as a reference against whichto stabilize the long-term laser frequency. Utilizing the sensor signals from (3–72) and(3–74), we have a first-order measure of the change in the laser frequency over the11This could be a voltage controlled oscillator, a function generator, or any relatively
stable MHz oscillator
12

5 ± 0.1 Gm over one year.

75

individual round-trip arm-length delays with a transfer function given by:
TAL:s (s) =L(∆s1 (t))
L(φ1 (t))

(4–14)

= φ1 (s) 1 − e−sτ .

(4–15)

where the functional variable, ’s,’ is the complex Laplace frequency s = σ + iω while ∆s1
refers to the round-trip delay to SCs . A filtered output of a single arm sensor signal13[53, 77, 78] or some filtered linear combination of the two individual arm signals14[79–81] can be used to control the local, φ1 (t), laser phase with the complete open-looptransfer function given by:
TAL:Comp (s) = TSum (s)Asum (s) + TDif (s)Adif (s).

(4–16)

Asum (t) = ∆21 (t) + ∆31 (t),(4–17)

Adif (t) = ∆21 (t) − ∆31 (t)(4–18)

where

resulting in a laser noise suppression given by the closed loop transfer function:
TClosed =1
1 + TAL:Comp (s).

(4–19)The implementation of arm-locking requires real-time construction of the sensor signals
as well as some method of actuating the laser frequency either through a PZT-mounted
cavity [82], offset phase-locked lasers [78], or side-band locking [83]. Special caremust be taken to estimate the Doppler shifts since integration of Doppler errors cause

13Single arm-locking

14

Dual-modified arm locking

76

a frequency-pulling effect which limits the noise suppression and causes the laser
frequencies to drift as a function of the integrated Doppler error [84].
The electronic components described in chapter. 6 in combination with The
University of Florida Laser Interferometry Simulator (UFLIS) benchtop were used by
the author and others to perform the first hardware implemented proof-of-concept single
arm-locking experiment (chapter. 6.3.1) revealing the yet-to-be-discovered frequency
pulling effects [41, 77, 84]. Expansions to the simulator have also proven dual andmodified arm-locking configurations along with frequency pulling effects. [54, 81].
4.3 TDI Theory
Prior to now, we have simply mentioned that we must form particular time-shifted
and time-compressed linear combinations to account for the time-changing unequalarm-lengths and cancel the dominant laser phase noise but we have not definedthese specific combinations. These linear combinations fall into two major categories:(1) TDI-Sagnac combinations and (2) TDI-X combinations. The three TDI-Sagnac
combinations are constructed by completing the laser transfer chain around counter
propagating directions of the LISA constellation. These combinations are significantly
less sensitive to gravitational waves, thus providing an estimation of non-laser noise
sources such as PD noise or scattered light effects. The TDI-X combinations areconstructed by completing the laser transfer chain along individual arms of theinterferometer canceling the laser phase noise and extracting the GW signals. As
previously stated, the ability to form these combinations depends directly on theaccuracy of the measured arm-lengths (chapter. 4.1.3), but at the same time, these
combinations can be exploited to estimate the arm-lengths (chapter. 4.4.2.1).
4.3.1 TDI Combinations
We begin our analysis by constructing the two major TDI combinations which
are shown diagrammatically in Figure 4-2. The TDI-X combination is representativeof a Michelson-type interferometer while the TDI-ζ combination is representative of77

a Sagnac-type interferometer. The TDI formulations have not been consistent within
the literature [55] such that in this description, we will use the same notation as thosedefined in chapter. 3.4.3.
The TDI-analysis and combinations used depend on the dynamics of the orbits.
The different TDI generations are derived based on the orbital dynamics approximation
used. The TDI − X0.0 combination assumes the arm-lengths are constant, dτq (t)/dt = 0,
non-rotating τq (0) = τq′ (0), and equal, τq (0) = τq (0)15 . The TDI − X1.0 combination
assumes the arm-lengths are constant, dτq (t)/dt = 0 and non-rotating τq (0) = τq′ (0),but unequal, τq (0) 6= τq (0). The TDI − X1.5 combination assumes the arm-lengths areconstant, dτq (t)/dt = 0, but rotating, τq (0) 6= τq′ (0), and unequal, τq (0) 6= τq (0). Finally,
the TDI − X2.0 combination assumes the arm-lengths are non-constant, dτq (t)/dt 6= 0,
rotating, τq (0) 6= τq′ (0), and unequal, τq (0) 6= τq (0). These are outlined in Table 4-1.
Thus, the TDI − X2.0 combination should completely account for the laser noise coupling
given the linear arm-length rate of change we have considered in the previous sections.
Assuming we have a continuous data-stream of the observables for SC acceleration
terms to have an effect on the data combinations, this could be accounted for withfurther expansion of the TDI combinations. This is usually un-necessary given thelikely-hood of a ’unbroken’ data-set, but none-the-less, we will see in chapter. 4.4that this can be accounted for, given the LISA-orbital dynamics, with segmented
data-analysis.

15

The τq refers to the ’opposite’ arm of the τq -arm vs. τq -arm Michelson-type
interferometer.

78

Table 4-1. Orbital dynamics approximations for TDI generations
Generation Michelson Arm-Length Counter-Propagating Delay
TDI 0.0
τq (t) = τq (t)τq (0) = τq′ (0)TDI 1.0
τq (t) 6= τq (t)τq (0) = τq′ (0)TDI 1.5
τq (t) 6= τq (t)τq (0) 6= τq′ (0)TDI 2.0
τq (t) 6= τq (t)τq (0) 6= τq′ (0)TDI 3.0+
τq (t) 6= τq (t)τq (0) 6= τq′ (0)Delay Dynamicsdτq (t)/dt = 0dτq (t)/dt = 0dτq (t)/dt = 0dτq (t)/dt = βqdτq (t)/dt = βq (t)Figure 4-2. Diagrammatic models of the TDI-X and Sagnac combinations: Here we
present the geometric representation of the LISA constellations in parallel tothose developed in [85]. The laser transfer chain in these diagramatic
representations show how to time-shift the observables to form the TDI
combinations and ensure that the laser noise sources will cancel when thelaser transfer chain is a closed loop.
4.3.2 Sagnac Combinations
4.3.2.1 TDI Six-Pulse Combinations
The first order six-pulse combination, named α are constructed from the θsr
combinations (3–77 and 3–79) from each of the respective SC and take the form:
α1.0 = θ21 − θ31(4–20)= φ1 − φ1;2′ 1′ 3′ − [φ1 − φ1;312 ]+ [h21 + h32;3′ + h13;1′ 3′ ] − [h31 + h23;2 + h12;12 ]= φ1;312 − φ1;2′ 1′ 3′ + [h21 + h32;3′ + h13;1′ 3′ ] − [h31 + h23;2 + h12;12 ].

79(4–21)

Generally16 , these combinations cancel the laser phase terms and leave six hsr GW
terms in the TDI − α1.0 approximation. They are named based on the fact that adelta-function gravitational wave input is replicated six times in the resulting combination.
If we extend this to the TDI − α2.0 approximation by accounting for time-changing
delays and un-equal counter-propagating light travel time-delays, we would have totrace the laser chain back to the starting point resulting in a 12 ssr -term expression and a
twelve-pulse GW response.
4.3.2.2 TDI Symmetric-Sagnac Combination
The first order Sangnac combination written as:
ζ1.0 = s31;1 + s12;2 + s23;3 − (s21;1′ + s32;2′ + s13;3′ )(4–22)

is free of laser phase noise in a non-rotating constellation and is orders of magnitudeless sensitive to GW signals. This could be used to discriminate between instrument
noises and GW stochastic background signals [86, 87]. This combination is showngeometrically in Figure 4-2; one notes that the rotation of the constellation in thegeometric representation causes an open laser transfer chain for a rotating constellation
which destroys the common mode laser phase cancellation the Sangac-1.5 combinations [85, 88]. Again, this is accounted for by the modified Sangac combination which
re-traces the time-delay path resulting in a 12-term TDI − ζ2.0 function.
4.3.3 Michelson X-combinations
The Michelson X combinations17 form three interferometers combinations whichcollectively form a basis in the 2-dimensional plane of the constellation for the h+ and

16

Ignoring clock-noise terms

17

Using other SC as our frame of reference, we can obtain the ’Y’ and ’Z’
combinations

80

h× strain. In the following sections, we construct the Michelson X combinations from theround-trip single-arm sensor signals, ∆sr , defined in (3–72) and (3–74).
4.3.3.1 TDI-X0.0
In the special case where the total round trip delay-times are equal ([τq (0) + τq′ (0)] =[τq (0) + τq′ (0)]) and the differential SC velocities are zero (dτq (t)/dt = βq = 0), thedifference of the sensor signals, (3–72) and (3–74),
X0.0 = ∆21 − ∆31 ,(4–23)

generates the standard equal-arm Michelson interferometer output, independent of laser
phase noise. Expanding explicitly,
X0.0 = φ1 − φ1 (t − τ3′ (0) − τ3 (0)) + h21 (t) + h12 (t − τ3′ (0))+

(4–24)

f12
f21
φClk−1(t) +φClk−2 (t − τ3′ (0))
fClk
fClk

− [φ1 − φ1 (t − τ2 (0) − τ2′ (0)) + h31 (t) + h13 (t − τ2 (0))
+

f31
f13
φClk−1 (t) +φClk−3 (t − τ2 (0))]
fClk
fClk

which reduces in this special case to:
X0.0 = h21 (t) + h12 (t − τ3′ (0)) − h31 (t) − h13 (t − τ2 (0))+

(4–25)

f21
f12
φClk−1 (t) +φClk−2(t − τ3′ (0))
fClk
fClk
f31
f13
−
φClk−1 (t) −φClk−3 (t − τ2 (0))
fClk
fClk

where we see the four-pulse GW response and clock noise coupling. Maintaining theβ = 0 assumption, the laser fields are not Doppler shifted and the beatnote frequencies
on opposite SC will be equal such that we can further simplify this combination and write81it as:X0.0 = h21 (t) + h12 (t − τ3′ (0)) − h31 (t) − h13 (t − τ2 (0))+

(4–26)

f21[φClk−1 (t) + φClk−2 (t − τ3′ (0))]
fClk
f31
−
[φClk−1 (t) + φClk−3 (t − τ2 (0))].
fClk

We notice that these clock noise terms are the same clock-noise transfer terms
described in (4–3). Rescaling the clock-noise transfer sideband measurements andsubtracting them from the TDI − X0.0 combination, we obtain a Michelson GW output
free of any other noise sources:
X0.0 = h21 (t) + h12 (t − τ3′ (0)) − h31 (t) − h13 (t − τ2 (0))

(4–27)

Now that we have shown how the clock-noise terms are accounted for, we will ignore
them in the rest of the TDI description. Light field Doppler shifts from inter-SC motion
change the beatnote to clock frequency ratio, (fsr ± fDop )/fClk, but can be subtracted fromthe sensor signals ∆sr before any of the following combinations are formed. That said,
we will revisit how these terms couple into the UFLIS-TDI simulations described in Ch. 7.
Unfortunately, despite all this work, the TDI−X0.0 combination is rarely a reasonable
laser phase cancellation technique since the LISA arm-lengths are almost always
un-equal. None-the-less, this serves as a first order example of how heterodyne GW
interferometry is performed. The considerations beyond this show the techniques toaccount for orbital dynamics and changes in the light-travel time-delays between the SC.
4.3.3.2 TDI-X1.0
The TDI-X1.0 combination [89], written as
X1.0 = ∆21 − ∆31 − ∆21;2′ 2 + ∆31;33′ ,(4–28)

replicates the equal-arm phase delays and cancels the common laser phase noise in thecase where ([τ2 + τ2′ ] 6= [τ3 + τ3′ ]) and β2 − β3 ≃ 0 as shown geometrically in Figure 4-2.
82

Expanding explicitly, but still allowing for non-zero β-values, we can write:
X1.0 = [φ1 − φ1;33′ ] − [φ1 − φ1;2′ 2 ] − [φ1 − φ1;33′ ];2′ 2 + [φ1 − φ1;2′ 2 ];33′(4–29)+ [h21 + h12;3′ ] − [h31 + h13;2 ] − [h21 + h12;3′ ];2′ 2 + [h31 + h13;2 ];33′= [φ1;33′ 2′ 2 − φ1;2′ 233′ ](4–30)+ [[h21 − h21;2′ 2 ] + [h12;3′ − h12;3′ 2′ 2 ]] − [[h31 − h31;33′ ] + [h13;2 − h13;233′ ]].
From the expansion we notice the TDI − X1.0 eight-pulse GW response. Failing toaccount for the SC-motion and time-changing delays, the TDI-X1.0 combination is limitedby the [φ1;33′ 2′ 2 − φ1;2′ 233′ ] terms which do not cancel completely since the time-delaytransformations are performed in a different order. Calculating the delay error as a resultof the transformation order using the leading (1 − β) terms18 we can write:δτ = τ (4[1 − β2 ] − 4[1 − β3 ]) = 4τ [β3 − β2 ](4–31)

where τ is the mean one-way delay time19 . Evaluating this the result in (4–6), we
obtain[90]:X̃1.0 > 4τ |β2 − β3 |φ̃˙ 1(4–32)

where φ̃˙ 1 is the time-differentiated laser phase spectrum. Given orbital characteristics
where this limit is large enough to restrain the IMS sensitivity, we must further expand to
the general TDI − X2.0 combination to account for this residual noise.

18

We maintain the assumption that β << 1, (1 − β) ≃ 1, and approximate (1 − β)n = 1.

19

τ ≃ 16.7 s

83

4.3.3.3 TDI-X2.0
The TDI-X2 combination, written as [91, 92],
X2.0 = ∆21 − ∆31 − ∆21;2′ 2 + ∆31;33′(4–33)− ∆21;33′ 2′ 2 + ∆31;2′ 233′ + ∆21;2′ 22′ 233′ − ∆31;33′ 33′ 2′ 2
is used to cancel the, [φ1;33′ 2′ 2 − φ1;2′ 233′ ], laser phase noise terms left in the TDI − X1.0
combination. This produces a 16-pulse response to gravitational waves, written explicitlyas:X2.0 = [[h21 + h12;3′ ] − [h31 + h13;2 ] − [h21 + h12;3′ ];2′ 2 + [h31 + h13;2 ];33′ ](4–34)− [[h21 + h12;3′ ];33′ 2′ 2 − [h31 + h13;2 ];2′ 233′
− [h21 + h12;3′ ];2′ 22′ 233′ + [h31 + h13;2 ];33′ 33′ 2′ 2 ]but, given the correct time-delays, cancels all the laser phase noise and accounts for the
independent linear-time-delays, τq (t), assuming d2 τ (t)/dt2 = 0. Annual changes in β, orSC acceleration terms (d2 τ (t)/dt2 6= 0), may be accounted for with the further expansionof these TDI combinations although, this is unnecessary as argued in following section(chapter. 4.3.4). The MATLAB code required to form these data-combinations using thessr data-sets is provided in Appendix B.
4.3.4 LISA Orbital Dynamics and TDI Data Analysis
In the following experiments, β is assumed to be constant and we will focus on
the TDI-X2.0 velocity corrections. Thus, in order to utilize the TDI-ranging methods
outlined in chapter. 4.4.2.1 for LISA-TDI data-analysis, the β value will have tobe adjusted to avoid the acceleration-dependent accumulated error. Although acontinuous measure and correction to the β values are possible, this can simply be
accomplished by segmenting the data-analysis, in the worst case LISA-like scenario,ppevery δτ Tyear /(β π) = 3.3 ns ∗ 3.15 × 107 s/(66 ns/s ∗ π) = 708 s [55]. Given thetime-frame for the acceleration effects to couple into the data-analysis, even in this84

worst-case scenario, it is easier to adjust the β-value in the data-analysis than it is toform a TDI − X3.0 , 32-pulse GW response, combination.
4.4 Ranging
As we have stated in chapter. 4.1.3 and as we can see explicitly from the ∆sr and
TDI − X1.0/2.0 combinations, the laser noise cancellation directly depends on our ability
to estimate the one-way inter-SC arm-lengths and form these linear combinations. From√
(4–6) we can calculate the ranging accuracy needed to cancel the 280 Hz/ Hz laser
pre-stabilization input noise.
Generally, the cancellation of the local φ1 (t) laser phase noise from the far s1s
signals depends on the accuracy of the out-going delay times, τ3 (t) and τ2′ (t), while
the cancellation of the far laser phase noise from the local ss1 signals depends on theaccuracy of the in-coming delay times, τ3′ (t) and τ2 (t). Assuming the lasers on separate
SC are independently stabilized such that φ1 6= φ2 6= φ3 , then each of the one-way
inter-SC time-delay functions, τq (t) = τq (0) + βq t, must be evaluated to a 1 meter (3.3 ns)
accuracy.
Exploiting the phase-locking techniques described in [71], we can phase-lock the
far lasers, φ2 and φ3 , to the delayed φ1 field from the master SC and transfer the stabilityof the master laser to these far lasers. This results in expressions for the far SC’s laser
phase noise given by:
φ3 (t) = φ1;2′ ,φ2 (t) = φ1;3 ,(4–35)

to within the tracking accuracy of the PLL. In this special case, the one-way delays
are accounted for by controlling the far sensor signals with a PLL such that, s1s =φPLL−Error ≃ 0. Taking the expressions for φ2 and φ3 described in (4–35), and evaluatingthem in ∆s1 (3–72 and 3–74), we see that these expressions simplify to:∆21 (t) = s21 (t),∆31 (t) = s31 (t).

85

(4–36)

Thus, since the s1s terms do not effect the noise cancellation in this phase-locked
configuration, we can reduce the delay constraints to two, round-trip delays given by:
τ2′ 2 (t) = [(1 − β2 )((1 − β2′ )(t − τ2′ (0)) − τ2 (0))](4–37)

τ33′ (t) = [(1 − β3′ )((1 − β3 )(t − τ3 (0)) − τ2′ (0))].
Meanwhile, the constraints on the individual one-way delays are greatly reduced
resulting from the relatively-stable20 PLL noise. These reduced ranging constraints
result from the fact that the ’input’ PLL noise used in (4–6), allows for large values of δτ
while still meeting the X-combination’s IMS sensitivity.
Two methods, Pseudo-random Noise (PRN) ranging [75, 93, 94] and TDI-Ranging[95] have been proposed to measure the inter-SC arm-lengths (light-travel times). The
PRN ranging method, described in the next chapter involves the use of additional opticalcomponents to modulate the laser carriers with PRN-codes; the cross-correlation of thesix inter-SC ranging codes with the local copies of each of these PRN codes provides
an independent, real-time measure of the one-way inter-SC distances. TDI-Ranging,on the other hand, requires no additional components and determines the inter-SC
ranging values in post-processing by exploiting the laser cancellation characteristics of
the TDI-combinations. Provided with the ease of implementation of the TDI-Ranging
technique, the TDI experiments described in chapter. 7 will attempt to experimentallydevelop and improve upon the TDI-Ranging methods proposed in [95].
4.4.1 Pseudo-random Noise (PRN) Code Ranging
Pseudo-random noise (PRN) code cross-correlation techniques are well understoodinter-device distance tracking methods which have been developed and verified for use
in global positioning satellites (GPS). The application of these methods in LISA involves

20√√
1µHz/ Hz − 1mHz/ Hz86modulating one of six local codes (one for each laser source) onto the carrier laser
field. After the inter-SC/intra-SC laser field transmission and detection, the PRN codesfrom the adjacent laser field is cross-correlated with the local PRN codes. The six PRN
codes have the designed property such that they do not correlate with each-other, but
do periodically correlate with themselves depending on the length of the PRN code. A
positive cross-correlation of the PRN code is used in conjunction with a delay-lock-loop
to actively track the ’incoming’ code from the adjacent benchtop and produce aninter-SC delay read-out. Assuming the measured delays correctly account for electronicand special relativistic delay terms, these delay read-outs can be directly used to form∆sr and TDI − X1.0/2.0 combinations in real time.
This ranging method has both benefits and complications. The real-time PRN delay
measurements allow us to form the ∆sr and TDI − X1.0/2.0 combinations in real-time on
the SC. The ∆sr terms can be used for arm-locking as described in chapter. 6.3.1. The
TDI − X1.0/2.0 combinations can be formed on-board from the ssr observables and sentin their pre-constructed TDI-form to Earth rather than having to transmit the 18 individual
χsr , bsr , and ssr observable signals.
On the other hand, as we’ll see comparatively in the next section, this ranging
method adds unnecessary complications to the LISA design. Electro-optical modulators(EOMs), which might introduce additional noise terms, must be used to modulate thelaser-field with the PRN codes before the inter-SC transmission, adding weight andcomplexity to the LISA design. Also, there is no guarantee that the delays measured
by the PRN ranging methods are equal to those needed to form the TDI combinations
since the laser noise cancellation in the TDI-combinations depends on the time-delay
from the laser field’s generation to observables’ detection.21 TDI-ranging, on the other21

Including PD, ADC, and phasemeter phase delay responses.

87

hand, does not require an additional EOM and determines the one-way delays, including
electronic phase delays, in post-processing.
4.4.2 Time-delay Interferometry Ranging (TDIR)
Time-delay interferometry ranging [95] is a method of determining the LISA signal
travel times in post-processing by minimizing the total root mean squared (RMS) powerin the LISA measurement frequency band. Taking the measured, ssr , combinations, weform the theoretical TDI-2.0 combinations using initial estimates of the six variable,22
light travel time delays. The time-delay parameters are scanned over the possible
values23 until the total RMS power in the TDI − X2.0 is minimized. The initial delay offset
parameters, τq (0), can vary by 0.66 seconds and must be measured to an accuracyof less than 3.3 nanoseconds (4–6). Thus, assuming no previous knowledge of thetime-delay values, we have a large parameter-space over which to scan with each of thefour time-delay offset parameters taking one of 2 × 108 possible values.24 Unless sometracking method is developed, this parameter search could be computationally intensive.
Although [95] proves the capabilities and limitations of the RMS minimization
ranging method, it does not provide a method of actively determining the timedependent
delay parameters. In addition, the RMS minimization method’s delay parameter
calculation introduces errors in the time-delay values caused by low-frequency
gravitational wave signals. In this experiment, we will present and employ a new methodof TDI-ranging by modulating the laser field with a ranging tone at a frequency outsideof the LISA measurement band, in this case, 1 − 1.5 Hz. Although this method has beenused for spacecraft ranging on active projects [96] and has been considered for use on
LISA [97], the author is not aware of a formal analysis in the literature. In the following

22

We assume that βq = βq′ and τq (0) 6= τq′ (0).

