LASER COMMUNICATIONS FOR LISA AND THE UNIVERSITY OF FLORIDA LISAINTERFEROMETRY SIMULATOR

By

DYLAN SWEENEY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2012

c⃝ 2012 Dylan Sweeney

2

I dedicate this work to my wife, Sandra Londoño, without you nothing I have achievedwould be possible. You gave me the strength and confidence I needed to finish this

thesis. You are my inspiration and you give meaning to everything I do.

I also dedicate this work to our son, Nicolas Sweeney. May you find everything you seekin life.

3

ACKNOWLEDGMENTS

I owe a debt of gratitude to all of the people who have assisted me in my work.

Thanks to Guido Mueller, my thesis advisor, for providing the expertise to guide my

research, the laboratory to work in, and prompt and helpful feedback on my progress.

Thanks to my fellow graduate students Alix Preston, Yinan Yu, Shawn Mitryk, Johannes

Eichholz, Aaron Spector, and Darsa Donelan, who have been excellent co workers.

Special thanks to Shawn Mitryk whose help using our digital signal processing

hardware was invaluable. Thanks to the post-docs in the lab Vinzenz Wand, Jose

Sanjuan Munoz, and Syed Azer Reza who were always available to help me. Thanks

to the undergraduate students Justin Cohen and Amanda Cordes, who assisted in the

development of my experimental set up.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER

1 BACKGROUND: GENERAL RELATIVITY AND GRAVITATIONAL WAVES . . . 12

1.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Gravitational Waves Sources . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.1 Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 Extreme Mass Ratio Inspirals . . . . . . . . . . . . . . . . . . . . . 201.3.3 Supernova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.4 Rotating Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.5 Stochastic Background . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4 Gravitational Wave Detectors . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.1 Resonant Mass Detectors . . . . . . . . . . . . . . . . . . . . . . . 221.4.2 Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.3 Ground Based Laser Interferometers . . . . . . . . . . . . . . . . . 231.4.4 Space Based Laser Interferometers . . . . . . . . . . . . . . . . . 27

2 THE LASER INTERFEROMETRY SPACE ANTENNA (LISA) . . . . . . . . . . 28

2.1 Disturbance Reduction System . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Interferometric Measurement System . . . . . . . . . . . . . . . . . . . . 30

2.2.1 IMS Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Noise Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.3 Heterodyne Interferometry . . . . . . . . . . . . . . . . . . . . . . . 322.2.4 LISA Sensor Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.5 Laser Noise Removal . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.6 Time Delay Interferometry . . . . . . . . . . . . . . . . . . . . . . . 352.2.7 Phasemeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Laser Communication System and Requirements . . . . . . . . . . . . . . 382.3.1 Clock Noise Transfers . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1.1 Electro-Optic Modulators . . . . . . . . . . . . . . . . . . 392.3.1.2 Frequency Synthesizers . . . . . . . . . . . . . . . . . . . 402.3.1.3 TDI with Clock Noise Removal . . . . . . . . . . . . . . . 412.3.1.4 Clock Transfer Chain Noise Requirement . . . . . . . . . 42

2.3.2 Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5

3 THE UNIVERSITY OF FLORIDA LISA INTERFEROMETRY SIMULATOR . . . 49

3.1 UFLIS Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Laser Bench Top Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Prestabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Electrical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Digital Signal Processing Hadware . . . . . . . . . . . . . . . . . . . . . . 533.6 Phasemeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.7 Electronic Phase Delay Unit . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8 Previous Experiments with UFLIS . . . . . . . . . . . . . . . . . . . . . . 563.9 Goals of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 VERIFICATION OF INTER SPACE CRAFT CLOCK TRANSFER . . . . . . . . 59

4.1 Tests of Frequency Synthesizers . . . . . . . . . . . . . . . . . . . . . . . 594.1.1 Stanford Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.2 Custom Phase Lock Loop . . . . . . . . . . . . . . . . . . . . . . . 604.1.3 Rupptronik Frequency Synthesizers . . . . . . . . . . . . . . . . . 614.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Tests of Electro-Optic Modulators . . . . . . . . . . . . . . . . . . . . . . . 634.2.1 MHz Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.2 GHz Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Frequency Synthesizer and EOM Combination . . . . . . . . . . . . . . . 684.4 Electronic Test of Clock Noise Transfer . . . . . . . . . . . . . . . . . . . . 70

5 DEVELOPMENT OF PSEUDO-RANDOM NOISE CODE RANGING . . . . . . 73

5.1 Delay-Locked Loop (DLL) Design . . . . . . . . . . . . . . . . . . . . . . . 735.1.1 Phasemeter Transfer Function . . . . . . . . . . . . . . . . . . . . . 735.1.2 Filtering the Double Frequency Term . . . . . . . . . . . . . . . . . 745.1.3 Linearized Delay-Locked Loop Model . . . . . . . . . . . . . . . . . 825.1.4 Choice of Filter Parameters . . . . . . . . . . . . . . . . . . . . . . 845.1.5 Noise Term Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1.5.1 Phasemeter Out of Band Noise . . . . . . . . . . . . . . . 875.1.5.2 Local PRN code Interference . . . . . . . . . . . . . . . . 88

5.1.6 Rounding of Tracking Code Delay . . . . . . . . . . . . . . . . . . . 905.2 Delay-Locked Loop Simulations . . . . . . . . . . . . . . . . . . . . . . . . 915.3 Optical Delay-Locked Loop Tests . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.1 Tracking PRN Code Generated with Local Clock . . . . . . . . . . 965.3.2 Tracking PRN Code Generated With a Separate Clock . . . . . . . 995.3.3 Delay-Locked Loop Tracking With Interfering PRN Code . . . . . . 103

6 ELECTRONIC TEST OF TDI WITH PRN RANGING . . . . . . . . . . . . . . . 108

6

7 FUTURE INTEGRATION OF UFLIS AND LASER COMMUNICATION . . . . . 112

7.1 PRN Ranging and the Phasemeter Delay Unit . . . . . . . . . . . . . . . . 1127.2 Clock Noise Transfers and UFLIS . . . . . . . . . . . . . . . . . . . . . . . 113

8 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7

LIST OF FIGURES

Figure page

1-1 Basic and advanced Michelson interferometers . . . . . . . . . . . . . . . . . . 24

2-1 The orbit of the three LISA SC . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2-2 Diagram of the LISA satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2-3 A simplified version of the layout of the LISA optical bench . . . . . . . . . . . . 35

2-4 The three step approach to canceling the intrinsic laser phase noise in LISA . . 36

2-5 An IQ tracking phasemeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2-6 Simplified model of LISA that UFLIS is based on . . . . . . . . . . . . . . . . . 43

2-7 The laser modulation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2-8 The PRN code and the Manchester encoding scheme . . . . . . . . . . . . . . 45

2-9 Forming the PRN error signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2-10 The delay lock loop architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3-1 How the EPD unit models the long travel times of the LISA arms . . . . . . . . 50

3-2 The previous set up of the laser bench top for UFLIS . . . . . . . . . . . . . . . 51

3-3 The current set up of the laser bench top for UFLIS . . . . . . . . . . . . . . . . 51

3-4 The formation of optical beat notes in the current UFLIS set up . . . . . . . . . 52

3-5 The electronic portion of UFLIS . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3-6 The digital signal processing system . . . . . . . . . . . . . . . . . . . . . . . . 54

3-7 Spectral results of two measurements of the UFLIS phasemeter system . . . . 55

3-8 Spectral results of a test of the sample and hold EPD unit . . . . . . . . . . . . 57

4-1 Experimental set up to test the frequency synthesizers . . . . . . . . . . . . . . 60

4-2 The PLL used to lock the 2 GHz VCO to a 10 MHz signal . . . . . . . . . . . . 61

4-3 Set up and spectral results of a test of frequency down converters . . . . . . . 62

4-4 Spectral results of the three frequency up-conversion measurements . . . . . . 63

4-5 The experimental set up for the test of the EOM’s phase stability at 5 MHz . . . 65

4-6 Spectral results of the EOM’s phase stability at 5 MHz . . . . . . . . . . . . . . 66

8

4-7 The experimental set up for the test of the EOM’s phase stability at 2 GHz . . . 67

4-8 Spectral results of the EOM’s phase stability at 2 GHz . . . . . . . . . . . . . . 67

4-9 Combined test of both the frequency synthesizers and the EOMs . . . . . . . . 69

4-10 Linear spectral density of combined synthesizer and EOM measurement . . . 69

4-11 The set-up for the electronic test of the clock noise transfer concept using frequencysynthesizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4-12 Linear spectral density of the results of the electronic test of the clock noisetransfer concept using frequency synthesizers . . . . . . . . . . . . . . . . . . 72

5-1 Diagram of the LISA phasemeter and Delay-Locked Loop . . . . . . . . . . . . 74

5-2 Phasemeter with filter to remove the 2ν term. . . . . . . . . . . . . . . . . . . . 77