23

(16.33 s < τq (0) < 17 s), (−66ns/s < β < 66ns/s)

24

0.66 s/3.3 ns = 2 × 108
88

sections, we attempt to present this formal analysis of the TDI ranging-tone application
and realization, including a time-delay search algorithm developed by the author.
4.4.2.1 TDI Ranging Tone
The TDI-ranging tones can be placed on each of the six LISA laser fields by adding
a fTone − Hz sinusoidal modulation to the piezo-electric transducer which controls thelaser frequency. This is significantly more straight-forward than the EOMs required to
implement the PRN-ranging method. On the other hand, the real-time measure of thearm-lengths provided by the PRN ranging method has the advantage of producing thesensor and TDI variables in real-time which is beneficial towards the implementation ofarm-locking and reducing the satellite-to-Earth data transmission demands.
Extending the RMS minimization concept, the application of a ranging tone
effectively increases the laser noise by introducing more signal power at the specified
ranging tone frequencies. As shown by (4–6), this results in a better estimation of theranging error, δτ , than the inherent laser noise cancellation would provide. In addition,
because we are only interested in the power minimization near the ranging tone
frequency, this ranging method should not be affected by low frequency gravitational
waves.
The ranging tone is optimally modulated onto the laser field at a frequency mid-way
between the frequency-domain zeros of the inter-SC sensor signals, ∆sr , to avoid
inherent tone cancellation along a single arm25 :N+1
N< fTone <.
τq (0) + τq′ (0)τq (0) + τq′ (0)(4–38)

Also, to avoid confusion between the individual laser modulations, each of the Lasersr
fields should be modulated at a different frequency, fTone−sr . Using the reverse argumentof the ranging requirement derivation, the cancellation of the local φ1 (t) tone from the far

25

This can be seen graphically in Figure 7-4.

89

s1s signals constrains the out-going delay times, τ3 (t) and τ2′ (t), while the cancellation of
the far laser tones from the local ss1 signals constrains the in-coming delay times, τ3′ (t)and τ2 (t), resulting in all four one-way time-delay functions.
Revisiting the LISA-model where the far lasers are phase-locked to the delayed
master laser field, (4–35), the delayed local laser modulation is transmitted back to to
local SC. This results in two round-trip time-delay function constraints, (4–37), using thecancellation of the local tones in the TDI combinations. In spite of this, the phase-locked
lasers could still be modulated with their own ranging tones restoring all four, one-way
constraints.
4.4.2.2 TDI Ranging Parameter Search Algorithm
The method used to determine the six-parameters which define the four one-way
delay-functions is outlined in Figure 4-3. The first step involves providing an estimateof the time-delay functions. The estimate does not need to be accurate; any estimation
error will be corrected by the convergent properties of the search algorithm. The βestimates are then used to time-scale the ssr signals by the appropriate factors as
defined by the TDI − X2.0 combination26 . The time-scaled data set is then broken into
’N’ sections which are individually used to determine the four time-delay offsets, τq (0),for each section.
As we’ve previously stated, the span of possible values of the time-delay offsets incomparison with the required time-delay ranging accuracy results in a large parameter
space over which these values can vary. Brute force scanning and computing the TDI
combination for all 1032 values is effectively impossible. Instead, we begin by evaluating
the TDI combination for each-time delay in the range of possible values from 16.2 s to
17.2 s with a precision of 0.1 s. In this specific case, the delay-segmentation results

26

The TDI-X2 combination is used in favor of the the TDI-X1 combination because of
the possible inherent constraints of the TDI-X1 : (4–32)

90

in 10 possible delays values for each of the four dimensions of the one-way delay
times, resulting in 104 possible delay permutations. The RMS power of the TDI-X2.0
combination in the fTone Hz ± 1 mHz frequency band is calculated for each of the 104
possible values. The set of delays which contains the minimum RMS power is used asa new reference point. The time-delay parameter space around this minimum (±0.1 s)
is, broken into more precise delay values (0.01 s). The RMS power minimization scanin this new parameter space is performed to, again, determine the delay parameters
to a better precision. The process is repeated improving the precision by a factor of 10
each time until the four delay-time offsets are constrained to a 1 ps precision and theranging tone is dominated by instrument noise sources.27 The delay segmentation
precision, in this case a factor of 10 for each iteration, is chosen to avoid the possibilityof converging on a local minima and obtaining the wrong delay-time estimate. A more
effective method might include the evaluation of a surface gradient which converges
on the RMS-minimized delay times which has been fitted to the time-delay gridthrough Monte-Carlo analysis; this method could accelerate the data-analysis, assist
in confirming the result is not a local minima, and ensure that the actual optimized
delay-times have been determined.
Once the four time-delay offsets are calculated for each of the N data-segments, alinear regression of the offsets is performed to obtain a functional definition of all four ofthe one-way time delays:τ2 (t) = (1 − β2 )(t − τ2 (0)),τ3′ (t) = (1 − β3 )(t − τ3′ (0)),τ2′ (t) = (1 − β2 )(t − τ2′ (0)),τ3 (t) = (1 − β3 )(t − τ3 (0)).

27(4–39)

Although we scan the delay parameters to a 1 ps precision, the actual delay error isdetermined based on the ranging tone cancellation which is limited by instrument noise
sources.

91

The calculated slope provides a more accurate evaluation of β-values for this section ofdata than the original estimation. Assuming the ranging-tone cancellation and time-delay
offset calculation was limited by (4–32), the improvement in the β estimation increases
the precision of the time-delay offsets in the second iteration of the ranging-tonecancellation algorithm. Generally, after three iterations the one-way delay functions
are optimally evaluated. The first iteration determines the β values. The second
iteration determines the delay-offsets. The third iteration optimizes the values overthe entire data-set. Once the four one-way delay functions are derived, they are used
to calculate the TDI-X2 combination for the entire data-set. In addition, the variance on
the linear-regression provides us with a means of determining the ranging precision incomparison with the precision defined by the ranging tone cancellation using (4–6). The
process is simplified to a four-parameter (β2 , β3 , τ2′ 2 (0), τ33′ (0)) estimation when the farlaser are slave phase locked to the master local laser.

92

93
Figure 4-3. Flow chart of the ranging-tone minimization process: The process depicted by this flow chart minimizes theranging tone and maximally constrains the six variable light travel time delays. The results of this process forthe different experimental configurations are presented in Table 7-1 and Table 7-2

CHAPTER 5
THE LISA PHASEMETER
5.1 Digitial Signal Processing (DSP) System
The phasemeter read-outs, digital control systems, and electronic inter-SC delay
simulation components were developed using programmable DSP equipment acquired
from Pentek, Inc. The Pentek system is constructed from three basic components. The
62561 is a four-channel, 14-bit, analog to digital signal converter (ADC) sampling card
containing a field programmable gate array (FPGA) capable of high-speed real-time
fixed-point data-processing. Meanwhile, the 62282 is a four-channel, 16-bit, digital toanalog converter (DAC) read-out card with the same FPGA capabilities. These two
daughter cards are connected through the master processing and control card, the4205.3 The 4205 handles the data transfers between the daughter cards and can becontrolled through a serial based user interface. The 4205’s processor can perform
floating point calculations on the measured data and store the results in 1 GByte ofsynchronous dynamic random-access memory (SDRAM) or send the data through an
Ethernet transfer to an external data-storage computer. Both the input ADCs and output
DACs can be externally clocked either by the same source or by two independent clock
sources.
5.2µCycle Phase Measurements

The phasemeter, programmed to the 6256’s FPGA, is designed to measure the√phase of a 2 − 20 MHz PD beatnote signal with an accuracy of 1 µcycle/ Hz (Table 3-1).
This device is used, not only to measure LISA-like science observables, but also to
generate the phase data for the electronic simulation of inter-SC field transmission delay

1

Model 6256 Dual/Quad 105 MHz A/D w/Virtex-II Pro FPGA - VIM-2

2

Model 6228 4-Ch. D/A, Digital Up-converter & FPGA VIM-2 Module

3

Model 4205 VIM/PMC Carrier and MPC7457 PowerPC VME Board

94and phase modulation effects. It can also be used to generate high-speed frequency
read-outs for real-time digital control systems such as arm-locking. In the following
sections we outline the design, characteristics, noise sources, and performance of the
University of Florida phasemeter.
in(t) = Ain(t) sin(in(t))

cos(fb(t))

Ierr(t) = [Ain(t) / 2] cos[in(t)-fb(t)] ≃ [Ain(t) / 2]
Qerr(t) = [Ain(t) / 2] sin[in(t)-fb(t)] ≃ 0

ffb(t)

fb(t)

ferr(t)sin(fb(t))
out(t) = Aout(t)sin(2foutt + out(t))

err(t) ≃ [Ain(t) / 2] [in(t)-fb(t)]foffsetFout(t)Figure 5-1. A model of the LISA phasemeter: Here we show how a digital offset phaselock loop is locked to the sampled input signal. Meanwhile, the feedback
frequency signal, ffb(t), and the multiplier demodulated outputs, Q(t) andI(t), are recorded to reconstruct the phase and amplitude of the input signal.
The details and theoretical analysis of the phasemeter operation can befound in the text (chapter 5.2.1)
5.2.1 Design
The phasemeter design is modeled off of a standard offset phase lock loop (PLL)
tracking controller. The name ’phasemeter’ is misleading since the PM core actually
records a 64-bit frequency-proportional feedback signal in a digital PLL which is thenintegrated in post-processing to generate the phase. As shown in Figure 5-1, an input
signal, Φin (t) = Ain (t)sin(φin (t)), is sampled with a 14-bit accuracy at a sampling95

frequency, fs .4 The digitized beatnote is then mixed with sine and cosine components
of a digital feedback signal with a constant amplitude, Afb , and a time varying feedbackRphase, φfb (t) = dt(ffb (t) + foffset ), given a user-defined offset frequency, foffset . The
feedback sinusoids, Afb sin(φfb (t)) and Afb cos(φfb (t)), are generated with numericallycontrolled oscillators (NCOs) using a ΩLUT = 28-bit look-up table (LUT). Oncethe feedback and input signals are multiplied,5 the signal is filtered with a N stage
cascaded-integrating-comb (CIC) filter [98]. The data is down-sampled by a factor of R
and low-pass filtered by the CIC-transfer function such as those shown in Figure 5-2.
The transfer function of the CIC filter in the Laplace domain takes the form:
G(s) = R1−N

(1 + z−R )N(1 − z−1 )Nswhere z = e− fs .

(5–1)

An example of the CIC transfer functions for fs = 50.0 MHz, Rfb = 16, and N =2, 4, 8 stage filters are plotted in Figure 5-2. fs is chosen at a frequency with a lower
bound defined by the Nyquist sampling frequency, fs > 2fNy , where fNy is the signals
largest frequency component of interest, and with an upper bound defined by thetiming constraints of the FPGA and ADCs. A larger down-sampling factor, R, reduces
the tracking loop’s update rate, reducing the timing requirements of the FPGA, butalso results in an increase of the PLL’s in-loop phase delay, reducing the tracking
bandwidth. The number of stages, N, defines by what factor the high-frequency data issuppressed before down-sampling and aliasing the high-frequency information into themeasurement. With regards to the LISA mission, a nice benefit of using the CIC filter forthis down-sampling process is that the ’zeros’ of the transfer function are aliased to DC,
significantly suppressing any out-of-band information before it is aliased into the LISA
science data frequency band (DC - 1 Hz). This is discussed further in chapter 5.2.2.2.

4

fs = 40 − 100 MHz

5

The 14-bit ADC out and the 28-bit LUT sinusoid result in a ΩQ = 42-bit I/Q precision

96

Suppression Magnitude

0

10

−20

10

−40

2 Stage
4 Stage
8 Stage

0

0.5

10

400

1.5

2

2.5
3
Frequency (Hz)

3.5

2.5
3
Frequency (Hz)

3.5

4

4.5

5
7

x 10

2 Stage
4 Stage
8 Stage

300
Radians

1

200
100
0
0

0.5

1

1.5

2

4

4.5

5
7

x 10

Figure 5-2. CIC filter transfer functions: The magnitude and phase response of theN = 2, 4, 8 stage CIC filters using a data-rate, fs = 50 MHz, and adown-sampling factor, R = 16, is plotted. A larger number of stages,
although increasing the suppression of higher-order aliased frequency
bands, also causes an increase in the filter’s phase delay.
Switching our focus to the PM’s PLL tracking loop, if we assume that fin =dφin (t)/dt ≃ foffset , and suppress the sum term of the demodulated signal with a
CIC low-pass filter, it results in the sine and cosine of the difference phase, φerr (t) =φin (t) − φfb (t). The CIC filter also reduces the tracking loop’s data rate to fCore = fs /Rfb.
As a result of the sine/cosine multiplication and filtering we obtain two signals which areproportional to the in-phase and quadrature components of the feed-back error signal,
Afb Ain (t)sin(φerr (t))
2
Afb Ain (t)cos(φerr (t)).
I(t) ∝ G(s)[Ain (t)sin(φin (t))Afbsin(φfb (t))] =2

Q(t) ∝ G(s)[Ain (t)sin(φin (t))Afb cos(φfb (t))] =

(5–2)
(5–3)

Using Q(t) as the error signal for the PLL, it is filtered using fixed-point bit-shifters andcascaded accumulators which are constructed to produce the feed-back controller, H(s),97given by:H(s) = 2

−N1

+2

f2
+ 2−N3 s2ss−N2 fs



.

(5–4)

The N1,2,3 values are the bit-shifting factors in the P-I-I2 feed-back filter, where N1 <N2 < N3 ,6 defines the gain and frequency-zeros of the P-I-I2 filter. This filter generates
the phase-proportional feed-back frequency signal, ffb (t), with a 60-bit precision7 . The
user-defined offset frequency, foffset , is added to this feed-back signal, integrated with afixed-point accumulator8 to generate the feed-back phase, φfb(t), and used to drive the
NCO/LUT mentioned above. Assuming the system is ’locked’ the feedback signal should
track the input signal such that, within the bandwidth of the controller:
Q(t) ≃ 0

φfb(t) ≃ φin (t).

(5–5)

Since we are interested in the phase fluctuations, the frequency feedback signal,
ffb (t), will be integrated to generate the in-band phase output, φin−band (t). Using thefeed-forward, G(s), and feed-back, H(s), transfer functions in combination with basic
control theory [99], we can calculate the expected closed-loop in-band and out-of-band
transfer functions based on the feed-forward and feed-back transfer functions as defined
above:"#Ain H(s)G(s)≃1s+ Ain H(s)G(s)fs#"
Ain G(s)δQ(s)≃ 0.
=Afb s+ Ain H(s)G(s)δφin (s)fsδφfb (s)=Afbδφin (s)6

For most designs, N1 ≃ 11, N2 ≃ 18, N3 ≃ 28

7

Ωfb = ΩQ + N3 = 42 + 28 = 70 bits, which are then truncated to 60 bits.

8

TAccum (s) = fs /s

98

(5–6)
(5–7)

The time-changing input amplitude coupling, analyzed in chapter 3.4.2.5, isassumed as being constant for this feed-back analysis such that amplitude fluctuationsof the input will have little affect on the transfer function. From the definitions in (5–6)
and (5–7), we can interpret these as the in-band and out-of-band frequency information
based on these closed-loop transfer functions:φin−band (t) = φfb (t)φout−of−band (t) = atan



Q(t)
I(t)



= φerr (t).

(5–8)

Thus, we can reconstruct the entire input signal’s phase information by simply adding
these signals:
φOut (t) = φin−band (t) + φout−of−band (t) ≃ φin (t)(5–9)

We can also calculate the input amplitude:
AOut (t) =p

I(t)2 + Q(t)2 ≃ Ain (t)(5–10)The expected bandwidth, depending on the amplitude of the input signal and thetiming delays of the FPGA, should be at least a few kHz such that in the LISA frequency
band, the phase information of the input signal is completely contained in the in-band
information and, thus, we can ignore the out-of-band information:fout−of−band (t) ≃0φOut (t) =φfb (t) ≃ φin (t)(5–11)(5–12)

This eliminates the need for the sine component of the mixer multiplication and reduces
the constraints on the FPGA design if amplitude measurements are not needed.
To generate the phasemeter data, the Ωfb = 60-bit feedback signals, ffb(t),for each of the four ADC inputs and PM outputs are packed into a series of 32-bit
values, transfered through the DSP system to the 4205, and communicated to a99

data-storage computer through an Ethernet connection. This provides us with afrequency quantization precision of δf = fs /2Ωfb (Figure 5-3).
In the following chapters the sampling rate and down-sampling factor vary basedon the experiment being performed and the computational demands on the FPGA.
The primary clock rates used are 40 MHz, 62.5 MHz, and 100 MHz9 . To get an ideaof the frequency precision, taking fs = 50.0 MHz, we produce frequency data with aquantization precision of 50.0 MHz/260 = 54 pHz. This is unnecessarily precise, butsince the FPGA and processor perform computations based on 32-bit ’words,’ and the32-bit precision, 62.5 MHz/260 = 14.5 mHz, is not sensitive enough to meet the LISA
requirements, we are required to use two ’words’ per data-point which allows for up-to
64-bit fixed point frequency outputs.10 This is explored more in chapter 5.2.2.1.
5.2.2 Phasemeter Readouts
Now that we have presented the design and capabilities of the PM core, the 60-bit,
fCore = fs /Rfb frequency output data can be used for three purposes: (1) to record
LISA-like science data of PD observables, (2) to generate the phase/frequency data
for simulating the inter-SC phase/frequency delays, (3) for high-speed digital feedback
control systems such as arm-locking and phase-locking. Maintaining the LISA precision
throughout the system, we keep the 60-bit frequency precision for all possible LISA-likeusages of the PM core. The read-out rate, on the other hand, varies depending on
the application. LISA science measurements call for 3-10 Hz PD phase readouts [29].
Simulations of the inter-SC delays require a relatively high data-rate to maintain thephase data-rate for accurate data-interpolation (chapter 6.2.2.1) and to prevent aliasing
into the measurement band when electronically replicating the LISA-like laser field

9

LISA is expected to use a 50 MHz ultra stable oscillator (USO) clock [29].

10

4 bits are used to label the frequency value with an indexing number based on theassociated ADC channel.

100

delays. PM readouts for digital control systems such as arm-locking use the maximum
allowed data-rate to minimize the phase delays applied by the digital controller which
reduce the tracking band-width.
To achieve the desired data-rate for LISA-science measurements and electronic
phase delay (EPD) simulations, the PM frequency readout data is down-sampled again
by CIC filters. The CIC filters down-sample the data to a fast, fPLL ≃ 400 kHz, medium,
fEPD ≃ 50 kHz, or slow, fLISA ≃ 10 Hz, readout rate, which vary slightly depending on the
down-sampling factor, R, and the clock rate, fs . The important point here is to consider
the aliasing effects of the CIC filter and the quantization noise of the frequency read-out.
5.2.2.1 Phase Quantization Noise
We can calculate the phasemeters’ quantization phase noise based on the
bit-resolution and readout rate of the frequency information from the same arguments
used to derive, (3–24), and is written explicitly as:
δφDig (ω) =1 fs 2−Ωfb Cycles√√
ω 6fdata Hz(5–13)

where Ωfb is the bit precision of the frequency feedback data and fdata is the CIC
down-sampled output data rate. To see how this compares to the LISA phasemeter
precision requirements, the low-frequency phase quantization limit, scaled to a clock
rate of fs = 50 MHz, is plotted for two different read-out rates, 3 Hz and 10 Hz, and threedifferent bit precisions, 47, 48, and 49 bits, in Figure 5-3. Assuming we have a 3 Hz
data-rate, we require 49-bits scaled to the 50 MHz clock. Using a 10 Hz data-rate we canreduce this to a 48-bit precision. We also plot the quantization precision for the 10 Hz,
60-bit UF-LISA science phasemeter. Despite that the 60-bit PM precision is far beyond
the LISA requirements, we maintain this high bit-precision simply as a result of the DSP
system’s capabilities.

10147−bits, 10Hz
48−bits, 10Hz
49−bits, 10Hz
60−bits, 10Hz
47−bits, 3Hz
48−bits, 3Hz
49−bits, 3Hz
LISA Req−4

Cycles/rt(Hz)

10

−6

10

−8

10

−10

10

−4

10

−3

10

−2

−1

10

10

0

10

1

10

Frequency (Hz)Figure 5-3. Theoretical PM readout digitization limitations: The frequency bit-precisionand data read-out rate couple to limit the digitization-limited PM
measurement sensitivity through quantization noise (5–13). Although thisspecifies how many bits are needed in the PM tracking loop, the data
precision could be reduced for satellite-to-Earth transmission once it has
been read out of the PM tracking loop by removing the frequency offset andretaining only the dynamic bits.
5.2.2.2 CIC Downsampling and Aliasing
The CIC filter is used to down-sample the data because of the relative ease of
programming on an FPGA since it consists of cascaded differentiators and integrators
which may be constructed using fixed-point accumulators and subtracters. In addition,
as shown in Figure 5-4, aliased terms have an infinite suppression at all frequencies
which are aliased to DC. Despite this feature, we must ensure that the CIC aliased
phase-noise is suppressed beyond the LISA requirement in the LISA measurement
band, up to 1 Hz. As an example, we take the pre-stabilized laser input noise of√
280 Hz/ Hz and plot the magnitude of the pass-band and first aliased frequency bandfor a 10 Hz data rate down-sampled with N = 2 and N = 6 stage CIC filters in Figure 5-4.
We can see that, given this data-rate and a 6-stage CIC down-sampling filter, we obtaina first-aliased band suppression which meet the LISA requirements. This is verified
by an experimental measurement as shown in Figure 5-14. A finite impulse response102

(FIR) filter with a flat pass-band and significant suppression in the stop-band to prevent
aliasing could be used to improve the performance near the sampling rate and reducethe data rate to 3 Hz but this is computationally demanding for a fixed-point FPGA.
6

10

4

10

2

Cycles/rt(Hz)

10

0

10

Pre−stabilized Laser (280 Hz/rt(Hz))
2−Stage Pass−band
2−Stage 1st−aliased band
2−Stage Limit
4−Stage Pass−band
4−Stage 1st−aliased band
4−Stage Limit
6−Stage Pass−band
6−Stage 1st−aliased band
6−Stage Limit
LISA phasemeter Requirement

−2

10

−4

10

−6

10

−8

10 −4
10

−3

10

−2

−1

10

10

0

10

1

10

Frequency (Hz)Figure 5-4. Phasemeter noise caused by CIC filter aliasing: The down-sampling filters
used to read out the phasemeter data must ensure that the noise in thealiased frequency bands are sufficiently suppressed before being folded into
the measurement band. The first aliased band of the N = 2, 4, 6 stage CIC
filters used to down-sample the PM core data in the UF-phasemeter areplotted as a function of the expected input laser noise. Using a 10 Hz
data-rate, the CIC-filter down-sampling filters require at least N=6-stages tomeet the LISA requirements.
5.2.3 Phasemeter Test-Measurements
We begin the verification and noise analysis process by performing software
simulations in MATLAB-Simulink using the fixed-point Xilinx-DSP toolkit and hardware
simulations with a voltage controlled oscillator (VCO) test input.11 In both the software

11

Our VCOs have similar noise characteristics as the pre-stabilized laser beatnotes
and is used as a test input for many verification measurements as shown in Figure 6-2.

103

and hardware simulations, we record the 60-bit feedback, ffb (t), and the 30-bit
quadrature error, Q(t), and in-phase, I(t), signals.
In−Band Phase
Phase (Radians)

0
−0.2
−0.4
−0.6
−0.8
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2
−3

x 10
Phase (Radians)

Out−of−Band Phase
0.2
0.1
0
−0.1
−0.2
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2
−3

x 10
Phase (Radians)

Corrected Phase
Input Phase
Measured Phase

0.6
0.4
0.2
0
−0.2
0

0.2

0.4

0.6

0.8

1
Time (sec)

1.2

1.4

1.6

1.8

2
−3

x 10

Figure 5-5. Software verification of PM performance: The in-band (top) and out-of-band(middle) phase information is plotted for a simulated laser-like noise input for
2 ms. The bottom curve shows the input phase compared against thesummed in-band and out-of-band phase information. The initial tracking
transients are seen at the beginning of the simulation and the curves matchas expected once the phasemeter has locked onto the input signal.
The time-series output of a software simulation is shown in Figure 5-5 while thespectral results of a hardware experiment using a VCO input source is shown in
Figure 5-6. From the hardware experiment we can see that the out-of-band phaseerror is well below the LISA requirement, justifying (5–11). The ’difference’ terms referto the subtraction of two different ADCs, but which sample the same input source being
electronically split, effectively showing the combined un-common ADC and phasemeter
phase noise limitation. This will be discussed extensively in the following chapters. At
this point, it simply verifies that both ADC’s and phasemeters are measuring the same√
signals to a 0.1 µcycle/ Hz accuracy, at least in the high frequency range.

104Linear Spectral Density210In−band Phase
Out−of−Band Phase
Total Phase
In−band Difference
Out−of−band Difference
1pm LISA Requirement

0

10

−2

Cycles/sqrt(Hz)

10

−4

10

−6

10

−8

10

−10

10

0

10

1

10

2

3

10

4

10

10

5

10

Frequency (Hz)Figure 5-6. Hardware verification of PM performance: The high-frequency noise
spectrum of the hardware tested phasemeter using a VCO input source isplotted. The in-band√ phase (blue curve) matches the VCO input noise to alevel of 0.1µcycles/ Hz (cyan curve) when performing a differential ADC
measurement (Figure. 5-9). The out-of-band error (green curve) immediately
drops below the LISA requirement and continues to track the input noise to abetter precision at lower frequencies due to the P-I-I2 feedback transfer
function (Figure. 5-7). Thus, including the out-of-band error in the measured
phase does little to improve the phase precision (purple curve).
Dividing the spectra of the frequency feedback signal, f̃fb (ω), by the quadrature error
spectra, Q̃(ω), as shown in Figure 5-7, we obtain the expected feedback transfer function
based on the programmed FPGA controller design. More on the design and verification
aspects of the phasemeter can be found by referencing the groundwork experiments
performed by Ira Thorpe [41].
5.2.4 Phasemeter Noise Model
Taking into consideration phasemeter, ADC, and clock noise sources, we can write
the sampled phasemeter signal as:
φPM,i = φin + δφPM + δφADC +

105

fin
δφClk
fs

(5–14)

6

10

TheorySimulation
Hardware Test

4

|H(s)|

10

2

10

0

10

−2

10

−1

10

0

10

1

10

2

10

3

10

4

10

5

10

Frequency (Hz)Figure 5-7. Verification of the PM feedback transfer function: Dividing the spectra of the
feedback frequency signal by the out-of-band error signal, we obtain the
expected feedback transfer function (5–6) for both the software andhardware verification tests.
whereδφPM = δφCIC,o + δφQuant,i(5–15)

δφClk = δφCLK,o + δφCLK,i(5–16)

δφADC = δφADC,o + δφADC,i(5–17)

so that we may differentiate between noise terms which are common between all
the channels, δφx,o , and terms which are independently applied to the ’i-th’ channel,
δφx,i . The CIC noise is labeled with an ’o’ since it correlates with the input noise. The
quantization noise, on the other hand, is applied to each channel independently. In
chapter 5.3, we will further investigate the ADC noise terms. To distinguish between theclock noise and ADC noise terms, we will define the clock noise sources as any signal

106

which scales proportionately with the input frequency, such as timing jitter, while ADC
noise sources are defined as absolute, frequency-independent additive noise signal.
In an attempt to model these different noise sources, a diagram of where they are
introduced in the measurement process is shown in Figure 5-8.
5.2.5 Differential and Entangled Measurements
To distinguish between these terms, we take three types of measurements, as
shown in Figure 5-9, with each attempting to probe a different noise source as outlined in
the following sections.
5.2.5.1 Digitally Split Differential NoiseThe first measurement uses a 1 MHz VCO test input which is sampled at a rateof fs =62.5 MHz with a single ADC. The ADC sampled data is passed to two different
phasemeter cores with two different PLL offset frequencies. This results in two
data-streams which are only limited by the phasemeter core’s measurement anddigitization precision, δφQuant,i , in (5–14). The measurements are taken at four different
rates then plotted together to span the 100 µHz to 10 kHz frequency range as shown in√
Figure 5-10. The ’quantization noise’ limit is at the 100 pcycle/ Hz level for frequencies
above 10 mHz and equals the PM phase quantization limit at frequencies below 10 mHz.
Using the amplitude-phase quantization noise level as given by (3–24) and considering
the fin−loop = 62.5/128 MHz in-loop data rate, we can solve for the ADC-noise-freeeffective in-loop phase quantization, Ωfb :pcycles21−Ωfb,≃ 100 √
δxADC Amp.−Quant. (ω) = p
6fin−loopHz(5–18)

resulting in an effective phasemeter amplitude quantization precision of Ωfb = 23.5-bits.
Again, these noise sources, being well below the LISA measurement requirements,
are not of concern to us. Although, the digitization precision calculations can be useful indetermining the amount of data that must be transmitted to Earth for data-processing of
the science PM read-outs.

107Figure 5-8. Phasemeter noise model: A model of the ADC inputs and clock distribution
is depicted. Common and independent noise terms are added to the ADC
input and clock input signals in to represent the δφClk and δφADC terms in
5–14. The ADC itself has a ground reference which produces an additive
phase noise and independent voltage references for the ADC-comparators
which represent the VGnd (t) and G(t) (3–25) terms respectively anddiscussed in chapter 3.4.2.3. These terms are also representative of the
AOff (t) and δA(t) (3–38) terms in chapter 3.4.2.5.

108

Figure 5-9. Experimental models of ADC noise estimation measurements: The
quantization noise experiment (left) maintains the common ADC noise terms
but distinguishes between independent PM core quantization noise terms.
The differential noise experiment (middle) distinguishes between
independent ADC noise terms but cancels common ADC noise terms as
explained in chapter 5.2.5.2. The entangled phase measurement (right)
provides a measure of the common-mode ADC noise terms(chapter 5.2.5.3).
5.2.5.2 Electronically Split Differential NoiseThe next measurements are taken to probe the un-common clock, ADC, andphasemeter noise sources. Splitting a demodulated 1 MHz VCO output using anelectronic radio-frequency (RF) transformer splitter, we measure the signal phase using
two different ADCs, sampled with the same clock source. In this case, the common
noise sources, δφCIC,o , δφADC,o, and δφClk,o cancel from the measurement while theun-common sources, δφADC,i and δφClk,i, do not. We expect the δφClk,i to be dominatedby timing-jitter terms while δφADC.i might introduce phase and amplitude noise terms
outlined in chapter 3.4.2.5. A complete spectrum of the 1 MHz input and the two-channel
differential output is plotted in Figure 5-10. The measurement precision is limited at
low frequencies, below 10 Hz, by a combination of timing-jitter and ADC/transformer
dispersion. The precision is limited at high frequencies, above 10 Hz, by the couplingof the amplitude-sampling noise terms described in chapter 3.4.2.3. This is verified
by varying the signal frequency and amplitude, then comparing the PM measurement
sensitivity with the changes in the measurement characteristics.

109

Figure 5-10. PM/ADC Quantization and differential noise: Here we plot the results of thequantization and differential noise measurements modeled in Figure 5-9
from 100 µHz to 10 kHz. The quantization noise shows an in-loop whiteΩfb = 23.5-bit quantization phase noise for frequencies above 10 mHz the
expected 60-bit frequency quantization PM read-out limitation forfrequencies above 10 mHz. The differential measurement is limited by theΩδV = 13.78-bit effective amplitude quantization noise level for frequencies
above 10 Hz and a timing jitter limited noise given by (5–19) for frequencies
below 10 Hz.
In order to probe the timing jitter terms, we take long-term, low-frequency measurements of input beatnote frequencies ranging from 1.0 MHz to 79.5 MHz. The measurement at 79.5 MHz is aliased into the measurement-band by the fs =60 MHz clock,producing a measurable oscillation at 19.5 MHz. The results, Figure 5-11, show thatthe measured phase noise decreases as the measurement frequency is decreased
from 79.5 MHz, to 12.0 MHz, to 8.0 MHz. Based on the noise level and scaling of thesesignals, we estimate a timing jitter noise of,
40 fs
δtJit (ω) = √ √ .
f Hz

110

(5–19)

Confirming this estimate numerically we obtain a phase noise of:δφJit (ω) = fin δt(0.1 Hz)(5–20)

31.8 µCycles√= (79.5 × 106 ) · δt(0.1 Hz) = √Hzf4.8µCycles= (12.0 × 106 ) · δt(0.1 Hz) = √ √Hz
f
3.2µCycles= (8.00 × 106 ) · δt(0.1 Hz) = √ √fHz
1.6µCycles= (4.00 × 106 ) · δt(0.1 Hz) = √ √Hzfall of which match with the measured phase noise for frequencies larger than 8 MHz.
Unexpectedly, lower frequencies (4.0, 2.0, and 1.0 MHz) result in a higher phase noise
precision and a reduced measurement sensitivity. The increased noise at these lower
frequencies is caused by temperature correlated phase dispersion introduced by the RFtransformers used to AC couple the ADC signal input; see chapter 5.4.1 for a descriptionof the experiments and methods used to correct for this noise.
In an attempt to evaluate (3–25), we vary the input amplitude of a demodulated
10 MHz VCO input. Using two peak-to-peak input amplitudes, 200 mV and 600 mV, weobserve a factor of 3 improvement in the measurement sensitivity at high frequencies
when the signal is not limited by timing jitter. The tracking bandwidth for this PM design
also increases from ≃ 3 kHz to ≃ 10 kHz. The amplitude noise coupling allows us toevaluate the G(ω)VBias factor in (3–25). Based on the maximum peak-to-peak input
amplitude of 2000 mV, we can estimate this amplitude-to-phase noise coupling as:
δφAmp (ω) =2000 mV
nCycles· 7.5 √
VPk−Pk:in (mV)Hz

(5–21)

Note that the amplitude noise is not a result of aliased amplitude noise from 2ω(chapter 3.4.2.5) since these would not scale with the signal power. We have also
accounted for the out-of-band phase terms (tan−1 (Q/I)) in these measurements, whichindicates that this is not a result of variations in the signal’s amplitude as described by

111

−2

10

ADC Noise 79.5MHz
ADC Noise 12MHzADC Noise 8MHzADC Noise 4MHzADC Noise 2MHzADC Noise 1MHz
1 pm Level−3

Cycles/sqrt(Hz)

10

−4

10

−5

10

−6

10

−7

10 −4
10

−3

10

−2

−1

10

10

0

10

1

10

Frequency (Hz)Figure 5-11. ADC timing jitter noise limitations: The phase noise due to the timing jitterscales inversely with the fin MHz input frequency (3–27) and matches thetheoretically calculated jitter values for fin > 8, MHz for δtJit (ω) (5–19). The
phase measurement for input frequencies with fin < 8 MHz are limited bytemperature dependent dispersion due to the RF transformers(chapter 5.4.1).
chapter 3.4.2.5. Using the inverse argument used to derive (3–26), we can calculate theeffective quantization precision of ΩδV = 13.78 which is nearly equal to the full 14-bit
ADC precision.
We will have an independent measure of this noise source using a different
experimental setup in chapter 5.3.
At this point, we claim the measurement is limited by a combination of timing
jitter and RF transformer phase dispersion in the low-frequency range and limited byamplitude noise in the high-frequency range. Since the ADC’s, although independent
integrated circuits, are on the same printed circuit board they may have some common
temperature dependent phase delay or common amplitude-voltage reference noise.
In order to differentiate between these common terms, we take an entangled phase

112

0

10

Input
ADC Noise Only (High)ADC Noise Only (Low)
Expected Amplitude Noise (200 mV Pk−Pk)
Expected Amplitude Noise (600 mV Pk−Pk)
Expected Timing Jitter−2

Cycles/sqrt(Hz)

10

−4

10

−6

10

−8

10

−10

10

0

10

1

10

2

3

10

10

4

10

5

10

Frequency (Hz)Figure 5-12. ADC amplitude noise limitations: The amplitude noise scales inversely with
the input amplitude and indicates that the noise floor is due to the ADC
quantization bit-limitation. Using this assumption, we find an effective
number of bits ΩδV = 13.78 which is nearly equal to the specified 14-bit
ADC precision.
measurement involving 3 different signals which combine to cancel the input and clock
noise sources but leave the common ADC and phasemeter noise sources.
5.2.5.3 Entangled Phase Noise
Finally, we perform what is commonly known as an entangled phase measurement.
This measurement involves mixing three independent VCO signals (φVCO1 (t), φVCO2 (t),φVCO3 (t)) with frequencies (fVCO1 , fVCO1 , fVCO3 ) and measuring the differential noise
between each of the VCOs with three ADCs and PMs to obtain three signals (φA (t),φB (t), φC (t)) given by:
fVCO1 − fVCO2
φClk (t) + φADC,A (t) + φPM,A (t)fClk
fVCO2 − fVCO3
φB (t) = φVCO2 (t) − φVCO3 (t) +φClk (t) + φADC,B (t) + φPM,B (t)fClk
fVCO1 − fVCO3
φClk (t) + φADC,C (t) + φPM,C (t)φC (t) = φVCO1 (t) − φVCO3 (t) +fClk

φA (t) = φVCO1 (t) − φVCO2 (t) +113(5–22)(5–23)(5–24)

Taking a linear combination of the differential noise measurements, namely:
φEnt (t) = φA (t) + φB (t) − φC (t)(5–25)

we cancel the VCO and clock noise terms in the final combination but are left with noisedue to the individual phasemeter and ADCs.
8

10

LISA Requirement 1pm
LISA Requirement 10pm
Input VCOElectronically Split
Entangled Phase

6

10

4

10

Cycles/sqrt(Hz)

2

10

0

10

−2

10

−4

10

−6

10

−8

10 −4
10

−3

10

−2

10

−1

10

0

10
Frequency (Hz)

1

10

2

10

3

10

4

10

Figure 5-13. Entangled phase measurement results: The entangled phasemeasurements show an increased noise at low frequencies by up to afactor of 10 due to the independent RF transformer phase dispersion terms(chapter 5.4.1). The measurement couples in aliased noise near the
sampling rate due to the CIC down-sampling filters (chapter 5.2.2.2).
The results plotted in Figure 5-13, show up to a factor of 10 increased noise at low
frequencies due to a common ADC noise which cancels in the 2-channel differential
measurement. The additional low-frequency error in comparison to the differential
measurement results from common-mode temperature dependent phase dispersionof the RF transformers (chapter 5.4.1). We also notice an increased noise level near
the sampling rate because of the aliasing limitations of the CIC down-sampling filter
described in chapter 5.2.2.2. Verifying the CIC filter aliasing error, we perform the same114

measurement with a 2-stage, 4-stage, and 6-stage, CIC down-sampling filters. As we
can see in Figure 5-14, the 2-stage CIC down-sampling filter will not meet the LISA
requirements at frequencies above 10 mHz.
8

10

Input Noise2−Stage CIC
4−Stage CIC
6−Stage CIC
LISA Requirement

6

10

4

10

Cycles/sqrt(Hz)

2

10

0

10

−2

10

−4

10

−6

10

−8

10 −5
10

−4

10

−3

10

−2

10
Frequency (Hz)

−1

10

0

10

1

10

Figure 5-14. N-stage CIC filter aliasing and entangled phase measurements: The
entangled phase measurements are performed using a PM-core which
down-samples the measured PM data with a N = 2, 4, 6 stage CIC filter. As
described in chapter 5.2.2.2 and shown in Figure 5-4, the CIC filter
down-sampling filter aliases noise into the measurement band, limiting thephase precision of the measured input.
5.2.6 Applications in LISA and LIGO
Although the LISA phasemeter design was motivated by the need to measure
the phase of LISA’s heterodyne beatnotes to a 1 µCycle accuracy, the phasemeterhas applications in many heterodyne and homodyne interferometry schemes. LISA
will use the PM to measure not only the LISA observables, but also the clock-noise
sidebands and inter-SC data communication signals. The PM frequency readouts
are required to perform arm-locking controls and is also applicable in Mach-Zehnder115laser pre-stabilization and digital heterodyne phase-locking. The high-frequency PM
sensitivity can also evaluate thermal coating noise for LIGO [100].
5.3 ADC Noise Estimation
We employ a novel ADC phase-noise measurement technique to obtain a better
understanding of the noise coupling added by the ADCs. Using the clock as the input
source as shown in Figure 5-15, we adjust the length of the cable feeding the input
signal, ∆L, to change the phase relationship between the input signal and the clock
source.
When there is an offset phase of 0 between the input and the clock trigger, we
sample the clock itself at it’s zero crossing. In this case, we are sensitive to the linear
combination of the ground-reference, VGnd (t) in (3–25), timing jitter noise, δ t̃Jit (ω) in(3–27), and other un-accounted-for white-noise terms, φThermal (t). Assuming the signal
has a slope of ψ = 2πfClk VClk , measured in [V/s] with VClk being the clock amplitude, we
can write the sampled output of this measurement as:V∆φ=0 (t) = VGnd (t) + ψ(fClk )δtJit (t) + φThermal (t).

(5–26)

When there is an offset phase of π/2 between the input and clock trigger,
we sample the signal’s peak. In this case, we are sensitive to the ADC amplitude
fluctuations, the G(t)VBiaso term in (3–25), the clock amplitude fluctuations, VClk (t), andthe ground offset term, VGnd (t), as in the zero-crossing measurement. Assuming theclock oscillator source has an amplitude of VClk (t), we can now write the sampled output
as:V∆φ=π/2 (t) =VGnd (t) + VClk (t)G(t)VBiaso(5–27)

Splitting the clock input signal and performing the same prescription for a second
ADC, we can compare the two ADC measurements to cancel common noise terms suchas the clock’s amplitude fluctuations, VClk (t), or common VGnd (t) terms.

116Figure 5-15. Model for estimating ADC phase and amplitude noise: Splitting a clock
source and adjusting the phase relationship between the clock and the
ADC input, we effectively sample the clock with itself canceling thecommon clock noise. When the phase relationship between the clock andthe input, ∆φ = 0, we sample at the zero-crossing which couples in timing
jitter terms with a proportionality factor given by the ’slope’, ψ. When thephase relationship between the clock and the input, ∆φ = π/2, we sample
the peak of the clock oscillation; differential measurements will cancel thecommon clock amplitude noise and be limited by the ADC amplitude noise
factor, G(t) (3–25).

117

Figure 5-16. Common source ADC phase and amplitude noise results: Thezero-crossing measurements (left column) show a strong ADC noisecorrelation which, when differential measurements are performed, cancel toreveal the thermal and timing jitter noise sources (top-left) as explicitly
shown in Figure 5-17. The individual ADC noise sources correlate strongly
with the temperature at frequencies below 2 mHz indicating the coupling oftemperature dependent RF dispersion. The individual peak-samplingmeasurements (right column) also show a strong low-frequency correlation
with the temperature. The loss of correlation between the individual ADC
samples below 2 mHz (bottom-left), indicates that the sampling precision isdominated by voltage bias reference noise, VBiaso G(t) in (3–25).