5-3 The PRN code after demodulation from the carrier with and without the 2ν filter 78

5-4 Error signals with and without the 2ν filter at beat notes of 1, 2, and 2.5 MHz . 79

5-5 Error signals with and without the 2ν filter at beat note of 3, 5, and 10 MHz . . 80

5-6 Error signals with and without the 2ν filter at beat notes of 13.5, 15, and 20 MHz 81

5-7 The Delay-Locked Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5-8 Error signal formed with early-late correlation of Manchester encoded PRN. . . 84

5-9 Linearized version of the Delay-Locked Loop . . . . . . . . . . . . . . . . . . . 85

5-10 Digital implementation of a single pole low pass filter. . . . . . . . . . . . . . . 86

5-11 Simulated tracking signal error caused by interfering PRN code. . . . . . . . . 89

5-12 Possible relative positions between the incoming PRN code and the trackingcode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5-13 Spectral results of the DLL tracking simulation with noise at 1 µcycles/√Hz . . 93

5-14 Spectral results of the DLL tracking simulation with noise at 10 µcycles/√Hz . . 94

5-15 Optical set up of the PRN ranging experiments. . . . . . . . . . . . . . . . . . . 97

5-16 Power spectral densities of beat notes used in optical DLL tests. . . . . . . . . 98

5-17 Plot of measured pseudo ranges with 1 µcycle/√Hz noise while tracking PRN

generated on local clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5-18 Spectral results of measured pseudo ranges with 1 µcycle/√Hz noise while

tracking PRN generated on local clock . . . . . . . . . . . . . . . . . . . . . . . 99

9

5-19 Plot of measured pseudo ranges with 10 µcycle/√Hz noise while tracking PRN

generated on local clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5-20 Spectral results of measured pseudo ranges with 10 µcycle/√Hz noise while

tracking PRN generated on local clock . . . . . . . . . . . . . . . . . . . . . . . 101

5-21 Plots of measured pseudo ranges with 1 µcycle/√Hz noise while tracking PRN

generated on a separate clock . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5-22 Spectral results of measured pseudo ranges with 1 µcycle/√Hz noise while

tracking PRN generated on la separate clock . . . . . . . . . . . . . . . . . . . 102

5-23 Plot of measured pseudo range with 10 µcycle/√Hz noise while tracking a

PRN code generated on a separate clock . . . . . . . . . . . . . . . . . . . . . 102

5-24 Long term tracking of PRN code generated on a separate clock with 10 µcycle/√Hz

noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5-25 Spectral results of long term tracking of PRN code generated on a separateclock with 10 µcycle/

√Hz noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5-26 Time series of DLL tracking with interfering code at a clock offset of 2.3 Hz . . 105

5-27 Spectral results of DLL tracking with interfering code at clock offset of 2.3 Hz . 106

5-28 Time series of DLL tracking with interfering code at clock offset of 52.3 Hz . . . 106

5-29 Spectral results of DLL tracking with interfering code at clock offset of 52.3 Hz 107

6-1 Experimental set up of the electronic test of TDI with PRN ranging . . . . . . . 109

6-2 Spectral results of the electronic test of TDI with PRN ranging . . . . . . . . . . 110

7-1 Modifications to the phasemeter EPD unit to include PRN code delays. . . . . 113

7-2 Spectral results of the error in the PRN code timing using the phasemeter EPDunit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7-3 Preliminary test of UFLIS with clock noise transfers . . . . . . . . . . . . . . . . 116

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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 COMMUNICATIONS FOR LISA AND THE UNIVERSITY OF FLORIDA LISAINTERFEROMETRY SIMULATOR

By

Dylan Sweeney

December 2012

Chair: Guido MuellerMajor: Physics

The Laser Interferometer Space Antenna (LISA) is a proposed joint space based

gravitational wave detector between the National Aeronautics and Space Administration

(NASA) and the European Space Agency (ESA). The LISA mission uses laser

interferometry to measure fluctuations in the path length between three spacecraft

caused by gravitational waves. LISA is designed to be sensitive to gravitational waves in

the frequency band 30 µ Hz to 1 Hz complementing ground based detectors which are

sensitive at higher frequencies. There are many components to the LISA mission, all of

which must be tested prior to launch. One such component is the laser communication

subsystem which is used to transfer clock signals, measure the range, and to share

recorded data between the spacecraft using the laser links between the spacecraft.

Researchers at the University of Florida have constructed a simulator of the LISA

interferometry (UFLIS). This simulator has been used to verify the efficacy of several

LISA technologies including time delay interferometry and arm locking. This dissertation

describes work done at the University of Florida to test individual components of the

laser communication subsystem and to integrate them into UFLIS.

11

CHAPTER 1BACKGROUND: GENERAL RELATIVITY AND GRAVITATIONAL WAVES

1.1 General Relativity

The principle of Galilean relativity, that all uniform motion is relative, is a key feature

of Newtonian mechanics and was once thought to be a universal principle. However, the

laws of classical electrodynamics, which were discovered through experiments, were

not invariant under a Galilean transformation of coordinates from one inertial frame to

another. In 1905 Einstein published his special theory of relativity. By requiring that

uniform motion be relative and that the speed of light be the same in all inertial frames,

Einstein boldly claimed that all the laws of physics should be invariant under Lorentz

transformations, not Galilean transformations. Interestingly special relativity implies

that the distance between points and the time between events are not the same for all

observers. Instead the spacetime interval

�s2 = −�t2 + �x2 + �y 2 + �z2 (1–1)

is the same for all observers in inertial reference frames in special relativity (in this

chapter we will take units with c = G = 1). Since this quantity contains both time and

space, special relativity is formulated in terms of a spacetime geometry. In this geometry

events in spacetime are labeled by four-vectors

xµ = (t, x , y , z) (1–2)

This geometry can be characterized by a metric tensor which determines how the

invariant interval is calculated. In special relativity the metric is

ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(1–3)

12

The spacetime interval can be calculated as

�s2 = ηµνxµxν = xµx

ν (1–4)

using the Einstein summation convention in which repeated indices are summed

(aµbµ = �3µ=0aµbµ). This is further shortened by lowering an index by (aµ = ηµνaν). Greek

letters are taken to be the combined time and spatial indices 0-3, while Latin letters are

taken to be just the spatial indices 1-3.

While the laws of classical electrodynamics are invariant under Lorentz transformations,

Newton’s law of gravity is not. In 1917 Einstein published his general theory of relativity

which replaced Newtonian gravity with a theory consistent with special relativity. In

general relativity gravity is not a force acting on masses, but the result of the curvature

of spacetime due to the presence of matter/energy. This curvature is expressed by the

spacetime metric which, in general relativity, is a dynamical variable changing over both

space and time. In general relativity ηµν is replaced with gµν and the spacetime interval

must be calculated as

�s =

∫gµνdx

µdxν (1–5)

because gµν itself depends on the coordinates.

Given a metric gµν , the motion of matter is given by

d2xµ

dτ 2+ �µνρ

dxν

dτ

dxρ

dτ= 0 (1–6)

This is called the geodesic equation. �µνρ is the Chrisoffel symbol and is defined by

�µνρ ≡1

2(−∂µgνρ + ∂ρgνµ + ∂νgµρ) (1–7)

The metric is given by the Einstein field equations

Rµν −1

2gµνR = 8πGTµν (1–8)

13

Rµν is the Ricci curvature tensor and is given by

Rµν = ∂ρ�ρµν − ∂ν�ρµρ + �ρµν�σρσ + �σµρ�ρνσ (1–9)

R is the Ricci Scalar and is given by contracting both indices of the Ricci curvature

tensor. The Ricci curvature tensor is a measure of the curvature of the geometry.

It represents the curvature of spacetime as a measure of the tendency of matter to

converge or diverge over time.

Tµν is the energy momentum tensor, it is a measure of the flux of the µ component

of the four momentum across a unit area of the xν = constant surface. Therefore the

energy momentum tensor has components corresponding to energy density (T00),

momentum density(T0i ), energy flux(Ti0), shear stress(Tij i ̸= j), and pressure(Tii ). One

can think of the geodesic equation, which describes the motion of matter in a spacetime

metric, as analogous to the Lorentz force law, which describes the motion of charged

particles in electric and magnetic fields. Likewise one can think of the Einstein field

equations, which describe the spacetime metric in the presence of matter/energy, as

being analogous to Maxwell’s equations, which describe the electric and magnetic fields

in the presence of charged particles.

1.2 Gravitational Waves

In classical electrodynamics Maxwell’s equations lead directly to wave solutions

of the electric and magnetic fields in free space where there are no sources. Since

Einstein’s equations are nonlinear they must first be linearized in order to get wave

solutions. To do so it is assumed that the spacetime metric only contains small

perturbations to flat spacetime

gµν = ηµν + hµν (1–10)

14

where hµν

other will follow the null geodesic

ds2 = gµνdxµdxν = −dt2 + (1 + h+ (t)) dx2 = 0 (1–14)

If the light leaves the x = 0 mass at t0 and arrives at the x = L mass at tL then the travel

time is ∫ tLt0

dt =

∫ L0

√1 + h+ (t (x))dx (1–15)

This equation can be approximated to first order in h as

tL − t0 =∫ L0

(1 +

1

2h+ (t (x))

)dx (1–16)

Since the gravitational wave changes the coordinate speed of light, t (x) is not known.