118

The following zero-crossing and peak-sampling measurements use a 62.5 MHz
clock input which has a amplitude of VClk = 0.45 V. The phasemeter core is replacedwith CIC down-sampling filters to produce the sampled voltage ADC output at a 14.9 Hz
down-sampled rate. At the same time, we measure the temperature of the ADC’s to see
if there is any thermal noise correlations in the measurements.
The results of the zero-crossing measurement, as shown in the left column of
Figure 5-16, show a strong correlation between the two individual ADC measurements(red and blue curve in top-left, pink in bottom-left) which cancels in the difference
between the signals (black curve in top-left). This individual ADC noise is strongly
temperature correlated at frequencies at and below 1 mHz and indicates common mode
phase dispersion. Subtracting the signals we obtain a voltage noise estimated by (green
curve in top-left of Figure fig:ADCPhaseAmpNoise):r∆Ṽ∆φ=0 (ω) = 5 × 10−6

1 Hz Volts√fHz(5–28)

Scaling this by ψ, we obtain a timing jitter of:∆Ṽ∆φ=0 (ω)ψ
r
1 Hz Voltss1√= 5 × 10−6·f
2πfClk VClk Volt
Hzr
1
1 Hz Volts√·= 5 × 10−6f
2π 62.5 MHz 0.45 VoltsHz
r
1 Hz fs
√ .
≃ 28
f
Hz

δ t̃(ω) =(5–29)(5–30)

This is smaller, yet, comparable to the timing jitter measured in chapter 5.2.5.2. If this isactually the timing jitter term it should scale with respect to the clock frequency throughthe ψ dependence. To probe this dependence, we take the same measurement at
50 MHz, 75 MHz, and 100 MHz. The results (Figure 5-17), show the same noise level
for the 50 MHz and 75 MHz signals as we observe here. This indicates that the noise
source we are measuring is not timing jitter, but rather, a phase error which is likely due

119to the internal electronic thermal noise background, φThermal (t). The 100 MHz signal
shows a slight increase in the phase noise error indicating that we have now made
ψ large enough to dominate over the thermal background terms in (5–26). Given thisassumption, we re-evaluate the jitter level:∆Ṽ∆φ=0 (ω)ψ
r1
1 Hz Voltss√·= 9 × 10−6f
2πfClk VClk Volt
Hzr
1
1 Hz Volts√·= 9 × 10−6f2π100.0 MHz0.45 VoltsHz
r
1 Hz fs
√ .
≃ 32
f
Hz

δ t̃(ω) =(5–31)(5–32)

which is closer to the estimate given by (5–19). Scaling this 100 MHz zero-crossing,voltage noise measurement to a 10 MHz input signal and comparing it against a 10 MHz
measurement, the results match the measured jitter as shown in Figure 5-18.
Moving on to the peak-sampling measurements as shown in the right column of
Figure 5-16, we observe some common amplitude noise at frequencies below 1 mHz
and at frequencies above 10 mHz. The amplitude noise at these frequencies is likely
due to common mode fluctuations in the input signal or in the voltage bias reference,
G(t)Vbias , although, without an independent measure of the input amplitude, there is
no way to differentiate between these terms. The differential ADC noise shows sometemperature correlation at frequencies between 2 and 20 mHz. This is most probably
due to temperature dependent noise in the voltage bias, G(t)Vbias , since it is not likelythat the amplitude of the clock source correlates with the temperature of the ADCs. Thatsaid, these noise sources on this absolute voltage scale are smaller than the timing jittersources and are scaled by the inverse amplitude before coupling into the measured
phase as explained in chapter 3.4.2.5. Although, because of the ’white’ nature of the√voltage noise in this 1-20 mHz region of ≃ 5 × 10−5 Volts/ Hz, it indicates that we shouldbe about to calculate the effective number of bits in this low-frequency range. Assuming

120

Figure 5-17. Zero-crossing timing jitter estimate: The differential zero-crossing timing
jitter measurements provide us with the ability to directly probe the ADC
timing jitter and it’s frequency dependence. The input frequency
dependence of Ψ allows us to differentiate between φThermal (t) andψ(fClk )δtJit (t) in (5–26). The 50 and 75 MHz measurements show the same
noise level and are dominated by φThermal (t). The 100 MHz measurement
increases ψ(fClk ) to allow us to probe δtJit (t) (5–32).
(3–24) still holds, we calculate the effective number of bits from,|VBias | 21−Ω
5 × 10−5 Volts√ =√|VIn | 6fs
VFull−Scale Hz
5 × 10−5 Volts
0.5 Volts
21−Ω
√
√ .
=
0.225 Volts 6 · 14.9 Hz
1.0 Volts Hz(5–33)(5–34)The solution gives a bit-resolution of Ω = 13.2 and, thus, it seems that thermal
fluctuations have decreased the effective number of bits in this frequency range. These

121

−4

10

−5

Cycles/rt(Hz)

10

−6

10

−7

10

−8

10 −4
10

Zero−Crossing:50 MHz
Zero−Crossing:75 MHz
Zero−Crossing:100 MHz
Differential ADC, 10.0 MHz
10MHz*40 fs/sqrt(f)
−3

−2

10

10

−1

10
Frequency (Hz)

0

10

1

10

Figure 5-18. The timing jitter estimate obtained from Figure 5-17 is scaled andcompared directly to an experimental PM measurement of a 10 MHz input
signal. The results indicate that we have √obtained√ an accurate estimate of
the ADC timing jitter given by ≃ 30 − 40/ f fs/ Hz which will limit the LISA
measurement
precision such that the PMs and ADCs do not meet the√
1 µcycle/ Hz measurement requirement by up to an order of magnitude
near the 3 mHz corner frequency.
low-frequency amplitude-phase fluctuation terms will become apparent once the timing
jitter terms have been extracted in the next chapter.
Now that we know that timing jitter is the limiting noise source of these measurements, we attempt to estimate and correct for these terms. Also, we must further
investigate and correct non-jitter limitations of the phasemeter in the 1-8 MHz frequency
range, of which, these zero-crossing and peak measurements do not probe. That said,
the correlation between the ADC input phase and the temperature at low frequencies
in Figure 5-16 (bottom left) indicate a temperature dependent phase response of the
ADCs. If the slope of this phase-response is large enough, it might over-come anddominate the timing jitter terms.
5.4 Timing Jitter Extraction
Timing jitter correction is based off the concept that, although the timing jitter terms
are ADC independent, a second modulation, added to the ADC input signal, can beused used as a reference to measure and correct for the timing jitter. As shown in
Figure 5-19, a second oscillator is added using an RF transformer-splitter. The summed122

output is split again into two different ADCs. The phase of the input and the ranging toneon each ADC is measured with it’s own phasemeter. Measuring the input signals, we
obtain PM outputs which take the form:
fIn(−φClk (t) + fClk δtADC−1 )fClk
fIn
φIn:ADC2 (t) = φIn (t) +(−φClk (t) + fClk δtADC−2 )fClk
φIn:ADC1 (t) = φIn (t) +(5–35)(5–36)

Repeating this for the added reference-tone signals, we obtain:
fTone(−φClk (t) + fClk δtADC−1 )
fClk
fTone
φTone:ADC2 (t) = φTone (t) +(−φClk (t) + fClk δtADC−2 )fClk

φTone:ADC1 (t) = φTone (t) +(5–37)(5–38)

Using the these, we are able to probe the differential timing jitter terms to correct for
the input signal’s timing jitter. Note that a higher frequency reference tone is desirable
since the timing jitter scales with this frequency. In fact, the jitter tone can be placed at afrequency above the sampling rate; measuring the aliased oscillation results in stronger
jitter couplings as shown in Figure 5-11.
Subtracting the two input signals and the two reference signals, we are able toevaluate the differential timing jitter at each frequency, fin and fTone :∆φ (t) = fIn [δtADC−1 − δtADC−2 ](5–39)∆Tone (t) = fTone [δtADC−1 − δtADC−2 ](5–40)

Rescaling and subtracting these signals we cancel the differential ADC timing jitterand obtain a calibrated result which cancels all the noise sources we have considered:∆φCalib. = ∆φ (t) +

fIn
fTone

∆Tone (t)(5–41)

Note that this differential timing jitter correction method will not give a measure of
the absolute, individual timing jitter terms from which we can obtain a true measurement123

of φIn (t). A few possible methods of performing this absolute jitter correction are
discussed in chapter 5.4.2.

Figure 5-19. Timing jitter calibration model: Here we show a model of how the input
signal, in this case, a VCO, is added to a calibration signal[oscillator/function generator(FG)] with an inversely driven splitter. The twosummed oscillations are split and sampled with two independent ADCs butwith the same clock source. The PM cores measure the phase of the
VCO-input and calibration signals which are each contain the timing jitter of
their respective ADCs scaled by the magnitude of the measured input
frequency as described in chapter 5.4.
A series of initial tests performed well when the ranging tone and input signal
were relatively equal in frequency as shown in the right column of Figure 5-20,√
nearly obtaining the 1 µcycle/ Hz requirement. The temperature measurements andcorrelations for the ADC noise estimation of chapter 5.3 are also used here to see if themeasured phase correlates with the ADC temperature. The top-right plot of Figure 5-20
showed no correlation with either input signal or the differential timing jitter indicatingthat temperature effects play little role in the noise coupling at these high frequencies.
When the timing jitter correction is attempted using a low-frequency (4 MHz) input signal
and a high-frequency (15 MHz) tone, we obtain no noticeable improvement in the phasemeasurement precision.

124

Figure 5-20. Temperature dependent phase dispersion: Placing the input oscillation at
4 MHz and the jitter calibration input at 15 MHz, the timing jitter extraction islimited by jitter-uncorrelated noise in the input oscillation. The differential
phase noise of this 4 MHz input signal correlates strongly with temperatureof the RF transformers which introduce dispersion terms that are notaccounted for in the timing jitter extraction. Placing the input oscillation andjitter calibration tone close to each-other, 14 and 15 MHz respectively, we
reduce the temperature dependent dispersion and the timing jitter
extraction works as explained in chapter 5.4
Correlating each of these with the measured temperature, we see a strong
correlation between the low-frequency input and the measured temperature as shownin the top-left plot of Figure 5-20. This indicates that there is some phase loss whichchanges as a function of temperature at these low frequencies. This low-input-frequency
limit had already been indicated to some extent by the results plotted in Figure 5-11. To
see how strong the temperature coupling effect is, we perform the timing jitter extraction
for three different frequencies across the 2-20 MHz LISA input-frequency range (2 MHz,125

8 MHz, and 16 MHz) and attempt to calibrate them against a (19 MHz) oscillator tone.
The results are shown in the top plot of Figure 5-22.
5.4.1 Phase Dispersion Mitigation
A detailed look into the Pentek design indicates that the RF transformers used
to AC couple the input signal, the ADT4-5WT, have a strong input return loss and,thus, a steep phase dependence at these low frequencies as shown in the left columnof Figure 5-21. Temperature variations change the transfer function response andphase loss for these RF transformers which couple into the low-frequency measured
phase (Figure 5-11). Mini-circuits, the integrated-circuit production company whichmakes the ADT4-5WT also produces another model, the ADT1-6T, which has asignificantly more constant input return loss, and thus, a smaller phase change inthe LISA-frequency range. After the risks and costs where weighed, we decided toreplace the RF transformers on the Pentek boards to better suit the 2 − 20 MHz LISA
input frequency range of interest.

Figure 5-21. The stock ADT4-5WT has a bandwidth between 300 kHz and 500 MHz.
The replacement ADT1-6T’s bandwidth is from 30 kHz to 125 MHz,
providing a better suited AC coupler to make PM measurements at
LISA-like frequencies. [101, 102]126

Thankfully, changing in RF transformers did not damage the Pentek’s electronics
and improved the low-frequency performance as shown by the improved timing jitter
result in Figure 5-22. At this point, we can see that provided with the reference tone’s
measure of the timing jitter, we should be able to correct for the timing jitter of themeasured input phase across the entire 2 − 20 MHz LISA input-frequency range.

Figure 5-22. Replacement RF transformer dispersion mitigation: Repeating the timing
jitter extraction tests with the old (ADT4-5WT) and new (ADT1-6T) RF
transformers, we find the RF transformer phase dispersion has beencorrected and the timing jitter extraction has been significantly improved.
The jitter extraction results using the new ADT1-6T√ RF transformers meet
the LISA measurement requirement of 1 µcycle/ Hz for all frequencies in
the 2-20 MHz range. Once the timing jitter has been removed, the phase
precision is limited by the signal-amplitude dependent quantization noise
floor; chapter 5-12 for f > 20 mHz, (5–33) for f < 20 mHz, over thefrequencies for which the RF transformer dispersion no longer limits theresults.
5.4.2 Absolute Timing Jitter Extraction
Performing absolute timing jitter extraction is somewhat more difficult since we need
the same reference phase noise, φTone (t), at two different frequencies, fTone1 and fTone2 ,127such that we can form the combination:
φIn−Calib (t) = φIn:ADC1 (t) −fIn[φTone1:ADC1 (t) − φTone2:ADC1 (t)]fTone1 − fTone2(5–42)

fIn(−φClk (t) + fClkδtADC−1 )
fClk


fIn
fTone1 − fTone2−(−φClk (t) + fClk δtADC−1 )fTone1 − fTone2
fClk= φIn (t) +≃ φIn (t)This would be accomplished by up-converting or down-converting a common signal√
with a 1 µcycle/ Hz accuracy, or rather, an accuracy better than the timing jitter noise
limitation, which may or may not be feasible based on the noise characteristics of theelectronics and methods used to perform the frequency-conversion.
5.5 Phasemeter Performance Review
Compiling the results of all of these measurements, we have a well-defined modelof the limiting noise sources in the PM measurement process. The regions of which overthese terms couple is shown in Figure 5-23. The frequency range from 10 Hz to 10 kHz
is dominated by white amplitude noise which couples into the phase from a combinationof variations in the input beatnote’s amplitude (chapter 5.2.5.2) and quantization noise(chapter 3.4.2.5). Increasing the signal voltage decreases the quantization noise andreduces the measured phase noise proportionately.
The LISA measurement band, 0.1 mHz to 1 Hz, is dominated by timing jitter andtemperature dependent phase dispersion. The timing jitter scales proportionately witha decrease in the signal frequency and can be corrected for by the methods mentioned
in chapter 5.4 and chapter 5.4.2. Once the timing jitter has been removed, the limiting√
noise sources is suppressed to the amplitude noise limit with a 1/ f flicker-type noise
sources coupling in at frequencies below 10 mHz.
The phase dispersion from the RF transformers plays a large roll in limiting thephasemeter phase sensitivity. Internal or external heating of any filtered element,

128

Figure 5-23. Phasemeter noise model, estimation, and limits: The primary phase
precision limiting √effects have been identified and plotted in comparison
with the 1 µcycle/ Hz phasemeter requirement. The frequency range
above 10 Hz is dominated by amplitude-quantization noise and scales
inversely with the input√amplitude
from 2 Volts peak-to-peak to 200 mVolts√
peak-to-peak. The 40/ f fs/ Hz timing jitter dominates the frequencies
below 1 Hz and scales proportionately with the 2-20 MHz input frequency.
This assumes that any temperature-correlated low-frequency phasedispersion, which scales based on the slope of the transfer functions’phase response, has been mitigated; that said, a constant phase responsein the 2-20 MHz range is desirable.
including photo-detectors, transformer-splitters or AC couplers, and mixers, will havea transfer function with some defined phase response. If the phase response is notflat over the 2 to 20 MHz LISA heterodyne beatnote frequency band, the temperaturedependent electronics will produce a time-changing phase loss which couples intomeasured phase. The dominate heating source in the 1 to 100 mHz frequency range
in these experiments resulted from internal heating of the ADCs upon power up moreso than environmental temperature changes. Thus, despite the quiet temperature

129

8

10

LISA Requirement 1pm
LISA Requirement 10pm
Input VCO Noise
10 MHz, Electronically Split
Entangled Phase Noise
Jitter Calibrated Phase Noise

6

10

4

10

Cycles/sqrt(Hz)

2

10

0

10

−2

10

−4

10

−6

10

−8

10 −4
10

−3

10

−2

10

−1

10

0

10
Frequency (Hz)

1

10

2

10

3

10

4

10

Figure 5-24. Comparison of the PM noise characteristics: The 10 MHz differential phase(chapter 5.2.5.2), entangled phase (chapter 5.2.5.3), and timing jitter
extraction (chapter 5.4) results are plotted together for a direct comparisonof PM experimental results and noise performance.
environment of the LISA mission, these noise sources will still be present from electronic
heating and cooling.

130

CHAPTER 6
THE UNIVERSITY OF FLORIDA LISA INTERFEROMETRY SIMULATOR
6.1 The LISA Laser Test-bench
The UFLIS experimental benchtop as pictured in Figure 6-1 consists of four
controllable Nd:YAG, 1064 nm lasers, LaserRef , Laser1 , Laser2 , and Laser3 , which
generate laser fields with phase signals given by φRef (t), φ1 (t), φ2 (t), and φ3 (t). We
interpret each individual Laseri as the laser on SCi producing the laser field φ(t) as
described in chapter 3.4.3.4. Heterodyned PD signals between LaserRef and the otherthree laser fields produce differential measurements of the laser phases:
φ1R = φ1 − φRef

φ2R = φ2 − φRef

φ3R = φ3 − φRef

(6–1)

such that combinations of these signals will cancel the common LaserRef noise. Thus,
the reference laser is used as a global reference with which the other three laser fields
can be measured. Reproducing the expected LISA pre-stabilized laser noise, we
PDH lock LaserRef and Laser1 to a ULE cavity. The following section reports on thephasemeter measurements of this pre-stabilized laser noise. The next section describes
the methods used to simulate the inter-SC electronic delay including the multi-second
time-changing laser phase delay, MHz laser field Doppler shifts, and µCycle GW
modulations. The last section describes how these three components are combined toperform advanced arm-locking and TDI simulations.
Reference Cavity Stabilization.
The University of Florida laser benchtop PDH locks two lasers, LaserRef (or RL)
and Laser1 (or L1), to two different cavities, 26.0 cm and 22.5 cm in length respectively,
resulting in a laser beat-notes between the two lasers with a frequency stability given by
the relationship:δfRL/L1 (ω) =p
δfRL (ω)2 + δfL1 (ω)2 .

131

(6–2)

Figure 6-1. UFLIS laser benchtop: The laser benchtop, pictured here, PDH
pre-stabilizes the lasers and forms all the necessary PD signals to performthe LISA interferometry simulations. The four lasers, three representing theindividual SC and one acting as as optical clock, can be seen in the bottom
left corner. The vacuum tank on the right contains the ULE cavities andprovides the necessary temperature and pressure shielding required totransfer the cavity stability to the laser frequency. The phasemeter dataacquisition and EPD data processing DSP electronics can be seen in the
background near the top-left of the image.

132

The stability of these lasers depends highly on the feedback electronics who’s gain can√be adjusted to achieve a ≃ 30 − 300Hz/ Hz stability and are limited by the temperaturestability of the cavity length [52]. Phasemeter measurements of the free-runninglaser stability in comparison with the cavity stabilized laser stability are plotted in
Figure 6-2 from which we observe an improvement in the laser phase noise by 4 ordersof magnitude at 10 mHz. Also, as a comparison, we plot the VCO noise level used formany of the previous phasemeter verification measurements. In LISA, the lasers canbe individually locked to their own reference cavity or one master cavity stabilized laser
can be transfered to the other lasers by phase-locking the differential heterodyned laser
fields. Thus, from this point on, we assume that Laser1 is the master stabilized field on
SC1 and that the other laser fields are phase-locked as described in [71].
Linear Spectral Density

8

10Free Running Laser
Cavity Stabilized Laser
VCO

6

10

4

10

Cycles/sqrt(Hz)

2

10

0

10

−2

10

−4

10

−6

10

−8

10 −4
10

−3

10

−2

10

−1

10

0

10
Frequency (Hz)

1

10

2

10

3

10

4

10

Figure 6-2. Measurements of commonly used sources: Here we have plotted thephasemeter measurements of the spectral noise of the free-running, cavity
stabilized, and VCO noise sources used in the UFLIS simulations. Cavity
pre-stabilization provides us with 4 order of magnitude free-running noise
suppression at 10 mHz. The VCO provides similar noise characteristics tothe cavity stabilized lasers below 10 Hz and is used as a test input for many
simulations.