Assuming that the coordinate speed of light is c leads to t (x) = t0 + x . Taking the real

part of Equation 1–13

�t = L+

∫ L0

1

2h+cos (2πf (t0 + x)) dx (1–17)

By integrating and using a trigonometric identity, the perturbation in the light travel time

due to the gravitational wave is

�t =1

2h+L ·

sin (πfL)

2πfL· cos

(2πf

(t0 +

L

2

))(1–18)

This change in travel time can be detected with the use of laser interferometry.

If the laser is split with one half traveling between masses along the x-axis and the

other half traveling between masses along the y-axis, when the light is recombined an

interference pattern will form that is proportional to the travel time difference between the

two interferometer arms. This interference pattern can be written in terms of the phase

of the laser light as

ϕ(t) =2π

λ(�tx − �ty) (1–19)

16

where λ is the wavelength of the laser light. For the + polarization of a wave traveling

along the z-axis, the perturbations in the light travel time along the x and y axes will be

out of phase, causing them to add in Equation 1–19. This will lead to a modulation in the

laser phase at the frequency of the gravitational wave at an amplitude of

ϕ ≈ 2πλh+L (1–20)

1.2.2 Generation

Using the retarded Green’s function and setting R2 = (x i − y i)(xi − yi) the general

solution to Equation 1–11 is

hµν(t, x i

)= 4

∫1

RTµν

(t − R, y i

)d3y (1–21)

where x i is the field position, y i is the source position, t is the time, and t − R is the

retarded time (the time it takes the gravitational signal to propagate from the source to

the field point). Assuming that the source of the gravitational wave is near the origin and

that the field is measured far away from the source, r 2 = x ixi >> y iyi , and assuming that

the source is traveling much slower than the speed of light, ddty i

1.3 Gravitational Waves Sources

There are several different types of astrophysical sources that are predicted to

emit gravitational waves that will be detectable by current and future gravitational wave

detectors. Among them are binaries of compact objects (white dwarfs, neutron stars, or

black holes), extreme mass ratio inspirals, supernova collapse, spinning neutron stars,

and the stochastic background of gravitational waves left over from the big bang.

1.3.1 Binary Systems

Binary systems of compact objects are the best hope of detecting gravitational

waves. There have been many direct detections of white dwarf - white dwarf (WD/WD)

binaries. Also, via radio wave observations of pulsars (neutron stars that emit periodic

pulses of radio waves) NS/NS binaries are known to exist as well. In 1974 Hulse and

Taylor discovered the binary pulsar PSR1916+16 [1]. By observing the modulation of

the rate of received pulses for over twenty years they were able to show that this binary

system was losing energy in accordance with general relativity.

Since binaries of compact objects are known to exist there has been a considerable

amount of research into predicting the gravitational waveforms they will emit. Binaries

are thought to undergo three stages in their evolution. The first stage is inspiral.

During this stage the stars orbit each other in an approximately Newtonian orbit while

adiabatically moving closer to each other in order to compensate for the energy lost to

gravitational waves. As the stars move closer to each other both the amplitude and the

frequency of the emitted gravitational waves will increase. For a circular orbit

ωGW = 2

√G(M1 +M2)

R3(1–24)

hGW = 2G 2

c4M1M2

r · R(1–25)

18

Here R is the radius of the binary system and r is the distance from the binary to the

detector. By measuring the amplitude, the frequency, and the rate of change of the

frequency of the waves from a binary inspiral it is possible to determine its distance

from the earth. This information can be used to determine Hubble’s constant [2]. Using

post-Newtonian approximations the gravitational waveforms for the inspiral stage have

been computed out to order 7/2 [3].

The second stage is merger. During this stage the stars are no longer able to

maintain their orbits and they rapidly fall into each other. The result is a highly relativistic

collision. The gravitational waves from the merger stage must be computed numerically

[4]. The merger stage will be an invaluable test of general relativity in the strong field

limit. By monitoring the inspiral stage the initial conditions for a binary merger can be

determined and used in a numerical simulation. The results from the simulation can then

be compared to the measured waves from the actual merger.

The third stage is ringdown. During this stage the newly merged stars have formed

one object but have yet to settle down into an equilibrium state. During the ringdown

phase the emitted gravitational waves are exponentially damped oscillations. In the case

of a black hole binary system the two black holes will eventually settle down into a single

Kerr black hole, characterized only by its total mass and spin. The ringdown stage can

be modeled as linear perturbations to the Kerr spacetime.

In order to maximize a gravitational wave detector’s ability to detect binary

coalescences the signal processing method of matched filters is employed [5]. Matched

filters work by cross correlating the detector output with a pre-computed gravitational

waveform. Because the particular parameters of the binary coalescence are not known

ahead of time, an entire family of templates must be created with various values for the

masses, spins, and eccentricity of the orbit. In order to achieve an accurate template,

the waveforms from the inspiral, merger, and ringdown stages must be carefully put

together [6].

19

1.3.2 Extreme Mass Ratio Inspirals

Another source of gravitational waves are extreme mass ratio inspirals. Supermassive

black holes (104 to 107 solar masses) are thought to exist in the center of many galaxies.

When a smaller compact object orbits a supermassive black hole it loses energy to

gravitational radiation and inspirals into the supermassive black hole. One critical

difference between extreme mass ratio inspirals and roughly equal mass binaries

is that the small compact object spends much more time in the near vicinity of the

supermassive black hole, emitting 100,000’s or more cycles of gravitational waves.

These waves will contain detailed information about the strong field region surrounding

the black hole [7] and the value of the supermassive black hole’s multiple moments

[8]. This information will provide a test of the Kerr hypothesis, that all black holes are

uniquely determined by their mass and spin angular momentum. The gravitational

waves from extreme mass ratio inspirals are expected to be weak so the method of

matched filters must be used. To create templates for matched filters, the small compact

object is treated as a small perturbation to the supermassive black hole.

1.3.3 Supernova

The supernova of massive stars are an expected source of gravitational waves.

Gravitational waves are thought to be the strongest from type-II supernova, which is the

collapse of the core of a massive star to form a NS or BH. Most of the energy that is

lost in a supernova is carried away by neutrinos, but it is possible that enough energy

could be lost to gravitational waves to create a detectable signal. Gravitational waves

must come from the non symmetric portion of the collapse as spherical motions have

no quadrupole moment, and hence do not emit gravitational waves. The non spherical

portion of the supernova could be caused by the initial rotation of the star [9], low mode

convection [10], or anisotropic neutrino emission [11].

20

1.3.4 Rotating Neutron Stars

It is supposed that rapidly rotating stars could possibly emit detectable gravitational

waves via several mechanisms. When a neutron star is formed it is extremely hot,

rotating rapidly, and can be idealized as an axially symmetric perfect fluid. Perturbations

to this idealization result in a family of normal modes called the r-modes. The loss of

energy and angular momentum to gravitational radiation makes these modes unstable

[12] and the loss of energy and angular momentum will reduce the rotation rate of the

neutron star until it becomes stable. It is estimated that the rotation rate of a neutron star

may reduce to approximately 100 Hz one year after its formation [13]. The energy loss

to gravitational radiation should be sufficient to detect a newly formed spinning neutron

star in the Virgo Cluster [14].

An accreting neutron star could also be a source of gravitational waves. In a binary

system of a neutron star and a gas giant, the neutron star accretes mass from the gas

giant by tidal forces. This accretion increases the spin of the neutron star. One would

expect to find a wide variety of spin rates for accreting neutron stars, however, most

accreting neutron stars have spin rates close to 300 Hz. An explanation for this is that

asymmetries in the accretion could lead to gravitational waves that radiate away angular

momentum at the same rate as the accretion [15].

A rapidly rotating neutron star might not be perfectly symmetrical. Asymmetries, due

to either a strain in the star’s solid crust or due to the star’s magnetic field, could produce

detectable gravitational waves [16]. If a neutron star cools while undergoing oscillations

of the star’s fluid near the surface, it is possible that deformations could be ”frozen” into

the crust. If the interior of a neutron star is superconducting then the magnetic field will

be confined to the outer crust resulting in a much stronger magnetic field inside the crust

than outside the star. This strong magnetic field may be able to support large enough

deformations to produce gravitational waves.