133

6.2 The Electronic Phase Delay (EPD) Unit
The electronic phase delay (EPD) unit is constructed out of all three Pentek
components described in chapter 5.1. Generally described, the Pentek-6256 phasemeter front-end produces frequency information at a 10-100 kHz data rate which is thenpassed to the Pentek-4205. The frequency information is stored on the Pentek-4205 ina buffer who’s length is based on the data-rate and the desired hold time.1 Once thedelay is applied the frequency information is interpolated to introduce the time-changing
delay. The interpolated data is added to a large MHz Doppler modulation and small mHz
GW modulation. The processed frequency information is then sent to the Pentek-6228
where it is integrated to generate the phase and used to drive a numerically controlled
oscillator (NCO). The NCO signal is reconstructed using the Pentek-6228’s DACs. The
details of the design and noise are described in the following sections.
6.2.1 Design
A diagrammatic model of a single channel of the EPD unit is shown in Figure 6-3.
The Pentek-6256 has 4 ADC’s and two FPGA’s which are connected to four independent phasemeters producing 60-bit frequency data at a CIC down-sampled rate offEPD = 10 − 400 kHz depending on the application and clock rate. Assuming the EPD
measures a signal, Asin(2πfin (t)t), with a clock given by Asin(2πfClk (t)t), the phasemeter
output takes the form:
fEPD [n] = fin [n] +finfClk [n]
fClk

(6–3)

with n = t/fEPD due to the down-sampled EPD data-rate.
This data is packed into 16, 32-bit packets which take the form shown in Figure 6-4.
The last two bits of each packet are used as a tag with which to ensure proper
data-communication. Data-space for 60 bits of Doppler and 60 bits of GW information

1

delay time (τ )134

CLK(t) ∝ sin(2(fs+fs t) t)

in(t) = Ain(t) sin(in(t))

out(t)Figure 6-3. Model of the electronic phase delay (EPD) unit: Here we have depicted amodel of EPD unit, which is constructed using the DSP systems described inchapter 5.1. The input signal is measured with a PM core and the frequency
data is stored in memory for a pre-defined storage time representing theinter-SC light travel time delay. The delayed frequency data is linearlyinterpolated to simulate a time-changing delay (not pictured) and added to a
Doppler offset/modulation and the simulated GW signal. A numericallycontrolled oscillator (NCO) reconstructs the delayed frequency information
and an DAC regenerates the signal with the same clock used to sample theinput.
is left open in the packing scheme which is be populated with data on the Pentek-4205
once the frequency data has been read from the 6256. Depending on the desired delay,
τ , the storage space for the τ /fEPD data-points must be reserved for each channel.
After the frequency data is stored for the proper amount of time, it is linearlyinterpolated to apply a time-changing delay. It is important that the interpolation takethe right form; LISA will have a linear change in the phase rather than a linear change
in the frequency. As a result, assuming we are trying to interpolate the data such thatt → t(1 − β), a factor of (1 − β) must be introduced to account for the time-integration
nature of the frequency to phase conversion. That is, where one might normally use:f[n(1 + β)] = (1 − βn)f[n] + f[n + 1]

135

(6–4)

Figure 6-4. EPD unit data-packing scheme: The 60-bit frequency, Doppler, and GW
modulation data of the two independent EPD channels is multiplexed into a
16 point data-stream of 32-bit values. Numerical flags are added to themost-significant-bits of the frequency data to ensure that the data is correctlyinterpreted and data-points are not missed.

136

to apply a linear interpolation between f[n] and f[n + 1], we will use:f[n(1 + β)] = (1 − β)(1 − βn)f[n] + f[n + 1]

(6–5)

to account for the frequency to phase conversion.
Once the linear interpolation is applied the frequency data is packed with a 1 µHz to
1 mHz GW modulation signal and a ±20 MHz Doppler signal. It is possible to modulate
the Doppler signal to account for the breathing of the LISA constellation but in thefollowing experiments, we will always use a constant Doppler shift. The GW signal
can be arbitrarily generated using the on-board processor or the GW data can be
uploaded to the EPD unit from the data-processing computer before the delay process
is begun. Once the GW, fGW (t), and Doppler, fDop , information is packed along withthe time-interpolated frequency data, again, as shown in Figure 6-4, the information isadded together, integrated to generate the phase information, and generated using the
NCO/DAC output described above.
Zero-Delay EPD Unit (Frequency Control Filtering).
Bypassing the Pentek 4205 data-storage and, instead, filtering and transmitting thePM generated data directly to the NCO, we can design high-speed frequency control
filters for heterodyne locking, phase locking, and arm-locking experiments. The 60-bit
phasemeter frequency data, fPM [n], is passed through a digital fixed point finite impulse
response (FIR) filter; the filtered data can be reconstructed by an NCO with less than a
1 ms delay2 and used to control the laser frequency. For more detail on these filteringand locking methods, refer to [54] and chapter 6.3.1.

2

Optimal performance results in a ∼results in a 17 kHz feed-back bandwidth.

0.06 ms filter delay. A first order approximation

1376.2.2 Verification
The EPD unit’s noise can be written in terms of the input-frequency data, fin (t), thedelay, τ , and the sampling (φClk,ADC (t)) and regeneration (φClk,DAC (t)) clocks3 as:
φEPD (t) = φin (t − τ (t)) + hGW (t)
+

(6–6)

fin
fin ± fDop
φDAC,Clk(t) −φADC,Clk(t − τ (t))
fClk
fClk

from which we can see the clock noise terms coupling into the measurement. Interestingly, these clock noise terms take the same form as those described in (3–70) and(4–3). In this way, the EPD unit produces the differential clock noise terms which cancelin (3–70) and would normally need to be accounted for by clock noise transfers.
Testing the EPD performance, we use a VCO signal as a phase noise source andmeasure the original VCO and and EPD-delayed VCO signals with phasemeters. This
produces two signals of the form:
φVCO,PM (t) = φVCO (t) −fVCO
φPM,Clk(t),fClk

fVCO ± fDop
φDAC,Clk(t)
fClk
fVCO
fVCO ± fDop−
φADC,Clk(t − τ (t)) −φPM,Clk(t).
fClk
fClk

(6–7)

φVCO,EPD,PM (t) = φVCO (t − τ (t)) + hGW (t) +(6–8)

Time shifting the phasemeter measurement and subtracting it from the EPD measurement we obtain a differential result given by:∆φEPD (t) = φVCO,EPD,PM(t) − φVCO,PM (t − τ )= hGW (t) +
−

(6–9)

fVCO ± fDop
(φDAC,Clk(t) − φPM,Clk(t))
fClk

fVCO
(φADC,Clk(t − τ (t)) − φPM,Clk(t − τ (t))).
fClk

3(6–10)

Although these are generally the same source, they may have non-common terms
which do not cancel.

138

If we assume that the clock used to drive the EPD input ADC and output DAC are basedon the same source, φEPD,Clk(t), we can write the expected noise level as:∆φ̃EPD (t) = h̃GW (t) +fVCO 1 − e−2πiτ ± fDop[φ̃EPD,Clk(ω) − φ̃PM,Clk(ω)].
fClk(6–11)

Thus, we couple in the differential clock noise scaled by the arm transfer function,
1 − e−2πiτ . Taking measurements of this form with independent clock sources,
phase-locked clock sources, and split clock sources, and comparing the noise levels
to the differentially subtracted EPD measurements as shown in Figure 6-5 we seethat the phase noise of the measurement (solid lines) equals the clock noise (dottedlines) scaled by the arm transfer function, 1 − e−2πiτ , with τ = 16.6 s, as expected by(6–11). Obviously we will use the split clock source to drive both the EPD units and PM
measurements to reduce this clock noise coupling.
Using a 4 MHz VCO input source and a split clock source we take a long-term
EPD measurement to see the over-all noise level of the EPD unit. The results, with andwithout a 7.2 mHz GW modulation are plotted in Figure 6-6 and compared against thephasemeter noise level. Thus, it is reasonable to say that the EPD unit reproduces the√inter-SC light travel time delay to within a 10 µCycle/ Hz accuracy4 .
Although the EPD unit meets the requirement, we must check to see how the clock
noise sources couple into a heterodyned differential arm measurement. To reproduce
the sensor signals described in (3–72) or (3–74) we take the EPD delayed signal, (6–6),
and electronically mix it with the same VCO used as the input to the EPD unit. The

4

The minimized time-delay is found using the methods described in Chapters. 4.4.2.1
and 4.4.2.2

139

Linear Spectral Density

2

10

Input
Delay Unit (Same Clock)
Clock Splitter Noise
Delay Unit (Unlocked Clock)
Clock Noise
Delay Unit (Locked Clock)
Clock PLL NoiseLISA 12pm Requirement0

Cycles/sqrt(Hz)

10

−2

10

−4

10

−6

10

−3

10

−2

10

−1

10
Frequency (Hz)

0

10

1

10

Figure 6-5. Noise limitations of the EPD unit: The differential clock noise between the
EPD unit and the phasemeter used to verify the EPD noise limits themeasured performance derived in (6–11) for fDop = 0. The clock noise(dotted lines) for the three different cases, (1) free-running, unlocked clocks(2) phase locked clocks, and (3) electronically split clocks are estimated bytaking a differential measurement of the same 1 MHz VCO source using
different clock sources. The EPD noise scales with this differential clock
noise multiplied by the delayed-comparison (or sensor) transfer function,
1 − e−2πiτ , as expected.
mixed output of the prompt and delayed signals takes the form5 :∆φArm (t) = φVCO (t) − φEPD (t)(6–12)= φVCO (t) − φVCO (t − τ (t)) − hGW (t)−5fVCO − fDop
fVCO
φDAC,Clk(t) +φADC,Clk(t − τ (t))
fClk
fClk

We assume the Doppler shift is negative in this case.

140

8

10

VCO Input
EPD Unit Noise Performance
EPD Unit with GW Modulation
1pm Requirement
12pm Requirement

6

10

4

10

Cycles/sqrt(Hz)

2

10

0

10

−2

10

−4

10

−6

10

−8

10 −4
10

−3

−2

10

10
Frequency (Hz)

−1

10

0

10

Figure 6-6. EPD Unit’s phase-noise transmission accuracy: When using the same clock
source for the EPD sampling, EPD regeneration and PM measurement
systems, the common clock noise terms cancel to within the accuracy of theclock distribution and the EPD unit results in the plotted transmission
replication accuracy. A 7.2 mHz GW modulation is added to one of themeasurements to verify that the hGW (t) terms in (6–11) do not cancel.
Measuring this with a phasemeter:∆φArm,PM (t) = φVCO (t) − φVCO (t − τ (t)) − hGW (t)−(6–13)

fVCO
fVCO − fDop
φDAC,Clk(t) +φADC,Clk(t − τ (t))
fClk
fClk
fDop
−
φPM,Clk(t).
fClkwhich, reduces in the case where φPM,Clk(t) = φDAC,Clk(t) = φADC,Clk(t) to:∆φArm,PM (t) = φVCO (t) − φVCO (t − τ (t)) − hGW (t)−(6–14)fVCO(φDAC,Clk(t) + φADC,Clk(t − τ (t))).
fClk

where we have left φDAC,Clk(t) and φADC,Clk(t) shown explicitly with which we cancompare these directly to (3–72) and (3–74). Again, this measured EPD unit signal

141

acquires the same input and clock noise scaling as the clock noise corrected inter-SC
observables, (3–70).
6.2.2.1 Time-changing Time Delay
To verify the capabilities of the electronic replication of the time-changing delay,
we take the prompt measurement of a VCO signal and compare it with the an EPDprocessed signal as described in the previous section. The experimental delay is
set to τ (t) = 16.6 s + 100t ns/s. If we simply minimize the noise without considering
the time-changing delay, we will obtain the average delay for the measurement as
the calculated offset delay time. The difference between the signals as plotted in
Figure 6-7 shows that the of the difference is in-phase with the prompt signal at the
beginning of the measurement and out-of-phase with the prompt signal near the endof the measurement. This obviously limits the noise cancellation capabilities as shown
explicitly in Figure 6-8
Accounting for the time-changing delay, we perform a the fractional delay interpolation of the input signal using the methods defined in chapter 4.1.4 and 4.4.2.2. Oncethe prompt data has been correctly interpolated we subtract it from the EPD processedsignal. Comparing the results with the base-line EPD noise performance as shown in
Figure 6-8, we see that once the time-changing delay has been accounted for, the noise
performance matches the EPD noise level.
6.2.2.2 Gravitational Wave Injection
To test the GW injection accuracy we inject a frequency modulation with apower-function increasing frequency enveloped by a Gaussian amplitude modulation
given by:fGW (t) = sin

2π(10−3 Hz)tt
3/8
(1 − 3.7∗10
4 s)

(t−to )2e− 2∗107 s2(6–15)

Granted, this is not a realistic GW modulation source but, none the less, it provides
a test of the ability to reconstruct the signals injected by the EPD unit. Using a VCO

142

Figure 6-7. Time series of the interpolated delay difference: The time-series of the input
signal subtracted from the time-delay interpolated signal shows that, at first,
the differential phase leads the input phase due to a small positive timing
error but towards the end of the measurement the differential phase lags the
input phase due to a small negative timing error causing an apparent phase
inversion. Once the time-changing delay has been accounted for thedifferential phase cancels to the EPD precision floor.
source we time-shift the EPD delayed signal and subtract it from the post-processedtime-shifted measurement of the prompt signal. The time-series of the injected input
modulation, the measured GW modulation, and their difference is plotted in Figure 6-9.
From the spectrum of these measurements, we see that the noise level of the GW
injected EPD unit matches the base-line EPD noise level once the expected GW signal
has been removed from the EPD processed measurement.
Another GW source of concern is the low-frequency binary confusion noisebackground. Estimates of this background noise vary [46, 103] such that we will
generally estimate the binary confusion noise level as:h̃GW−Background (ω) =≃This is plotted explicitly in Figure 8-1.

143

.01

mHz

√ .
s
+ .001 Hz
2π(6–16)

Figure 6-8. Interpolated delay differential measurements and corrections: The
time-changing delay causes a spectral noise floor which scales with thedelay error δτ = βT/2 where T is the measurement time as defined by (4–6).
If we time-scale the measured input signal using fractional delay
interpolation and subtract it from the EPD processed time-changing delay
signal, we account for the additional delay error and cancel the noise towithin the EPD unit’s noise floor precision.
6.3 UFLIS Simulations
Using the laser bench-top, EPD unit, control filters, and phasemeter measurements,
we are able to perform many LISA-like tests of the interferometry system including
advanced arm-locking and TDI simulations. Although a more refined and in-depth
analysis of the arm-locking tests and results can be found in [54], we present a first testof the UFLIS electronics by performing a single-arm arm-locking experiment as outlined
in chapter 4.2.2. Next we outline the time-delay interferometry simulations and set up the
TDI characteristics which are to be explored in the next chapter.
6.3.1 Arm-Locking Stabilization
The UFLIS electronic components were first used to test the single arm-locking
methods being presented in [53, 77]. Since a much more in-depth analysis has
been presented since these simulations were performed we will briefly discuss the144

Figure 6-9. Time series of an arbitrary gravitational wave: The gravitational wave
modulation defined by (6–15) is injected into the EPD unit and extracted witha differential input measurement using the appropriate time-shift to cancel
the noise. The extracted signal and the theoretically injected signal agree
well; subtracting the measured GW signal from the theoretical signal, we
completely cancel the injected GW signal to within the EPD unit precision.
experimental setup, the results obtained, and the un-anticipated limitations. The single
arm-locking experiment is presented at the top of Figure 6-11. The PDH pre-stabilized
LaserL1 is heterodyned with LaserPL which is offset phase-locked to LaserRL . The
frequency off-set in the phase-lock is driven by an oscillator modulated with thearm-locking control signal. The arm-locking control signal is generated by taking theLaserL1 /LaserPL beat-note and forming the sensor signals as defined in (3–72) using
an EPD unit with a Doppler offset, fDop , and time delay of τ = 1 s. This sensor signal isfiltered with the frequency feed-back controller described in chapter 6.2.1. The feed-backfilter is defined in terms of the frequency as:
TAL (f) =ha

a3 a2 a14
+ 3 + 2 +
4
f
f
f
f

i

a0
+√ .
f

(6–17)

The f 1/2 provides the an additional phase advance in the region of the locking frequency.
This is because the sensor signal’s phase swings between −π and π with respect to the145Figure 6-10. Spectral correction of an arbitrary gravitational wave: The EPD unit with a
VCO input is injected with the GW modulation defined by (6–15); the EPD
output and input are measured with phasemeter and then fractional delay
filtered to cancel the common VCO noise. The measured GW signal(green) matches the spectral density of the injected wave-form (red)exactly. Subtracting out the GW modulation (cyan) we are left with a noiselevel which equals the EPD precision (purple). The GW modulated curve(cyan) is low-pass filtered so that the GW modulation is apparent in thetime-series (Figure. 6-9)
input laser phase at frequencies above the first zero, 1/τ . Meanwhile, the low-frequency
integrators kick in at frequencies below the first zero, 1/τ , and scale with the coefficientssuch that a4 < a3 < a2 < a1 to maintain locking stability. The results, shown in
Figure 6-11, show an additional five orders of magnitude noise suppression of thelaser frequency noise at 10 mHz and significantly reduced frequency fluctuations in thetime-series. The low-frequency stabilization limit is defined by the accuracy of the control
electronics [54].
Advanced Arm-Locking Controllers.
The primary complication with achieving the locking condition described in theprevious chapter was the user-defined Doppler and phasemeter frequencies. Unlessthese values were exactly equal the experiment encountered an integration of the146

Figure 6-11. Long-arm hardware-based single-arm-locking experiment: The model at
the top depicts the experimental setup of the first hardware based
heterodyne single-arm-locking experiment. The RL/L1 beatnote provides ameasure of the ’input’ noise while the PL (phase-locked)/RL beatnote isstabilized according to the proposed arm-locking techniques [53]. The
low-frequency noise is suppressed by 5 orders of magnitude at frequencies
below 10 mHz. The measurement is limited by quantization noise in the
EPD unit. Refer to [54] for more information on the details andadvancements to this simulation in performing LISA-like tests of thearm-locking capabilities. These simulations provided a first test of the
UFLIS/DSP system’s capabilities to perform LISA-like simulations.
frequency error which caused system instability. This indicated a frequency pulling effectwhich has since been described in [79] and [84].
Since the single arm-locking experiment was performed, more in-depth experiments
have been conducted by the author and colleagues. A detailed presentation of thesingle arm-locking experiment and results can be found in [78]. The same arm-locking
methods were used in combination with a PDH side-band locking scheme to prove thecapabilities of laser frequency control in LISA [83]. These components were also used to147

Table 6-1. TDI experimental characteristics: Five 40000 s experiments are performed
with increasingly more complicated, yet more LISA-like, characteristics. The
transponder experiment provides us with a baseline measure of theexperimental setup’s noise performance. The ’dynamic’ experiments
demonstrate the ability to determine and account for the time-changing
delay-times. The ’LISA-like’ experiment proves the ability to remove
independent SC noise sources and constrain one-way delay times. Finally,
the confusion noise experiment verifies that the TDI-ranging capability will not
be limited by low-frequency LISA noise sources.
Simulation Name
s1r
Inter-SC Velocity
Injectedβ (ns/s)
GW Signal
Static Transponders1r ≃ 0β2 = 06.227 mHz
(2-Way)β3 = 0
BinaryDynamic Transponders1r ≃ 0β2 = −100
6.227 mHz
(2-Way)β3 = +150
BinaryStatic LISA-like
s1r = φPLLr √β2 = 06.227 mHzβ3 = 0
Binary(4-Way)≃ (1.0/f)mCycle/ Hz
Dynamic LISA-like
s1r = φPLLr √β2 = −100
6.227 mHzβ3 = +150
Binary(4-Way)≃ (1.0/f)mCycle/ Hz
Confusion-Noises1r = φPLLr √β2 = −100
6.227 mHzβ3 = +150
Binary + CN
(4-Way)≃ (1.0/f)mCycle/ Hz
verify the dual and dual-modified arm-locking schemes presented in [79] and validated
in [84] and [81].
6.3.2 TDI Simulation Outline
The TDI measurements the following sections are generated using the 2-4 sensor
observables, s1r and/or ss1 , defined in (3–70) over a 10-12 hour simulations. The five
TDI experiments we will perform are outlined in Table. 6-1. Each stage will introduce a
new noise source into the measurement such that we can decipher the noise couplings
based on the results of each simulation.
We begin with a ’static transponder’ measurement in which we will use constant, but
unequal, τ2′ 2 6= τ33′ 6 , arm-lengths to test the TDI-X1.0 combinations. The transponder6[τ2 ≃ τ2′ ≃ 16.55 s] 6= [τ3′ ≃ τ3 ≃ 16.75s]

148

measurements, as shown in Figure 7-1, will simulate LISA in a noise-less far-SC
configuration, as though the ’outgoing’ master laser fields are reflected perfectly off
the far SC and where s1r = 0. The ranging methods described in chapter 4.4.2.2 areused to calculate the round-trip delay time7 and cancel the laser phase noise. Next, we
introduce time-changing arm-lengths with the β2 and β3 values outlined in Table. 6-18 9
to validate the capabilities of the TDI-X2.0 combinations through ’dynamic’ simulations.
Next, we introduce the noise coupling from the far SC signals, s1r , by phase locking
the lasers on the far SC to the delayed ’out-going’ laser fields and measuring theresulting PD sensor signals as shown in the experimental model (Figure 7-7)10
Again, we repeat the ’static’ and ’dynamic’ simulations, including the additionallaser phase noise signals, s1r , in the combinations; meanwhile, we are able to test
the ranging-accuracies of the one-way delay times using the methods described inchapter 4.4.2.2. Finally, we introduce a low-frequency confusion-noise backgroundsignal to investigate the limitations, if any, of this additional low-frequency noise on theranging accuracies.

7

The one-way delay times are un-defined in these simulations.