21

1.3.5 Stochastic Background

In the same way the cosmic microwave background is electromagnetic radiation left

over from the beginnings of the universe, it is postulated that there may be an analogous

background of gravitational radiation. The cosmic microwave background only lets us

directly probe the universe back to the point when the universe first became transparent

to electromagnetic radiation, 380,000 years after the big bang. Since gravitational

waves interact very weakly with matter the detection of primordial gravitational waves

will allow us to directly probe the very early stages of the universe. One mechanism

for the creation of a stochastic background of gravitational waves left over from the

early universe is inflation. Since the measured anisotropies in the cosmic microwave

background agree so well with inflationary models of the universe it is expected that the

gravitational waves predicted by inflation should also be present [17].

1.4 Gravitational Wave Detectors

There are several different methods for detecting gravitational waves: resonant

mass detectors, pulsar timing, ground based laser interferometers, and space based

laser interferometers.

1.4.1 Resonant Mass Detectors

The first gravitational wave detector was a resonant mass detector built by John

Weber in the 1960s. A resonant mass detector is essentially a large metal cylinder that

will vibrate when a gravitational wave stretches it along its axis. A transducer is used

to convert the mechanical oscillations into an electrical signal which is then amplified

and measured. Since the amplifier adds noise to the signal the bars are constructed

as to have a narrow band resonance. This will allow a gravitational wave signal near

the resonance of the bar to build up to an amplitude large enough to overcome the

amplifier noise. Typical resonance frequencies are in the range 500-1500 Hz. In order to

overcome thermal noise fluctuations, modern resonant mass detectors are cryogenically

cooled to temperatures below 100 mK.

22

Even though the sensitivity of resonant mass detectors is so narrow in frequency,

it has been thought that they might be able to detect some kind of burst of gravitational

radiation. Such a burst could come from a supernova or the merger of two compact

objects. Detection should be possible if the burst wave carries enough energy in

the resonant mass detector’s sensitivity band. There have been several coincident

searches for burst events between resonant mass detectors [18] [19] and even between

resonant mass detectors and laser interferometers [20] [21]. While the search results

have all been negative so far they have placed upper limits on the rate of burst events.

Collaborations between resonant mass detectors have also been able to place an upper

limit on the gravitational-wave stochastic background [22].

1.4.2 Pulsar Timing

Millisecond pulsars emit radio waves at incredibly regular intervals. The pulsar

PSR J0437-4715 has been observed to have a root-mean-square error in the timing of

its pulses of 200 ns over a measurement time of ten years [23]. Pulsar timing is done

by comparing the time of arrival of the pulses with a prediction based on the various

parameters of the pulsar. The observed deviations from the prediction are called the

residuals. Pulsar timing can be used to search for gravitational waves by correlating the

residuals from the observations of several pulsars. While the effect of the gravitational

wave at each pulsar will be uncorrelated, the effect at the Earth will be.

Pulsar timing is most sensitive to gravitational waves in the very low frequency band

of 10−9 to 10−7 Hz. Sources at these frequencies include nearby supermassive massive

black hole binaries of 109 or more solar mass [24] and a stochastic background caused

by the large number of solar mass black hole binaries at redshifts of z = 2 [25].

1.4.3 Ground Based Laser Interferometers

Ground based laser interferometers are based on the simple design of a Michelson

interferometer. A coherent light source is split and sent down two perpendicular and

equal length arms, the light is reflected off of mirrors at the end of the arms, recombined

23

Figure 1-1. The lay out of a basic Michelson interferometer (left) and the lay out of anadvanced ground based gravitational wave interferometer (right) withFabry-Perot cavities, power recycling, and signal recycling mirrors. Notshown are mode cleaning optics after the laser and before thephotodetector.

at the beam splitter, and finally measured on a photodetector. A small difference in arm

lengths will result in a phase shift of the signal at the photodetector. In order to detect

a gravitational wave, the mirrors of the interferometer must be free to move along the

direction of the arms. From Equation 1–18 a gravitational wave will result in a length

change of

�lGW ≈1

2hL (1–26)

The largest ground based interferometers are built with 4 km arms and are expected

to observe gravitational waves of amplitude 10−19/√Hz . This corresponds to a change

in length measurement of 10−16meters/√Hz . In order to reach this sensitivity several

sources of noise must be accounted for. For example minute changes in the refractive

index of air or even the collision of air molecules with the interferometer optics would

dwarf the gravitational wave signals, so the entire interferometer is placed inside a

vacuum. Also the effective length of the interferometer can be increased by replacing the

mirrors of a Michelson interferometer with a Fabry-Perot cavity. The increase in effective

arm length is proportional to the finesse of the cavity. Ground based interferometers are

designed to operate at frequencies from approximately 10 Hz to 10 kHz.

24

Ground vibrations limit the performance of ground based interferometers at low

frequencies. The interferometer optics can be shielded from seismic noise by being

suspended by multiple pendulums. Since the pendulums are resonant at around 1 Hz,

higher frequency vibrations in the measurement band are suppressed.

Thermal noise within the interferometer optics will also be a limiting factor. Thermal

noise in the suspension system can cause the mirrors to move. The suspension system

is made out of a material with a high Q factor in order to confine the thermal noise to

a narrow frequency region. The optics themselves also fluctuate due to thermal noise.

The largest source of thermal noise is Brownian motion in the mirror coatings.

At high frequencies interferometers are limited by shot noise. Shot noise comes

from the quantization of light; random fluctuations in the arrival time of photons can

mimic a phase shift caused by a gravitational wave. Shot noise can be mitigated by

using more light. Increasing the number of photons averages out more of the random

fluctuations and increases the signal to noise ratio. By placing a mirror in front of the

beam splitter the outgoing light from the interferometer can be reflected back into the

arms. This technique is known as power recycling. Using this technique the light power

in the interferometer arms can reach hundreds of kilowatts. The intensity of the light

cannot be increased without limit, however, as increasing the intensity increases the

radiation pressure noise. Radiation pressure noise comes from uncorrelated quantum

fluctuations in the power in each arm of the interferometer. These small changes in the

power result in changes to the force applied to the mirrors by the light itself. This results

in spurious accelerations of the mirrors that act like gravitational waves. The radiation

pressure can be reduced by increasing the mass of the mirrors.

There are four large gravitational wave detector groups currently operating

worldwide, TAMA 300, GEO 600, VIRGO, and LIGO. TAMA 300, officially known as

the 300m Laser Interferometer Gravitational Wave Antenna, is a prototype detector

located in Japan. It has reached a sensitivity of 3 × 10−21/√Hz at 1.3 kHz [26] and

25

produced upper limits on the rate of gravitational wave bursts in a coincident search with

the LIGO detectors [27].

GEO 600 is a 600 m detector located in Germany. It is designed to incorporate

the latest advancements in seismic isolation and signal recycling. The latest seismic

isolation systems contain active control loops to sense and reduce the seismic noise.

Signal recycling is done by placing a mirror in front of the interferometer’s photodetector.

This forms a cavity through which the signal must pass. This resonance of this cavity

can be tuned to the frequency of an expected gravitational signal. This allows the

interferometer to operate in a narrow band mode with an improved performance in a

narrow but tunable frequency band.

VIRGO is a 3 km detector located in Italy. VIRGO is designed to have the best

performance of all ground based interferometers at low frequencies. It achieves this

performance by suspending all of the interferometer optics from a superattenuator. The

superattenuator is essentially a seven stage pendulum. However, in the superattenuator

the first five suspended masses are a large drum shaped mass known as a mechanical

filter. These mechanical filters help achieve excellent suppression of the vertical seismic

noise. Additionally, the top stage is connected to the ground by means of an inverted

pendulum with a resonant frequency of approximately 40 mHz [28]. The superattenuator

system allows the VIRGO interferometer to push the contribution of seismic noise down

to approximately 3 Hz.

LIGO is a network of two detectors, both in the United States, one in Washington

and the other in Louisiana. The site in Louisiana operates a 4 km interferometer while

the Washington site operates a 4km and a 2 km interferometer in parallel. LIGO is a

good complement to VIRGO in that is has better sensitivity at high frequencies while

VIRGO has better sensitivity at low frequencies. LIGO has produced several significant

astrophysical results including upper limits on the rate of low mass binary coalescences

[29], the rate of continuous gravitational wave signals (e.g. from rotating neutron stars)

26

[30], and the energy loss due to gravitational waves associated with gamma ray bursts

[31].

Both the VIRGO and LIGO detectors are currently being upgraded to their

Advanced VIRGO and Advanced LIGO configurations. These upgrades include a

large increase in laser power, monolithic suspensions made out of fused silica, heavier

mirrors, improved coatings, and signal recycling. The aim of these upgrades is to

achieve an improvement in sensitivity by a factor of ten. This increase in sensitivity

would lead to a 1000 times increase in the volume of space observed by the detectors.

It is predicted that together Advanced VIRGO and Advanced LIGO should be able to

observe approximately 40 gravitational wave events per year [32].