8

β2 = −100.0ns/s ns/s(v2 = −30.0 m/s); β3 = +150.0ns/s ns/s(v2 = 45.0 m/s)

9|β2 − β3 | = 250.0 ns/s

10

The far SC sensor signals, s1r , are used as the input signals for phase-locking the far
SC’s lasers such that the sensor signal is equal to the PLL’s error signal, φPLLr , as shownin Figure 7-7 and Table. 6-1.
149

CHAPTER 7
TIME-DELAY INTERFEROMETERY SIMULATONSThe following chapter present the details and results of the experiments describedin chapter 6.3.2. In all of these measurements we will inject a 6.277 mHz GW binary
with a strain amplitude of 4.35 × 10−20 m/m 1 to simulate the RXJ0806.3+1527 binary
described in the verification binary table (Table 2-1). Although this strain amplitude is afactor of 100 larger than what is expected in a actual LISA-like detection, it is used toprove the TDI combinations’ ability to cancel the laser phase noise without canceling themodulated GW signals.
7.1 Transponder TDI Simulations
We first establish a baseline ranging and measurement precision with static
arm-lengths (β = 0) and without noise being introduced by the ’far’ SC. Referring to
the experimental model shown in Figure 7-1, we take the LaserRef /Laser1 beatnote anduse it as the input for the TDI simulator. The PD beatnote is electronically mixed with afrequency modulation fMod (t) = Asin([ωo + 500sin(2 ∗ pi ∗ t)]t) to apply the ranging tone2
and to shift the laser beatnote to a PM measurable frequency, in this case, ωPD − ωo
= 7 MHz. This mixed output is electronically split, measured with a phasemeter, andprocessed by EPD units to simulate the ’out-going’ inter-SC laser field transfer. Constant
EPD delays of τ2 (0) ≃ τ2′ (0) ≃ 16.75 s and τ3 (0) ≃ τ3′ (0) = 16.55 s are programmed to
the EPD units while Doppler shifts of fDop2 = +3 MHz and fDop3 = −2 MHz are applied to
the frequency signals. The delayed ’out-going’ signals are directly connected to another
EPD unit to simulate the ’returning’ inter-SC laser field transfer. This noise-less transfer
between the EPD units, in a LISA model for example, will behave as though the laser1ν(t) = 2 sin(2π(6.227 mHz)t) µHz
δφGW (t) ≃ 4ν(t)/(2π(6.227 mHz)) = 204.4µcyc/Hz
hGW (t) = δφGW (t)λ/L = 4.3 × 10−20 Hz−12fmod = 1 Hz, Amod = 500 Hz

150

Figure 7-1. Model of the TDI-Transponder experimental benchtop: This TDI-Transponder
model shows how the individual pre-stabilized laser fields are combined toform the input noise signal and how this signal is processed by the EPD
units to replicate a LISA-like inter-SC laser link and produce the ssr
observables []. The input PD beatnote is demodulated to a PM-measurable
frequency using an oscillator with a 1 Hz phase modulation. The
PM-read-out signals are delayed by the τ2′ (t) and τ3 (t) inter-SC light travel
times, added with GW modulations, and regenerated with a Doppler offset.
In these transponder measurements the delayed signals are immediately
injected into the ’return’ trip EPD units τ2 (t) and τ3′ (t) as though the laser
fields were reflected off mirrors at the far SC. GW modulations and Doppler
offsets are added to the return-trip EPD units also. The round-trip delay
signals are electronically mixed with the local LaserRef /Laser1 beatnote toproduce the s21 and s31 sensor signals.
fields were reflected off mirrors on the far SC such that the far sensor observables are
signal-free, s1r = 0, in the TDI combinations. Finally, the ’return’ field EPD simulated
output is heterodyned3 with the local laser phase, in this case the LaserRef /Laser1

3

Electronically mixed

151

beatnote, to produce the local s21 , s31 PD observables. The PD signals are measured for40000 s by phasemeters with the 60-bit precision at a 14.9 Hz data-rate.
We then introduce the time-changing arm-lengths defined in Table 6-1 such thatthe inter-SC time delays take the form defined in (4–39). Despite this definition, as we
described in chapter 4.4, without any noise being introduced at the far SC we will only beable to constrain the round-trip delay times found in, (3–72) and (3–74) defined explicitlyas:τ2′ 2 (t) = [(1 − β2 )((1 − β2′ )(t − τ2′ (0)) − τ2 (0))](7–1)

τ33′ (t) = [(1 − β3′ )((1 − β3 )(t − τ3 (0)) − τ2′ (0))].

(7–2)

Fortunately, at the same time, this means we only have to scan over the 2-dimensional,
τ2′ 2 , τ33′ , basis. We will also verify the TDI 1.0 limitations defined in chapter 4.3.3.2.
7.1.1 Static-Arm Transponder Simulation
Utilizing the PD measurements of s21 and s31 , the 40000 s data-set is broken into
40, 1000 s segments. The iterative formation of the TDI combination for each segment
minimizes attempts to minimize the ranging tone and calculate the time-delay functions
through linear regression as described in chapter 4.4.2.2 and shown in Figure 4-3. The
first and only iteration produces a slope error (constraint on the arm-length velocities), of|2β| < 50 fs/s and a variance (round-trip ranging accuracy) of less than 0.6 ns (∼ 0.18 m)
as shown in Table 7-14 . In this experiment, we note that the TDI-X1 combination’s
ranging-tone minimization produces the same estimation to within the measurement
error of the round-trip light travel time as the TDI-X2 combination. If β 6= 0, thedifferent TDI combinations would produce different estimations since the ranging
tone minimization using the TDI-X1 combination would be limited by (4–32) and will4

Refer to Figure 7-4 to see the results of the ranging-tone cancellation near the 1 Hz
frequency bin.

152

8

10

6

Ranging Tone

10

4

10

Arm−2 Sensor (∆ (t))
22Arm−3 Sensor (∆3(t))

0Phasemeter Precision
TDI 1.0 Combination
TDI 2.0 Combination

Cycles/√Hz

10
10

−2

10

−4

Pre−Stabilization
Requirement

6.227 mHz Binary

IMS Requirement
PM Requirement

10

−6

10

Ranging Limit (0.6 ns)
−8

10 −4
10

−3

10

−2

10
Frequency (Hz)

−1

10

0

10

Figure 7-2. Raw static transponder experimental results: The sensor signals(s21 (t) = ∆21 (t), s31 (t) = ∆31 (t)) are used with the ranging values found in
Table 7-1 to generate the raw spectra of TDI-X1 and TDI-X2 combinations.
The phasemeter measurement limitation for the 7 MHz input beatnote
frequency is also plotted for comparison. The TDI-combinations drop below
the measurement limitation due to the inherent TDI transfer functions but theGW signals are also suppressed by the TDI transfer function. The 6.277 mHz
gravitational wave is only revealed once the laser noise has been removed.
The expected ranging limitation is almost 2 orders of magnitude below the
TDI combinations indicating that all the input laser noise has been
suppressed beyond the measurement sensitivity.
estimate the light-travel time-delay as the mean delay for a particular data-segment.
Thus, since each of these TDI-ranging methods produce the same result, even in thecase where β = 0, we will use the TDI-X2.0 combinations for all TDI-ranging estimations.
Using the ranging-calculated values (Table 7-1) we form the TDI-X1 and TDI-X2
combinations for the entire data-set. The linear spectral density of the TDI combinations,as plotted in Figure 7-2, show the laser noise cancellation and reveal the phasemodulated GW binary at 6.277 mHz. The expected ranging limitation based on the 0.6 nsvariance calculated above and evaluated with (4–6) is well below the measurement
sensitivity. We can also see the difference between the 8-pulse and 16-pulse response153transfer functions of the TDI-X1 and TDI-X2 combinations. Rescaling by the average
inverse TDI response transfer functions given by:
T1.0 (s) =|1 − e−stau2′ 2 | + |1 − e−stau33′ |2

T2.0 (s) = T1.0 (s) 1 − e−stau2′ 233′

(7–3)
(7–4)

we see the TDI-X1 and TDI-X2 combinations result in identical effective sensitivities(Figure 7-3), as expected for a static interferometer. From this plot we can also see thatboth measurements are limited by the EPD unit’s phase-transmission precision as waspreviously plotted in Figure 6-6. From this point forward we will interpret these static
TDI-X1 and TDI-X2 combinations to represent the baseline precision of the UF-LISA-TDI
simulator. We calculate the effective laser suppression magnitude by dividing the input
spectrum by the input noise by the re-scaled TDI spectra; from Figure 7-6 we obtaingreater than 1010 noise cancellation at 1 mHz.
7.1.2 Dynamic-Arm Transponder Simulation
Expanding the simulation characteristics, we include the time-dependent armlengths into the experiment with the β-values defined in Table 6-1 resulting in an
expected TDI-X1 limitation as defined by (4–32) with |∆β| = 250 ns/s. Initially assuming
β = 0, the 40000 s measurements of the s21 and s31 signals are broken into 40, 1000 s
segments and processed using the methods described in chapter 4.4.2.2. These
data segments are used to minimize the ranging tone and calculate the round-trip
time-delay offsets, τ2′ 2 (0) and τ33′ (0), for each segment as shown in Figure 4-3. The
linear regression of these time-delay offsets provides a first-order measure of the β toan accuracy of 100ps/s as shown in Table 7-1. The process also calculates a first ordermeasure of the round-trip time delay with a ranging precision of < 7.5 µs (∼ 1.7 km) but
due to the incorrect β = 0 assumption, these values tend to equal the average delay
for the data-segment. A second iteration improves the β accuracy to 80 fs/s and theranging precision to < 5.9 ns (∼ 1.8 m) providing a much more accurate measure of

154

155

Table 7-1. Transponder TDI-ranging estimation: The method outlined in Figure 4-3 and the experiments described incapters 7.1.1 and 7.1.2 are used to calculate estimates of the inter-SC delay functions, (4–37), and ranging
errors. The estimates verify the TDI-ranging capabilities and are used to generate the TDI-combinations for
Figures 7-3 and 7-5. The ’return’ delay times tend to be longer than the ’outgoing’ delay times by 250 ms as aresult of internal delays within the DSP system’s EPD units.
δτ22′ , δτ33′ (σ)
Iterationβ (Slope)
τ22′ (0), τ33′ (0) (Offset)
TDI 1.0
Transponder
1 (TDI 1.0) 2β2 = −44.5 fs/s ± 20.9 fs/s
τ22′ (0) = 33.55204887148 s ± 0.23 ns δτ22′ = 0.54 ns
2β3 = −46.3 fs/s ± 12.5 fs/s
τ33′ (0) = 33.15222859583 s ± 0.14 ns δτ33′ = 0.32 ns
1 (TDI 2.0) 2β2 = −41.0 fs/s ± 21.2 fs/s
τ22′ (0) = 33.55204887151 s ± 0.24 ns δτ22′ = 0.55 ns
2β3 = −46.3 fs/s ± 12.7 fs/s
τ33′ (0) = 33.15222859579 s ± 0.14 ns δτ33′ = 0.33 ns
TDI 2.0
Transponderδτ22′ = 7.5 µs
12β2 = −200.247 ns/s ± 100 ps/s
τ22′ (0) = 33.5518847 s ± 2.3 µs
δτ33′ = 7.0 µs
2β3 = +300.056 ns/s ± 95 ps/s
τ33′ (0) = 33.1525027 s ± 2.2 µsδτ22′ = 5.9 ns22β2 = −199.9998668 ns/s ± 80 fs/s τ22′ (0) = 33.5519484187 s ± 1.8 ns
2β3 = +300.0001130 ns/s ± 23 fs/s τ33′ (0) = 33.1523897572 s ± 0.51 ns δτ33′ = 1.7 ns
32β2 = −200.0000058 ns/s ± 8.9 fs/s τ22′ (0) = 33.55194832884 s ± 0.20 ns δτ22′ = 0.65 ns
2β3 = +300.0001361 ns/s ± 4.5 fs/s τ33′ (0) = 33.15238977691 s ± 0.10 ns δτ33′ = 0.33 ns

810Pre−Stabilization
Requirement
6

10

Pre−Stabilized Laser Noise φ1(t)Simulator Baseline (EPD Noise)
Corrected TDI 1.0 Combination
Corrected TDI 2.0 Combination

4

10

2

Cycles/√Hz

10

TDI 1.0/2.0
0

10

−2

10

−4

10

IMS Requirement
PM Requirement−610

Ranging Limit (0.6 ns)
−8

10 −4
10

−3

10

−2

10
Frequency (Hz)

−1

10

0

10

Figure 7-3. Corrected static transponder experimental results: The input signal, φ1 (t), isplotted along with the corrected TDI-X1 and TDI-X2 spectra which have beenscaled by the inverse TDI transfer functions (7–3). Both TDI spectra matchexactly once the inverse TDI-combination transfer function has beenaccounted for. The increased laser phase noise around 100 mHz is a resultof an over-coupled temperature tracking in the PDH controller which may
result in a ranging-limited sensitivity loss near 100 mHz. The EPD unit’s
phase transmission accuracy is plotted and is identified as the primary TDI
simulation limiting noise source. This static TDI-transponder result will act as
the simulator’s ’baseline’
performance for the rest of the chapter. The results√
meet the 18 pm/ Hz IMS requirement (3–5).
the time-dependent arm-lengths, τ2′ 2 (t) and τ33′ (t). The final iteration optimizes the βprecision to 8.9 fs/s and the ranging precision to less than 0.65 ns (∼ 0.2 m). (Table 7-1)
Applying the calculated round-trip functional values from the third iteration of theranging procedure, the entire data-set is used to produce the linear power spectral
density for the TDI-X1 and TDI-X2 combinations as plotted in Figure 7-5. The TDI-X1
combination is limited, as anticipated, by (4–32) with τ ≃ 16.6 s and |∆β| = 250 ns/s. The
TDI-X2 combination’s correction terms account for this dynamic arm-length limitation
and remove the velocity dependent laser phase noise resulting in a sensitivity equal tothe experiment’s baseline noise. This result meets the IMS requirement defined by the156Baseline

4

4

10

2

0

TDI 1.0 Phase−Locked0

TDI 2.0 Phase−Locked

2

10

10

0

10

4

10

2

10

10

4

10

2

10

Cycles/√Hz

TDI 2.0 Transponder

10

0

10

10

φ (ω)
1

∆2(ω)

−2

−2

∆ (ω)

10

−2

10

3

−2

10

10

Simulator Baseline
TDI−X (ω)
2

PLL Noise

−4

10

−4

−4

10

−6

−6

10

−8

Frequency (Hz)

1.05

10
0.95

−6

10

−8

1

10

−6

10

10
0.95

−4

10

10

−8

1

1.05

−8

10
1.45

Frequency (Hz)

1.5

Frequency (Hz)

1.55

10
0.95

1

1.05

Frequency (Hz)Figure 7-4. Ranging tone cancellation spectral results: The ranging tone modulated
input are plotted along with the resulting TDI-X2 combinations for the four
primary experiments (Table 6-1). The decreased accuracy of the ∆2 arm ascompared to the ∆3 arm in Table 7-1 is likely due to the proximity of thenearest arm zero. Note that the ranging tone for the TDI 1.0 withphase-locked lasers simulation is at 1.5 Hz instead of 1 Hz which results in a
reduced one-way ranging accuracy (Table 7-2) from the reduced PLL noise.
LISA mission concept design. (Table 3-1) The 0.65 ns (∼0.2 m) ranging precision, as
calculated in Table 7-1 and plotted in Figure 7-5, is not expected to be a limiting noise
source.
The TDI-X1 ’s noise suppression (Figure 7-6) equals the theoretical inter-SC
velocity dependent limit (4–32) while the TDI-X2 ’s noise suppression equals that of
the simulator’s baseline suppression characteristics.
7.2 LISA-like (Master-Slave Phase Locked Laser) TDI Simulations
At this point, we include the phase-locking and transmission of the Laser2/3 signals
on the far spacecraft as shown in Figure 7-7. Thus, this experiment will generate andmeasure all four ssr 5 beatnote observables as defined in (3–70). The two far sensor

5

Two local-SC measurements: s21 , s31 ; Two far-SC measurements: s12 , s13

157

8

10

Pre−Stabilized Laser Noise φ (t)Pre−Stabilization
Requirement

1

TDI 1.0 Combination
TDI 1.0 Limit (∆β=250 ns/s)
Simulator Baseline
Corrected TDI 2.0 Combination

6

10

4

10

2

TDI 1.0

Cycles/√Hz

10

0

10

−2

10

TDI 2.0−4

10

IMS Requirement
PM Requirement−610

Ranging Limit (0.65 ns)
−8

10 −4
10

−3

10

−2

10
Frequency (Hz)

−1

10

0

10

Figure 7-5. Dynamic transponder experimental results: The suppression of the TDI X1
combination is limited by the arm-length time-dependence as defined by(4–32) with ∆β = 250 ns/s. The TDI X2 combination removes the additional
time-dependent-coupled laser noise and reveals the 6.277 mHz GW signal.
As with the static case (Figure 7-3), the EPD unit’s phase transmission
accuracy is the primary limiting noise source in the TDI combinations
although, some sensitivity loss may occur due to a limited ranging capability
around 100 mHz (Figure 7-9.
signals, s12 , s13 , are used for, and are equal to, the phase noise of the PLL laser-locking
error signal. Meanwhile, the heterodyned PD output represents the local laser phaseand is used as the input for the ’return’ field simulation of inter-SC light transmission.
These signals are used to construct the TDI combinations and minimize the ranging
tone using the iterative process described in chapter 4.4.2.2 and shown in Figure 4-3
through a 4-dimensional sweep of the individual light-travel times, τ2 (t), τ2′ (t), τ3 (t),τ3′ (t).
7.2.1 Static-Arm LISA-like Simulation
The reader should note that these measurements were taken earlier than boththose presented in chapters 7.1.1, 7.1.2, and 7.2.2. As a result, the ranging tone was

158

−3

10

TDI 1.0 ∆β=0
TDI 2.0 ∆β=0
TDI 1.0 ∆β=250ns/s
TDI 2.0 ∆β=250ns/s
Expected TDI 1.0 Limit (∆β = 250ns/s, τ = 16.6s)

Suppression Magnitude (|X(ω)|/|ssr(ω)|)

−4

10

−5

10

−6

10

−7

10

−8

10

−9

10

−10

10

−11

10

−4

10

−3

10

−2

−1

10

10

0

10

Frequency (Hz)Figure 7-6. TDI laser phase noise suppression: The achieved laser noise suppression
magnitudes for the transponder simulations of the static and dynamic TDI-X1
and TDI-X2 combinations are plotted. The TDI-X1 combination’s noise
suppression equals the theorized limit (4–32) in the dynamic-arm simulation.
The TDI-X1 and TDI-X2 result in the similar laser suppression characteristics
in the static-arm simulation. The dynamic-arm TDI-X2 combination’s laser
noise suppression equals that of the static case, verifying that the velocity
coupled laser noise has been completely accounted for and removed.
placed at a different frequency (1.5 Hz) and the EPD data-rate was higher. Despite thesechanges, the effects are only noticeable through the reduced PLL ranging accuracy.
These same results have also been published in peer-reviewed literature [104].
The optimized outcome of the ranging process is found after a single iteration
and results in a measure of 2β to an accuracy better than ∼ 70 fs/s and a round-trip
ranging precision of less than ∼ 5.0 ns (∼ 1.5 m) as shown in Table 7-2. Applying a linear
regression to the calculated one-way delay times we find a one-way ranging error of∼ 100 µs (∼ 30 km). The outgoing and return delay times are un-equal by ∼ 250 ± 0.1 ms,

159

Figure 7-7. Model of the LISA-like TDI experimental benchtop: This TDI-PLL modelshows how the individual pre-stabilized laser fields are combined to form the
input noise signal and how this signal is processed by the EPD units andused to phase-lock the ’far’ SC lasers, thus replicating all four LISA-likeinter-SC laser link and produce all four ssr observables. The setup is thesame as is defined in Figure 7-1 expect that the ’transmitted’ EPD outputs
are used to phase-lock the far Laser2 and Laser3 . The PLL error signals areexactly equal to the far SC sensor signals, s12 and s13 . The LaserRef /Laser2/3
beatnotes are used as the input signals for the ’return’ field light transmission
simulation.
proving the TDI−X1.5 combination’s6 ability to extract individual phase errors despite
un-equal delays along a single arm (τq (0) 6= τq′ (0)).
Using the calculated one-way functional time-delays we form the TDI−X1.0 and
TDI−X2.0 combinations; re-scaling the TDI spectral density by the inverse arm transfer
functions in (7–3), we plot each TDI combinations’ phase noise in Figure 7-8 resulting
in the same measured sensitivity for both combinations. This noise level is likely due

6Table 4-1

160

8

10

Pre−Stabilized Laser Noise φ (t)1Total PLL NoiseTDI 1.0 Combination
Simulator Baseline
Corrected TDI 2.0 Combination

6

10

4

10

TDI 1.02

Cycles/√Hz

10

0

10

’Far’ PLLs
−2

10

−4

10

−6

10

Laser Ranging Limit (3 ns)
One−Way Ranging Limit (1 ms)

−8

10 −4
10

−3

10

−2

10
Frequency (Hz)

−1

10

0

10

Figure 7-8. Static LISA-like laser experimental results: The TDI spectral results andcomparisons with the input laser phase noise and PLL noise of thestatic-arm TDI simulation with phase-locked lasers is plotted. The corrected
TDI-2.0 combination using the values from the TDI-ranging estimation(Table 7-2) matches with the baseline performance from Figure 7-3. The
laser and PLL noise suppression as a result of the estimated ranging
precision is not expected to be a limiting noise source which is verified bycross-correlating the TDI combination with the input signals as shown in
Figure 7-9. If the PLL noise is not accounted for, the TDI combination islimited by these additional PLL noise sources (not pictured).
to the coupling of multiple un-correlated EPD clock noise sources, defined in (6–11),
into the measurement. Based on the poor cross-correlations between the input noise
and the the PLL noise sources with the resulting TDI combinations we conclude thatthe ranging precision is not a limiting noise source and that all the known sources havebeen removed from the final combination. We also note that the expected ranging
precision using the round-trip and one-way variances along with (4–6) is well below the
TDI combination as plotted in Figure 7-8.

161

Cxy (Magnitude Squared Coherence)

TDI Coherence Functions
1

φ vs. TDI−X (Static Transponder)
1

0.8

φ

PLL2

0.6

2

φ1 vs. TDI−X2 (Static LISA−like)vs. TDI−X (Static LISA−like)2

φPLL3 vs. TDI−X2(Static LISA−like)

0.4
0.2
0 −3
10

−2

−1

10

10

0

10

Frequency (Hz)Figure 7-9. Cross-correlated TDI combinations - Static Transponder and LISA-like: The
magnitude squared cross-correlation of the measured laser phase noise and
PLL noise show a poor correlation with the TDI-X2 combination forfrequencies below for all frequencies in the static TDI-PLL simulation (4-way
delay estimate: chapter 7.2.1). The laser noise correlates with the calculated
TDI-X2 combination for frequencies above 100 mHz in the static
TDI-transponder measurement (2-way delay estimate: chapter 7.1.1). The
poor correlation of the input noise sources with the TDI combinations verify
that the input noises have been sufficiently removed using the TDI-ranging
arm-length estimate.
7.2.2 Dynamic-Arm LISA-like Simulation
After three iterations of the ranging process, the optimized estimation of theinter-SC delay times results in a measure of 2β to an accuracy better than ∼ 70 fs/s
and a round-trip ranging precision of ∼ 5.0 ns (∼ 1.5 m) as shown in Table 7-2. The
constraint’s on the one-way delay times through the residual PLL noise removal can
not be applied until the precision of the round-trip delay times are accurate enoughto remove the input laser noise from the TDI combinations to reveal the residual PLL
noises in the TDI combinations. Thus, it requires at least one iteration of the ranging
process until the one-way delay times begin to be constrained. At the same time,
the accuracy of the constrained one-way delay times will be significantly reducedin comparison to the constraints on the round-trip delay times resulting from thecomparatively reduced noise suppression magnitude of the PLL noise (4–6). Applying

162

a linear regression to the calculated one-way delay times after the third iteration we finda one-way ranging error of about ∼ 100 µs (∼ 30 km). Again, the outgoing and returndelay times are un-equal by ∼ 250 ± 0.1 ms, proving the capability to extract individual SC
phase error with un-equal delays along each arm (τq (0) 6= τq′ (0)).
Applying the optimized one-way functional values from the ranging procedure,
outlined in Table 7-2, we generate the TDI-X1 and TDI-X2 combinations (Figure 7-10).
Rescaling by the inverse arm response function, the TDI-X1 combination meets the
expected limitation (4–32). The TDI-X2 combination meets the LISA IMS requirement
to within a factor of 4 and is, again, limited by a combination of multiple EPD clock
noise coupling sources resulting in a reduced sensitivity above the simulator’s baseline
performance. Again, the poor correlation of the input noise sources with the final
combination, Figure 7-11, leads us to conclude that the ranging precisions are not alimiting noise source and that all known noise sources have been removed from the final
combination. Further supporting this claim, the expected ranging precision using theround-trip and one-way variances along with (4–6) is well below the TDI combination asplotted in Figure 7-10.
Binary Confusion Noise Injection.
Finally, we introduce the low frequency simulated ’confusion noise’ into themeasurement to ensure that these low-frequency terms do not limit the ranging
precision. The confusion noise background given by:
hBackground (ω) ≃

.01

mHz

√
s
+ .001 Hz
2π

(7–5)

where ’s’ is the complex Laplace frequency, and 6.277 mHz mono-chromatic binary
with a single-arm strain amplitude of 3.5 × 10−20 are simultaneously injected. The
injected background noise level is set to match most estimations in the 0.1 mHz
to 1.5 mHz but does not roll off as quickly at frequencies above 1.5 mHz as mostconfusion noise estimates do (Figure 8-1) [46, 103]. This is modified from the expected

163

164

Table 7-2. LISA-like TDI-ranging estimation: The method outlined in Figure 4-3 and the experiments described incapters 7.2.1 and 7.2.2 are used to calculate estimates of the inter-SC delay functions, (4–37 and 4–39), andthe associated ranging errors. The estimates verify the TDI-ranging capabilities and are used to generate the
TDI-combinations for Figures 7-8 and 7-10.
δτ22′ , δτ33′ : δτ2 , δτ2′ , δτ3 , δτ3′ (σ)
Iterationβ (Slope)
τ22′ (0), τ33′ (0) (Offset)
TDI 1.0 PLL
1 (TDI 1.0)
2β2 = −66.1 fs/s ± 45.4 fs/s
τ22′ (0) = 33.5508007898 s ± 1.01 ns δτ22′ = 3.2 ns(Round-trip)
2β3 = 58.7 fs/s ± 37.4 fs/s
τ33′ (0) = 33.2152647154 s ± 0.83 ns δτ33′ = 2.6 ns
1 (TDI 1.0)
β2 = −4.07 ns/s ± 13.6 ns/s
τ2 (0) = 16.694054656 s ± 301 µs
δτ2′ = δτ2 = 1.12 ms∗(One-Way)τ2′ (0) = 16.8567461 s ± 301 µs
β3 = 26.1 ns/s ± 15.9 ns/s
τ3 (0) = 16.51799231 s ± 352 µs
δτ3 = δτ3′ = 0.95 ms∗
τ3′ (0) = 16.6972724 s ± 352 µs
TDI 2.0 PLLδτ22′ = 0.895 µs
12β2 = −199.984 ns/s ± 12 ps/s
τ22′ (0) = 33.59821021 s ± 0.28 µsδτ33′ = 0.568 µs
2β3 = +300.052 ns/s ± 7.8 ps/s
τ33′ (0) = 33.21476669 s ± 0.18 µs
22β2 = −200.000015 ns/s ± 71 fs/s τ22′ (0) = 33.5982645303 s ± 1.6 ns δτ22′ = 5.2 ns
2β3 = +300.000013 ns/s ± 26 fs/s τ33′ (0) = 33.2146434958 s ± 0.58 ns δτ33′ = 1.9 ns
3 (TDI-2.0)
2β2 = −200.000028 ns/s ± 68 fs/s τ22′ (0) = 33.5982645401 s ± 1.5 ns δτ22′ = 5.0 ns(Round-trip)
2β3 = +300.000020 ns/s ± 25 fs/s τ33′ (0) = 33.2146435166 s ± 0.58 ns δτ33′ = 1.9 ns
3 (TDI-2.0)
β2 = −103.3 ns/s ± 1.4 ns/s
τ2 (0) = 16.68021 s ± 31 µs
δτ2 = δτ2′ = 99 µs∗(One-Way)τ2′ (0) = 16.91805 s ± 31 µs
β3 = +152.24 ns/s ± 1.4 ns/s
τ3 (0) = 16.48824 s ± 32 µs
δτ3 = δτ3′ = 105 µs
τ3∗′ (0) = 16.72640 s ± 32 µs
Confusion NoiseTDI 2.0
3 (Round-Trip) 2β2 = −199.999991 ns/s ± 50 fs/s τ22′ (0) = 33.6012734891 s ± 1.1 ns δτ22′ = 3.7 ns
2β3 = +300.000137 ns/s ± 22 fs/s τ33′ (0) = 33.2100302983 s ± 0.49 ns δτ33′ = 1.6 ns
3 (One-Way)β2 = −96.81 ns/s ± 2.3 ns/s
τ2 (0) = 16.73546 s ± 53 µsδτ2 = δτ2′ = 169 µs
τ2∗′ (0) = 16.86582 s ± 53 µs
β3 = +149.439 ns/s ± 1.4 ns/s
τ3 (0) = 16.53994 s ± 32 µs
δτ3 = δτ3′ = 102 µs∗
τ3′ (0) = 16.67009 s ± 32 µs

810Pre−Stabilized Laser Noise φ (t)1Total PLL NoiseTDI 1.0 Combination
TDI 1.0 Limit (∆β=250 ns/s)
Simulator Baseline
Corrected TDI 2.0 Combination

6

10

4

10

Pre−Stabilized Laser
2

Cycles/√Hz

10

0

10

’Far’ PLLs

−2

10

TDI 2.0−410Laser Ranging Limit (5 ns)
−6

10

−8

One−Way Ranging Limit (100 µs)

10 −4
10

−3

10

−2

10
Frequency (Hz)

−1

10

0

10

Figure 7-10. Dynamic LISA-like experimental results: The suppression of the TDI X1
combination is limited by the expected arm-length time-dependence
defined by (4–32) with ∆β = 250 ns/s. The TDI X2 combination removes the
input laser noise, the ’far’ PLL residual phase noise, and thetime-dependent coupled laser noise to reveal the 6.277 mHz GW signal.
The sensitivity limitation comes, most likely, as a result of multiple
unaccounted for EPD noise as a result of the time-changing delay. This isdetermined based on a comparatively improved performance using thestatic-PLL simulation and the fact that the time-changing delay is the majordifference between these measurements.
situation to ensure that the addition low-frequency noise does not limit the ranging
precision. The monocromatic GW phase modulation due to the RXJ0806.3+1527
binary is a factor of 100 smaller than the expected strain amplitude averaged overa 1 year measurement. That said, the GW phase modulation amplitude is simply
programmed to be out-of-phase when injected into the individual arms simulator andwith no consideration for the GW polarization or detector orientation over the course of
the year.
Again, the optimized ranging estimation places bounds on the β accuracy better
than ∼ 50 fs/s and a round-trip ranging precision of ∼ 3.7 ns (∼ 1.1 m). Thus, this165

Cxy (Magnitude Squared Coherence)

TDI Coherence Functions
1

φ vs. TDI−X (Dynamic Transponder)
1

0.8

φ

PLL2

0.6

2

φ1 vs. TDI−X2 (Dynamic LISA−like)vs. TDI−X (Dynamic LISA−like)2

φPLL3 vs. TDI−X2 (Dynamic LISA−like)

0.4
0.2
0 −3
10

−2

−1

10

10

0

10

Frequency (Hz)Figure 7-11. Cross-correlated TDI combinations - Dynamic Transponder and LISA-like:
The magnitude squared cross-correlation of the measured laser phasenoise and PLL noise show a poor correlation with the TDI-X2 combination
for frequencies below for all frequencies in the dynamic TDI-PLL simulation(4-way delay estimate: chapter 7.2.2). As with the static case (Figure 7-9),
the laser noise correlates with the calculated TDI-X2 combination forfrequencies above 100 mHz in the dynamic TDI-transponder measurement(2-way delay estimate: chapter 7.1.2). The poor correlation of the input
noise sources with the TDI combinations verify that the input noises havebeen sufficiently removed using the TDI-ranging arm-length estimate.
confusion noise result achieves a ranging precision on the same order as the simulator’sphase-locked performance, proving that low-frequency noise has little to no effect on theranging tone cancellation or the measured arm-lengths.
Finally, once the long-term residual quadratic phase drifts have been removed
from the strain measurement, the time-domain comparisons between the TDI-X2
outputs of the phase-locked and confusion noise experiments are plotted in Figure 7-12
to show the additional low-frequency noise. These TDI−X2 noise spectra with andwithout the confusion noise are plotted, Figure 8-1, in particular, in terms of the actual√
LISA length strain in cycles/ Hz along with the low-frequency acceleration noise,
mid-frequency shot-noise, and scaled by the high-frequency sensitivity loss. We
also plot the theoretical ’1-year’ averaged strain of a few actual gravitational wave
sources and the expected GW confusion background [103] in comparison with the

166

−3

4

x 10

3

Phase−Locked (Monochromatic Binary)Phase−Locked (Monochromatic Binary + Confusion Noise)

2

Cycles

1
0
−1
−2
−3
−4
0

1000

2000

3000

4000

5000 6000
time (s)

7000

8000

9000

10000

Figure 7-12. Confusion noise time-series comparison: The time series of the dynamic
LISA-like simulations with and without a confusion noise background(chapters 7.2.2 and 7.2.2) are plotted explicitly after low-pass filtering thealiased noise near the sampling-frequency. The added low-frequency
confusion noise is readily apparent when comparing the results of themeasured 6.277 mHz binary with the binary plus confusion noise (7–5).
injected background (7–5) and GW signal. The deviation in the spectral amplitude√(≃ 7 mCyc/ Hz) is caused by spreading of the GW into approximately 3 frequency
bins7 .

7204µcycles/Hz ×√√√
10000 s = 20.4 mCyc/ Hz ≃ 3 × 7 mCyc/ Hz
167

CHAPTER 8CONCLUSION
In this dissertation we have presented how LISA and other space-based interferometers will achieve the strain sensitivities required to measure astronomically interesting
GW sources using advanced stabilization techniques such as cavity locking and arm
locking as well as through the application of post-processing time-delay interferometry
combinations. We have considered how the LISA-IMS science observables areformed and how the differential proof-mass length is constructed from these individual
heterodyne PD observables. We have presented the design, phase precision, and likely
low-frequency limiting noise sources of LISA-like phasemeters and how they are used inheterodyne interferometry. A method of using the phasemeters to electronically simulate
the inter-SC laser field transmission characteristics including the time-changing delay,
Doppler shift, and GW modulation was developed, tested, and used to perform some of
the first hardware based arm-locking experiments.
Time-delay interferometry simulations using µcycle phasemeters, multi-second delay
EPD units, and the UFLIS laser benchtop were developed and compared to the noise
sources expected in LISA. The simulations tested the ability of time-delay interferometry
combinations to cancel the laser phase noise and extract the GW information with,both, static (constant arm-length) and dynamic (time-changing arm-length) LISA-likecharacteristics. We have also presented a simple but powerful method of estimating
the time-dependent arm-lengths using a TDI-ranging reference tone modulated onto
the laser field at frequencies above the GW frequency measurement band. The
post-processing formation of the TDI-X2 combination which optimally cancels theranging tone provides a functional constraint on the time-changing inter-SC delays toan accuracy better than that required to remove the laser phase noise from the TDI
combinations.

168

0

10

Simulator Baseline
Phase−Locked 4−Way TDI 2.0
Confusion Noise 4−Way TDI 2.0
Injected Confusion Noise Limit
Theoretical Confusion Noise Limit
RX J0806
V407 Vul
ES CetAM CVn−1

10−2

Cycles/√Hz

10

−3

10

−4

10

−5

10

−6

10

−4

10

−3

10

−2

10

−1

10

0

10

Frequency (Hz)Figure 8-1. Compiled results and comparison with TDI for LISA: In this figure we havecompiled all the results of the TDI simulations and attempt to make a directcomparison with the expected LISA strain sensitivity. The baseline spectral
noise of the UFLIS simulator (grey-blue) from the TDI-Transponder(chapter.7.1.1) measurements is plotted. The velocity corrected TDI-X2.0
spectrum of the dynamic arm TDI simulation with (cyan) and without (blue)
the injected binary confusion noise (dotted-magenta) is plotted incomparison with the IMS sensitivity requirement. The three TDI simulations
are scaled by the frequency factors of (3–2) to account for thehigh-frequency GW-sensitivity loss expected in LISA. The DRS and IMS
requirement are root-square summed and scaled by (3–2) to produce theeffective
p single-link LISA sensitivity with (black) and without (dotted-black)the 2 5/3 sensitivity factor. An estimate of the confusion noise limit isplotted (dotted-red) along with the four brightest verification binaries√ rescaled
from a 1-year averaged strain sensitivity to noise spectra in cycles/ Hz. The
strain magnitude of the 1 year averaged RX-J0806 binary and the 10000 s
EPD injected GW have amplitudes such that they result in similar LSD
amplitudes in this figure. The results of these experiments verify the ability to
account for a ≃ 0.4 s (0.1 Gm) un-equal arm-length mis-match and 75 m/svelocity coupled laser noise by TDI-ranging the inter-SC distance to an
accuracy of 5 ns (1.5 m) and forming the TDI-X2 combination to extract the
GW information to within a√factor of 5 of the single-link
LISA sensitivity.
√
√
−22
−18
RX-J0806 ∼ 4 × 10 /Hz √1 year = 2.2 × 10 / Hz
=
10.5
mCyc/
Hz
√
√
EPD-GW ∼ 4.3 × 10−20 /Hz 10000 s = 4.3 × 10−18 / Hz = 20.5 mCyc/ Hz

169

The experimental results show that more than 10 orders of magnitude of laser
frequency noise can be canceled using appropriately time-shifted data streams with
the TDI−X2 data combination. We have also verified that the ability to cancel the laser
frequency noise using a TDI−X1 data combination is limited by the relative velocities
between the spacecraft. Meanwhile, we have also demonstrated that the TDI-X2
combination accounts for this inter-SC velocity limited TDI−X1 combination. The
simulations have also verified the removal of the residual phase lock loop noise addedat the far spacecraft and have showed that the PLL noise extraction is possible withrelaxed one-way ranging requirements due as compared to the round-trip requirements.
The results of these TDI experiments demonstrated the ability to account for a ≃ 0.4 s(0.1 Gm) un-equal arm-length mis-match and 75 m/s velocity coupled laser noise by
TDI-ranging the inter-SC distance to an accuracy of 5 ns (1.5 m) by forming the TDI-X2
combination to extract the GW information to within a factor of 5 of the single-link LISA
sensitivity.
In the process of developing the UFLIS-TDI simulator, we have developed andtested data analysis tools which use the raw phasemeter data streams to extract
the light-travel time function and generate the TDI-X2 data streams. We have also
added a confusion noise GW-background to the TDI simulations and verified that thislow-frequency background does not interfere with our ranging capabilities.
The combination of these experiments, validating the multi-second time-changing
inter-SC characteristics, and the experiments performed in [105], verifying the opticalnoise couplings, represent the essential characteristics of a LISA-like interferometry
measurement system and the post-processing cancellation of laser phase noise using
the appropriate TDI combinations. Future UFLIS experiments should include real,
LISA-like GW signals using data-sets generated with LISA-tools like Synthetic LISA[55]. The use of independent clock sources and the verification that clock noise sources
can be accounted for in the TDI combinations would also be useful. Finally, simulations

170

with three independently stabilized lasers might also be valuable towards verifying theconstrains on the one-way ranging capabilities.

171

APPENDIX ATIME VARYING FRACTIONAL DELAY INTERPOLATION FUNCTION

function [xOut,tOut] = TVFDfilter(x,fs,beta,Tau,N)% [xOut,tOut] = TVFDfilter(x,fs,beta,Tau,N)%% Computes a time-stretched/compressed and time-shifted data-set% with an N-point Lagrange fraction delay filter as defined in% "Post-processed time-delay interferometry for LISA" by Shaddock et. al.
%% INPUTS:%
x = time series input [amplitude]%
fs = sampling frequency [Hz]%beta = fractional shift rate (in s/s)%
Tau = absolute time-shifting value in seconds%
N = interpolation length (number of points to use in%
Fractional Delay Filter)%% OUTPUTS:%
xOut = output amplitude vector%
tOut = output time vector%% Ira Thorpe, Shawn Mitryk% Updated 2-18-12
L = length(x);M = (N-1)/2;
k = (-(N-1)/2):1:((N-1)/2);
offset = floor(Tau*fs);
Dfrac = -(Tau-(offset/fs))*fs;
xOutShift(1:L-N+1) = 0;
for i=1:L-N+1
DTaufrac = beta*i;
td = (N-1)/2+Dfrac-DTaufrac;b1 = gamma(td+1)/(gamma(N+1)*gamma(td-N+1));
b2 = gamma(N)./(gamma(k+(N-1)/2+1).*gamma(N-k-(N-1)/2));w = ((pi*N)/(sin(pi*td)))*b1*b2;
h = sinc(Dfrac-DTaufrac-k).*w;
xOutShift(i) = sum(h.*(x(i:i+N-1)’));end172

xOut = xOutShift(N+M+1:L-N+1)’;
tOut = (0:length(xOut)-1)’/fs;
An up-to-date version of these functions and their application in producing theresults demonstrated in these experiments can be obtained from [106].

173

APPENDIX B
TDI 2.0 COMBINATION FUNCTION

function [TDI2Out, TDI1Out] = TDIComb(fs, s21, s31, s12, s13, tau33,
tau13, beta3o, tau22, tau12, beta2o)% [TDI2Out, TDI1Out] = TDIComb(fs, s21, s31, s12, s13, tau33,%
tau13, beta3, tau22, tau12, beta2)%% Computes the TDI 1.0 and TDI 2.0 Combinations based on the 6-variable% arm-length delays. Based on "Data combinations accounting for LISA
% spacecraft motion" by Shaddock et. al.
%% INPUTS:%
fs = Sample frequency%%
s21 = Sensor signal (sending: SC2 receiving: SC1)%
s31 = Sensor signal (sending: SC3 receiving: SC1)%
s12 = Sensor signal (sending: SC1 receiving: SC2)%
s13 = Sensor signal (sending: SC1 receiving: SC3)%%
tau33 = Round-trip delay from SC1 through SC2 (Arm3)%
tau13 = One-way delay time from SC1 to SC2 (Arm3)%
beta3 = Arm3 time compression factor%
beta = (velocity between SC1 & SC2/speed of light)%
tau22 = Round-trip delay from SC1 through SC3 (Arm2)%
tau12 = One-way delay time from SC1 to SC3 (Arm2)%
beta2 = Arm2 time compression factor%% OUTPUTS:%TDI2Out = TDI2.0 Combination%
TDI1Out = TDI1.0 Combination%% Shawn Mitryk% Updated 2-18-12
L1 = length(s12);
N = 51;M = (N-1)/2;
alpha2 = (1-2*beta2o);alpha3 = (1-2*beta3o);
gamma2 = (1-beta2o);
gamma3 = (1-beta3o);174beta3 = 1-alpha3;
beta2 = 1-alpha2;% TDI 2.0 shift values
Tau23 = (tau22)*alpha3+(tau33)
beta23 = 1-(alpha2*alpha3)
Tau32 = (tau33)*alpha2+(tau22)
beta32 = 1-(alpha3*alpha2)
Tau232 = alpha2^2*tau33 + alpha2*tau22 + tau22
beta232 = 1-(alpha2^2*alpha3)
Tau323 = alpha3^2*tau22 + alpha3*tau33 + tau33
beta323 = 1-(alpha3^2*alpha2)% Calculate the sensor signals from the one-way signals[chan12.data, chan12.t] = TVFDfilter(s12,fs,beta2o,tau12,N);[chan13.data, chan13.t] = TVFDfilter(s13,fs,beta3o,tau13,N);
chan21.data = s21(2*N:L1-M);
chan31.data = s31(2*N:L1-M);
chan22.data =chan33.data =chan21.data + chan12.data;chan31.data + chan13.data;% Form the TDI Combinations based on the 6-variable delays
L2 = length(chan22.data);[chan2_2s.data,chan2_2s.t]= TVFDfilter(chan22.data,fs,beta2,tau22,N);[chan3_3s.data,chan3_3s.t]= TVFDfilter(chan33.data,fs,beta3,tau33,N);[chan22_23s.data,chan2_23s.t]= TVFDfilter(chan22.data,fs,beta23,Tau23,N);[chan33_32s.data,chan3_32s.t]= TVFDfilter(chan33.data,fs,beta32,Tau32,N);[chan22_232s.data,chan2_232s.t]= TVFDfilter(chan22.data,fs,beta232,Tau232,N);
175

[chan33_323s.data,chan3_323s.t]= TVFDfilter(chan33.data,fs,beta323,Tau323,N);
chan22a.data = chan22.data(2*N:L2-M)’;
chan33a.data = chan33.data(2*N:L2-M)’;
chan22_2a.data = chan22_2s.data;chan33_3a.data = chan33_3s.data;chan22_23a.data = chan22_23s.data;chan33_32a.data = chan33_32s.data;chan22_232a.data = chan22_232s.data;chan33_323a.data = chan33_323s.data;
tdi1a.data = chan22a.data + chan33a.data- chan22_2a.data - chan33_3a.data;
tdi1b.data = - chan22_23a.data - chan33_32a.data+ chan22_232a.data + chan33_323a.data;
tdi2.data = tdi1a.data + tdi1b.data;
TDI1Out = tdi1a.data;
TDI2Out = tdi2.data;
An up-to-date version of these functions and their application in producing theresults demonstrated in these experiments can be obtained from [106].