1.4.4 Space Based Laser Interferometers

Ground based interferometers are limited to observing gravitational waves above

10 Hz. However, there are many interesting sources of gravitational waves at lower

frequencies, such as binary coalescences of supermassive black holes, extreme mass

ratio inspirals, and the initial inspiral of white dwarf and neutron star binaries. By placing

a detector in space the low frequency seismic noise can be avoided entirely and much

longer arms can be used. While there are several options concerning the length of the

arms and the satellite configuration and orbits, the focus will be on the specific design

known as the Laser Interferometer Space Antenna (LISA). This is the subject of Chapter

2.

27

CHAPTER 2THE LASER INTERFEROMETRY SPACE ANTENNA (LISA)

The Laser Interferometer Space Antenna (LISA) is a joint mission between NASA

and ESA to observe gravitational radiation in the frequency range 10−4 to 1 Hz. Like

other detectors it will measure the fluctuations in the phase that laser light picks up

as it travels between free falling test masses. Unlike any other detector LISA will be

located in outer-space, enabling much greater separation between the test masses and

removing the seismic noise which limits ground based detectors at low frequencies.

LISA will consist of three spacecraft (SC) separated by 5 million km arranged in an

approximately equilateral triangle. Each SC will be in an independent heliocentric orbit

such that the LISA constellation will lag behind the Earth by 20 degrees and the plane

formed by the SC will be at a 60 degree angle to the elliptic plane (Figure 2-1). The LISA

interferometry will be complicated by the fact that the length of each of the LISA arms

will be different and changing by approximately one percent [33]. These changes to the

lengths of the arms are mainly due to the difference in the gravitational pull of the Earth

on the SC. The LISA constellation will rotate as it orbits the sun, causing a Sagnac effect

whereby the light travel time is different in opposite directions in the same arm.

There are two main components to the LISA mission. First the test masses must

be shielded from any disturbance from perfect free-fall motion. This is the job of the

Disturbance Reduction System (DRS). It consists of control electronics which measure

and correct for changes in the distance between the SC and the test masses. The

second is the Interferometric Measurement System (IMS). Its job is to measure the

fluctuations in the distance between the test masses that are caused by gravitational

waves via heterodyne laser interferometry.

2.1 Disturbance Reduction System

Each SC will house two free-falling test masses. The interferometry system will

measure the distance between three pairs of test masses on different SC. The purpose

28

Figure 2-1. The orbit of the three LISA SC. The center of the SC constellation will lagbehind the earth at an angle of 20 degrees. The plane formed by the threeSC is at a 60 degree angle with respect to the plane formed by the orbit ofthe Earth. The satellites are arranged in an approximately equilateraltriangle 5 million km apart from each other.

of the Disturbance Reduction System is to ensure that the test masses follow, as

close to possible, geodesic motion. Any spurious acceleration due to non gravitational

forces will mimic the effect of gravitational waves. These forces could come from solar

radiation, the interplanetary magnetic field, temperature variations on the SC, and

electrostatic and magnetic fields on the SC. While the SC will be specially designed to

minimize these forces on the test masses, these forces cannot be removed completely

so the test masses themselves will be designed to minimize the effect. The test masses

will be made out of a gold-platinum alloy specially designed to minimize their magnetic

susceptibility [34]. This has the added benefit of making the test masses heavy, reducing

any non-gravitational acceleration. Any excess electrostatic charge that builds up on the

test masses will be removed with UV-light via the photo-electric effect.

While the test masses will be shielded from non gravitational forces the SC will not.

In order to keep the SC from crashing into the test masses, the test mass positions will

be measured with capacitive sensors and micro-Newton thrusters will be used to correct

the SC’s position and orientation.

The Disturbance Reduction System will be tested on LISA Pathfinder which is

a precursor to the LISA mission scheduled to be launched in 2014. LISA Pathfinder

will house two test masses inside of one SC and will measure their motion with laser

29

interferometry. The masses will be shielded from any disturbances with a system very

similar to LISA’s. Pathfinder will allow the disturbance reduction system to be tested in

space before the main LISA mission.

2.2 Interferometric Measurement System

The Interferometric Measurement System is responsible for measuring the

fluctuations in the distance between the test masses that are caused by gravitational

waves. It is composed of optical components such as lasers, beam splitters, electro-optic

modulators (EOMs), photodetectors, and optical cavities. It is also composed of

electronic components such as MHz oscillators, frequency up-converters, analog

and digital control loops, and phasemeters. It will measure gravitational waves by

measuring the fluctuations in the phase accumulated by laser light as it travels between

freely falling test masses.

2.2.1 IMS Overview

Each SC will have two lasers which will send light to the other two SC (Figure

2-2). The laser light will be used to measure changes in the distance between the

test masses located inside the SC. The laser fields which travel counter clockwise

are denoted with an asterisk. Each arm is denoted with the number of the SC that

is opposite to that arm. Since the LISA constellation is rotating the travel time of the

light will be different in opposite directions along the same arm. So light travel times

in the counter clockwise direction are denoted with a prime. Each SC contains two

optical benches on which light is combined from the far SC, the local laser, and from the

opposite laser on the same SC. The entire optical bench will be fabricated form Zerodur

ceramic glass with each optical component attached using hydroxy catalysis bonding

[35]. Before the light is sent to the far SC it is expanded in a 40 cm diameter telescope.

2.2.2 Noise Requirement

The lasers will have a wavelength of 1064 nm. With an arm length of 5 million

km, a gravitational wave with strain 10−21 will create oscillations of amplitude 5 pm.

30

Figure 2-2. Diagram of the LISA satellites, IMS system, and the laser links between SC.Each SC contains two test masses, two optical benches, and two telescopesfor transmitting the outgoing laser beam. The received light is captured withthe same telescope. Not shown is the back link fiber which connectsadjacent optical benches on the same SC. Also shown is the namingconvention for the lasers and light travel times. Each laser shares a numberwith the SC it is located on. The lasers that transmit in the counter clockwisedirection are denoted with an asterisk. The travel times are named with thenumber of opposite SC with travel times in the counter clockwise directiondenoted with a prime.

31

This corresponds to 5 × 10−6 cycles of the laser light. LISA is limited by two noise

sources. The first is the shot noise of the lasers on the photodetectors. The light power

will be reduced to around 100 pW after traveling between spacecraft. This gives a

white noise of approximately 4 × 10−5 cycles√(Hz)

[36]. The second is the disturbances from

free fall in the motion of the test masses. The spurious test mass accelerations are

expected to be constant in frequency leading to a displacement noise that is inversely

proportional to frequency squared. At 2.8 mHz the displacement noise due to spurious

accelerations will become larger than the shot noise. The sum of the noise from all of

the subsystems which make up the IMS must perform better than the combination of

these two sources. So the requirement on any one subsystem of the IMS is taken to be

an order of magnitude lower. In this dissertation the following is taken to be the LISA

requirement for an individual system.

10−6

√1 +

(2.8mHz

f

)4cycles√Hz

(2–1)

2.2.3 Heterodyne Interferometry

Since the laser light will not be reflected back to the original SC, heterodyne

interferometry is used. We can model the electric field of the laser light with a complex

exponential

E1(t) = A1ei(2πν1t+ϕ1(t)) (2–2)

where A1 is the amplitude of the laser light, ν1 is the frequency, and ϕ1(t) is the time

varying phase of the laser light. The phase may contain terms due to the intrinsic laser

noise, the gravitational wave signals, photodetector noise, etc. Combining two such

laser fields on a photodetector yields

PD = |E1(t) + E2(t)|2 = A21 + A22 + 2A1A2cos (2πν12t + ϕ12(t)) (2–3)

32

where ν12 = ν1 − ν2 and ϕ12(t) = ϕ1(t) − ϕ2(t). LISA will measure the time varying

phase of the AC part of this signal. So we can write the photodetector signal as just the

phase

PD = ν12t + ϕ12(t) (2–4)

where the phase is in cycles. Heterodyning two lasers simply results in the difference of

their frequencies and phases.

2.2.4 LISA Sensor Signals

The incoming light is combined with the local laser on the science photodetector

(called PDSC see Figure 2-3). We can write the phase of laser 1 as

ν1t + ϕ1(t) (2–5)

and the phase of laser 2* from SC 2 as

ν2∗t + ϕ2∗(t − τ3′) + SC1(t) + h3′(t) (2–6)

where ν is the frequency of the lasers, ϕ(t) is the intrinsic phase noise of the lasers,

τ3′ is the time delay of the light traveling from SC 2 to SC 1 (along arm 3), SC1(t) is the

phase noise caused by the motion of the optical bench on SC 1 with respect to the test

mass, and h3′(t) is the phase change caused by gravitational waves while traveling from

SC 2 to SC 1. Combining the two laser fields on a single photodetector results in the

difference of their phases. The phase of the signal at PDSC is

PDSC = ν12∗t + ϕ1(t)− ϕ2∗(t − τ3′)− SC1(t)− h3′(t) (2–7)

This will form a signal in the range of 2-20 MHz. This signal contains the phase

fluctuations caused by the motion of the SC and the phase fluctuations accumulated

by the light as it travels between the SC due to gravitational waves. The light from the

adjacent optical bench is used to form two beat notes, one with the light directly from the

33

local laser and the other with the light from the local laser after it has been reflected off

of the test mass (PDA and PDTM).

PDA = ν11∗t + ϕ1(t)− ϕ1∗(t)− ϕ�ber(t) (2–8)

PDTM = ν11∗t + ϕ1(t)− ϕ1∗(t) + SC1(t)− ϕ�ber(t) (2–9)

Subtracting these signals removes the common laser and fiber noise between them

and leaves the phase fluctuations due to the motion between the SC and the test mass.

Finally, combining all three photodetector signals gives

PDSC + PDTM − PDA ≡ s21 = ν12∗t + ϕ1(t)− ϕ2∗(t − τ3′)− h3′(t) (2–10)

which only contains the differential laser noise and the gravitational wave signal. s21

is the signal formed on SC 1 with laser light sent from SC 2. There will be six of these

signals, one for each test mass. The extraction of the gravitational wave signals from the

laser noise is discussed in the next section.

2.2.5 Laser Noise Removal

If the laser phase noise was less than the requirement in Equation 2–1 then no

removal of the laser noise would be necessary. However, a typical free running laser is

fifteen orders of magnitude more noisy than the LISA noise limit. The removal of this

noise poses a serious technical challenge. The first step in the laser noise removal

is prestabilization. In the current configuration of LISA one laser is designated as the

master laser, which will be stabilized to a local reference on the SC. The other lasers

will be phase locked to the master. Several laser prestabilization methods have been

demonstrated including locking to an optical cavity [37] [38], a molecular resonance [39],

or an unequal arm heterodyne interferometer [40].

The next step in the laser phase noise removal is to stabilize the master laser using

the LISA arms as a reference. This technique is known as arm locking. Since the lasers

34

Figure 2-3. A simplified version of the layout of the LISA optical bench. There are twooptical benches on each SC. There is an optical fiber that transmits the lightto and from the adjacent bench. Photodetector SC measures a laser beatnote between the local laser and the received laser light from the far SC. Itcontains the signal due to the gravitational waves (picked up by the incominglight as it traveled from the far SC) and the signal due to the relative motionbetween the two SC. Photodetector TM is a beat note between the locallaser after it has been reflected off of the test mass and the laser from theadjacent bench. It contains the signal from the motion of the SC relative tothe test mass. Photodetector A is a beat note between the local laser andthe laser from the opposite bench. It is used to remove the laser noise fromthe measurement of the position of the SC relative to the test mass.

on the other SC are phase locked to the master laser, the light from the far SC can be

treated (to within the phase lock loop error) as being a mirror reflection of the master

laser delayed by the round trip travel time of the arms. The heterodyne signals between

the local and received laser light can be used to form an error signal in a control loop,

further stabilizing the master laser. Arm locking has been experimentally demonstrated

using a long cable length to simulate the lisa arms [41], as well as using electronic

delays [42] [43].

2.2.6 Time Delay Interferometry

The final step to remove the laser noise is a post processing technique known

as time delay interferometry (TDI). TDI removes the laser noise by forming linear

combinations of time delayed versions of the s data streams (Section 2.2.4). There are

35

Figure 2-4. The three step approach to canceling the intrinsic laser phase noise in LISA.The lasers are prestabilized by locking the master laser to an optical cavityand then phase locking each other laser to the master. Arm locking furtherreduces the noise of the master laser by locking it to the very stable armlength between the SC. The final step, TDI, removes the laser noise byforming suitably time delayed combinations of the sensor signals in postprocessing.

several combinations which can in principle remove the laser noise to arbitrary precision

assuming a rigid LISA constellation [44]. These combinations are known as TDI 1.0.

One such combination is the Michelson X

X ≡ s21 − s31 − s21,2′2 + s31,33′ − s12,3′ + s13,2 + s12,3′2′2 − s13,233′ (2–11)

where the numbers after the comma denote a delay by the travel time of the LISA

arm(s) and the time dependence has been dropped for convenience, e.g. s13,233′ ≡

s13(t − τ2 − τ3 − τ3′). Once this algorithm is applied, only the gravitational wave signals

will be left. Of course the Michelson X combination will also contain noise from the

photodetectors, spurious motions of the test masses, and from the various subsystems

of the IMS. However, all of these noise sources added together will be smaller than the

36

requirement in 2–1. There are several factors which limit our ability to suppress the laser

noise with TDI algorithm to arbitrary precision. First we must know what delays to use

in order to apply the TDI algorithm. An small timing error in the delays will lead to an

imperfect cancelation of the laser noise. In the frequency domain this error will scale as

ϕ(t)− ϕ(t − �t) → ~ϕ(f )[1− e−i2πf �t

]≈ ~ϕ(f )(−i2πf �t) (2–12)

The requirement on the accuracy of the measured range is taken to be 1 meter (3 ns).

At a frequency of 1 mHz this corresponds to roughly 11 orders of magnitude noise

suppression. This is chosen so that TDI can be effectively applied in the case that either

pre-stabilization or arm-locking are used without the other.

Another limitation on the efficacy of TDI is the finite sampling rate at which the

LISA sensor signals are measured. The sensor signals will be sampled at a very low

rate (3-10 Hz). However, to apply the TDI algorithm the measured sensor signals must

be delayed in time with a precision of 3 ns. In order to shift the data stream with this

precision we must accurately interpolate between the data points which, at a 10 Hz rate,

are separated by 0.1 seconds. This can be done efficiently using fractional delay filters.

It has been found that only the surrounding 15 data points will be needed to interpolate

to the accuracy required by LISA when using a Lagrange interpolation [45].

In the actual LISA mission the arm lengths will be changing at a rate of up to 10

m/s and the rotation of the SC will cause a difference in the light travel time in opposite

directions along the same arm. TDI 2.0 takes these effects into account [46]. While

TDI 2.0 cannot be applied to arbitrary precision it will be sufficient to suppress the laser

noise below the LISA requirement.

2.2.7 Phasemeters

The phasemeter used in LISA is an IQ phasemeter with a tracking local oscillator

[42]. The beat note is mixed with a local oscillator forming an error signal that is used

in a control loop. This allows the local oscillator to match the phase noise of the beat

37

Figure 2-5. An IQ tracking phasemeter. Also shown is the Q output being filtered andsent to the delay lock loop for pseudo random noise (PRN) ranging.

note within the bandwidth of the control loop. The in band phase is added to the out of

band phase to form the phase measured by the phasemeter. This measurement must

be accurate to the order of the LISA requirement (Equation 2–1).

The phasemeter can accurately measure the phase, but it is driven by an imperfect

clock. The measurements made by the phasemeter are not perfectly periodic when

compared to a perfect clock. A timing error in the clock, �tclk , is converted into a phase

error in the measurement by �ϕ = �tclkν. The timing error of the clock is related to the

clock’s phase by �t = ��clkνclk . So the phase noise introduced by the clock noise is

given by

�ϕ = − ννclk

�clk (2–13)

When the timing error is early the phase will be smaller than it should be resulting in

the minus sign.

2.3 Laser Communication System and Requirements

There are several tasks necessary for the success of LISA that require communications

between the SC. These are a transfer of the clock noise between SC, a measurement

of the range of the LISA arms, and a transfer of recorded data. These tasks could all be

38

done with traditional radio communications, but to reduce power consumption and the

mass of the SC they are done using the laser links between the SC.

2.3.1 Clock Noise Transfers

Each spacecraft will measure the phase of the beat notes and the phase lock loop

error relative to its own clock. In Section 2.2.4 the noise sources and gravitational waves

were written as compared to a perfect clock. In Section 2.2.7 it was shown that the clock

noise introduces phase noise in all the phasemeter measurements. With the clock noise

terms the sensor signals are

s21 = ν1∗2t + ϕ1∗(t)− ϕ2(t − τ3′)− h3′(t)−ν1∗2νclk

�1 (2–14)

s13 = ν13∗t + ϕ1(t − τ2′)− ϕ3∗(t) + h2′(t)−ν13∗νclk

�2 (2–15)

where �1 is the phase noise of the clock on SC 1 etc. The additional sensor signals

(s12 and s31) can be formed by cyclic permutation of the indicies. Inserting these into

Equation 2–11 produces

X = −ν12∗νclk

[�1 −�2,3′ +�1,2′2 −�2,3′2′2] +ν1∗3νclk

[�1 −�3,2 +�1,33′ −�3,233′] (2–16)

where the gravitational waves have been dropped for clarity. In order to cancel these

terms we must measure �1 − �2,3′ and �1 − �3,2. These terms are the difference

between the clock signals from the far SC delayed by the light travel time and the clock

signal of SC 1. To do so we modulate the clock signal onto the laser beams using an

electro optic modulator (EOM).

2.3.1.1 Electro-Optic Modulators

An EOM is an optical device used to modulate the amplitude or phase of laser light.