176

REFERENCES[1] I. S. Newton, Philosophiae Naturalis Principia Mathematica, S. Pepys, Reg. Soc.
Praeses, London, 1687.
[2] A. Einstein, “Relativitätsprinzip und die aus demselben gezogenen folgerungen,”Jahrbuch der Radioaktivität, vol. 4, pp. 411–462, 1907.
[3] A. Einstein, “Grundlage der allgemeinen relativitätstheorie,” Annalen der Physik,
vol. 4, no. 49, pp. 769–822, 1916.
[4] D. Blokhintsev and F. Gal’perin, “Gipoteza neitrino i zakon sokhraneniya energii[neutrino hypothesis and conservation of energy],” Pod Znamenem Marxisma 6,
vol. 67, pp. 147–157, 1934.
[5] C. Cutler, W. A. Hiscock, and S. L. Larson, “Lisa, binary stars, and the mass of thegraviton,” Phys. Rev. D, vol. 67, pp. 024015, Jan 2003.
[6] F. Zwicky, “On the masses of nebulae and of clusters of nebulae,” AstrophysicalJournal, vol. 86, pp. 217–+, Oct. 1937.
[7] J. H. Oort, “Observational evidence confirming Lindblad’s hypothesis of a rotationof the galactic system,” Bulletin Astronomical Institute of the Netherlands, vol. 3,
pp. 275, Apr. 1927.
[8] “Nobel prize in physics 2011 - scientific background - the accelerating universe,”
2012, http://www.nobelprize.org/nobel prizes/physics/laureates/2011/advanced.html.
[9] F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflectionof light by the sun’s gravitational field, from observations made at the total eclipseof may 29, 1919,” Philosophical Transactions of the Royal Society of London.
Series A, Containing Papers of a Mathematical or Physical Character, vol. 220,
pp. pp. 291–333, 1920.
[10] A. Einstein and N. Rosen, “On gravitational waves,” Journal of the Franklin
Institute, vol. 223, no. 1, pp. 43–54, 1937.
[11] R. A. Hulse and J. H. Taylor, “Discovery of a pulsar in a binary system,” Astrophysical Journal, vol. 195, pp. L51–L53, Jan. 1975.
[12] J. Weber, “Detection and generation of gravitational waves,” Phys. Rev., vol. 117,
pp. 306–313, Jan 1960.
[13] B. P. Abbott, R. Abbott, et al., “Ligo: the laser interferometer gravitational-wave
observatory,” Reports on Progress in Physics, vol. 72, no. 7, pp. 076901, 2009.
[14] A. Abramovici et al., “Ligo: The laser interferometer gravitational-wave
observatory,” Science, vol. 256, no. 5055, pp. 325–333, 1992.

177

[15] C. L. Carilli and S. Rawlings, “Motivation, key science projects, standards andassumptions,” New Astronomy Review, vol. 48, pp. 979–984, Dec. 2004.
[16] B. J. Burt, A. N. Lommen, and L. S. Finn, “Optimizing pulsar timing arrays tomaximize gravitational wave single-source detection: A first cut,” AstrophysicalJournal, vol. 730, no. 1, pp. 17, 2011.
[17] E. Mauceli, Z. K. Geng, W. O. Hamilton, W. W. Johnson, S. Merkowitz, A. Morse,
B. Price, and N. Solomonson, “The Allegro gravitational wave detector: Dataacquisition and analysis,” Phys. Rev. D, vol. 54, pp. 1264–1275, July 1996.
[18] C. Frajuca et al., “Searching for monochromatic signals in the allegro gravitational
wave detector data,” Journal of Physics: Conference Series, vol. 228, no. 1, 2010.
[19] L. Gottardi, A. de Waard, O. Usenko, G. Frossati, M. Podt, J. Flokstra, M. Bassan,
V. Fafone, Y. Minenkov, and A. Rocchi, “Sensitivity of the spherical gravitational
wave detector minigrail operating at 5 k,” Phys. Rev. D, vol. 76, pp. 102005, Nov
2007.
[20] B. Abbott, R. Abbott, et al., “An upper limit on the stochastic gravitational-wave
background of cosmological origin,” Nature, vol. 460, pp. 990–994, Aug 2009.
[21] B. Abbott, R. Abbott, et al., “Setting upper limits on the strength of periodic
gravitational waves from psr j1939+2134 using the first science data from the geo
600 and ligo detectors,” Phys. Rev. D, vol. 69, pp. 082004, Apr 2004.
[22] B. Abbott, R. Abbott, et al., “Limits on gravitational-wave emission from selectedpulsars using ligo data,” Phys. Rev. Lett., vol. 94, pp. 181103, May 2005.
[23] K. Glampedakis and S. Babak, “Mapping spacetimes with LISA: inspiral of a test
body in a ’quasi-Kerr’ field,” Class. Quantum Grav., vol. 23, pp. 4167–4188, June
2006.
[24] L. Barack and C. Cutler, “Using lisa extreme-mass-ratio inspiral sources to test
off-kerr deviations in the geometry of massive black holes,” Phys. Rev. D, vol. 75,
pp. 042003, Feb 2007.
[25] C. L. MacLeod and C. J. Hogan, “Precision of hubble constant derived using blackhole binary absolute distances and statistical redshift information,” Phys. Rev. D,
vol. 77, pp. 043512, Feb 2008.
[26] J. S. Bloom, D. E. Holz, et al., “Astro2010 Decadal Survey Whitepaper:
Coordinated Science in the Gravitational and Electromagnetic Skies,” ArXiv
e-prints, Feb. 2009.
[27] T. Prince et al., “Lisa: Probing the universe with gravitational waves,” Tech. Rep.,
March 2009, http://list.caltech.edu/lib/exe/fetch.php?media=documents:science:lisa science case v1 2.pdf.

178

[28] T. Prince et al., “Laser interferometer space antenna for the detection ofgravitational waves, pre-phase a report,” Tech. Rep., Max-Plank-Institute forQuantum Optics, Garching, 1998.
[29] “Lisa assessment study report (yellow book),” Tech. Rep., Feb. 2011,
http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48364#.
[30] O. Jennrich et al., “Ngo assessment study report (yellow book): Esa/sre(2011),”
Tech. Rep., 2012, http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=49839#.
[31] S. Kawamura et al., “The japanese space gravitational wave antennadecigo,”
Class. Quantum Grav., vol. 23, no. 8, pp. S125, 2006.
[32] “Lisa - laser interferometer space antenna - media gallery - lisa
desktop backgrounds,” March 2012, reprinted with permission
from ’Using NASA Imagery and Linking to NASA Web Sites http://www.nasa.gov/audience/formedia/features/MP Photo Guidelines.html’,
http://lisa.nasa.gov/gallery/index.html.
[33] K. Danzmann, T. Prince, et al., “Lisa science requirements document, v3.0,” Tech.
Rep., 2005, http://list.caltech.edu/lib/exe/fetch.php?media=documents:requirements:srd v3 0 sb.pdf.
[34] D. Shaddock and K. M. others, “Lisa frequency control white paper,” Tech. Rep.,
July 2009, unpublished.
[35] G. Giampieri, R. W. Hellings, M. Tinto, and J. E. Faller, “Algorithms forunequal-arm michelson interferometers,” Optics Communications, vol. 123,
no. 4–6, pp. 669–678, 1996.
[36] “Request for information - concepts for the nasa gravitational-wave mission,” Sept
2011, http://pcos.gsfc.nasa.gov/studies/gravwave/-gravitational-wave-mission-rfis.php.
[37] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Twenty First Printing, 1
edition, 1998.
[38] S. M. Carroll, An Introduction to General Relativity: Spacetime and Geometry,
Addison Wesley, 1301 Sansome St., San Francisco, CA 94111, 1 edition, 2004.
[39] J. D. Jackson, Classical Electrodynamics, 3rd Ed, Wiley, 2007.
[40] C. W. F. Everitt, D. B. DeBra, et al., “Gravity probe b: Final results of a spaceexperiment to test general relativity,” Phys. Rev. Lett., vol. 106, pp. 221101, May
2011.

179

[41] J. I. Thorpe, Laboratory Studies of Arm-Locking Using Tthe Laser Interferometry Space Antenna Simulator at the University of Florida, Ph.D. dissertation,
University of Florida, 2006.
[42] A. Stroeer and A. Vecchio, “The lisa verification binaries,” Class. Quantum Grav.,
vol. 23, no. 19, pp. S809, 2006.
[43] J. M. Weisberg and J. H. Taylor, “The Relativistic Binary Pulsar B1913+16:
Thirty Years of Observations and Analysis,” in Binary Radio Pulsars, F. A. Rasio& I. H. Stairs, Ed., July 2005, vol. 328 of Astronomical Society of the PacificConference Series, p. 25.
[44] D. R. Lorimer, “Binary and millisecond pulsars,” Living Reviews in Relativity, vol.
11, no. 8, 2008.
[45] A. G. Lyne, M. Burgay, M. Kramer, A. Possenti, R. Manchester, F. Camilo, M. A.
McLaughlin, D. R. Lorimer, N. D’Amico, B. C. Joshi, J. Reynolds, and P. C. C.
Freire, “A double-pulsar system: A rare laboratory for relativistic gravity andplasma physics,” Science, vol. 303, no. 5661, pp. 1153–1157, 2004.
[46] J. A. Edlund, M. Tinto, A. Królak, and G. Nelemans, “White-dwarf white-dwarf
galactic background in the lisa data,” Phys. Rev. D, vol. 71, pp. 122003, Jun 2005.
[47] G. Fabbiano, J. Wang, M. Elvis, and G. Risaliti, “A close nuclear black-hole pair inthe spiral galaxy NGC3393,” Nature, vol. 477, pp. 431–434, Sept. 2011.
[48] C. Hogan and P. Binetruy, “Gravitational Waves from New Physics,” in astro2010:
The Astronomy and Astrophysics Decadal Survey, 2009, vol. 2010 of ArXiv
Astrophysics e-prints, p. 129.
[49] M. Armano, M. Benedetti, et al., “Lisa pathfinder: the experiment and the route tolisa,” Classical and Quantum Gravity, vol. 26, no. 9, pp. 094001, 2009.
[50] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley,and H. Ward, “Laser phase and frequency stabilization using an opticalresonator,” Applied Physics B: Lasers and Optics, vol. 31, pp. 97–105, 1983,
10.1007/BF00702605.
[51] E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,”
American Journal of Physics, vol. 69, pp. 79–87, Jan. 2001.
[52] R. J. Cruz, Development of the UF LISA Benchtop Simulator for Time Delay
Interferometry, Ph.D. dissertation, University of Florida, 2006.
[53] B. S. Sheard, M. B. Gray, D. E. McClelland, and D. A. Shaddock, “Laser frequency
stabilization by locking to a lisa arm,” Physics Letters A, vol. 320, no. 1, pp. 9–21,
2003.

180

[54] Y. Yu, Arm Locking for Laser Interferometer Space Antenna, Ph.D. dissertation,
University of Florida, 2011.
[55] M. Vallisneri, “Synthetic lisa: Simulating time delay interferometry in a model lisa,”
Phys. Rev. D, vol. 71, pp. 022001, Jan 2005.
[56] J. W. Conklin, S. Buchman, et al., “LAGRANGE: LAser GRavitational-wave
ANtenna at GEo-lunar Lagrange points,” ArXiv e-prints, Nov. 2011.
[57] P. Amaro-Seoane, S. Aoudia, et al., “eLISA: Astrophysics and cosmology in the
millihertz regime,” ArXiv e-prints, Jan. 2012.
[58] R. F. Voss and J. Clarke, “Flicker (1/f) noise: Equilibrium temperature andresistance fluctuations,” Phys. Rev B, vol. 13, pp. 556–573, Jan. 1976.
[59] M. Mansuripur, The physical principles of magneto-optical recording, Cambridge
University Press, 1998.
[60] J. Franz and V. Jain, Optical communications components and systems: analysis,design, optimization, application, CRC Press, 2000.
[61] H. Nyquist, “Thermal agitation of electric charge in conductors,” Phys. Rev., vol.
32, pp. 110–113, Jul 1928.
[62] L. G. Kazovsky and D. A. Atlas, “Miniature Nd:YAG lasers: noise and modulation
characteristics,” Journal of Lightwave Technology, vol. 8, pp. 294, 1990.
[63] C. C. Harb, T. C. Ralph, E. H. Huntington, D. E. McClelland, H.-A. Bachor, and
I. Freitag, “Intensity-noise dependence of nd:yag lasers on their diode-laser pumpsource,” J. Opt. Soc. Am. B, vol. 14, no. 11, pp. 2936–2945, Nov 1997.
[64] F. Isen, DSP for MATLAB and LabVIEW.: Fundamentals of discrete signal processing, Synthesis lectures on signal processing. Morgan & Claypool Publishers,
2009.
[65] I. Biļinskis, Digital alias-free signal processing, John Wiley, 2007.
[66] R. Lyons, Understanding Digital Signal Processing, Prentice Hall, 2010.
[67] D. Dallet and J. Silva, Dynamic characterisation of analogue-to-digital converters,
Kluwer international series in engineering and computer science. Springer, 2005.
[68] W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. Shaddock, “Clock noise
removal in lisa,” AIP Conference Proceedings, vol. 873, no. 1, pp. 312–318, 2006.
[69] R. W. Hellings, “Elimination of clock jitter noise in spaceborne laserinterferometers,” Phys. Rev. D, vol. 64, pp. 022002, Jun 2001.
[70] D. A. Shaddock, “Space-based gravitational wave detection with lisa,” Classicaland Quantum Gravity, vol. 25, no. 11, pp. 114012, 2008.

181

[71] M. Tinto, D. A. Shaddock, J. Sylvestre, and J. W. Armstrong, “Implementation oftime-delay interferometry for lisa,” Phys. Rev. D, vol. 67, pp. 122003, Jun 2003.
[72] T. Laakso, V. Valimaki, M. Karjalainen, and U. Laine, “Splitting the unit delay [fir/all
pass filters design],” Signal Processing Magazine, IEEE, vol. 13, no. 1, pp. 30–60,
jan 1996.
[73] D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri, “Postprocessedtime-delay interferometry for lisa,” Phys. Rev. D, vol. 70, pp. 081101, Oct 2004.
[74] S. Barke, M. Tröbs, B. Sheard, G. Heinzel, and K. Danzmann, “Eomsideband phase characteristics for the spaceborne gravitational wave detector
lisa,” Applied Physics B: Lasers and Optics, vol. 98, pp. 33–39, 2010,10.1007/s00340-009-3682-x.
[75] D. Sweeney, Laser Communications for the LISA and the University of Florida
LISA Laser Interferometry Simulator, Ph.D. dissertation, University of Florida,
2012, unpublished.
[76] W. Press, Numerical recipes in FORTRAN: the art of scientific computing,
FORTRAN Numerical Recipes. Cambridge University Press, 1992.
[77] J. Thorpe and G. Mueller, “Experimental verification of arm-locking for lisa using
electronic phase delay,” Physics Letters A, vol. 342, no. 3, pp. 199–204, 2005.
[78] V. Wand, Y. Yu, S. Mitryk, D. Sweeney, A. Preston, D. Tanner, G. Mueller, J. I.
Thorpe, and J. Livas, “Implementation of armlocking with a delay of 1 second in
the presence of doppler shifts,” Journal of Physics: Conference Series, vol. 154,
no. 1, pp. 012024, 2009.
[79] A. Sutton and D. A. Shaddock, “Laser frequency stabilization by dual arm locking
for lisa,” Phys. Rev. D, vol. 78, pp. 082001, Oct 2008.
[80] K. McKenzie, R. E. Spero, and D. A. Shaddock, “Performance of arm locking in
lisa,” Phys. Rev. D, vol. 80, pp. 102003, Nov 2009.
[81] Y. Yu, S. J. Mitryk, and G. Mueller, “Experimental validation of dual/modified dualarm locking for lisa,” Classical and Quantum Gravity, vol. 28, no. 9, pp. 094009,
2011.
[82] A. Preston, Stability of Materials for Use in Space-based Interferometric Missions,Ph.D. dissertation, University of Florida, 2010, unpublished.
[83] J. C. Livas, J. I. Thorpe, K. Numata, S. Mitryk, G. Mueller, and V. Wand,“Frequency-tunable pre-stabilized lasers for lisa via sideband locking,” Classical and Quantum Gravity, vol. 26, no. 9, pp. 094016, 2009.
[84] Y. Yu, V. Wand, S. Mitryk, and G. Mueller, “Arm locking with doppler estimation
errors,” Journal of Physics: Conference Series, vol. 228, no. 1, pp. 012044, 2010.

182

[85] M. Vallisneri, “Geometric time delay interferometry,” Phys. Rev. D, vol. 72, pp.
042003, Aug 2005.
[86] M. Tinto, J. W. Armstrong, and F. B. Estabrook, “Discriminating a gravitational
wave background from instrumental noise in the lisa detector,” Phys. Rev. D, vol.
63, pp. 021101, Dec 2000.
[87] N. J. Cornish, “Detecting a stochastic gravitational wave background with the laserinterferometer space antenna,” Phys. Rev. D, vol. 65, pp. 022004, Dec 2001.
[88] D. A. Shaddock, “Operating lisa as a sagnac interferometer,” Phys. Rev. D, vol. 69,
pp. 022001, Jan 2004.
[89] J. W. Armstrong, F. B. Estabrook, and M. Tinto, “Time-delay interferometry forspace-based gravitational wave searches,” Astrophysical Journal, vol. 527, no. 2,
pp. 814, 1999.
[90] N. J. Cornish and R. W. Hellings, “The effects of orbital motion on lisa time delay
interferometry,” Class. Quantum Grav., vol. 20, no. 22, pp. 4851–4860, 2003.
[91] D. A. Shaddock, M. Tinto, F. B. Estabrook, and J. W. Armstrong, “Datacombinations accounting for lisa spacecraft motion,” Phys. Rev. D, vol. 68,
pp. 061303, Sep 2003.
[92] D. A. Shaddock, “Space-based gravitational wave detection with LISA,” Class.
Quantum Grav., vol. 25, no. 11, pp. 114012, jun 2008.
[93] J. J. Esteban, I. Bykov, A. F. G. Marı́n, G. Heinzel, and K. Danzmann, “Opticalranging and data transfer development for lisa,” Journal of Physics: Conference
Series, vol. 154, no. 1, pp. 012025, 2009.
[94] A. Sutton, K. McKenzie, B. Ware, and D. A. Shaddock, “Laser ranging andcommunications for lisa,” Opt. Express, vol. 18, no. 20, pp. 20759–20773, Sep
2010.
[95] M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging
for space-borne gravitational-wave detectors,” Phys. Rev. D, vol. 71, pp. 041101,
Feb 2005.
[96] R. J. Coates, “Tracking and Data Acquisition for Space Exploration,” Space Sci.
Rev., vol. 9, pp. 361–418, May 1969.
[97] S. E. Pollack and R. T. Stebbins, “A demonstration of lisa laser communication,”
Classical and Quantum Gravity, vol. 23, no. 12, pp. 4201, 2006.
[98] E. Hogenauer, “An economical class of digital filters for decimation andinterpolation,” IEEE Transactions on Acoustics Speech and Signal Processing, vol. 29, no. 2, pp. 155–162, 1981.

183

[99] A. Oppenheim, A. Willsky, and S. Nawab, Signals and systems, Prentice-Hallsignal processing series. Prentice Hall, 1997.
[100] Y. Levin, “Internal thermal noise in the ligo test masses: A direct approach,” Phys.
Rev. D, vol. 57, pp. 659–663, Jan 1998.
[101] “Rev. d m112648 adt4-5wt,” reprinted with permission from Samuel A. Baylis,
http://www.minicircuits.com/pdfs/ADT4-5WT.pdf.
[102] “Rev. c m112648 adt1-6t,” reprinted with permission from Samuel A. Baylis,
http://www.minicircuits.com/pdfs/ADT1-6T.pdf.
[103] P. L. Bender and D. Hils, “Confusion noise level due to galactic and extragalactic
binaries,” Classical and Quantum Gravity, vol. 14, no. 6, pp. 1439, 1997.
[104] S. J. Mitryk, V. Wand, and G. Mueller, “Verification of time-delay interferometry
techniques using the University of Florida LISA interferometry simulator,” Class.
Quant. Grav., vol. 27, pp. 084012, 2010.
[105] G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A.
Shaddock, “Experimental demonstration of time-delay interferometry for thelaser interferometer space antenna,” Phys. Rev. Lett., vol. 104, pp. 211103, May
2010.
[106] S. Mitryk,“Lisa-matlab data-analysis functions,”
http://www.phys.ufl.edu/research/lisa/DOCUMENTS/LISAFUNC/.

184

BIOGRAPHICAL SKETCH
Shawn Mitryk was born in Willingboro, NJ in the summer of 1983. Growing up
in close contact with a large family, Shawn was always surrounded by people who
were more than happy to satisfy his curiosity. After moving to Florida, Shawn became
interested in many aspects of both the arts and sciences, and eventually, found his nichein physics. After obtaining an engineering background education from the University
of Florida in 2006, Shawn continued to obtain a PhD. in experimental physics with theLaser Interferometer Space Antenna Project in May of 2012.

185