LISA will use phase modulators. In an EOM the laser light passes through a crystal that

exhibits a linear electro-optic effect (e.g. lithium niobate). This crystal is placed between

two electrodes. The electro-optic crystal experiences a change in its refractive index that

39

is proportional to the applied voltage to the electrodes. If a sinusoidal signal is applied to

the EOM the phase of the light is modulated as

e i [2πνt+ϕ(t)] → e i [2πνt+ϕ(t)+msin(2πFt+�(t))] (2–17)

where m is the amplitude of the modulation (in radians), F is the frequency of the signal,

and � is the time varying phase of the signal. This can be expanded in terms of Bessel

functions as

e i [2πνt+ϕ(t)+msin(2πFt+�(t))] = e i [2πνt+ϕ(t)]∞∑

n=−∞

Jn(m)ein[2πFt+�(t)] (2–18)

In the case that the amplitude of the modulation (m) is small the above expression

reduces to

J0(m)ei [2πνt+ϕ(t)] + J1(m)e

i [2π(ν+F )t+ϕ(t)+�(t)] − J1(m)e i [2π(ν−F )t+ϕ(t)−�(t)] (2–19)

The result of the modulation is to create three distinct frequency components, a carrier

and an upper and lower sideband. The main carrier is at the original frequency of

the laser light and contains the original phase signal, but has been attenuated by the

factor J0(m). The two sidebands are separated in frequency from the main carrier by

the frequency of the modulation. The upper sideband contains the sum of the original

phase signal and the modulation phase signal, while the lower sideband contains their

difference. The sidebands are attenuated by J1(m). The modulation amplitude will be

chosen so that the sidebands contain 10 percent of the laser power [47].

2.3.1.2 Frequency Synthesizers

In order to suppress the clock noise terms to a level below the requirement

(Equation 2–1) the entire clock noise transfer chain must not add any phase noise

to the clock signals above the requirement. This includes the cables that transfer the

clock signal to the EOMs, the EOMs themselves, and the measurement of the phase

of the sidebands carrying the clock signals. Since the sideband signals contain less

40

power than the carrier they will have a lower signal to noise ratio than the main carrier

measurement. To overcome this noise, the phase of the clock signals will be amplified

by frequency up-conversion with a frequency synthesizer before they are modulated on

the EOMs. The result of a frequency up-convertersion is to multiply both the frequency

and phase of a signal by a factor α

sin [2πνt + ϕ(t)] → sin [2πανt + αϕ(t)] (2–20)

Each clock signal, �i will be up-converted by a factor of αi to a frequency in the

GHz range.

2.3.1.3 TDI with Clock Noise Removal

Figure 2-6 is a simplified version of the LISA mission. In this simplified version each

SC only has one laser and we don’t consider the laser links between SC 2 and SC 3.

The University of Florida Interferometry Simulator (UFLIS), which will be discussed in

Chapter 5, is based upon this simplified model of LISA. When the modulated light from

both the far and local lasers are combined on the photodetector they will form three

signals (Figure 2-7), the beat note between the lower sidebands (SSL), the beat note

between the carriers (CC ), and the beat note between the upper sidebands (SSU). All

other beat notes (e.g. between a lower sideband and the carrier) will be at frequencies

greater than the bandwidth of the photodetector. The upper and lower sideband signals

are given by

s21L = (ν12 + (α2 − α1) νclk) t + · · · − α1�1 + α2�2,3 −(ν12 + (α2 − α1) νclk)

νclk�1 (2–21)

s21U = (ν12 + (α1 − α2) νclk) t + · · ·+ α1�1 − α2�2,3 −(ν12 + (α1 − α2) νclk)

νclk�1 (2–22)

The phase noise of the carrier-carrier beat (s31) has been omitted for clarity. There are

similar equations for the s31 sidebands. We can combine the upper and lower sidebands

41

to obtain the clock noise.1

s∗21 ≡s21L − s21U

2α2= �2,3 −�1 (2–23)

s∗31 ≡s31L − s31U

2α3= �3,2 −�1 (2–24)

These signals are precisely what we need to measure to cancel the clock noise in

Equation 2–16. We can now modify the Michelson X combination to remove the clock

noise

X = s21 − s31 − s21,22 + s31,33 − s12,3 + s13,2 + s12,322 − s13,233

− ν21νclk

[s∗21 − s∗21,22

]+

ν31νclk

[s∗31 − s∗31,33

](2–25)

2.3.1.4 Clock Transfer Chain Noise Requirement

The combination s21L − s21U will contain shot noise at the photodetectors, and noise

from each part of the clock noise transfer chain such as the frequency synthesizers and

the EOMs. This noise will be multiplied by ν21νclk2α2

as in equations 2–23 and 2–25. The

noise multiplied by this factor must be lower than the LISA requirement. Thus the noise

requirement on the clock noise transfer chain is

νclk2α2ν21

10−6

√1 +

(2.8mHz

f

)4cycles√Hz

= 2× 10−4√1 +

(2.8mHz

f

)4cycles√Hz

(2–26)

where we have used the current baseline design of upconverting to 2 GHz and assumed

the worst case scenario of a 20 MHz beat note.

42

Figure 2-6. The simplified model of LISA that UFLIS is based on. In this model each SConly has one laser and the laser link between SC 2 and 3 is not considered.There are also no test masses as the each SC is assumed to be stable tothe level of the gravitational waves. This way the s terms (Section 2.2.4) aresimply the photodetector output on each SC. Each laser passes through anEOM where it is modulated with both a PRN code and a frequencyup-converted version of the local clock.

2.3.2 Laser Ranging

In order to measure the distance between the spacecraft pseudo random noise

(PRN) codes are employed in a way similar to those used in GPS. A PRN code is a

series of ones and negative ones that has all the properties of a truly random process

1 The same result could be obtained by subtracting the carrier-carrier from one of thesideband-sideband beats. However, the term would be divided by α instead of 2α.

43

Figure 2-7. The laser modulation scheme. Each laser is modulated with a frequencyup-converted signal from the local clock producing sidebands at frequenciesαiνclk away from the carrier. The sidebands will be modulated with anamplitude designed to produce ten percent of the laser power in thesidebands. When combined with the light from the far SC three beat notesare formed, between the lower sidebands, the carriers, and the uppersidebands. The PRN code is also modulated onto the lasers causing aspread spectrum around the laser peaks. The PRN amplitude is chosen sothat one percent of the laser power is in the spread spectrum. This will be solow that the PRN spread spectrum will be buried under the shot noise level.

except that it is finite in length (Figure 2-8). A PRN code will be modulated onto the

phase of each laser with the same EOM used to modulate the clock signals. The code

will be recovered by the phasemeter on the far spacecraft. These sudden jumps in the

phase of the signal spread some of the power of the carrier into nearby frequencies

(Figure 2-7).

The codes used in LISA will be 1024 chips long and have a chip rate of approximately

1.5 MHz (Figure 2-8). The code period will then be about 0.6 ms or 1.8 × 105 m. Such

a code will contain frequency components all the way down to the inverse of the code

period, which would be 1.5 kHz. Such frequency components are within the bandwidth

of the phasemeter and would be tracked and suppressed by the phasemeter. To remove

44

Figure 2-8. The PRN code and the Manchester encoding scheme. The chosen PRNcodes have 1024 chips that can be either positive or negative. The code isdesigned to mimic true noise in that is has very little correlation with itselfunless it is within one chip of itself. Since the PRN codes are produced andanalyzed digitally each chip has a finite number of samples. Each chip willhave 32 samples. The code is Manchester encoded by replacing everypositive chip with a rising edge and every negative chip with a falling edge.

these lower frequency components a Manchester encoding scheme is applied. Each +1

chip is replaced by a rising edge and each -1 chip is replaced by a falling edge (Figure

2-8). The Manchester encoding attenuates these lower frequency components.

In LISA, the laser light will be modulated with both the clock noise and the PRN

code before the optical bench. Each laser will be modulated with its own code. The

correlation will be done with the code from the far SC, but the code from the local SC

will also be present. The codes must be chosen such that there is no cross correlation

between the different codes so that the local code does not interfere with the generation

of the error signal. Six such codes were created by numerical optimization at the AEI

and were graciously loaned to us [48].

The PRN code is correlated with a locally generated version of the code. The

correlation is nearly zero unless the delay between the signals is within one chip length

45

Figure 2-9. The auto-correlation of the PRN code is only positive if the delay is between+1 and -1 chip. Subtracting a half chip late correlation with a half chip earlycorrelation creates an error signal.

Figure 2-10. The delay lock loop architecture.

of an integer multiple of the length of the sequence. If two versions of the local code

are used, with one of them delayed by one chip with respect to the other, an error

function can be created by subtracting the correlation of the first local code and the

received code from the correlation of the delayed local code and the received code as in

Figure 2-9. This error signal can be used to maintain a delay lock between the local and

received codes. The delay used to maintain the lock will be proportional to the distance

between the spacecraft. This is known as a delay lock loop (DLL).

46

The architecture for the delay lock loop is shown in Figure 2-10. The error signal

is scaled and sent to the local code generator to advance or delay the local code. This

control signal is integrated to give the pseudo-range. This is the pseudo-range and not

the true range for two reasons. First the length of the LISA arms is much greater than

the correlation time. LISA will use the deep space network to determine the range to an

integer multiple of the correlation time/distance, then the pseudo range will be the left

over fraction of the correlation time. Second the start of the codes on different spacecraft

may not be synchronized. If the code at SC 1 starts at t1 and the code at SC 2 starts at

t2 then the pseudo ranges T1 and T2 will be

T1 =t2 + τ − t1 (2–27)

T2 =t1 + τ − t2 (2–28)

where τ is the fractional remainder of the delay time between the spacecraft. The

average of the pseudo ranges plus the rough estimate from the deep space network

gives the true range. The difference between the two pseudo ranges can be used to

determine the start time difference between the two codes. This will be important in

order to synchronize the sampling of data between SC.

When the code is sampled by the phasemeter it is digitized in its amplitude and

also in time. Thus the start of the code will be rounded to the next sample time. At

a sampling rate of 50 MHz, the sample time is 20 ns, leading to a measurement

limitation of 6 meters. A much better precision will be achieved due to the dithering

of the received PRN code by the noise present in the measurement. The shot noise

at the photodetector will cause the error signal to drive the local code away from the

incoming code. If the actual arrival of the incoming code is between samples of the local

signal processing, then the error caused by the shot noise will be biased toward the

closer sample time. The averaging of the range measurements over many correlations

47

will smooth out the random error caused by the shot noise and give a measurement

more precise than the sample time. While the length of the LISA arms will be changing

(up to 10 m/s) the change in the rate will be so slow that it can be taken as constant for

the purposes of measuring the range. Since the rate is constant a linear interpolation

will fit the data.

The amplitude of the PRN code is specified to be 0.14 rad/√Hz to make the power

in the spread spectrum one percent of the carrier’s power. At the high frequencies of the

PRN code the laser noise will be dominated by shot noise at the photo detectors. In fact

the shot noise will completely bury the PRN code as in Figure 2-7.

Another way to measure the arm lengths is to simply apply the TDI combinations in

post processing for a variety of delays until the laser noise in the LISA band has been

minimized [49]. The concern is that by minimizing the signal in the LISA band one might

minimize the gravitational wave signals instead of the laser noise. Another scheme is to

apply a modulation tone to each laser just outside of the measurement band, at say 1.1

Hz. The minimization could be applied to just this frequency band instead of the entire

laser noise in the LISA band.

48

CHAPTER 3THE UNIVERSITY OF FLORIDA LISA INTERFEROMETRY SIMULATOR

The University of Florida LISA Interferometry Simulator (UFLIS) is a hardware

simulator of the LISA Interferometric Measurement System (IMS). The purpose of UFLIS

is to provide an apparatus that can be used to test various aspects of LISA technology.

UFLIS is unique in that it has the capability to reproduce realistic light travel time delays

using an electronic phase delay (EPD). The EPD also allows for the simulation of

Doppler shifts in the frequency of the laser light. UFLIS has been used to test TDI [37]

[50], arm locking [43] [51], and TDI with time changing delays [52].

3.1 UFLIS Concept

The basic concept of UFLIS is to model as closely as possible the simplified

LISA interferometry in Figure 2-6. In this simplified version of LISA each SC only

has one laser, there are no test masses, and the laser links between SC 2 and 3 are

not considered. The laser on SC 1 is the master laser and is stabilized to an optical

reference while the lasers on the other two SC are phase locked to the master. In UFLIS

the long (16 s) delay time between the SC are simulated by measuring the phase of the

laser light, storing it in memory, and electronically regenerating it. In order to measure

the phase of the laser light, each of the three lasers in UFLIS (representing each SC) is

heterodyned with a fourth laser known as the reference laser (Figure 3-1). In this way

the frequency of the laser light is brought down to the MHz range allowing the phase

to be measured, stored, and regenerated. Since each laser is heterodyned with the

reference laser before it is delayed or combined with any other laser, the only effect of

the reference laser is to replace the phase noise of each laser with

ϕ1(t) → ϕ1R(t) ≡ ϕ1(t)− ϕR(t) (3–1)

Since the reference laser is independent of each other laser it only increases the

laser noise by a factor of√2.

49

Figure 3-1. How the EPD unit models the long travel times of the LISA arms. Thereference laser is used to form a beat note with both Laser 1 and 2. TheLaser 1 beat note is recorded, stored in memory, and regenerated. Thedelayed signal is then combined with the Laser 2 beat note with an electronicmixer.

3.2 Laser Bench Top Set-Up

The original layout of the laser bench top of UFLIS is shown in Figure 3-2 and

is described in [37]. In the original layout the lasers were combined in free space. In

the current layout all four lasers are coupled into polarization maintaining fibers via

fiber couplers from Schäfter+Kirchhoff (model 60SMS-1-4-A11-03). Each laser is sent

through a fiber-optic EOM from Jenoptik (model PM1060HF) in order to upgrade UFLIS

to include clock noise transfers and PRN ranging (Figure 3-2). In both layouts the

reference laser and laser 1 are both stabilized to reference cavities housed in a vacuum

tank. In the previous layout mode matching lenses were used to match the mode of

each of the numbered lasers to the reference laser before they were measured at the

photodetectors. In the current layout polarization maintaining fiber splitter/combiners

from Opto-Link Corporation are used to split the reference laser three ways and combine

50

Figure 3-2. The previous set up of the laser bench top for UFLIS. Not shown are theFaraday isolators after each laser, the mode matching lenses, theelectro-optic modulator, quarter wave plate, and other optics associated withthe Pound-Drever-Hall locking scheme.

Figure 3-3. The current set up of the laser bench top for UFLIS. Not shown are theFaraday isolators after each laser, the mode matching lenses and mirrorsused to couple the laser light into the fibers, the electro-optic modulator,quarter wave plate, and other optics associated with the Pound-Drever-Halllocking scheme. The fibers used to form the beat notes and thephotodetectors are shown in Figure 3-4.

it with lasers 1, 2, and 3 (Figure 3-4). The fiber components are connected using

mating sleves from Thorlabs (model ADAFC2-PMW). The combined laser fibers are

coupled directly to photodetectors from Electro-Optic Technology Incorporated (model

ET-300A-FC-DC).

51

Figure 3-4. The formation of optical beat notes in the current UFLIS set up. A three wayfiber splitter is used to split the reference laser, while three two way fibercombiners are used to form beat notes between each of the numberedlasers and the reference laser. The fibers containing the beat notes aredirectly attached to the photodetectors.

3.3 Prestabilization

Both the reference laser and Laser 1 are frequency stabilized to Zerodur optical

cavities housed in a vacuum tank. The lasers are locked to the optical cavities by the

Pound-Drever-Hall method [53]. This prestabilization set-up was designed to emulate

as realistically as possible the baseline design for LISA. The UFLIS prestabilization was

shown to meet the original frequency stabilization requirement of 30Hz/√Hz [54] [37].

3.4 Electrical Components

The photodetector signals of the beat notes between the numbered lasers and the

reference laser are used to simulate the individual laser signals in LISA (Figure 3-5). In

LISA each laser signal is split, with ¡ 0.1 percent used to form the local beat note and the

other part sent to the opposite SC. In UFLIS each simulated laser signal is electronically

split, with one half sent to the EPD unit (simulating the light travel time between SC) and

the other half being electronically mixed (Mini Circuits ZAD-6+) with the electronically

delayed simulated laser signal from the other SC. The electronic mixers (and low pass

filters) simulate the photo detectors in LISA on which the local laser and far laser are

beat together. The outputs of the mixers simulate the sij signals (Equation 2–10) and are

sent to a phasemeter to be measured.

52

Figure 3-5. The electronic portion of UFLIS. The photodetector signal from the RL-L1beat note is split four ways, two portions are sent to the EPD unit to bedelayed and combined with the signals from the other photodetectors. Theother two portions are electronically mixed with the delayed signals from theother photodetectors.

3.5 Digital Signal Processing Hadware

Both the phasemeter and the EPD unit used in UFLIS require high speed digital

signal processing. UFLIS uses such a system from Pentek Corporation consisting of

their 4205, 6256, and 6228 models (Figure 3-6). The 4205 model motherboard contains

a 1 GHz 32 bit microprocessor and is able to interface with two daughter cards via

velocity interface mezzanine (VIM) connectors. The 6256 daughter card has four 14-bit

analog-to-digital converters (ADCs). The ADCs can sample an analog signal at rates

up to 105 MHz. The 6256 daughter card contains two Xilinx Virtex-II FPGAs both of

which recieve data from two of the four ADCs. The two sets of ADCs and FPGAs can

be clocked independently. The 6228 daughter card has four 16-bit digital-to-analog

converters (DACs) all of which receive data from a single FPGA and are triggered by a

single clock input. The FPGAs on th